From Complex Analysis to Operator Theory: A Panorama: In Memory of Sergey Naboko (Operator Theory: Advances and Applications, 291) [1st ed. 2023] 3031311388, 9783031311383

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Operator Theory Advances and Applications 291

Malcolm Brown, Fritz Gesztesy, Pavel Kurasov, Ari Laptev, Barry Simon, Gunter Stolz, Ian Wood, Editors

From Complex Analysis to Operator Theory: A Panorama In Memory of Sergey Naboko

Operator Theory: Advances and Applications Volume 291

Founded in 1979 by Israel Gohberg Series Editors: Joseph A. Ball (Blacksburg, VA, USA) Albrecht B¨ottcher (Chemnitz, Germany) Harry Dym (Rehovot, Israel) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Wolfgang Arendt (Ulm, Germany) Raul Curto (Iowa, IA, USA) Kenneth R. Davidson (Waterloo, ON, Canada) Fritz Gesztesy (Waco, TX, USA) Pavel Kurasov (Stockholm, Sweden) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Ilya Spitkovsky (Abu Dhabi, UAE)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) J.William Helton (San Diego, CA, USA) Marinus A. Kaashoek (Amsterdam, NL) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Bernd Silbermann (Chemnitz, Germany)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Orange, CA, USA) Birgit Jacob (Wuppertal, Germany) Andr´e C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (G¨ottingen, Germany)

Malcolm Brown • Fritz Gesztesy • Pavel Kurasov • Ari Laptev • Barry Simon • Gunter Stolz • Ian Wood Editors

From Complex Analysis to Operator Theory: A Panorama In Memory of Sergey Naboko

Editors Malcolm Brown (deceased) School of Computer Science & Informatics Cardiff University Cardiff, UK

Fritz Gesztesy Department of Mathematics Baylor University Waco, TX, USA

Pavel Kurasov Department of Mathematics Stockholm University Stockholm, Sweden

Ari Laptev Department of Mathematics Imperial College London London, United Kingdom

Barry Simon Division of Physics, Mathematics and Astronomy California Institute of Technology Pasadena, CA, USA

Sirius Mathematical Center Sochi, Russia

Ian Wood School of Mathematics, Statistics and Actuarial Sciences University of Kent Canterbury, UK

Gunter Stolz Department of Mathematics University of Alabama at Birmingham Birmingham, AL, USA

ISSN 0255-0156 ISSN 2296-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-031-31138-3 ISBN 978-3-031-31139-0 (eBook) https://doi.org/10.1007/978-3-031-31139-0 Mathematics Subject Classification: 01A65, 47B40, 47B28, 47A45, 47B25, 46E22, 47A55, 47B32, 35P15, 81Q12, 81Q10, 46N50, 47B80, 47A20, 81Q35, 34L25, 47F05, 35Q61, 35Q74, 34E05, 34E13, 58C40, 35J10, 37D05, 47A10, 47B36, 52C23, 58J51, 35J15, 35J70, 47B44, 35K65, 34L20, 34L40, 45P05, 37K40, 35Q53, 37K45, 35Q15, 47A48, 47A06, 47B50, 30E99, 35P20, 49R05, 81Q20, 37K15, 47B35, 33C45, 39A70, 47A40, 47B39, 34D10, 34L15, 30H20, 3010, 47N50 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkh¨auser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Contents

Sergey Naboko: A Life in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxi

List of Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxvii

Photographs (Private and Academic Life) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii Part I On Some of Sergey’s Works Working with Sergey Naboko on Boundary Triples. . . . . . . . . . . . . . . . . . . . . . . B. Malcolm Brown, Marco Marletta, and Ian Wood

3

Operator-Valued Nevanlinna–Herglotz Functions, Trace Ideals, and Sergey Naboko’s Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fritz Gesztesy

11

Mathematical Heritage of Sergey Naboko: Functional Models of Non-Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander V. Kiselev and Vladimir Ryzhov

27

On Crossroads of Spectral Theory with Sergey Naboko. . . . . . . . . . . . . . . . . . Pavel Kurasov

43

Sergey Naboko’s Legacy on the Spectral Theory of Jacobi Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luis O. Silva and Sergey Simonov

49

On the Work by Serguei Naboko on the Similarity to Unitary and Selfadjoint Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dmitry Yakubovich

61

v

vi

Contents

Part II Research Contributions Functional Models of Symmetric and Selfadjoint Operators . . . . . . . . . . . . Sergio Albeverio, Volodymyr Derkach, and Mark Malamud

75

Schr¨odinger Operators with .δ-potentials Supported on Unbounded Lipschitz Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jussi Behrndt, Peter Schlosser, and Vladimir Lotoreichik

123

Improved Lieb–Thirring Type Inequalities for Non-selfadjoint Schr¨odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sabine B¨ogli

151

Ballistic Transport in Periodic and Random Media. . . . . . . . . . . . . . . . . . . . . . . Anne Boutet de Monvel and Mostafa Sabri On the Spectral Theory of Systems of First Order Equations with Periodic Distributional Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kevin Campbell and Rudi Weikard Asymptotic Analysis of Operator Families and Applications to Resonant Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kirill D. Cherednichenko, Yulia Yu. Ershova, Alexander V. Kiselev, Vladimir A. Ryzhov, and Luis O. Silva

163

217

239

On the Number and Sums of Eigenvalues of Schr¨odinger-type Operators with Degenerate Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Claude Cuenin and Konstantin Merz

313

Gap Labelling for Discrete One-Dimensional Ergodic Schr¨odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Damanik and Jake Fillman

341

Degenerate Elliptic Operators and Kato’s Inequality. . . . . . . . . . . . . . . . . . . . . Tan Duc Do and A. F. M. ter Elst

405

Generalized Indefinite Strings with Purely Discrete Spectrum . . . . . . . . . . Jonathan Eckhardt and Aleksey Kostenko

435

Soliton Asymptotics for the KdV Shock Problem of Low Regularity . . . Iryna Egorova, Johanna Michor, and Gerald Teschl

475

Realizations of Meromorphic Functions of Bounded Type . . . . . . . . . . . . . . . Christian Emmel and Annemarie Luger

501

Spectral Transition Model with the General Contact Interaction . . . . . . . Pavel Exner and Jiˇr´ı Lipovsk´y

523

Weyl’s Law under Minimal Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rupert L. Frank

549

Contents

vii

Weyl–Titchmarsh M-Functions for .ϕ-Periodic Sturm–Liouville Operators in Terms of Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fritz Gesztesy and Roger Nichols

573

On Discrete Spectra of Bergman–Toeplitz Operators with Harmonic Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Golinskii, S. Kupin, J. Leblond, and M. Nemaire

609

One Dimensional Discrete Schr¨odinger Operators with Resonant Embedded Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wencai Liu and Kang Lyu

619

On the Invariance Principle for a Characteristic Function . . . . . . . . . . . . . . Konstantin A. Makarov and Eduard Tsekanovskii

637

A Trace Formula and Classical Solutions to the KdV Equation . . . . . . . . . Alexei Rybkin

667

Semiclassical Analysis in the Limit Circle Case . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimitri R. Yafaev

679

Sergey Naboko: A Life in Mathematics

In memoriam

©Ekaterina Vasilyeva

Sergey Naboko (.Serge˘ i Nikolaeviq Naboko) died in St. Petersburg on December 24, 2020, 1 week before his 71st birthday. Sergey was a talented mathematician with many remarkable results who influenced the development of analysis, especially its applications to mathematical physics—a branch of mathematics well-represented in Leningrad (St. Petersburg). The closest relatives are his wife Ekaterina Vasilyeva (Kate) and three older siblings (Natalia, Olga, and Vasily) with families. Sergey was born on January 2, 1950, in Vyritsa (.Vyrica) near Leningrad (St. Petersburg) in a family that was strongly influenced by the tradition of Sergey’s maternal great-grandfather, Saint Serafim Vyritsky (.prepodobny˘ i Serafim .Vyricki˘ i), who was canonised by the Orthodox Church in 2000. Sergey’s grandmother Eugeniya Lubarskaya (.Evgeni Lubarska) came from Moscow jewish petite bourgeoisie. Sergey had two older sisters (Natalia and Olga) and one older brother (Vasily), all extremely close to their mother Margarita Nikolaevna (.Margarita Nikolaevna), granddaughter of Serafim Vyritsky. All children received an excellent education ix

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and lived secular lives so that even the closest friends could not guess the family’s background, which of course was known to competent authorities.

Academic Career Already as a young man, Sergey showed great talent and he started primary school 1 year ahead of his contemporaries. After high school, he studied at the Department of Mathematical Physics of Leningrad University with outstanding teachers such as Vladimir Smirnov, Olga Ladyzhenskaya, Ludvig Faddeev, Mikhail Birman, and Boris Pavlov. This was an excellent environment for studying applications of analysis in mathematical and theoretical physics, especially, in connection with the spectral theory for differential operators. Sergey had close ties with the analysis group at the Faculty of Mathematics and Mechanics of Leningrad University, especially with N.K. Nikolskii and his seminar. Sergey received his PhD (.kandidat nauk) in 1977 with Boris Pavlov as supervisor. The dissertation, which dealt with non-Hermitian operators, was judged to be brilliant. Mark Krein even stated that the quality of the dissertation was on the level of the Russian doctoral degree (.Doktor Nauk). Subsequently, when evaluating new doctoral dissertations that fell into the purview of Mark Krein, one could hear from him [1] Yes, not bad, but can it be compared with Naboko’s Ph.D. thesis?

Sergey’s background in Christian circles combined with Jewish origin was of course not exactly a springboard in the Soviet society, still, in 1991, Sergey became a professor at the famous Department of Mathematical Physics, where he had studied originally. Before perestroika, Sergey could not travel abroad, given his background, but in 1991 he was allowed to visit Sweden followed by visits and longer stays at many foreign countries: Belgium, France, Germany, Ireland, Israel, Mexico, New Zealand, Norway, Poland, Portugal, Sweden, UK, and USA.

Sergey worked for extended periods as a visiting professor at the – University of Alabama at Birmingham (2004–2009); – University of Kent at Canterbury (2013–2015); – Stockholm University (2017–2019). Being an experienced tourist, he established permanent base camps at Cardiff, Bordeaux, Krakow, Lund–Stockholm, Mexico-City

—places he visited on a regular basis. Sergey remained faithful to his university in St. Petersburg, where he returned regularly after every trip, despite numerous proposals to accept a permanent position abroad (see, e.g., section The Swedish Connection). In addition to personal reasons, he felt comfortable as a prominent representative of the St. Petersburg mathematical school, attracting talented students whom he felt he could not abandon.

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Fig. 1 At Birmingham (Al) with Yulia Karpeshina and Gunter Stolz

Sergey liked to visit colleagues, learn new ideas, and expand mathematical knowledge (his own and his host’s). Everywhere he made close contacts with likeminded people who valued mathematical jewels higher than regular jewelry. One meets mathematicians having benefited from collaboration with Sergey everywhere in the world. He had an infectious enthusiasm and sheer endless energy for the subject. Sergey was ready to discuss mathematics at any time of the day and shared his deep knowledge with colleagues, often without requiring any acknowledgment. One could often see him explaining after somebody’s lecture how the subject can be seen from a general perspective, and in which possible directions one may proceed. He frequently understood the lecture better than the lecturer himself.

Personality In spite of mathematics having been Sergey’s passion, he was interested in all other areas of culture: literature, history, cinema, music, etc. He liked to share his knowledge and preferences with colleagues and friends: each time visiting a new place his luggage contained not only kilograms of chocolates, but also books which he was distributing following individual interests of his friends. When hosting guests in St. Petersburg, Sergey was always keen to show off his city, helping organize visits to palaces, concerts, or the ballet and inviting people home for a Russian meal.

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Sergey cared very much about other people and had an extraordinary ability to understand people down to their core—he claimed that this ability was inherited from his great-grandfather. While he could find something attractive in almost any person, he could not stand betrayal, hypocrisy, and meanness—some people who lost his trust had a difficult time gaining it back. Sergey had good relations with many people, but he did not permit people to come too close; being his close friend was a hard task due to extremely high standards he applied to people, including himself. He could easily see the needs of others and could often find a good solution. This ability had an unexpected side: if Sergey was convinced that somebody needed his help, it was absolutely impossible to convince him otherwise. It was better to let it slide. Similarly, Sergey had difficulties accepting presents despite having distributed numerous presents himself.

Fig. 2 Sergey at Gregynog Hall with Boris Pavlov (viewer’s right) and Pavel Kurasov (left). ©Ekaterina Vasilyeva

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University Professor Sergey Naboko was an extraordinary lecturer. Even students not formally registered for the course were attracted to his lectures. Highly non-trivial material was presented in such an elegant and clear way that it was hard to see the obstacles; therefore, he used to illustrate possible difficulties via explicit examples. Sergey always used very simple clothing, but students immediately saw the highly non-trivial and attractive underlying personality, a combination of an oldfashioned gentleman (really, a gentle man) and a modern human being interested in most facets of life.

Mathematical Research Sergey once listed his main research interests (the references refer to the list of Sergey’s publications) • Scattering theory for self-adjoint and nonself-adjoint operators, perturbation theory for continuous spectrum. Spectral shift function and scattering matrix properties [2, 3, 6, 9, 19, 26, 32, 44, 70, 71, 102, 122] • Embedding of point into continuous spectrum, in particular existence of embedded dense point spectrum [13, 16, 30, 31, 36, 39, 61, 79, 94, 104, 109] • Investigation of the singular spectrum [10, 12, 14, 17, 18, 25, 29, 34, 35, 37, 84] • Nevanlinna class operator-functions in perturbation theory [11, 12, 14, 20, 22–24, 27, 28, 33, 37, 38, 86, 113, 114] • Dissipative operators [100, 105, 117, 121, 125] • Liapunov stability problems, stability problems from hydrodynamics [41] • Spectral theory of non-dissipative operators, theory of operators with almost Hermitian spectrum [4–8, 10, 14, 15, 17, 35, 64, 67, 76, 85] • Functional models and characteristic functions of operators [4, 6, 7, 9, 46, 51, 56, 117, 121, 123] • Wave propagation in inhomogeneous media • Localization for Anderson models in random Schr¨odinger operators theory [73, 75, 78, 98] • The spectral structure for the matrix-differential operators from magnetohydrodynamics [45, 52, 54, 62, 82] • Spectral analysis of Jacobi matrices [7, 40, 42, 43, 47–50, 53, 55, 60, 61, 63, 65, 68, 72, 77, 80, 84, 89–91, 93, 95, 104, 108, 109, 111] • Spectral analysis of the Boltzmann transport operators [46, 51, 56] • Semiclassical analysis of Regge trajectories for singular and nonsingular potentials [57–59, 66, 69, 92] • Spectral properties of Block Jacobi Operators [103, 110, 112, 116, 126, 127] • Spectral theory of quantum graphs [74, 99, 101, 118, 125]

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• Homogenization theory for self-adjoint operators with stiff and soft regimes [115, 123] We tried to describe Sergey’s mathematical achievements by inviting some of his coauthors and prominent experts to write review articles. Each of these accounts is devoted to one of the areas in mathematical analysis where Sergey’s contributions were seminal: – B. Malcolm Brown, Marco Marletta and Ian Wood, Working with Sergey Naboko on Boundary Triples – Fritz Gesztesy, Operator-Valued Nevanlinna–Herglotz Functions, Trace Ideals, and Sergey Naboko’s Contributions – Alexander V. Kiselev and Vladimir Ryzhov, Mathematical Heritage of Sergey Naboko: Functional Models of Non-Self-Adjoint Operators – Pavel Kurasov, On Crossroads of Spectral Theory with Sergey Naboko – Luis O Silva and Sergey Simonov, Sergey Naboko’s Legacy on the Spectral Theory of Jacobi Operators – Dmitry Yakubovich, On the Work by Serguei Naboko on the Similarity to Unitary and Self-Adjoint Operators These contributions are included in the current volume.

Working with Sergey It was an honor working with Sergey. His knowledge of mathematical analysis allowed him to understand almost immediately whether a problem was trivial, or impossible to solve at this stage. He surprised colleagues by fearlessly attacking difficult problems, often not approached by other mathematicians. This ability can be traced back to his first paper [4] devoted to the number of geodesics on a fundamental domain of the modular group, related to the idea of proving the Riemann hypothesis using L.D. Faddeev’s and B.S. Pavlov’s scattering theory for automorphic functions, extended later by P.D. Lax and R.S. Phillips [2, 3, 7]. Sergey’s approach was typical for the St. Petersburg mathematical school: Start with the simplest possible example still possessing the main features of the problem and proceed gradually to more sophisticated and complicated cases.

Working alongside this approach can be compared with the siege of a fortress, where simple examples remind one of the falling of certain bastions without which the fortress can no longer be defended. When Sergey passed away, he left piles of papers containing calculations of numerous simple examples—they were used to get intuition on what was going on, that is, how some of the bastions were conquered. But taking down bastions was just a beginning of a long journey, often leading to a complete solution of the problem, removing unnecessary assumptions. On the other hand, generality was never to contradict clarity of the presented mathematics.

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He was never satisfied until the result could be formulated in a clear and elegant way. Sergey liked to work on several different problems at the same time, sometimes with completely unrelated teams of collaborators. This gave him the unique opportunity to gain general insights into mathematics, but one cannot help wonder how he was able to keep control over the entire network simultaneously.

Sergey’s Students When working with students, Sergey was extremely generous. At every conference he co-organized, he invited several young lecturers, thereby helping talented mathematicians at the beginning of their career to find their way in science. After lectures, Sergey liked to provide detailed comments without requiring any acknowledgment of his help. His encyclopedic knowledge of mathematical analysis allowed him to find unexpected connections and relations. Every thesis supervised by Sergey contains non-trivial mathematical results, many of the Bachelor and Master theses led to scientific publications in reputable journals. Traveling around the world, Sergey helped his colleagues to supervise students: In fact, one can frequently find his traces in students’ works, where he was not an official supervisor. Unfortunately, we failed to be able to compile a complete list of all Bachelor and Master students who worked under Sergey’s supervision; picking just a few of them did not seem appropriate. Sergey supervised a total of ten Ph.D. students (see Sergey’s Curriculum Vitae). It is remarkable that most of these students remained in academia, despite sometimes difficult circumstances.

The Swedish Connection Leningrad mathematicians traditionally had a very special relation to Scandinavia, in general, and Stockholm University, in particular. This can be explained not only by geographical reasons, but also by the cooperation agreement between the Universities of Stockholm and Leningrad signed in 1981 by Boris Pavlov, at that time the Vice-Rector of Leningrad University, an agreement that has lasted over 40 years. Therefore, it is not surprising that Sergey Naboko published his first article abroad in Arkiv f¨or Matematik (a renowned journal published by the Institute Mittag-Leffler) in 1987 [5]. It should be emphasized that the manuscript was submitted already in October 1984, a few months before Gorbachev became General Secretary of the CPSU and perestroika started. It is not surprising that the publication took more than 2 years—the postal service between the former USSR and the West was unreliable and scientists usually asked colleagues to carry abroad manuscripts and article proofs in their luggage.

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Fig. 3 At Institut Mittag-Leffler Pavel Kurasov, Stanislav Kupin, Sergey Naboko, Vladimir Mazya, Tatyana Shaposhnikova, Brian Davies, Luis Silva, Frantisek Stampach, Vladimir Sloushch, Sergey Simonov. ©Ekaterina Vasilyeva

Sergey’s first visit abroad was to Stockholm University in the Spring of 1991 as a guest of Professor Jaak Peetre. Sergey valued Swedish mathematics very highly, especially mathematical analysis, which of course has a unique tradition in Sweden. Sergey arrived in Sweden when his professorship at Leningrad University was already decided, but not yet formally approved. He was a fellow at the Department of Mathematics at Stockholm University within the cooperation agreement mentioned above, but spent a lot of time at the Institut Mittag-Leffler, where Jaak Peetre together with Svante Janson organized a 1-year program Operator Theory and Complex Analysis. At least half of the participants in the program came from the former Soviet Union, and one could easily get an impression that Nikolskii’s seminar temporarily changed its location. Sergey fell in love with Sweden, which can partially be explained by the fact that half of the month he stayed at the Institut Mittag-Leffler with its wonderful working atmosphere, rich library, beautiful surroundings, and kind administration, taking care of all the guests. One of his deepest impressions was a visit to the Royal Opera House, to which he returned many times since. Sergey did not feel constrained by the beautifully dressed public

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and even met Maya Plisetskaya calling her Our Goddess (.Naxa bogin), long before the book with the same title was published [6]. During the 30 years since then, Sergey visited Sweden on a regular basis collaborating with Pavel Kurasov (Stockholm University, 1996–2001, 2011–2019, Lund University, 2001–2010), Ari Laptev, and Oleg Safronov (Royal Institute of Technology, 2001–2011). He became a true member of the Swedish mathematical community and tried to learn more about the history of Swedish mathematics, always emphasizing the contributions by Swedish mathematicians in his lectures. One of the outstanding Swedish mathematicians even encouraged him to apply for a permanent position, but Sergey did not want to leave his mother, with whom he was especially close, taking care of her when it became necessary. When Margarita Nikolaevna died, Sergey went to Sweden for the mourning period. In 2017 Sergey received the prestigious position as a Wallenberg Guest Professor, to be spent at Stockholm University, collaborating with Pavel Kurasov. Unfortunately, serious illness first forced him to postpone the position and then slowed down his activity. It became difficult for Sergey to use all the wonderful opportunities connected with the position. Sergey liked Sweden and Stockholm so much that he used this grant, originally intended for 1 year, to stay almost 2 years. He passed away in St. Petersburg while still being a Guest Professor at Stockholm University. In June of 2023, a Wallenberg Symposium will be organized in Sergey’s memory.

Fig. 4 At seashore in San Sebastian. ©Ekaterina Vasilyeva

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Malcolm Brown (1946–2022) During the time period in which this volume was collated, Malcolm Brown, one of the editors, died on January 14, 2022. Malcolm was born in Aberdare, South Wales, on January 11, 1946. He studied in Cardiff under the supervision of Des Evans and obtained his PhD in 1975 with a thesis on deficiency indices for ordinary differential operators. Following a spell working for the National Health Service, Malcolm returned to Cardiff University in 1986. He worked in the School of Computer Science, being promoted to a chair in 2003, and was an active member of staff up until his death. Despite his position in computer science, the main focus of his research remained in spectral theory and differential equations. Malcolm had a talent for bringing people together to create fruitful collaborations. Many people will fondly remember the numerous conferences in spectral theory he organized, particularly those in Gregynog between 1993 and 2014. Malcolm likely first met Sergey during a sabbatical he spent in Birmingham, Alabama, in 2000. They remained close over the next two decades with Sergey visiting Cardiff many times and several of Sergey’s former students spending extended periods in Cardiff. Their initial collaboration was on inverse resonance problems with Rudi Weikard. Following a successful EPSRC application together with Marco Marletta, which also brought Ian Wood to Cardiff as a postdoc in 2006, much of their later collaboration focused on boundary triples. This area of research led to Sergey’s EU Marie Curie Fellowship which he spent at the University of Kent 2013–2015 and is the topic of one of the articles in this volume.

References 1. V. Adamyan, Private communication (2022) 2. P.D. Lax, Functional analysis, in Pure and Applied Mathematics (WileyInterscience, New York, 2002). MR1892228 3. P.D. Lax, R.S. Phillips, Scattering Theory for Automorphic Functions. Annals of Mathematics Studies, No. 87 (Princeton University Press, Princeton, N.J., 1976). MR0562288 4. S.N. Naboko Estimates of the number of geodesics on a fundamental domain of the modular group. Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 37, 43–46 (1973) (Russian). Differential geometry Lie groups and mechanics. MR0390966 5. S.N. Naboko, Uniqueness theorems for operator-valued functions with positive imaginary part, and the singular spectrum in the selfadjoint Friedrichs model. Ark. Mat. 25(1), 115–140 (1987). https://doi.org/10.1007/BF02384438. MR918381 6. E. Obo˘ imina, Ma˘ i Plisecka . Bogin russkogo baleta, Algoritm (2020) (Russian)

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7. B.S. Pavlov, L.D. Faddeev, Scattering theory and automorphic functions. Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 27, 161–193 (1972) (Russian). Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 6. MR0320781

Curriculum Vitae

(based on the application to the Knut and Alice Wallenberg Foundation) Sergey Naboko (Vyritsa (.Vyrica), Russia, January 2, 1950 —St. Petersburg (.Sankt-Peterburg), Russia, December 24, 2020) Citizenship: Russian citizen. Educational qualification: MSc, 1972 (Dept. of Mathematical Physics, Faculty of Physics, Leningrad University) Academic degrees: • Ph.D. in Mathematical Physics, 1977 (Leningrad Branch of Steklov Math. Inst.) Supervisor: Prof. B.S.Pavlov Title: “Functional model of nonselfadjoint operator and its application to spectral analysis and scattering theory” Referees: – Prof. A.A.Kostuchenko (Moscow University) – Prof. N.K.Nikolskii (Leningrad Branch of Steklov Math. Inst.) – Prof. V.M.Adamyan (University of Odessa) • Senior Doctorate in Math. Analysis, 1987 (Leningrad Branch of Steklov Math. Inst.) Title: “Operator-valued R-functions and their applications to spectral analysis” Referees: – Acad. V.A.Marchenko (Kharkov Inst. of Low Temperatures) – Prof. V.M.Adamyan (University of Odessa) – Prof. N.K.Nikolskii (Leningrad Branchof Steklov Math. Inst.) xxi

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Professional appointments: • • • • •

1972–1974 1974–1977 1977–1986 1986–1991 1991–2019

Junior researcher, Leningrad Branch of Steklov Math. Inst. PhD student, Leningrad Branch of Steklov Math. Inst. Assistant professor, Leningrad State Univ. Associate professor, Leningrad State Univ. Professor, St.Petersburg (Leningrad) State Univ.

Visits abroad: • Visiting positions: – 2004–2009 Visiting professor at the University of Alabama at Birmingham (USA); – 2013–2015 Marie Curie Fellow at the University of Kent at Canterbury (UK); – 2019–2020 Wallenberg guest professor at Stockholm University (Sweden). • Selected long term visits: – Belgium: Solvay Inst. for Physics and Chemistry, Brussels; – France: Bordeaux University; Ecole Polytechnique, Paris; University ParisVII; University Paris-Sud; – Germany: University of Regensburg; – Ireland: Dublin Institute of Technology; – Israel: Weizmann Institute of Science, Rehovot; – Mexico: UNAM, Mexico City; – New Zealand: Auckland University; – Norway: NTNU, Trondheim; – Poland: Math.Inst.Acad.Sci., Warsaw, – Portugal: Lisbon Technical University; – Sweden: Lund Institute of Technology; Mittag-Leffler Institute, Djursholm, Royal Institute of Technology, Stockholm; Stockholm Univ. (reg visits 1998– 2001, 2011–2019); – UK: University of Bath; Cardiff University; – USA: University of Alaska at Fairbanks; Columbia University, Columbia; Caltech, Pasadena; Georgia Tech.; Clark Atlanta University. Research interests: 1. Scattering theory for self-adjoint and non-self-adjoint operators, perturbation theory for continuous spectrum. Spectral shift function and scattering matrix properties 2. Embedding of point spectrum into continuous spectrum. Investigation of the singular spectrum of self-adjoint operators 3. Nevanlinna class operator-functions in perturbation theory 4. Dissipatvie operators. Functional models and characteristic functions of operators. Spectral analysis of the Boltzmann transport operators 5. Liapunov stability problems, stability problems from hydrodynamics

Curriculum Vitae

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6. Spectral theory of non-dissipative operators, theory of operators with almost Hermitian spectrum 7. Wave propagation in inhomogeneous media 8. Localization for Anderson models in random Schr¨odinger operators theory 9. The spectral structure for the matrix-differential operators from magnetohydrodynamics 10. Spectral analysis of Jacobi matrices. Spectral properties of Block Jacobi Operators 11. Inverse problems in terms of the boundary triples. Detectable subspaces and their properties 12. Semiclassical analysis of Regge trajectories for singular and nonsingular potentials 13. Spectral theory of quantum graphs 14. Homogenization theory for selfadjoint operators with stiff and soft regimes Teaching activity: (at St.Petersburg State Univ., Univ. of Alabama at Birmingham, Univ. of Kent at Canterbury) • General lecture courses in various mathematical subjects, including: Mathematical Analysis, Linear Algebra, Mathematical Physics, Complex Analysis, etc. • Special (advanced) courses of lectures given in: scattering theory, functional analysis, spectral theory of selfadjoint operators, series courses on dissipative operators: I, II, III, IV, V, perturbation theory, operator theory, linear algebra, operator-valued R-functions and their applications, spectral theory of PDO, Sobolev spaces and PDO • Seminars given in: numerical methods and computing, mathematical analysis, ordinary differential equations, linear algebra, operator theory, integral operator theory mathematical physics • Numerous courses in Pure Mathematics and Applications (around 20) given at University of Alabama at Birmingham (USA) in 2005–2009 • 3 courses of lectures at the University of Kent at Canterbury (UK) in 2013–2015 PhD students 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Michail Faddeev MSc 1984, PhD 1993 Serguei I. Iakovlev Master 1987, PhD 1992 Alexander V. Kiselev PhD 2000 Roman Romanov PhD 1999 Vladimir Ryzhov MSc 1988, PhD 1994 Luis O Silva MSc 1996, PhD 2003 Sergey Simonov MSc 2007, PhD 2010 Vladimir Veselov MSc, PhD 1986 Christoph Fischbacher PhD 2017 (co-supervisor) Edmund Judge PhD 2017 (co-supervisor)

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Major Grants • • • • • •

1994–2017 1998–2004 2011–2017 1997–2018 2016–2018 2013–2015

Polish Academy of Sciences Royal Swedish Academy of Sciences Stockholm University Russian Foundation of Fundamental Researches Russian Science Foundation EU Marie Curie Research Fellowship Grant

Editorial Board for International Journals • • • •

Complex Analysis and Operator Theory, Associated editor Mathematical Physics: Analysis and Geometry (2004–20014) Opuscula Mathematica Selected issues of ‘Operator Theory: Advances and Applications’ (2003–2014)

Referee for Journals (Selected) Journal of Functional Analysis, Proceedings of the London Mathematical Society, Transactions of AMS, Mathematical Physics: Analysis and Geometry, Journal of Approximation Theory, Constructive Approximation, Letters in Mathematical Physics, Journal of Mathematical Physics, Saint Petersburg Mathematical Journal, Journal of Physics A: Math.and Theoretical, Mathematische Nachrichten, Communications in Math.Physics, Integral Equations and Operator Theory, Journal of Operator Theory, Opuscula Mathematica, Complex Analysis and Operator Theory, etc.

Organised Conferences • Founder (together with J. Janas) of the Workshop series” Spectral Theory and its Applications” at Banach Center (Warsaw): 1998, 1999, 2000, 2001, 2003, 2005, 2007 • Founder (together with J. Janas, P. Kurasov and A. Laptev) of the conference series: Operator Theory, Analysis and Mathematical Physics (OTAMP): – – – – –

OTAMP2002, Bedlewo, Poland, May 11–18, 2002 OTAMP2004, Bedlewo, Poland, July 6–12, 2004 OTAMP2006, Lund, Sweden, June 15–22, 2006 OTAMP2008, Bedlewo, Poland, June 15–22, 2008 OTAMP2010, Bedlewo, Poland, August 5–12, 2010

Curriculum Vitae

– – – –

OTAMP2012, CRM Barcelona, Spain, June 11–14, 2012 OTAMP2014, Stockholm Univ., Sweden, July 7–11, 2014 OTAMP2016, Euler Institute, St. Petersburg, Russia, August 1–7, 2016 OTAMP2020, UNAM, Mexico City, Mexico, January 8–14, 2020

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List of Publications

Sergey Naboko

Research Papers [1] S. N. Naboko, Estimates of the number of geodesics on a fundamental domain of the modular group, Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 37 (1973), 43–46 (Russian). Differential geometry Lie groups and mechanics. MR0390966 [2] S. N. Naboko, The nonselfadjoint Friedrichs model, Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 39 (1974), 40–58 (Russian). Investigations on linear operators and the theory of functions, IV. MR0343109 [3] S. N. Naboko, Analytic continuation to the second sheet of the Fredholm determinant of the resolvent of a Schr¨odinger operator in R3 , Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47 (1974), 81–89, 186, 192 (Russian, with English summary). Investigations on linear operators and the theory of functions, V. MR0382870 [4] S. N. Naboko, Absolutely continuous spectrum of a nondissipative operator, and a functional model. I, Zap. Nauˇcn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI) 65 (1976), 90–102, 204–205 (Russian, with English summary). Investigations on linear operators and the theory of functions, VII. MR0500225 [5] S. N. Naboko, On the spectral analysis of nonselfadjoint operators, Dokl. Akad. Nauk SSSR 232 (1977), no. 1, 36–39 (Russian). MR0438157 [6] S. N. Naboko, Wave operators for nonselfadjoint operators and a functional model, Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 129–135, 275 (Russian, with English summary). Boundary value problems of mathematical physics and related questions in the theory of functions, 10. MR0477816 [7] S. N. Naboko, Absolutely continuous spectrum of a nondissipative operator, and a functional model. II, Zap. Nauchn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI) 73 (1977), 118– 135, 232–233 (1978) (Russian, with English summary). Investigations on linear operators and the theory of functions, VIII. MR513172 [8] S. N. Naboko, The separability of spectral subspaces of a nonselfadjoint operator, Dokl. Akad. Nauk SSSR 239 (1978), no. 5, 1052–1055 (Russian). MR0482298 [9] S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, Trudy Mat. Inst. Steklov. 147 (1980), 86–114, 203 (Russian). Boundary value problems of mathematical physics, 10. MR573902 [10] S. N. Naboko, On the singular spectrum of a nonselfadjoint operator, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 149–177, 266 (Russian, with English summary). Investigations on linear operators and the theory of functions, XI. MR629838 xxvii

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[11] S. N. Naboko, Conditions for similarity to unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 16–27 (Russian). MR739086 [12] S. N. Naboko, Uniqueness theorems for operator-valued functions with positive imaginary part and the singular spectrum in the selfadjoint Friedrichs model, Dokl. Akad. Nauk SSSR 275 (1984), no. 6, 1310–1313 (Russian). MR746376 [13] S. N. Naboko, Schr¨odinger operators with decreasing potential and with dense point spectrum, Dokl. Akad. Nauk SSSR 276 (1984), no. 6, 1312–1315 (Russian). MR753371 [14] S. N. Naboko, Similarity problem and the structure of the singular spectrum of nondissipative operators, Lecture Notes in Math 1043 (1984), 147–151. [15] V. F. Veselov and S. N. Naboko, The determinant of the characteristic function of a nonselfadjoint operator, Funktsional. Anal. i Prilozhen. 19 (1985), no. 4, 80–81 (Russian). MR820090 [16] S. N. Naboko, On the dense point spectrum of Schr¨odinger and Dirac operators, Teoret. Mat. Fiz. 68 (1986), no. 1, 18–28 (Russian, with English summary). [17] V. F. Veselov and S. N. Naboko, The determinant of the char acteristic function and the singular spectrum of a nonselfadjoint operator, Mat. Sb. (N.S.) 129(171) (1986), no. 1, 20–39, 159, DOI 10.1070/SM1987v057n01ABEH003053 (Russian); English transl., Math. USSR-Sb. 57 (1987), no. 1, 21–41. MR830093 [18] S. N. Naboko, Uniqueness theorems for operator-valued functions with positive imaginary part, and the singular spectrum in the selfadjoint Friedrichs model, Ark. Mat. 25 (1987), no. 1, 115–140, DOI 10.1007/BF02384438. MR918381 [19] S. N. Naboko, Conditions for the existence of wave operators in the non-selfadjoint case, Wave propagation. Scattering theory (Russian), Probl. Mat. Fiz., vol 12, Leningrad. Univ., Leningrad, 1987, pp. 132–155, 258 (Russian). MR923975 [20] S. N. Naboko, On the structure of roots of operator-valued functions with positive imaginary part in the classes σp, Dokl. Akad. Nauk SSSR 295 (1987), no. 3, 538–541 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 1, 92–95. MR901846 [21] S. N. Naboko, Estimates in symmetrically normed ideals for the difference of powers of accretive operators, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 2 (1988), 40–45, 131 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 21 (1988), no. 2, 53–59. MR965100 [22] S. N. Naboko, On the boundary values of analytic operator-valued functions with a positive imaginary part, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 157 (1987), no. Issled. po Line˘in. Operator. i Teorii Funktsi˘i. XVI, 55–69, 179, DOI 10.1007/BF01463185 (Russian, with English summary); English transl., J. Soviet Math. 44 (1989), no. 6, 786–795. MR899274 [23] S. N. Naboko, Nontangential boundary values of operator R-functions in a halfplane, Algebra i Analiz 1 (1989), no. 5, 197–222 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1255–1278. MR1036844 [24] S. Ya. Naboko, Estimates in operator classes for the difference of functions from the Pick class of accretive operators, Funktsional. Anal. i Prilozhen. 24 (1990), no. 3, 26–35, 96, DOI 10.1007/BF01077959 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 3, 187–195 (1991). MR1082028 [25] S. N. Naboko and S. I. Yakovlev, Conditions for the finiteness of the singular spectrum in a selfadjoint Friedrichs model, Funktsional. Anal. i Prilozhen. 24 (1990), no. 4, 88–89, DOI 10.1007/BF01077344 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 4, 338– 340 (1991). MR1092812 [26] S. N. Naboko and M. M. Faddeev, Operators of the Friedrichs model that are similar to a selfadjoint operator, Vestnik Leningrad. Univ. Fiz. Khim. vyp. 4 (1990), 78–82, 114 (Russian, with English summary). MR1127406 [27] S. N. Naboko, The structure of singularities of operator functions with a positive imaginary part, Funktsional. Anal. i Prilozhen. 25 (1991), no. 4, 1–13, 96, DOI 10.1007/BF01080076 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 4, 243–253 (1992). MR1167715

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[28] S. N. Naboko, Uniqueness theorems for operator R-functions in the classes, R0 (Sp )p > 1, Differential equations. Spectral theory Wave propagation (Russian), Probl. Mat. Fiz., vol. 13, Leningrad. Univ., Leningrad, 1991, pp. 169–191, 308 (Russian, with Russian summary). MR1341638 [29] E. M. Dyn kin, S. N. Naboko, and S. I. Yakovlev, A finiteness bound for the singular spectrum in a selfadjoint Friedrichs model, Algebra i Analiz 3 (1991), no. 2, 77–90 (Russian); English transl., St. Petersburg Math. J. 3 (1992), no. 2, 299–313. MR1137522 [30] S. N. Naboko and S. I. Yakovlev, The point spectrum of a discrete Schr¨odinger operator, Funktsional. Anal. i Prilozhen. 26 (1992), no. 2, 85–88, DOI 10.1007/BF01075284 (Russian); English transl., Funct. Anal. Appl. 26 (1992), no. 2, 145–147. MR1173094 [31] S. N. Naboko and S. I. Yakovlev, The discrete Schr¨odinger operator. A point spectrum lying in the continuous spectrum, Algebra i Analiz 4 (1992), no. 3, 183–195 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 3, 559–568. MR1190777 [32] S. N. Naboko, On the conditions for existence of wave operators in the non-selfadjoint case [ MR0923975 (90b:47019)], Wave propagation. Scattering theory Amer. Math. Soc. Transl. Ser. 2, vol. 157, Amer. Math. Soc., Providence, RI, 1993, pp. 127–149, DOI 10.1090/trans2/157/09. MR1252228 [33] S. N. Naboko, The boundary behaviour of Sp -valued functions analytic in the half-plane with nonnegative imaginary part, Functional analysis and operator theory (Warsaw, 1992), Banach Center Publ., vol. 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994, pp. 277–285. MR1285614 [34] S. Naboko, On the singular spectrum of discrete Schr¨odinger operator, S´eminaire sur les ´ ´ Equations aux D´eriv´ees Partielles, 1993–1994, Ecole Polytech., Palaiseau, 1994, pp. Exp. No. XII, 11. MR1300908 [35] S. N. Naboko, Similarity problem and the structure of the singular spectrum of nondissipative operators, Lecture Notes in Math 1573 (1994), 394–398. [36] S. N. Naboko and A. B. Pushnitski˘ı, A point spectrum, lying on a continuous spectrum, for weakly perturbed operators of Stark type, Funktsional. Anal. i Prilozhen. 29 (1995), no. 4, 31–44, 95, DOI 10.1007/BF01077472 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 29 (1995), no. 4, 248–257 (1996). MR1375539 [37] S. N. Naboko, Singular spectrum perturbation theory and operator-valued R-functions, Spectral and evolutional problems, Vol. 4 (Sevastopol, 1993), Simferopol. Gos. Univ., Simferopol 1995, pp. 11–14. MR1379343 [38] S. N. Naboko, Zygmund’s theorem and the boundary behavior of operator R-functions, Funktsional. Anal. i Prilozhen. 30 (1996), no. 3, 82–84, DOI 10.1007/BF02509510 (Russian); English transl., Funct. Anal. Appl. 30 (1996), no. 3, 211–213 (1997). MR1435142 [39] Sergei N. Naboko and Alexander B. Pushnitski, On the embedded eigenvalues and dense point spectrum of the Stark-like Hamiltonians, Math. Nachr. 183 (1997), 185–200, DOI 10.1002/mana.19971830112. MR1434982 [40] Jan Janas and Serguei N. Naboko, On the point spectrum of some Jacobi matrices, J. Operator Theory 40 (1998), no. 1, 113–132. MR1642534 [41] S. N. Naboko and C. Tretter, Lyapunov stability of a perturbed multiplication operator, Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995), Oper. Theory Adv. Appl., vol. 106, Birkh¨auser, Basel, 1998, pp. 309–326. MR1729603 [42] Jan Janas and Serguei Naboko, Jacobi matrices with absolutely continuous spectrum, Proc. Amer. Math. Soc. 127 (1999), no. 3, 791–800, DOI 10.1090/S0002-9939-99-04586-4. MR1469415 [43] Jan Janas and Serguei Naboko, Jacobi matrices with power-like weights—grouping in blocks approach, J. Funct. Anal. 166 (1999), no. 2, 218–243, DOI 10.1006/jfan.1999.3434. MR1707753 [44] Fritz Gesztesy, Konstantin A. Makarov, and Serguei N. Naboko, The spectral shift operator, Mathematical results in quantum mechanics (Prague, 1998), Oper. Theory Adv. Appl., vol. 108, Birkh¨auser, Basel, 1999, pp. 59–90. MR1708788

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[45] Volker Hardt, Reinhard Mennicken, and Serguei Naboko, Systems of singular differential operators of mixed order and applications to 1-dimensional MHD problems, Math. Nachr. 205 (1999), 19–68, DOI 10.1002/mana.3212050103. MR1709162 [46] Yu. A. Kuperin, S. N. Naboko, and R. V. Romanov, Spectral analysis of a one-velocity transport operator, and a functional model, Funktsional. Anal. i Prilozhen. 33 (1999), no. 3, 47–58, 96, DOI 10.1007/BF02465204 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 33 (1999), no. 3, 199–207 (2000). MR1724269 [47] Jan Janas and Serguei Naboko, Asymptotics of generalized eigenvectors for unbounded Jacobi matrices with power-like weights, Pauli matrices commutation relations and Cesaro averaging, Differential operators and related topics, Vol. I (Odessa, 1997), Oper. Theory Adv. Appl., vol. 117, Birkh¨auser, Basel, 2000, pp. 165–186. MR1764960 [48] Jan Janas and Serguei Naboko, On subordination properties of one-dimensional discrete Schr¨odinger operator, Operator theoretical methods (Timis¸oara, 1998), Theta Found., Bucharest, 2000, pp. 187–195. MR1770323 [49] J. Janas and S. Naboko, Multithreshold spectral phase transitions for a class of Jacobi matrices, Recent advances in operator theory (Groningen, 1998), Oper. Theory Adv. Appl., vol. 124, Birkh¨auser, Basel, 2001, pp. 267–285. MR1839840 [50] Jan Janas and Serguei Naboko, Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes, Recent advances in operator theory and related topics (Szeged, 1999), Oper. Theory Adv. Appl., vol. 127, Birkh¨auser, Basel, 2001, pp. 387–397. MR1902812 [51] S. Naboko and R. Romanov, Spectral singularities, Sz˝okefalvi-Nagy-Foias functional model and the spectral analysis of the Boltzmann operator, Recent advances in operator theory and related topics (Szeged, 1999), Oper. Theory Adv. Appl., vol. 127, Birkh¨auser, Basel, 2001, pp. 473–490. MR1902818 [52] Reinhard Mennicken, Serguei Naboko, and Christiane Tretter, Essential spectrum of a system of singular differential operators and the asymptotic Hain-L¨ust operator, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1699–1710, DOI 10.1090/S0002-9939-01-06239-6. MR1887017 [53] Jan Janas and Serguei Naboko, Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries, J. Funct. Anal. 191 (2002), no. 2, 318–342, DOI 10.1006/jfan.2001.3866. MR1911189 [54] Pavel Kurasov and Serguei Naboko, On the essential spectrum of a class of singular matrix differential operators. I. Quasiregularity conditions and essential self-adjointness, Math. Phys. Anal. Geom. 5 (2002), no. 3, 243–286, DOI 10.1023/A:1020929007538. MR1940117 [55] A. Laptev, S. Naboko, and O. Safronov, Absolutely continuous spectrum of Jacobi matrices, Mathematical results in quantum mechanics (Taxco, 2001), Contemp. Math., vol 307, Amer. Math. Soc., Providence, RI, 2002, pp. 215–223, DOI 10.1090/conm/307/05283. MR1946032 [56] Yuri A. Kuperin, Serguei N. Nabokov, and Roman V. Romanov, Spectral analysis of the transport operator: a functional model approach, Indiana Univ. Math. J. 51 (2002), no. 6, 1389–1425, DOI 10.1512/iumj.200251.2180. MR1948454 [57] S. N. Naboko, Z. Felfli, N. B. Avdonina, and A. Z. Msezane, Novel analytical Regge poles trajectories calculation for singular potentials: the Lennard-Jones potential, Contemporary problems in mathematical physics (Cotonou, 2001), World Sci. Publ., River Edge, NJ, 2002, pp. 300–313, DOI 10.1142/9789812777560 0009. MR1952998 [58] N. B. Avdonina, S. Belov, Z. Felfli, A. Z. Msezane, and S. N. Naboko, Semi-classical approach for calculating Regge-pole trajectories for singular potentials, Phys. Rev. A (3) 66 (2002), no. 2, 022713, 7, DOI 10.1103/Phys-RevA.66.022713. MR1955150 [59] S. N. Naboko, Z. Felfli, N. B. Avdonina, and A. Z. Msezane, Regge poles calculation for singular potentials: the Lennard-Jones potential, Proceedings of Neural, Parallel, and Scientific Computations. Vol. 2, Dynamic, Atlanta, GA, 2002, pp. 293–298. MR2146708

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[60] S. N. Naboko and Ya. Yanas, Criteria for semiboundedness in a class of unbounded Jacobi operators, Algebra i Analiz 14 (2002), no. 3, 158–168 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 14 (2003), no. 3, 479–485. MR1921992 [61] A. Laptev, S. Naboko, and O. Safronov, On new relations between spectral properties of Jacobi matrices and their coefficients, Comm. Math. Phys. 241 (2003), no. 1, 91–110, DOI 10.1007/s00220-003-0924-3. MR2013753 [62] Pavel Kurasov and Serguei Naboko, Essential spectrum due to singularity, J. Nonlinear Math. Phys. 10 (2003), no. suppl. 1, 93–106, DOI 10.2991/jnmp.2003.10.s1.7. MR2063547 [63] Anne Boutet de Monvel, Serguei Naboko, and Luis O. Silva, Eigenvalue asymptotics of a modified Jaynes-Cummings model with periodic modulations, C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 103–107, DOI 10.1016/j.crma.2003.12.001 (English, with English and French summaries). MR2038094 [64] S. Naboko and R. Romanov, Spectral singularities and asymptotics of contractive semigroups. I, Acta Sci. Math. (Szeged) 70 (2004), no. 1-2, 379–403. MR2072711 [65] Jan Janas, Serguei Naboko, and Gunter Stolz, Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods, J. Comput. Appl. Math. 171 (2004), no. 1-2, 265–276, DOI 10.1016/j.cam.2004.01.023. MR2077208 [66] S. M. Belov, N. B. Avdonina, Z. Felfli, M Marletta, A. Z. Msezane, and S. N. Naboko, Semiclassical approach to Regge poles trajectories calculations for nonsingular potentials: Thomas-Fermi type. J. Phys. A 37 (2004), no. 27, 6943–6954, DOI 10.1088/03054470/37/27/006. MR2078324 [67] A. V. Kiselev and S. N. Naboko, Nonselfadjoint operators with an almost Hermitian spectrum: weak annihilators, Funktsional. Anal. i Prilozhen. 38 (2004), no. 3, 39–51, DOI 10.1023/B:FAIA.0000042804.88453.4c (Russian, with Rus-sian summary); English transl., Funct. Anal. Appl. 38 (2004), no. 3, 192–201. MR2095133 [68] Jan Janas and Serguei Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36 (2004), no. 2, 643–658, DOI 10.1137/S0036141002406072. MR2111793 [69] Z. Felfli, S. Belov, N. B. Avdonina, M. Marlette, A. Z. Msezane, and S. N. Naboko, Regge poles trajectories for nonsingular potentials. The Thomas-Fermi potentials. Contemporary problems in mathematical physics, World Sci. Publ., Hackensack, NJ, 2004, pp. 217–232, DOI 10.1142/9789812702487 0009. MR2441353 [70] A. Laptev, S. Naboko, and O. Safronov, A Szeg˝o condition for a multidimensional Schr¨odinger operator, J. Funct. Anal. 219 (2005), no. 2, 285–305, DOI 10.1016/j.jfa.2004.06.009. MR2109254 [71] A. Laptev, S. Naboko, and O. Safronov, Absolutely continuous spectrum of Schr¨odinger operators with slowly decaying and oscillating potentials, Comm. Math. Phys. 253 (2005), no. 3, 611–631, DOI 10.1007/s00220-004-1157-9. MR2116730 [72] B. M. Brown, S Naboko, and R. Weikard, The inverse resonance problem for Jacobi operators, Bull. London Math. Soc. 37 (2005), no. 5, 727–737, DOI 10.1112/S0024609305004674. MR2164835 [73] G¨unter Stolz, Michael Aizenman, Alexander Elgart, Sergey Naboko, and Jeffrey H. Schenker, Fractional moment methods for Anderson localization in the continuum, XIVth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2005, pp. 619–625. MR2227878 [74] Sergey N. Naboko and Michael Solomyak, On the absolutely continuous spectrum in a model of an irreversible quantum graph, Proc. London Math. Soc. (3) 92 (2006), no. 1, 251–272, DOI 10.1017/S0024611505015522. MR2192392 [75] Michael Aizenman, Alexander Elgart, Serguei Naboko, Jeffrey H. Schenker, and Gunter Stolz, Moment analysis for localization in random Schr¨odinger operators, Invent. Math. 163 (2006), no. 2, 343–413, DOI 10.1007/s00222-005-0463-y. MR2207021 [76] Alexander V. Kiselev and Serguei N. Naboko, Nonself-adjoint operators with almost Hermitian spectrum: matrix model. I, J. Comput. Appl. Math. 194 (2006), no. 1, 115–130, DOI 10.1016/j.cam.2005.06.017. MR2230973

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List of Publications

[77] Anne Boutet de Monvel, Serguei Naboko, and Luis O. Silva, The asymptotic behavior of eigenvalues of a modified Jaynes-Cummings model, Asymptot. Anal. 47 (2006), no. 3-4, 291–315. MR2233923 [78] A. Boutet de Monvel, S. Naboko, P Stollmann, and G. Stolz, Localization near fluctuation boundaries via fractional moments and applications, J. Anal. Math. 100 (2006), 83–116, DOI 10.1007/BF02916756. MR2303305 [79] Pavel Kurasov and Serguei Naboko, Wigner-von Neumann perturbations of a periodic potential: spectral singularities in bands, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 161–183, DOI 10.1017/S0305004106009583. MR2296400 [80] David Damanik and Serguei Naboko, Unbounded Jacobi matrices at critical coupling, J. Approx. Theory 145 (2007), no. 2, 221–236, DOI 10.1016/j.jat.2006.09.002. MR2312467 [81] Rupert L. Frank, Christian Hainzl, Serguei Naboko, and Robert Seiringer, The critical temperature for the BCS equation at weak coupling, J. Geom. Anal. 17 (2007), no. 4, 559– 567, DOI 10.1007/BF02937429. MR2365659 [82] Pavel Kurasov, Igor Lelyavin, and Serguei Naboko, On the essential spectrum of a class of singular matrix differential operators. II. Weyl’s limit circles for the Hain-L¨ust operator whenever quasi-regularity conditions are not satisfied, Proc. Roy Soc. Edinburgh Sect. A 138 (2008), no. 1, 109–138, DOI 10.1017/S0308210506000576. MR2388939 [83] Malcolm Brown, Marco Marletta, Serguei Naboko, and Ian Wood, Boundary triplets and M-functions for non-selfadjoint operators, with applications to el liptic PDEs and block operator matrices, J. Lond. Math. Soc. (2) 77 (2008), no. 3, 700–718, DOI 10.1112/jlms/jdn006. MR2418300 [84] Jan Janas, Serguei Naboko, and Gunter Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices, Int. Math. Res. Not. IMRN 4 (2009), 736– 764, DOI 10.1093/imrn/rnn144. MR2480099 [85] Alexander V. Kiselev and Serguei Naboko, Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure, Ark. Mat. 47 (2009), no. 1, 91–125, DOI 10.1007/s11512-007-0068-3. MR2480917 [86] Malcolm Brown, James Hinchcliffe, Marco Marletta, Serguei Naboko, and Ian Wood, The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum, Integral Equations Operator Theory 63 (2009), no. 3, 297–320, DOI 10.1007/s00020-009-1668-z. MR2491033 [87] Fritz Gesztesy, Mark Malamud, Marius Mitrea, and Serguei Naboko, Generalized polar decompositions for closed operators in Hilbert spaces and some applications, Integral Equations Operator Theory 64 (2009), no. 1, 83–113, DOI 10.1007/s00020-009-1678-x MR2501173 [88] B. Malcolm Brown, Serguei Naboko, and Rudi Weikard, The inverse resonance problem for Hermite operators, Constr. Approx. 30 (2009), no. 2, 155–174, DOI 10.1007/s00365-0089037-8. MR2519659 [89] J. Janas, S. Naboko, and E. Sheronova, Asymptotic behavior of generalized eigenvectors of Jacobi matrices in the critical (“double root”) case, Z. Anal. Anwend. 28 (2009), no. 4, 411–430, DOI 10.4171/ZAA/1391. MR2550697 [90] Serguei Naboko, Irina Pchelintseva, and Luis O. Silva, Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis, J. Approx. Theory 161 (2009), no. 1, 314–336, DOI 10.1016/j.jat.2008.09.005. MR2558158 [91] Serguei Naboko and Sergey Simonov, Spectral analysis of a class of Hermitian Jacobi matrices in a critical (double root) hyperbolic case, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 1, 239–254, DOI 101017/S001309150700106X. MR2579689 [92] Sergey Belov, Karl-Erik Thylwe, Marco Marletta, Alfred Msezane, and Serguei Naboko, On Regge pole trajectories for a rational function approximation of Thomas-Fermi potentials, J. Phys. A 43 (2010), no. 36, 365301, 19, DOI 10.1088/1751-8113/43/36/365301. MR2671719

List of Publications

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[93] Anne Boutet de Monvel, Jan Janas, and Serguei Naboko, Unbounded Jacobi matrices with a few gaps in the essential spectrum: constructive examples, Integral Equations Operator Theory 69 (2011), no. 2, 151–170, DOI 10.1007/s00020-010-1856-x. MR2765582 [94] Serguei Naboko and Sergey Simonov, Zeroes of the spectral density of the periodic Schr¨odinger operator with Wigner-von Neumann potential, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 1, 33–58, DOI 10.1017/S030500411100079X. MR2943665 [95] Marco Marletta, S. Naboko, R. Shterenberg, and R. Weikard, On the inverse resonance problem for Jacobi operators—uniqueness and stability, J. Anal. Math. 117 (2012), 221– 247, DOI 10.1007/s11854-012-0020-8. MR2944096 [96] Anne Boutet de Monvel, Jan Janas, and Serguei Naboko, Elementary models of unbounded Jacobi matrices with a few bounded gaps in the essential spectrum, Oper. Matrices 6 (2012), no. 3, 543–565, DOI 10.7153/oam-06-37. MR2987026 [97] J. Janas and S. Naboko, Estimates of generalized eigenvectors of Hermitian Jacobi matrices with a gap in the essential spectrum, Mathematika 59 (2013), no. 1, 191–212, DOI 10.1112/S0025579312000113. MR3028179 [98] Sergey Naboko, Roger Nichols, and G¨unter Stolz, Simplicity of eigenvalues in Andersontype models, Ark. Mat. 51 (2013), no. 1, 157–183, DOI 10.1007/s11512-011-0155-3. MR3029341 [99] P Kurasov, G. Malenov´a, and S Naboko, Spectral gap for quantum graphs and their edge connectivity, J. Phys. A 46 (2013), no. 27, 275309, 16, DOI 10.1088/17518113/46/27/275309. MR3081922 [100] Marco Marletta and Sergey Naboko, The finite section method for dissipative operators, Mathematika 60 (2014), no. 2, 415–443, DOI 10.1112/S0025579314000126. MR3229497 [101] Pavel Kurasov and Sergey Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory 4 (2014), no. 2, 211–219, DOI 10.4171/JST/67. MR3232809 [102] Fritz Gesztesy, Sergey N. Naboko, and Roger Nichols, On a problem in eigenvalue perturbation theory, J. Math. Anal. Appl. 428 (2015), no. 1, 295–305, DOI 10.1016/j.jmaa.2015.03.018. MR3326989 [103] Jan Janas and Serguei Naboko, On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach, J. Difference Equ. Appl. 21 (2015), no. 11, 1103– 1118, DOI 10.1080/10236198.2015.1066341. MR3416573 [104] Edmund Judge, Sergey Naboko, and Ian Wood, Eigenvalues for perturbed periodic Jacobi matrices by the Wigner–von Neumann approach, Integral Equations Operator Theory 85 (2016), no. 3, 427–450, DOI 10.1007/s00020-016-2302-5. MR3523644 [105] Christoph Fischbacher, Sergey Naboko, and Ian Wood, The proper dissipative extensions of a dual pair, Integral Equations Operator Theory 85 (2016), no. 4, 573–599, DOI 10.1007/s00020-016-2310-5. MR3551233 [106] B. M. Brown, M Marletta, S N. Naboko, and I. G. Wood, Detectable sub-spaces and inverse problems for Hain-L¨ust-type operators, Math. Nachr. 289 (2016), no. 17-18, 2108–2132, DOI 10.1002/mana.201500231. MR3583259 [107] B. M. Brown, M. Marletta, S. Naboko, and I. Wood, Inverse problems for boundary triples with applications, Studia Math. 237 (2017), no. 3, 241–275,DOI 10.4064/sm8613-11-2016. MR3633967 [108] Anne Boutet de Monvel, Jan Janas, and Serguei Naboko, The essential spectrum of some unbounded Jacobi matrices: a generalization of the Last-Simon approach, J. Approx. Theory 227 (2018), 51–69, DOI 10.1016/j.jat.2017.12.002. MR3763848 [109] Edmund Judge, Sergey Naboko, and Ian Wood, Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique, Studia Math. 242 (2018), no. 2, 179– 215, DOI 10.4064/sm170325-23-8. MR3778910 [110] Jan Janas, Sergey Naboko, and Luis O. Silva, Green matrix estimates of block Jacobi matrices I: Unbounded gap in the essential spectrum, Integral Equations Operator Theory 90 (2018), no. 4, Paper No. 49, 24, DOI 10.1007/s00020-018-2476-0. MR3820400

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[111] E. Judge, S. Naboko, and I. Wood Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach, J. Difference Equ. Appl. 24 (2018), no. 8, 1247– 1272, DOI 10.1080/10236198.2018.1468890. MR3851162 [112] S. Kupin and S. Naboko, On the instability of the essential spectrum for block Jacobi matrices, Constr. Approx. 48 (2018), no. 3, 473–500, DOI 10.1007/s00365-018-9436-4. MR3869449 [113] Fritz Gesztesy, Sergey N. Naboko, Rudi Weikard, and Maxim Zinchenko, Donoghue-type m-functions for Schr¨odinger operators with operator-valued potentials, J. Anal. Math. 137 (2019), no. 1, 373–427, DOI 10.1007/s11854-018-0076-1. MR3938008 [114] B. M. Brown, M. Marletta, S. Naboko, and I. G. Wood, The detectable sub-space for the Friedrichs model, Integral Equations Operator Theory 91 (2019), no. 5, Paper No. 49, 26, DOI 10.1007/s00020-019-2548-9. MR4025480 [115] K. D. Cherednichenko, Yu. Yu Ershova, A. V. Kiselev, and S. N. Naboko, Unified approach to critical-contrast homogenisation with explicit links to time-dispersive media, Trans. Moscow Math. Soc. 80 (2019), 251–294, DOI 10.1090/mosc/291. MR4082872 [116] Jan Janas, Sergey Naboko, and Luis O. Silva, Green matrix estimates of block Jacobi matrices II: bounded gap in the essential spectrum, Integral Equations Operator Theory 92 (2020), no. 3, Paper No. 21, 30, DOI 10.1007/s00020-020-02576-7. MR4103981 [117] Malcolm Brown, Marco Marletta, Serguei Naboko, and Ian Wood, The functional model for maximal dissipative operators (translation form): an approach in the spirit of operator knots, Trans. Amer. Math. Soc. 373 (2020), no. 6, 4145–4187, DOI 10.1090/tran/8029. MR4105520 [118] Pavel Kurasov and Sergei Naboko, Gluing graphs and the spectral gap: a TitchmarshWeyl matrix-valued function approach, Studia Math. 255 (2020), no. 3, 303–326, DOI 10.4064/sm190322-4-11. MR4142756 [119] S. Naboko and I. Wood, The detectable subspace for the Friedrichs model: applications of Toeplitz operators and the Riesz-Nevanlinna factorisation theorem, Ann. Henri Poincar´e 21 (2020), no. 10, 3141–3156, DOI 10.1007/s00023-020-00935-z. MR4153893 [120] Stanislas Kupin and Sergey Naboko, A version of Watson lemma for Laplace integrals and some applications, Partial differential equations, spectral theory, and mathematical physics—the Ari Laptev anniversary volume, EMS Ser. Congr. Rep., EMS Press, Berlin, [2021] ©2021, pp. 289–300, DOI10.4171/ECR/18-1/17. MR4331820 [121] S. N. Naboko and S. A. Simonov, Titchmarsh-Weyl formula for the spectral density of a class of Jacobi matrices in the critical case, Funct. Anal. Appl. 55 (2021), no. 2, 94–112, DOI 10.1134/s0016266321020027. MR4422781 [122] Mahamet Ko¨ıta, Stanislas Kupin, Sergey Naboko, and Belco Tour´e, On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices, Int. Math. Res. Not. IMRN 13 (2022), 10249–10278, DOI 10.1093/imrn/rnaa328. MR4447146

Research Papers on the Way [123] Malcolm Brown, Marco Marletta, Sergey Naboko, and Ian Wood, The spectral form of the functional model for maximally dissipative operators: A Lagrange identity approach, St.Petersburg Math. J. (2023), to appear. [124] Kirill Cherednichenko, Alexander Kiselev, Sergey Naboko, and Luis O. Silva, Scattering theory for non-selfadjoint operators arising from boundary value problems, 2023. [125] Yilia Ershova, Kirill Cherednichenko, Sergey Naboko, and Luis o. Silva, Functional model for generalised resolvents and its application to time-dispersive media, St.Petersburg Math. J. (2023), submitted.

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[126] Christoph Fischbacher, Sergey Naboko, and Ian Wood, Complete Non-Selfadjointness for Schr¨odinger Operators on the Semi-Axis, St.Petersburg Math. J. (2023), to appear. [127] Pavel Kurasov, Sergey Naboko, and Jacob Muller, Maximal dissipative operators on metric graphs, (2023). [128] Sergey Naboko and Luis O. Silva, Spectral analysis of a class of block Jacobi matrices, 2023. [129] Sergey Naboko and Sergey Simonov, Estimates of Green matrix entries of self-adjoint unbounded block Jacobi matrices, St.Petersburg Math. J. (2023), to appear.

Editorial Work [128] Jan Janas, Pavel Kurasov, Ari Laptev, and Sergei Naboko (eds.), Operator methods in mathematical physics, Operator Theory: Advances and Applications, vol. 227, Birkh¨auser/Springer Basel AG, Basel, 2013. MR3075435 [129] Jan Janas, Pavel Kurasov, Ari Laptev, Sergei Naboko, and G¨unter Stolz (eds.), Spectral theory and analysis, Operator Theory: Advances and Applications, vol. 214, Birkh´auser/Springer Basel AG, Basel, 2011. MR2841292 [130] Jan Janas, Pavel Kurasov, Ari Laptev, Sergei Naboko, and G¨uter Stolz (eds.), Methods of spectral analysis in mathematical physics, Operator Theory: Advances and Applications, vol. 186, Birkh¨auser Verlag, Basel, 2009. MR2766425 [131] Jan Janas, Pavel Kurasov, Ari Laptev, Sergei Naboko, and G¨onter Stolz (eds.), Operator theory, analysis and mathematical physics, Operator Theory: Advances and Applications, vol. 174, Birkh´auser Verlag, Basel, 2007. Lectures from the International Conference on Operator Theory and its Applications in Mathematical Physics (OTAMP 2004) held in Bedlewo, July 6–11, 2004. MR2330822 [132] J. Janas, P Kurasov, and S. Naboko (eds.), Spectral methods for operators of mathematical physics, Operator Theory: Advances and Applications, vol. 154, Birkh¨auser Verlag, Basel, 2004. MR2103669 [133] Pavel Kurasov, Ari Laptev, Sergey Naboko, and Barry Simon (eds.), Analysis as a tool in mathematical physics, Operator Theory: Advances and Applications, vol. 276, Birkh´auser/Springer, Cham, [2020] ©2020. In memory of Boris Pavlov. MR4181253

Photographs (Private and Academic Life)

Fig. 5 Early years, hiking in Altai. ©Naboko family archive

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Fig. 6 Walking in the wood. ©Ekaterina Vasilyeva

Fig. 7 Near Bordeaux. ©Ekaterina Vasilyeva

Photographs (Private and Academic Life)

Photographs (Private and Academic Life)

Fig. 8 With Kate. ©Ekaterina Vasilyeva

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Fig. 9 Paris. ©Ekaterina Vasilyeva

Photographs (Private and Academic Life)

Photographs (Private and Academic Life)

Fig. 10 In Kent with Ian Wood and Christoph Fischbacher. ©Ekaterina Vasilyeva

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Fig. 11 Visiting Mexico. ©Ekaterina Vasilyeva

Photographs (Private and Academic Life)

Photographs (Private and Academic Life)

Fig. 12 Admiring Novgorod churches. ©Ekaterina Vasilyeva

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Fig. 13 In Pushkin (close to St. Petersburg) with Kate. ©Ekaterina Vasilyeva

Photographs (Private and Academic Life) Fig. 14 Somewhere in USA. ©Ekaterina Vasilyeva

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Fig. 15 With Kate in Birmingham, AL, USA

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Part I

On Some of Sergey’s Works

Working with Sergey Naboko on Boundary Triples B. Malcolm Brown, Marco Marletta, and Ian Wood

Dedicated to the memory of our dear friend, mentor and collaborator Sergey Nikolaevich Naboko (1950–2020)

Abstract In this short article we review the contribution of Sergey Naboko to the theory of boundary triples and outline some of its uses in applied mathematics. We also describe the benefits to our own mathematical careers which came from working with Sergey Naboko, starting with a brief history of our collaborations with him.

1 Introduction In 2004, the first two authors, collaborating with W.D. Evans, secured a grant (EPSRC GR/S47229/01) to fund a visit by Sergey Naboko to Cardiff. Roman Romanov had recently given a course of lectures in Cardiff on selfadjoint boundary triples. Sergey Naboko was convinced that the corresponding non-selfadjoint theory for adjoint pairs could help us to generalize results on spectral approximation of non-selfadjoint Hamiltonian systems [9] and PDEs [10]. However we were concerned by the lack of papers describing concrete applications of boundary

B. Malcolm Brown died in January 2022. M. Marletta School of Mathematics, Abacws, Cardiff University, Cardiff, UK e-mail: [email protected] I. Wood () School of Mathematics, Statistics and Actuarial Sciences, Sibson Building, University of Kent, Canterbury, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_1

3

4

B. M. Brown et al.

triples to PDEs. We sought further funding to investigate these topics. EPSRC grant EP/C008324/1 allowed us to employ the third author. In our first paper with Sergey Naboko [11], using the ‘Vishik trick’ [31] to ‘regularise’ the normal derivative, we described how the theory of boundary triples for adjoint pairs, as developed by Malamud and Mogilievski [24], could be applied to elliptic PDEs. (Independently, for the symmetric case, Behrndt and Langer [4] avoided the Vishik trick by developing an alternative theory of quasi-boundary-triples.) The article [11] marked the start of our collaboration with Sergey Naboko on this topic, which lasted until his untimely death in December 2020. We were still actively working with him on several problems when he died. In the rest of this paper we shall describe a few of the results in [11] and mention, without details, a few other works in which we have used ideas from [11]. The perspective we give is personal, and is certainly not intended as a scientific review of the whole theory or its development.

2 Background The modern theory of boundary triples associated with symmetric operators has its origins in the work of Koˇcube˘ı [19] and Gorbachuk and Gorbachuk [17] and has been extensively researched in Russia and Ukraine. Particularly accessible accounts of this work are available the articles of Derkach and Malamud [12, 13, 23] and the recent book [6]. Let A be a densely defined symmetric operator on a domain in a Hilbert space H , having equal (possibly infinite) deficiency indices. Then there exists an auxiliary Hilbert space .K (the ‘boundary space’) and operators .1 , .2 defined on the domain of .A∗ , such that the abstract ‘integration by parts’ formula 

A∗ u, v

.

 H

  − u, A∗ v H = 1 u, 2 vK − 2 u, 1 vK

(1)

holds. From this identity an abstract Dirichlet-to-Neumann map may be defined, analytic on the resolvent set of one of the selfadjoint extensions of A. Much of the theory of Weyl M-functions [32] for ordinary differential equations, carries over to this new abstract setting [12]. In particular, in view of its relevance to Sergey Naboko’s research, we mention the work of Storozh [29], Kopachevski˘ı and Kre˘ın [20], Ryzhov [28] on abstract Green’s formulae and functional models [27]. For our work with Sergey Naboko, however, non-selfadjoint operators were at the heart of all our investigations. We required the theory of boundary triples for adjoint pairs, introduced by Vainerman [30], and many of the associated results of Lyantze and Storozh [22]. In this context, the identity (1) is replaced by  .

∗ u, v A

 H

  − u, A∗ v

H

    = 1 u,  2 v H − 2 u,  1 v K .

(2)

Working with Sergey Naboko on Boundary Triples

5

 and .A  are closed, densely defined operators in H with .A∗ ⊇ A ∗ ⊇ A; Here A, .A we have two potentially distinct boundary spaces .H and .K; and the four abstract trace operators, .

∗ ) → H, 1 : D(A

∗ ) → K, 2 : D(A

 1 : D(A∗ ) → K,

 2 : D(A∗ ) → H,

are bounded with respect to the corresponding graph norms. Within this setup, one ∗ and .A∗ described by abstract boundary conditions is interested in restrictions of .A 1 u − B˜  2 u = 0, respectively, in which of the form .1 u − B2 u = 0 or .  .B ∈ L(K, H) and .B ∈ L(H, K) are bounded operators between the relevant spaces: ∗ |ker(1 −B2 ) and A B := A∗ |ker( AB := A  1 −B 2 ) .

.

(3)

By suitable choice of .1 and .2 , any closed extension can be written in this form. B are extensions of the ‘minimal’ operators A and .A,  whose domains are AB and .A given by

.

∗ ) ∩ ker 1 ∩ ker 2 and D(A)  = D(A∗ ) ∩ ker  D(A) = D(A 1 ∩ ker  2 .

.

(4)

Many of the results for the symmetric case, such as characterising extensions and investigating spectral properties via the Weyl-M-function, are also available in the adjoint pairs setting, see Malamud and Mogilievski [24, 25]. For the case of dual pairs of contractions, there is a parallel theory developed by Langer and Textorius [21]. Arlinski [3] has also developed a theory of abstract boundary conditions for maximal sectorial extensions of sectorial operators. Together with Sergey Naboko, the third author supervised Christoph Fischbacher’s PhD thesis which considered dissipative extensions in the boundary triples framework, characterising proper dissipative extensions .A , i.e. .A ⊆ A ⊆ A˜ ∗ , in [15] and non-proper dissipative extensions (no longer satisfying .A ⊆ A˜ ∗ ) in [14].

3 Elliptic PDEs and Dirichlet-to-Neumann Maps 3.1 Abstract Preliminaries One may associate abstract ‘Robin-to-Dirichlet’ maps .MB (λ) and .M˜ B(λ) with the B in (3) by the formulae operators .AB and .A ∗ −λ), MB (λ) : Ran (1 −B2 ) → K, MB (λ)(1 −B2 )u = 2 u for all u ∈ ker(A

.

 B(λ)(  B(λ) : Ran ( 1 − B 2 ) → H, M 1 − B 2 )v =  2 v for all v ∈ ker(A∗ −λ). M

.

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B. M. Brown et al.

It may be shown that .MB (λ) and .M˜ B(λ) are well-defined and analytic wherever  − λ)−1 , respectively, are analytic. To obtain a converse result, (AB − λ)−1 and .(A B with Sergey Naboko [11] we introduced abstract unique continuation hypotheses (UCH):

.

.

∗ − λ) ∩ ker(1 ) ∩ ker(2 ) = {0}; ∗ − λ satisfies UCH ⇔ ker(A A ∗ 1 ) ∩ ker( 2 ) = {0}. A − λ satisfies UCH ⇔ ker(A∗ − λ) ∩ ker(

(5)

Theorem 1 Let .μ ∈ C be an isolated eigenvalue of finite algebraic multiplicity of ∗ − μ and the operator .AB . Assume the unique continuation hypothesis holds for .A ∗ .A − μ. Then .μ is a pole of finite multiplicity of .MB (·) and the order of the pole of −1 at .λ = μ is the same as the order of the pole of .M (·) at .μ. .(AB − λ) B The UCH has the advantage that for applications to PDEs, it can usually be checked by reference to the existing literature. An alternative approach, found in [24], is to assume that one is able to determine a further boundary condition operator C such that .μ does not lie in the union of the spectra of .AC and .AB+C . Although [11] also contains results concerning the behaviours of .(AB −λ)−1 and .MB (λ) near the essential spectrum, in general not much more than the following can be said: ∗ −k and .A∗ −k satisfy UCH, if the weak limits .MB (k ±i0)g := Proposition 1 If .A w − limε→0 MB (k ± iε)g exist for all .g ∈ Ran (1 − B2 ), and there exists some .f ∈ Ran (1 − B2 ) such that .MB (k + i0)f = MB (k − i0)f , then .k ∈ σess (AB ). In fact block matrix Hain-Lüst operators  ∗

A =

.

2

d − dx 2 + q(x) w(x) w(x) u(x)



 ∗

, A =

2

d − dx 2 + q(x) w(x) w(x) u(x)

 ,

(6)

with domains .H 2 (0, 1) × L2 (0, 1), and q, u, w in .L∞ (0, 1), may be constructed to give spectacular examples in which the M-functions remain analytic at points of essential spectrum. This happens e.g. if u is real-valued, .k = u(x0 ) for some .x0 ∈ (0, 1), and w vanishes in a neighbourhood of .x0 .

3.2 Fitting PDEs into the Framework The main obstacle to applying the theory of boundary triples to elliptic PDEs is the j , fact that if one chooses, in (2), trace and normal derivative to define the .j and . then both the inner products on the right hand side may be formally infinite (except on subdomains of smooth functions) with the infinities cancelling between the two terms. We now describe, with an example, how in [11] we fit elliptic PDEs into the ‘adjoint pairs’ boundary triple framework, using the Vishik trick [31] generalised by Grubb [18, §I.3]. A completely abstract version of this trick was also developed by Ryzhov [28].

Working with Sergey Naboko on Boundary Triples

7

Let .p ∈ (C ∞ ())n , where . is a smooth bounded domain with outer normal .ν, and let A =  + p · ∇,

.

 =  − div (p ·), A

 Define with domains .D(A) = H02 () = D(A). γ1 u =

.

  ∂u + (p · ν)u  , ∂ν ∂

  γ2 u = u

∂

,  γ1 v =

∂v   , ∂ν ∂

  γ2 v = v  

∂

Then for .u, v ∈ H 2 () we have ∗ u, v)L2 () − (u, A∗ v)L2 () = (γ1 u,  (A γ2 v)L2 (∂) − (γ2 u,  γ1 v)L2 (∂) .

.

(7)

∗ and .A∗ are bigger than .H 2 (): in fact, It is easy to check that the domains of .A .

∗ ) = {u ∈ L2 () : ( + p · ∇)u ∈ L2 ()}, D(A D(A∗ ) = {v ∈ L2 () : v − div (p v) ∈ L2 ()}.

Neither of these domains need even be contained in .H 1 ().  ∗  D := A∗  Let .AD := A and .A be the associated elliptic operators with ker γ ker  γ 2

2

Dirichlet boundary conditions. By elliptic regularity, .D(AD ) = H 2 () ∩ H01 () = D ). By Grubb D ). Without loss of generality, assume that .0 ∈ ρ(AD ) ∩ ρ(A D(A ∗ ) = D(AD ) + ker A ∗ and .D(A∗ ) = D(AD ) + ker A∗ . For [18, Lemma II.1.1], .D(A −1/2 (∂) define .m ϕ, m .ϕ ∈ H ˜ 0 ϕ ∈ H −3/2 (∂) by 0 .

∗ u = 0 subject to γ2 u = ϕ; m0 ϕ = γ1 u, where u solves A γ1 v, where v solves A∗ v = 0 subject to  γ2 v = ϕ. m 0 ϕ = 

∗ ) and .v ∈ D(A∗ ), let For .u ∈ D(A u := γ1 u − m0 γ2 u,   v :=  γ1 v − m 0  γ2 v.

.

 are well-defined (see [18, §III.1]). (Evidently, .m0 The operators .m0 , m 0 ,  and . ∗ and .A∗ (with .λ = and .m 0 are the Dirichlet-to-Neumann maps associated with .A  ‘regularise’ .γ1 and . γ1 in the following precise sense: 0).) Moreover . and . ∗ ) and .D(A∗ ) with the graph norm. Then Theorem 2 (Grubb 1968) Equip .D(A ∗ 1/2   : . : D(A ) → H (∂) is continuous and surjective. The same is true for . ∗ 1/2 ∗ ), .v ∈ D(A∗ ) we have, instead of (7), D(A ) → H (∂). For all .u ∈ D(A ∗ u, v)L2 () − (u, A∗ v)L2 () = (u,  (A γ2 v) 1 ,− 1 − (γ2 u,   v)− 1 , 1 ,

.

2

2

2 2

where .(·, ·)α,−α denotes the duality pairing between .H α (∂) and .H −α (∂).

(8)

8

B. M. Brown et al.

To obtain an abstract Green formula of the form (2), we introduce a unitary isomorphism .J : H 1/2 (∂) → L2 (∂). Then .(J ∗ )−1 : H −1/2 (∂) → L2 (∂) is also a unitary isomorphism and (f, g) 1 ,− 1 = (Jf, (J ∗ )−1 g)L2 (∂) .

.

2

2

(9)

1 := J  2 := (J ∗ )−1  Defining .1 := J , .2 := (J ∗ )−1 γ2 , .  , . γ2 , we see that (8) 2 and (9) yield (2) with .H = K = L (∂): ∗ u, v)L2 () − (u, A∗ v)L2 () = (1 u,  (A 2 v)L2 (∂) − (2 u,  1 v)L2 (∂) .

.

j in It may also be shown that all the other hypotheses required of the .j and . the abstract theory are satisfied (surjectivity, abstract inverse trace theorem, etc.).  ∗  Remark 1 Note that .A is the operator with Dirichlet boundary conditions ker 2  ∗  (the Friedrichs extension of A), while .A is the Kre˘ın extension. By exchangker 1 ing the roles of .1 and .2 it is also possible to express the Neumann boundary condition in the form .1 − B2 for bounded B.

4 Applications Following the publication of [11], Grubb visited Cardiff and established a collaboration with two of us, to study M-functions and extension theory for adjoint pairs of partial differential operators. The article [1], which also draws on Grubb’s particular expertise in pseudodifferential methods, obtains a full characterization, including Kre˘ın resolvent formulas, of the realizations of non-selfadjoint second 3 order operators on domains with .C 2 + boundaries, following the self-adjoint work of Gesztesy and Mitrea [16]. A history of the parallel lines of development between the PDE approaches to the characterisation of extensions, starting from [18], and the boundary-triple approach, is outlined in [7]. Brown and Wood also commenced a collaboration with Klaus, Malamud and Mogilievski; their article [8] studies Dirac systems in one dimension. The main result—existence of Weyl-type solutions—is proved using a reduction to the selfadjoint case by a technique based on dual pairs of operators. All three of the current authors were keen to explore the possibilty of an abstract theory of inverse problems based on boundary triples, and we published several articles on this topic with Sergey Naboko. Our last paper with Naboko in this area is [26], and also reviews much of our earlier work. Others have also engaged in this activity; in the symmetric case, some of these ideas are already explored in [28], while [5] applies boundary triple techniques to PDEs of arbitrary even order. However the abstract approach to inverse problems has one serious limitation: as currently conceived it cannot reflect the fundamental difference between one

Working with Sergey Naboko on Boundary Triples

9

dimension and higher. It is well known, for instance, that to recover the potential in a Schrödinger operator from the M-function, in one dimension it is necessary to know M as an analytic function of the spectral parameter. In two dimensions and higher, subject to regularity hypotheses on the potential and the boundary, it suffices to know .M(λ) for a single value of .λ. The reason for this difference can be understood from the Alessandrini identity [2].

References 1. H. Abels, G. Grubb, I.G. Wood, Extension theory and Kre˘ın-type resolvent formulas for nonsmooth boundary value problems. J. Funct. Anal. 266(7), 4037–4100 (2014) 2. G. Alessandrini, Stable determination of conductivity by boundary measurements. Appl. Anal. 27(1–3), 153–172 (1988) 3. Y. Arlinskii, Abstract boundary conditions for maximal sectorial extensions of sectorial operators. Math. Nachr. 209, 5–36 (2000) 4. J. Behrndt, M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains. J. Funct. Anal. 243, 536–565 (2007) 5. J. Behrndt, J. Rohleder, An inverse problem of Calderón type with partial data. Comm. Partial Differential Equations 37(6), 1141–1159 (2012) 6. J. Behrndt, S. Hassi, H. de Snoo , Boundary Value Problems, Weyl Functions, and Differential Operators. Monographs in Mathematics, vol. 108 (Springer, Birkhäuser, 2020) 7. B.M. Brown, G. Grubb, I.G. Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Math. Nachr. 282(3), 314–347 (2009) 8. B.M. Brown, M. Klaus, M. Malamud, V. Mogilevskii, I. Wood, Weyl solutions and j selfadjointness for Dirac operators. J. Math. Anal. Appl. 480(2), 123344, 33 (2019) 9. B.M. Brown, M. Marletta, Spectral inclusion and spectral exactness for singular non-selfadjoint Hamiltonian systems. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2036), 1987–2009 (2003) 10. B.M. Brown, M. Marletta, Spectral inclusion and spectral exactness for PDEs on exterior domains. IMA J. Numer. Anal. 24(1), 21–43 (2004) 11. B.M. Brown, M. Marletta, S. Naboko, I. Wood, Boundary triplets and M-functions for nonselfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. London Math. Soc. 77(2), 700–718 (2008) 12. V.A. Derkach, M.M. Malamud, On the Weyl function and Hermitian operators with gaps. Soviet Math. Doklady 35, 393–398 (1987) 13. V.A. Derkach, M.M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95, 1–95 (1991) 14. C. Fischbacher, The nonproper dissipative extensions of a dual pair. Trans. Am. Math. Soc. 370, 8895–8920 (2018) 15. C. Fischbacher, S. Naboko, I. Wood, The proper dissipative extensions of a dual pair. Integr. Equ. Oper. Theory 85(4), 573–599 (2016) 16. F. Gesztesy, M. Mitrea, A description of all selfadjoint extensions of the Laplacian and Kreintype resolvent formulas in nonsmooth domains. J. Anal. Math. 113, 53–172 (2011) 17. V.I. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Operator Differential Equations (Kluwer, Dordrecht, 1991) 18. G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Sc. Norm. Super. Pisa Cl. Sci. 22(3), 425–513 (1968) 19. A.N. Koˇcube˘ı, Extensions of symmetric operators and symmetric binary relations. Mathematical Notes of the Academy of Sciences of the USSR 17(1), 25–28 (1975)

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20. N.D. Kopachevski˘ı, S.G. Kre˘ın, An abstract Green formula for a triple of Hilbert spaces, and abstract boundary value and spectral problems. (Russian) Ukr. Mat. Visn. 1(1), 69–97 (2004); translation in Ukr. Math. Bull. 1(1), 77–105 (2004) 21. H. Langer, B. Textorius, Generalized resolvents of dual pairs of contractions, in Invariant Subspaces and Other Topics: 6th International Conference on Operator Theory, Timisoara and Herculane (Romania) (Timi¸soara, Herculane, 1981). Operator Theory: Advances and Applications, vol. 6 (Birkhäuser, Basel, 1982), pp. 103–118 22. V.E. Lyantze, O.G. Storozh, Methods of the Theory of Unbounded Operators (Russian) (Naukova Dumka, Kiev, 1983) 23. M.M. Malamud, On some classes of extensions of sectorial operators and dual pairs of contractions, in Recent Advances in Operator Theory (Groningen, 1998). Operator Theory: Advances and Applications, vol. 124 (Birkhäuser, Basel, 2001), pp. 401–449 24. M.M. Malamud, V.I. Mogilevskii, On extensions of dual pairs of operators. Dopov. Nats. Akad. Nauk Ukr. 1, 30–37 (1997) 25. M.M. Malamud, V.I. Mogilevskii, Kre˘ın type formula for canonical resolvents of dual pairs of linear relations. Methods Funct. Anal. Topology 8(4), 72–100 (2002) 26. S. Naboko, I. Wood, The detectable subspace for the Friedrichs model: applications of Toeplitz operators and the Riesz-Nevanlinna factorisation theorem, in Annales Henri Poincaré, vol. 21(10) (2020), pp. 3141–3156 27. V. Ryzhov, Functional model of a class of non-selfadjoint extensions of symmetric operators. Oper. Theory: Adv. Appl. 174, 117–158 (2007) 28. V. Ryzhov, A general boundary value problem and its Weyl function. Opuscula Math. 27(2), 305–331 (2007) 29. O.G. Storozh, On some analytic and asymptotic properties of the Weyl function of a nonnegative operator. (Ukrainian) Mat. Metodi Fiz.-Mekh. Polya 43(4), 18–23 (2000) 30. L.I. Vainerman, On extensions of closed operators in Hilbert space. Mathematical notes of the Academy of Sciences of the USSR 28, 871–875 (1980) 31. V.I. Vishik, On general boundary value problems for elliptic differential operators. Trudy Mosc. Mat. Obsv. 1, 187–246 (1952); Am. Math. Soc. Transl. 24(2), 107–172 (1963) 32. H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68(2), 220–269 (1910)

Operator-Valued Nevanlinna–Herglotz Functions, Trace Ideals, and Sergey Naboko’s Contributions Fritz Gesztesy

Dedicated to the memory of Sergey Naboko .(1950–2020 .): A virtuoso at the interface of operator and spectral theory and complex analysis.

Abstract After reviewing basic facts on bounded operator-valued Nevanlinna– Herglotz functions, we point out some of the highlights of Sergey Naboko’s results in connection with nontangential boundary values to the real axis for Nevanlinna– Herglotz functions taking values in various trace ideals. Keywords Operator-valued Nevanlinna–Herglotz functions · Trace ideals

1 Introduction Sergey Naboko was an exceptionally gifted mathematician whose work fundamentally influenced those working in spectral and scattering theory, operator theory, complex and harmonic analysis. He particularly touched all those who were fortunate to be counted among his many collaborators; we miss him dearly. The principal topic of this note centers around Sergey Naboko’s remarkable contributions to the theory of bounded operator-valued Nevanlinna–Herglotz functions lying in various trace ideals and their nontangential limits (resp., nonexistence thereof) to the real axis. Particular emphasis is given to the sharp transitions in connection with the Schatten–von Neumann .p -based trace ideals .Bp (H) as p

F. Gesztesy () Department of Mathematics, Baylor University, Waco, TX, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_2

11

12

F. Gesztesy

varies according to .p ∈ (0, 1), .p = 1, and .p ∈ (1, ∞), as amply demonstrated in [56–63]. In short, Sergey Naboko obtained the following fundamental results: (i) If .p ∈ (0, 1) and .M( · ) is a .Bp (H)-valued Nevanlinna–Herglotz function, then for (Lebesgue) a.e. .λ ∈ R, .M( · ) has nontangential boundary values .M(λ ± i0) in .Bp (H). (ii) If .p ∈ (1, ∞), there exist .Bp (H)-valued Nevanlinna–Herglotz functions .M( · ), such that for a.e. .λ ∈ R, .M( · ) has nontangential boundary values .M(λ ± i0) which are unbounded linear operators. (iii) If .M( · ) is a .B1 (H)-valued Nevanlinna–Herglotz function, then for a.e. .λ ∈ R, .M( · ) has nontangential boundary values .M(λ ± i0) in .Bp (H) for all .p ∈ (1, ∞). (iv) If a bounded operator-valued Nevanlinna–Herglotz function .M( · ) is the product of a trace class operator and an operator belonging to the Matsaev ideal, then for a.e. .λ ∈ R, .M( · ) has nontangential boundary values .M(λ ± i0) in .B1 (H). Basic facts on bounded operator-valued Nevanlinna–Herglotz functions are summarized in Sect. 2 and a variety of Sergey Naboko’s theorems for trace ideal valued Nevanlinna–Herglotz functions in addition to items .(i)–.(iv) above are surveyed in Sect. 3. Finally, we briefly comment on the notation used in this paper: Throughout, .H denotes a separable, complex Hilbert space with inner product and norm denoted by .( · , · )H (linear in the second argument) and . · H , respectively. The identity operator in .H is written as .IH . We denote by .B(H) (resp., .B∞ (H)) the Banach space of linear bounded (resp., compact) operators on .H and by .Bp (H) the .p based Schatten–von Neumann ideals of .B(H), .p ∈ (0, ∞) (see Sect. 3). The kernel (null space) of a linear operator will be denoted by .ker(·). By .B(R) we denote the collection of Borel subsets of .R, and we abbreviate .C± = {z ∈ C | ± Im(z) > 0}.

2 Basic Facts on Bounded Operator-Valued Nevanlinna–Herglotz Functions We start by reviewing some of the basic facts on (bounded) operator-valued Nevanlinna–Herglotz functions (also called Nevanlinna, Pick, R-functions, etc.). Definition 2.1 The map .M : C+ → B(H) is called a bounded operator-valued Nevanlinna–Herglotz function on .H (in short, a bounded Nevanlinna–Herglotz operator on .H) if M is analytic on .C+ and .Im(M(z)) ≥ 0 for all .z ∈ C+ . Here we follow the standard notation .

Im(M) = (M − M ∗ )/(2i),

Re(M) = (M + M ∗ )/2,

M ∈ B(H).

(2.1)

Operator-Valued Nevanlinna–Herglotz Functions

13

One notes that M is a bounded Nevanlinna–Herglotz operator-valued function if and only if the scalar-valued functions .(u, Mu)H are Nevanlinna–Herglotz for all .u ∈ H. As in the scalar case one usually extends M to .C− by reflection, that is, by defining M(z) = M(z)∗ ,

.

z ∈ C− .

(2.2)

  Hence M is analytic on .C\R, but .M C and .M C , in general, are not analytic − + continuations of each other. In contrast to the scalar case, one cannot generally expect strict inequality in .Im(M(·)) ≥ 0. However, the kernel of .Im(M(·)) has the following simple properties recorded in [37, Lemma 5.3] (whose proof was kindly communicated to us by Dirk Buschmann) in the matrix-valued context. The proof extends to the present operatorvalued setting as has been established in [38, Lemma A.2](see also [29, Proposition 1.2 .(ii)] for additional results of this kind): Lemma 2.2 Let .M(·) be a .B(H)-valued Nevanlinna–Herglotz function. Then the kernel .H0 = ker(Im(M(z))) is independent of .z ∈ C\R. Consequently, upon decomposing .H = H0 ⊕ H1 , .H1 = H0⊥ , .Im(M(·)) takes on the form  0 0 , . Im(M(z)) = 0 N1 (z) 

z ∈ C+ ,

(2.3)

where .N1 (·) ∈ B(H1 ) satisfies N1 (z) ≥ 0,

.

ker(N1 ) = {0},

z ∈ C+ .

(2.4)

Next we recall the definition of a bounded operator-valued measure (see, also [14, p. 319], [50, 67]): Definition 2.3 Let .H be a separable, complex Hilbert space. A map .Σ : B(R) → B(H), with .B(R) the Borel .σ -algebra on .R, is called a bounded, nonnegative, operator-valued measure if the following conditions .(i) and .(ii) hold: .(i) .Σ(∅) = 0 and .0 ≤ Σ(B) ∈ B(H) for all .B ∈ B(R). (ii) .Σ(·) is strongly countably additive (i.e., with respect to the strong operator topology in .H), that is,

.

Σ(B) = s-lim

.

N →∞

N 

Σ(Bj )

(2.5)

j =1

whenever B =



j ∈N

Bj , with Bk ∩ B = ∅ for k = , Bk ∈ B(R), k,  ∈ N.

14

F. Gesztesy

Σ(·) is called an .(operator-valued .) spectral measure (also, an orthogonal operator-valued measure) if additionally the following condition .(iii) holds: 2 .(iii) .Σ(·) is projection-valued (i.e., .Σ(B) = Σ(B), .B ∈ B(R)) and .Σ(R) = IH . .(iv) Let .f ∈ H and .B ∈ B(R). Then the vector-valued measure .Σ(·)f has finite variation on B, denoted by .V (Σf ; B), if .

V (Σf ; B) = sup

 N

.

 Σ(Bj )f H < ∞,

(2.6)

j =1

where the supremum is taken over all finite sequences .{Bj }1≤j ≤N of pairwise disjoint subsets on .R with .Bj ⊆ B, .1 ≤ j ≤ N. In particular, .Σ(·)f has finite total variation if .V (Σf ; R) < ∞. We recall that due to monotonicity considerations, taking the limit in the strong operator topology in (2.5) is equivalent to taking the limit with respect to the weak operator topology in .H. For relevant material in connection with the following result we refer the reader, for instance, to [1, 3, 4, 7, 8], [14, Sect. VI.5,], [18, Sect. I.4], [22–24, 27– 29, 44, 46, 47], [48, Lemmas 2.9, 2.10], [49, 50, 55, 64–66, 73, 76], and the detailed bibliography in [38] and [40]. Theorem 2.4 ([4, 18], Sect. I.4, [73]) Let M be a bounded operator-valued Nevanlinna–Herglotz function in .H. Then the following assertions hold .: .(i) For each .f ∈ H, .(f, M(·)f )H is a .(scalar.) Nevanlinna–Herglotz function. (ii) Suppose that .{ej }j ∈N is a complete orthonormal system in .H and that for some subset of .R having positive Lebesgue measure, and for all .j ∈ N, .(ej , M(·)ej )H has zero normal limits. Then .M ≡ 0. .(iii) There exists a bounded, nonnegative .B(H)-valued measure .Ω on .R such that the Nevanlinna representation .

ˆ

dΩ(λ) 1 + λz . R 1 + λ2 λ − z

ˆ 1 λ dΩ(λ) , z ∈ C+ , . − = C + Dz + λ − z 1 + λ2 R ˆ λ+ε  Ω((−∞, λ]) = s-lim dΩ(t) (t 2 + 1)−1 , λ ∈ R, .

M(z) = C + Dz +

.

ε↓0

−∞

 Ω(R) = Im(M(i)) − D = C = Re(M(i)),

ˆ

η↑∞

(2.8) (2.9)

dΩ(λ) (λ2 + 1)−1 ∈ B(H), .

(2.10)

1 M(iη) ≥ 0, iη

(2.11)

R

D = s-lim

(2.7)

 = holds in the strong sense in .H. Here .Ω(B) B(R).

´

2 −1 dΩ(λ), .B ∈ B 1+λ

Operator-Valued Nevanlinna–Herglotz Functions

15

(iv) Let .λ1 , λ2 ∈ R, .λ1 < λ2 . Then the Stieltjes inversion formula for .Ω reads

.

Ω((λ1 , λ2 ])f = π −1 s-lim s-lim

ˆ

λ2 +δ

.

δ↓0

ε↓0

λ1 +δ

dλ Im(M(λ + iε))f,

f ∈ H. (2.12)

(v) Any isolated poles of M are simple and located on the real axis, the residues at poles being nonpositive bounded operators in .B(H). .(vi) For all .λ ∈ R, .

.

s-lim ε Re(M(λ + iε)) = 0, .

(2.13)

ε↓0

Ω({λ}) = s-lim ε Im(M(λ + iε)) = −i s-lim εM(λ + iε). ε↓0

ε↓0

(2.14)

(vii) If in addition .M(z) ∈ B∞ (H), .z ∈ C+ , then the measure .Ω in (2.8) is countably additive with respect to the .B(H)-norm, and the Nevanlinna representation (2.8) and the Stieltjes inversion formula (2.12) as well as (2.13), (2.14) hold with the limits taken with respect to the . · B(H) -norm. .(viii) Let .f ∈ H and assume in addition that .Ω(·)f is of finite total variation. Then for a.e. .λ ∈ R, the normal limits .M(λ + i0)f exist in the strong sense and .

.

s-lim M(λ + iε)f = M(λ + i0)f = H (Ω(·)f )(λ) + iπ Ω  (λ)f, ε↓0

(2.15)

where .H (Ω(·)f ) denotes the .H-valued Hilbert transform ˆ H (Ω(·)f )(λ) = p.v.

∞

.

−∞

dΩ(t)f

1 = s-lim δ↓0 t −λ

ˆ |t−λ|≥δ

dΩ(t)f

1 . t −λ (2.16)

As usual, the normal limits in Theorem 2.4 can be replaced by nontangential ones. Using an approach based on operator-valued Stieltjes integrals, a special case of Theorem 2.4 was proved by Brodskii [18, Sect. I.4]. In particular, he proved the analog of the Herglotz representation for operator-valued Caratheodory functions. More precisely, if F is analytic on .D (the open unit disk in .C) with nonnegative real part .Re(F (w)) ≥ 0, .w ∈ D, then F is of the form ‰ F (w) = i Im(F (0)) + .

Re(F (0)) = Υ (∂ D),

dΥ (ζ ) ∂D

ζ +w , ζ −w

w ∈ D,

(2.17)

16

F. Gesztesy

with .Υ a bounded, nonnegative .B(H)-valued measure on .∂ D. The result (2.17) can also be derived by an application of Naimark’s dilation theory (cf. [4] and [32, p. 68]), and it can also be used to derive the Nevanlinna representation (2.7), (2.8) (cf. [4], and in a special case also [18, Sect. I.4]). Finally, we also mention that Shmul’yan [73] discusses the Nevanlinna representation (2.7), (2.8); moreover, certain special classes of Nevanlinna functions, isolated by Kac and Krein [45] in the scalar context, are studied by Brodskii [18, Sect. I.4] and Shmul’yan [73]. The next result recalls that .Im(M(z)) is boundedly invertible for all .z ∈ C+ if and only if this property holds for some .z0 ∈ C+ : Lemma 2.5 Let M be a bounded operator-valued Nevanlinna–Herglotz function in H. Then .[Im(M(z0 ))]−1 ∈ B(H) for some .z0 ∈ C+ .(resp., .z0 ∈ C− .) if and only if −1 ∈ B(H) for all .z ∈ C .(resp., .z ∈ C .). .[Im(M(z))] + − .

For a proof of Lemma 2.5 we refer, for instance, to [25, Theorem 1.2] (see also [38, Lemma A.5]). For a variety of additional spectral results in connection with operator-valued Nevanlinna–Herglotz functions we refer to [17] and [29, Proposition 1.2]. For a systematic treatment of operator-valued Nevanlinna–Herglotz families we refer to [25]. For a multitude of applications of operator-valued Herglotz functions, see, for instance, [1, 2, 5–7, 10–13, 17, 19–21, 23, 26–30, 33–40, 42, 43, 49–52, 68, 70, 71, 73], and the literature cited therein.

3 Trace Ideals and Sergey Naboko’s Contributions The question into the possible nature of the boundary values of .M(·+i0) for a.e. .λ ∈ R when .M(z) ∈ B1 (H), .z ∈ C+ , was initiated by de Branges [24] and by Birman and Èntina [15, 16]; it was subsequently taken up by Naboko [57–63] and we now turn to this circle of ideas. We start with a result due to Birman and Èntina [15, 16], see also Asano [8] (and [9, Sect. 3.4]), preceded in part by De Branges [24], that was derived in connection with their investigations in abstract scattering theory: Theorem 3.1 ([15, 16]) Let A be self-adjoint in .H, with .EA ( · ) its associated family of spectral projections, and let .D, D1 , D2 ∈ B2 (H). Then the following assertions .(i)–.(iv) hold .: (i) For .(Lebesgue .) a.e. .λ ∈ R,

.

K(λ) = d(D1 EA (λ)D2 )/dλ exists in the sense of convergence in B1 (H). (3.1)

.

(ii) For any .f ∈ H and a.e. .λ ∈ R, the strong derivative

.

d(DEA (λ)f )/dλ exists.

.

(3.2)

Operator-Valued Nevanlinna–Herglotz Functions

17

(iii) The limits

.

lim D1 (A − (λ ± iε)IH )−1 D2 := D1 (A − (λ ± i0)IH )−1 D2 ε↓0

.

(3.3)

exist in B2 (H)-norm for a.e. λ ∈ R. Moreover, .

  lim D1 (A − (λ + iε)IH )−1 D2 − D1 (A − (λ − iε)IH )−1 D2 ε↓0

  = D1 (A − (λ + i0)IH )−1 D2 − D1 (A − (λ − i0)IH )−1 D2

(3.4)

exists in B1 (H)-norm for a.e. λ ∈ R, and, for a.e. .λ ∈ R,   D1 (A − (λ + i0)IH )−1 D2 − D1 (A − (λ − i0)IH )−1 D2

.

= 2π iK(λ) ∈ B1 (H).

(3.5)

(iv) As .ε ↓ 0, the strong limits .s-limε↓0 D(A − (λ ± iε)IH )−1 f exist for any .f ∈ H and a.e. .λ ∈ R. Moreover,

.

.

s-lim D[(A − (λ ± iε)IH )−1 − (A − (λ ∓ iε)IH )−1 ]f ε↓0

= ±2π id(DEA (λ)f )/dλ.

(3.6)

Generally, the set of measure zero of values of .λ ∈ R in items .(ii) and .(iv) depends on the choice of .f ∈ H. In the following we turn to Naboko’s extraordinary extensions of these results to trace ideals beyond the trace (resp., Hilbert–Schmidt) class. To set the stage we recall some basic facts on Schatten–von Neumann ideals of .B(H). If T is compact on .H, that is, .T ∈ B∞ ((H), its singular values, denoted by .sj (T ), .j ∈ J , are given as the eigenvalues of .(T ∗ T )1/2 (choosing the unique nonnegative square root of .T ∗ T ≥ 0), ordered in nonincreasing manner, .s1 (T ) ≥ s2 (T ) ≥ · · · , with .J = J (T ) ⊆ N an appropriate index set. In case there are only finitely many nonzero singular values .s1 (T ), . . . , sN (T ), one agrees to employ the convention .sN +k (T ) = 0, .k ∈ N, and hence we will without loss of generality employ .J = N in the following. Since

.

sj (ST ) ≤ SB(H) sj (T ),

sj (T S) ≤ SB(H) sj (T ),

j ∈ N,

S ∈ B(H), T ∈ B∞ (H),

(3.7)

18

F. Gesztesy

the associated .p -based two-sided trace ideals of .B(H), .p ∈ (0, ∞), are then given by      sj (T )p < ∞ , Bp (H) = T ∈ B∞ (H)  j ∈N

.

p ∈ (0, ∞).

(3.8)



p 1/p = T  Moreover, . Bp (H) represents a norm if and only if j ∈N sj (T ) .p ∈ [1, ∞) (in particular, .Bp (H), .p ∈ [1, ∞) are symmetrically normed ideals), whereas .Bp (H) are quasi-normed spaces for .p ∈ (0, 1). Then (cf. [41, Sect. III.12], [72, Ch. IV], [74, Ch. 3]), B1 (H)∗  B(H),

Bp (H)∗  Bq (H), p ∈ (1, ∞) ∪ {∞}, p−1 + q −1 = 1. (3.9)

.

We also recall the Matsaev class .Bω (H), a separable, symmetrically normed ideal of .B(H), defined by      sj (T )/j = T Bω (H) < ∞ , Bω (H) = T ∈ B∞ (H)  j ∈N

.

(3.10)

and its dual, .BΩ (H), the nonseparable, symmetrically normed ideal of .B(H) given by        j j  BΩ (H) = T ∈ B∞ (H)  sup sk (T ) 1/ = T BΩ (H) < ∞ . j ∈N k=1 =1  Bω (H)∗ . (3.11) Then (cf. [41, Sect. III.15]), Bp (H) ⊂ Bω (H), p ∈ [1, ∞), B1 (H) ⊂ BΩ (H) ⊂ Bp (H), p ∈ (1, ∞) ∪ {∞}. (3.12)

.

Moreover, and more generally, given a binormalizing sequence .π = {πj }j ∈N (cf. [41, Sect. III.14]) such that π = {πj }j ∈N is nonincreasing, π1 = 1,  . πj = ∞, lim πj = 0, j →∞ j ∈N

(3.13)

Operator-Valued Nevanlinna–Herglotz Functions

19

one introduces the Lorentz class, .Bπ (H), a separable, symmetrically normed ideal of .B(H),      s .Bπ (H) = T ∈ B∞ (H)  (T )π = T  < ∞ , (3.14) j j B π (H )  j ∈N and its dual, .BΠ (H), the Marcinkiewicz class, a nonseparable, symmetrically normed ideal of .B(H) (cf. [41, Sects. III.14, III.15]),        j j   BΠ (H) = T ∈ B∞ (H)  sup sk (T ) π = T BΠ (H) < ∞ . j ∈N k=1 =1  Bπ (H)∗ . (3.15) In particular, .Bω (H) = Bπ (H), .BΩ (H) = BΠ (H) in the special case where .π = {1/j }j ∈N . Returning to the operator-valued Nevanlinna–Herglotz functons .M( · ) described in (2.7)–(2.11), we recall (cf. [77]) that as a consequence of Sz.-Nagy’s dilation theory (see, e.g., [75, Chs. I, II]), .M( · ) can be represented in the form,  M(z) = C + Dz + B 1/2 (IK + zA)(A − zIK )−1 B 1/2 H 

. = C + (B + D)z + 1 + z2 B 1/2 (A − zIK )−1 B 1/2 H ,

(3.16)

z ∈ C+ , where C = C∗,

.

D ≥ 0,

C, D ∈ B(H),

(3.17)

and .K ⊇ H is an auxiliary complex, separable Hilbert space such that A = A∗ , B = B ∗ ≥ 0 in K, B|KH = 0, B|H ∈ B(H),

.

(3.18)

with .B 1/2 ≥ 0 the unique nonnegative square root of .B ≥ 0. Since .M(i) = C + (B + D)i, if .S(H) is a trace ideal of .B(H) satisfying the monotonicity property (i.e., .0 ≤ S1 ∈ S(H), .0 ≤ S2 ≤ S1 , .S2 ∈ B(H), then .S2 ∈ S(H)), the property .M(i) ∈ S(H) implies .B, C, D ∈ S(H), and hence .M(z) ∈ S(H), .z ∈ C+ . Thus, in order to study nontangential boundary values to the real axis of .M( · ) in the trace ideal .S(H), one may, without loss of generality, study of operators of the restricted type,  N (z) = B 1/2 (A − zIK )−1 B 1/2 H ,

.

z ∈ C+ ,

(3.19)

20

F. Gesztesy

where A = A∗ , B = B ∗ ≥ 0 in K, B|KH = 0, B|H ∈ S(H).

.

(3.20)

To describe some of Naboko’s spectacular results in this context, we start with a general Fatou-type theorem: Given an analytic family .T ( · ) of .B(H)-valued functions on .C+ , one calls .T ( · ) nontangentially bounded at .λ0 ∈ R, if, for some .α > 0, .h > 0, there exists a cone .Cα,h (λ0 ) ⊂ C+ with vertex at .λ0 , Cα,h (λ0 ) = {z = x + iy | y ∈ (0, h); |x − λ0 | < αy},

.

(3.21)

such that sup

.

z∈Cα,h (λ0 )

T (z)B(H) < ∞.

(3.22)

Theorem 3.2 ([61]) Let .T ( · ) be a .B(H)-valued analytic function on .C+ , nontangentially bounded a.e. on .R. (i) Then there exists a measurable operator .T (μ) ∈ B(H), .μ ∈ R, such that for a.e. .λ ∈ R, .s-lim z→λ T (z) = T (λ) for any cone .Cα,h (λ) ⊂ C+ with vertex

.

z∈Cα,h (λ)

at .λ ∈ R. .(ii) Assume in addition that .T (z) ∈ B∞ (H), .z ∈ C+ and .T (λ) ∈ B∞ (H) for a.e. .λ ∈ R. If .[IH + T (z0 )]−1 ∈ B(H) for some .z0 ∈ C+ .(and hence1 for all .z0 ∈ C+ except for a countable set of isolated points accumulating only on .R.), then .[IH + T ( · )]−1 is nontangentially bounded a.e. on .R. In this context we also refer to [48, Lemma 6.2], [78, Theorem 1.8.5]. The following result shows that the condition .T (λ) ∈ B∞ (H) for a.e. .λ ∈ R in Theorem 3.2 .(ii) is crucial and the result fails without this assumption: Theorem 3.3 ([61]) There exists a .B(H)-valued Nevanlinna–Herglotz function T ( · ) on .C+ that satisfies the following conditions .(i)–.(v) .:

.

supz∈C+ T (z)B(H) < ∞. For all .p ∈ (1, ∞), .T (z) ∈ Bp (H), .z ∈ C+ . For all .z ∈ C+ , .[IH + T (z)]−1 ∈ B(H). −1 β, .z ∈ C , for some self-adjoint operators .β, A in .H .T (z) = β(A − zIK ) + and .K, respectively .(with .K ⊇ H .), and .β 2 ∈ Bp (H) for any .p ∈ (1, ∞). .(v) The operator .[IH + T (λ)] is not boundedly invertible for a.e. .λ ∈ R, in particular, .[IH + T ( · )]−1 is not nontangentially bounded a.e. on .R.

.(i) (ii) .(iii) .(iv) .

1 This

.

follows from the analytic Fredholm theorem, see [18, App. II], [69, Theorem VI.14], [78, Theorem 1.8.2], which asserts in addition that the principal part at a pole of .[IH +T ( · )]−1 consists of finite-rank operators.

Operator-Valued Nevanlinna–Herglotz Functions

21

The next result shows that for .p ∈ (1, ∞), normal boundary values to the real axis of .Bp (H)-valued Nevanlinna–Herglotz functions in general do not exist and can be unbounded operators a.e. on .R. Theorem 3.4 ([58, 62]) Let .0 ≤ B = B ∗ ∈ Bp (H) for some .p ∈ (1, ∞). Then there exists a self-adjoint operator A in .H such that N (z) = B 1/2 (A − zIH )−1 B 1/2 ,

.

z ∈ C+

(3.23)

(again, .0 ≤ B 1/2 the unique nonnegative square root of .B ≥ 0 .), has weak boundary values .N (λ±i0) that are unbounded operators for a.e. .λ ∈ R in the following sense: There exists a fixed dense set of vectors .ϕ ∈ H such that the limit

.

.

lim(ϕ, N (λ ± iε)ϕ)H = (ϕ, N (λ ± i0)ϕ)H ε↓0

(3.24)

exists. Naboko’s next result shows that the situation changes drastically if .p ∈ (1, ∞) is changed into .p ∈ (0, 1): Theorem 3.5 ([61, 62]) Let .p ∈ (0, 1) and suppose that .M( · ) ∈ Bp (H) is a Nevanlinna–Herglotz function. Then for a.e. .λ ∈ R, the nontangential boundary values .M(λ ± i0) exist in the topology of .Bp (H). Next, we turn to the most intricate case .p = 1: Theorem 3.6 ([57, 62]) Suppose that .M( · ) ∈ B1 (H) is a Nevanlinna–Herglotz function of the type (3.19), (3.20). Then for a.e. .λ ∈ R, the nontangential boundary values .M(λ ± i0) exist in .Bp (H)-norm for all .p ∈ (1, ∞), in particular, for a.e. .λ ∈ R,  .M(λ ± i0) ∈ Bp (H). (3.25) p∈(1,∞)

Theorem 3.7 ([57, 58]) For every positive sequence .d = {dj }j ∈N such that .dj ↓ 0 as .j → ∞, there exist self-adjoint operators .0 ≤ α and A in .H, with .α 2 ∈ B1 (H), such that for .N (z) = α(A − zIH )−1 α, .z ∈ C+ , its nontangential boundary values satisfy for a.e. .λ ∈ R,

.

lim sup j →∞

 j

 [dj ln(j + 1)] = ∞.

sk (N (λ ± i0))

(3.26)

k=1

Thus, the nontangential boundary vaulues to the real axis of .B1 (H)-valued Nevanlinna–Herglotz functions, generally, cannot lie in .B1 (H), and cannot belong to a Marcinkiewicz class .BΠ (H) smaller than .BΩ (H)  Bω (H)∗ .

22

F. Gesztesy

Finally, in connection with Lorentz and Marcinkiewicz classes we recall the following results: Theorem 3.8 ([58, 62, 63]) Suppose that .M( · ) ∈ B1 (H) is a Nevanlinna–Herglotz function.  .(i) Then the condition . j ∈N πj /j < ∞ is necessary and sufficient for .M( · ) to have nontangential boundary limits .M(λ ± i0) in .Bπ (H)-norm for a.e. .λ ∈ R.  1/(j bj ) < ∞. .(ii) Let .b = {bj }j ∈N such that .bj ↑ ∞ as .j → ∞ and . j ∈N Then for a.e. .λ ∈ R, the nontangential boundary limits .M(λ ± i0) exist in the j

 norm .supj ∈N k=1 sk ( · ) bj .  .(iii) For every nonnegative sequence .a = {aj }j ∈N with . j ∈N aj < ∞,  .

aj

j ∈N

 j

 sk (M(λ ± i0))

ln(j + 1) < ∞ for a.e. λ ∈ R.

(3.27)

k=1

(iv) For a.e. .λ ∈ R,

.

.

lim inf j sj (M(λ ± i0)) = lim inf j →∞

j →∞

 j

 sk (M(λ ± i0))

ln(j + 1) = 0.

k=1

(3.28) We conclude with a result exhibiting trace class nontangential boundary values to the real axis: Theorem 3.9 ([58, 63]) Suppose that .N ( · ) ∈ B1 (H) is a Nevanlinna–Herglotz function of the type (3.19), (3.20). In addition, suppose that .B|H ∈ B1 (H)×Bω (H), that is, .B|H is the product of a trace class operator and an operator in the Matsaev class. Then for a.e. .λ ∈ R, the nontangential boundary values .N (λ ± i0) exist in .B1 (H)-norm. Remark 3.10 (i) While we focused on Nevanlinna–Herglotz functions of the type .B 1/2 (A −  −1 1/2 zIK ) B H , .z ∈ C+ , and its nontangential boundary values .B 1/2 (A−(λ±  i0)IK )−1 B 1/2 H , for a.e. .λ ∈ R, where .0 ≤ B and A are self-adjoint in .K ⊇ H, and .B|KH = 0, it is clear  that these considerations extend to operators of the form .α ∗ (A − zIK )−1 α H and their nontangential boundary values to the real line by dropping nonnegativity and self-adjointness of .α ∈ B(K), given ∗ .α α ∈ S, a trace ideal. Moreover, employing a polarization-type identity [16,  p. 396], this further extends to operators of the form .β ∗ (A − zIK )−1 α H , ∗ ∗ .α, β ∈ B(K), .α α, β β ∈ S, and their nontangential boundary values to the real line.  ∗ −1 α  to spectral and scattering problems (via .(ii) The relevance of .β (A − zIK ) H Birman–Schwinger-type operators), is self-evident. While Theorems 3.2–3.9 .

Operator-Valued Nevanlinna–Herglotz Functions

23

only capture the proverbial tip of the iceberg of Naboko’s various results on existence and completeness of wave operators, on the Friedrichs model, on functional models, and on characteristic functions of non-self-adjoint operators, etc., related to the pair of operators .(A, A + αβ ∗ ), we refer to [54– 63] (see also [31, 53]) for a wealth of additional results and applications. .(iii) In view of the sharp transition of Naboko’s results described in this section regarding .0 < p < 1, .p = 1, and .1 < p < ∞, it is amusing to quote de Branges [24, p. 544] who writes the following in connection with his proof of Hilbert–Schmidt nontangential boundary values to the real axis of trace class valued Nevanlinna–Herglotz functions: “Perhaps through ignorance we find it necessary to suppose that the value of .μ are of trace class for a meaningful construction.” (In this context, see [24, eq. (2), on p. 545].) With hindsight, one concludes that de Branges’ trace class hypothesis was, in fact, the perfect setting for this problem. . Acknowledgments We are indebted to Konstantin A. Makarov, Mark Malamud, and Eduard Tsekanovskii for a variety of helpful suggestions and remarks.

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61. S.N. Naboko, On the conditions for existence of wave operators in the nonselfadjoint case, in Wave Propagation: Scattering Theory, ed. by M.Sh. Birman. American Mathematical Society Translations, Series 2, vol. 157 (1993), pp. 127–149 62. S.N. Naboko, The boundary behavior of Sp -valued functions analytic in the half-plane with nonnegative imaginary part, in Functional Analysis and Operator Theory, Banach Center Publications, vol. 30 (Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1994), pp. 277–285 63. S.N. Naboko, Zygmund’s theorem and the boundary behavior of operator R-functions. Funct. Anal. Appl. 30, 211–213 (1996) 64. F.J. Narcowich, Mathematical theory of the R matrix. II. The R matrix and its properties. J. Math. Phys. 15, 1635–1642 (1974) 65. F.J. Narcowich, R-operators II. On the approximation of certain operator-valued analytic functions and the Hermitian moment problem. Indiana Univ. Math. J. 26, 483–513 (1977) 66. F.J. Narcowich, G.D. Allen, Convergence of the diagonal operator-valued Padé approximants to the Dyson expansion. Commun. Math. Phys. 45, 153–157 (1975) 67. A.I. Plesner, V.A. Rohlin, Spectral theory of linear operators. Uspehi Matem. Nauk (N. S.) 1(11), 71–191 (1946). (Russian) Engl. transl. in Amer. Math. Soc. Transl. (2) 62, 29–175 (1967) 68. A. Posilicano, Boundary triples and Weyl functions for singular perturbations of self-adjoint operators. Meth. Funct. Anal. Topology 10, 57–63 (2004) 69. M. Reed, B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis, revised and enlarged edition (Academic Press, New York, 1980) 70. V. Ryzhov, A general boundary value problem and its Weyl function. Opuscula Math. 27, 305– 331 (2007) 71. S.N. Saakjan, Theory of resolvents of a symmetric operator with infinite defect numbers. Akad. Nauk. Armjan. SSR Dokl. 41, 193–198 (1965) (Russian) 72. R. Schatten, Norm Ideals of Completely Continuous Operators (Springer, Berlin, 1960) 73. Y.L. Shmul’yan, On operator R-functions. Siberian Math. J. 12, 315–322 (1971) 74. B. Simon, Trace ideals and their applications, in Mathematical Surveys and Monographs, vol. 120, 2nd edn. (American Mathematical Society, Providence, RI, 2005) 75. B. Sz.-Nagy, C. Foias, H. Bercovici, L. Kerchy, Harmonic Analysis of Operators on Hilbert Space, 2nd revised and enlarged edn. (Universitext, Springer, New York, 2010) 76. E.R. Tsekanovskii, Accretive extensions and problems on the Stieltjes operator-valued functions realizations, in Operator Theory and Complex Analysis, ed. by T. Ando, I. Gohberg. Operator Theory: Advances and Applications, vol. 59 (Birkhäuser, Basel, 1992), pp. 328–347 77. E.R. Tsekanovskii, Y.L. Shmul’yan, Questions in the theory of the extension of unbounded operators in rigged Hilbert spaces. J. Sov. Math. 12, 283–310 (1979) 78. D.R. Yafaev, Mathematical Scattering Theory. General Theory. Translations of Mathematical Monographs, vol. 105 (American Mathematical Society, Providence, RI, 1992)

Mathematical Heritage of Sergey Naboko: Functional Models of Non-Self-Adjoint Operators Alexander V. Kiselev and Vladimir Ryzhov

Abstract The paper surveys the area of functional models for dissipative and non-dissipative operators, and in particular the contributions made in this area by Sergey Naboko, to include: an explicit model construction, spectral analysis of the absolutely continuous subspace, the functional model approach to the scattering theory, and the work on the singular spectral subspace. We note that functional models for non-self-adjoint operators were a favourite brainchild of Sergey’s, which he devoted his time to for over 45 years of his illustrious research career. Keywords Functional models · Non-self-adjoint operators · Scattering theory

1 Dilation Theory for Dissipative Operators Functional model construction for a contractive linear operator T acting on a Hilbert space K is a well developed domain of the operator theory. Since pioneering works by B. Sz.-Nagy, C. Foia¸s [40], P. D. Lax, R. S. Phillips [16], L. de Branges, J. Rovnyak [7, 8], and M. Livšic [17], this research area attracted many specialists in operator theory, complex analysis, system control, gaussian processes and other disciplines. Multiple studies culminated in the development of a comprehensive theory complemented by various applications, see [9, 10, 29, 30, 32] and references therein. The underlying idea of functional model is the fundamental theorem of B. Sz.Nagy and C. Foia¸s stating that for a dissipative operator L under the assumption .C− ⊂ ρ(L) (dissipative operators satisfying this condition are called maximal), A. V. Kiselev () Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, UK e-mail: [email protected] V. Ryzhov Unity Technologies, San Francisco, CA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_3

27

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there exists a selfadjoint dilation of L, which is a selfadjoint operator .L on a wider space .H ⊃ K such that (L − zI )−1 = PK (L − zI )−1 |K ,

.

z ∈ C− ,

(1)

where .PK is an orthogonal projection from .H onto K. In applications, such a dilation .L should be minimal: it should not contain any reducing selfadjoint parts not related to the operator L. Mathematically the minimality condition is expressed as the equality .



clos

(L − zI )−1 |K = H,

z∈R /

where .H is the dilation space .H ⊃ K. Construction of a dilation satisfying this condition is a non-trivial task successfully solved for contractions by Sz.-Nagy and Foia¸s [40] with the help of Neumark’s theorem [28], and by B. Pavov [33, 34] for two important cases of dissipative operators arising in mathematical physics and successfully extended later to a general setting (more on this in the following sections). The functional model theory of non-selfadjoint operators studies operators L which have no non-trivial reducing selfadjoint parts. Such operators are called completely non-selfadjoint or, using a less accurate term, simple. In what follows, all non-selfadjoint operators are assumed closed, densely defined and simple, with regular points in both lower and upper half-planes.

1.1 Additive Perturbations Let .A = A∗ be a selfadjoint unbounded operator on a Hilbert space K and V a bounded (for simplicity) non-negative operator .V = V ∗ = α 2 /2 ≥ 0, where 1/2 . Let .L = A + i α 2 . The operators A and .V = α 2 /2 are the real and .α = (2V ) 2 imaginary parts of L defined on .dom(L) = dom(A). Following Pavlov, denote .E = clos ran α and define the dilation space as the direct sum of K and the equivalents of incoming and outgoing channels of the LaxPhillips scattering theory, see [16], .D± = L2 (R± , E), H = D− ⊕ K ⊕ D+ .

.

(2)

Elements of .H are represented as three-component vectors .(v− , u, v+ ) with .v± ∈ D± and .u ∈ K. The action of .L on the channels .D± is defined by .L :

, 0, iv ). The self-adjointness of .L = L ∗ and the require(v− , 0, v+ ) → (iv− + ment (1) lead to the form of dilation .L suggested in [33], ⎞ ⎛ ⎞ ⎛ − i dv v− dx ⎟ ⎜ α , (3) .L ⎝ u ⎠ = ⎝Au + 2 [v+ (0) + v− (0)]⎠ + v+ i dv dx

Functional Models of Non-Self-Adjoint Operators

29

defined on the domain .

dom(L ) = (v− , u, v+ ) ∈ H | v± ∈ W21 (R± , E), u ∈ dom(A), v+ (0) − v− (0) = iαu}

The “boundary condition” .v+ (0) − v− (0) = iαu can be interpreted as a coupling between the incoming and outgoing channels .D± , realised by the imaginary part of L acting on E. The characteristic function of L is the contractive operator-valued function defined by the formula S(z) = IE + iα(L∗ − zI )−1 α : E → E,

.

z ∈ C+ .

(4)

Owing to the general theory [40], the operator L is unitary equivalent to its model in the spectral representation of .L in accordance with (1). Due to the operator version of Fatou’s theorem [40], non-tangential boundary values of the function S exist in the strong operator topology almost everywhere on the real line. Denote .S = S(k) = s-limε↓0 S(k + iε), a. e. .k ∈ R. Similarly, let .S ∗ = S ∗ (k) := s-limε↓0 [S(k + iε)]∗ , which exists for almost all .k ∈ R. The symmetric form of the functional model is obtained by factorisation and completion of the dense linear set of vector-valued functions from the space .L2 (E) ⊕ L2 (E) with respect to the norm



   ˆ 

g˜ 2 g˜ g˜ I S∗



. dk ,

g := S I g g E⊕E H R

(5)

Note that the elements of .H are not individual functions from .L2 (E) ⊕ L2 (E), but rather equivalence classes formed after factorization over elements with zero g˜  .H –norm, followed by completion [30, 31]. It is easily seen that for each . g ∈H the expressions .g+ := S g˜ + g and .g− := g˜ + S ∗ g are in fact usual square summable vector-functions from .L2 (E).  ∗ The space .H = L2 SI SI with the norm defined by (5) turns out to be the spectral representation space of the self-adjoint dilation .L of the operator L. Henceforth we will denote the corresponding unitary mapping of .H onto .H by .Φ. It means that the operator of multiplication by the independent variable acting on .H , i.e., the operator .f (k) → kf (k), is unitary equivalent to the dilation .L .       Hence, for .z ∈ C \ R, the mapping . gg˜ → (k − z)−1 gg˜ , where . gg˜ ∈ H is unitary equivalent to the resolvent of .L and therefore L is mapped to its functional model (with the symbol . denoting unitary equivalence),   (L − zI )−1 PK (k − z)−1 

.

K

,

z ∈ C−

(6)

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The incoming and outgoing subspaces of the dilation space .H admit the form  + H2 (E) , .D+ := 0

0 , D− := H2− (E) 

K := H [D+ ⊕ D− ]

where .H2± (E) are the Hardy classes of .E-valued vector-functions analytic in .C± and .D± = ΦD± . As usual [36], the functions from vector-valued Hardy classes .H2± (E) are identified with their boundary values existing almost everywhere on the real line. They form two complementary mutually orthogonal subspaces, so that .L2 (E) = H2+ (E) ⊕ H2− (E). The image .K of K under the spectral mapping .Φ of the dilation space .H to .H is the subspace   g˜ − + ∗ .K = ∈ H | g˜ + S g ∈ H2 (E), S g˜ + g ∈ H2 (E) g The orthogonal projection .PK from .H onto .K is defined by formula (7). Note that the following definition has to be understood on the dense set of functions from 2 2 .L (E) ⊕ L (E) in .H .   g˜ g˜ − P+ (g˜ + S ∗ g) .PK = , g g − P− (S g˜ + g)

g˜ ∈ L2 (E), g ∈ L2 (E),

(7)

where .P± are the orthogonal projections from .L2 onto the Hardy classes .H2± .

2 Naboko’s Functional Model for a Family of Additive Perturbations The model approach to the analysis of dissipative operators outlined above relies exclusively on the knowledge of a characteristic function of a dissipative completely non-selfadjoint operator L. The properties of the operator are expressed in terms of its characteristic function, i. e., in the language of analytic operator-valued functions theory. This represents the true value of the functional model approach: all the abstract results obtained using model techniques become immediately available, once the characteristic function of the operator is known. Successful applications of the functional model approach for contractions and dissipative operators have inspired the search for models of non-dissipative operators. The attempts to follow the blueprints of Sz.-Nagy-Foias and Lax-Philitps meet serious challenges rooted in the absence of a self-adjoint dilation for such operators. The breakthrough came in the late seventies with the publication of papers [18, 19] and especially [20] by S. Naboko, who found a way to represent a nondissipative operator in a model space of a suitably chosen dissipative one. Apart

Functional Models of Non-Self-Adjoint Operators

31

from the model construction, his works largely contributed to the development of various areas in the non-self-adjoint operator theory. In contrast to the earlier results, his model representation does not rely on the uniqueness (up to a unitary equivalence) of the characteristic function of a completely non-selfadjont operator. Based on the dilation (3), the paper [20] provides an isometry between the dilation space (2) and the model space (5) in an explicit form. This explicitness plays a crucial rôle in passage to the model representation for non-dissipative operators using nothing more than Hilbert resolvent identities. All the building blocks of the method are clearly presented in terms of the original problem, which is especially appealing from the applications’ perspective. We next give a brief overview of the key ideas presented in [18–20].

2.1 Isometric Map Between the Dilation and Model Spaces Consider a non-self-adjoint operator L = A + iV

(8)

.

acting in the Hilbert space K, where .A = A∗ and .V = V ∗ is A-bounded with the relative bound less than 1. The domains of A and L coincide and the operator √ L is closed. Note that V can be written in the form .V = αJ2 α with .α = 2|V |, .J := sign V : E → E defined according to the functional calculus of self-adjoint operators. Like in (4), .E := clos ran α. The characteristic function of L admits the form (see, e.g., [39]) Θ(z) = IE + iJ α(L∗ − zI )−1 α : E → E,

.

z ∈ ρ(L∗ ).

(9)

Alongside with L introduce the dissipative operator .L || on the same domain .dom(L || ) = dom(L) as follows: L || := A + i|V | = A + i

.

α2 . 2

(10)

The operator .L || is precisely the dissipative operator of the preceding Section. The work [20] contains the model construction, the definition of the isometry .Φ : H → H from (a dense set in) the dilation space (2) to the model space (5) of .L || , which is a preliminary step towards the model for its additive perturbations of the form (8). Note that the characteristic function S of .L || is given by the expression (4) where L is replaced by .L || : S(z) = IE + iα(L−|| − zI )−1 α,

.

z ∈ ρ(L−|| ),

L−|| := (L || )∗ .

(11)

The argument of [18] shows that the characteristic functions of L and .L || are related via the Potapov-Ginzburg operator linear-fractional transformation, or PG-

32

A. V. Kiselev and V. Ryzhov

transform [3]. This fact is essentially geometric. It relates contractions on Kre˘ın spaces (i. e., the spaces with an indefinite metric defined by the involution .J = J ∗ = J −1 ) to contractions on Hilbert spaces. The PG-transform is invertible and the following assertion pointed out in [18] holds. Proposition 2.1 The characteristic function (9) of .L = A + iV is J -contractive on its domain and the PG-transform maps .Θ to the contractive characteristic function of .L || = A + i|V | defined by (11), as follows: Θ → S = −(χ + −Θχ − )−1 (χ − −Θχ + ),

.

S → Θ = (χ − +χ + S)(χ + +χ − S)−1 ,

(12)

where .χ ± = 12 (IE ± J ) are orthogonal projections onto the subspaces .χ + E (.χ − E, respectively). It appears somewhat unexpected that two operator-valued functions connected by formulae (12) can be explicitly written down in terms of their “main operators” L and .L−|| . This relationship between the characteristic functions of L and .L || goes in fact much deeper, see [2, 3]. In particular, the self-adjoint dilation of .L || and the J –self-adjoint dilation of L are also related via a suitably adjusted version of the PG-transform. Similar statements hold for the corresponding linear systems or “generating operators” of the functions .Θ and S, see [2, 3]. This fact is crucial for the construction of a model of a general closed, densely defined non-self-adjoint operator, see [38]. Assume as usual that the operator .L || is completely non-self-adjoint, and let .L be the minimal self-adjoint dilation of .L || of the form (3). Theorem 2.2 ([20], Theorem 2) There exists a mapping .Φ from the dilation space .H onto Pavlov’s model space .H defined by (5) with the following properties. 1. 2. 3. 4. 5.

Φ is isometric.   g˜ + S ∗ g = F+ h, .S g˜ + g = F− h, where . gg˜ = Φh, .h ∈ H −1 = (k − z)−1 ◦ Φ, .Φ ◦ (L − zI ) z∈C\R .ΦH = H , .ΦD± = D± , .ΦK = K −1 = (k − z)−1 ◦ Φ, .Φ ◦ (L − zI ) z ∈ C \ R. . .

Here the bounded maps .F± : H → L2 (R, E) are defined by the formulae 1 F+ : h → − √ α(L || − k + i0)−1 u + S ∗ (k)vˆ− (k) + vˆ+ (k), 2π 1 F− : h → − √ α(L−|| − k − i0)−1 u + vˆ− (k) + S(k)vˆ+ (k), 2π

.

where .h = (v− , u, v+ ) ∈ H and .vˆ± are the Fourier transforms of functions .v± ∈ L2 (R± , E).

Functional Models of Non-Self-Adjoint Operators

33

2.2 Model Representation of Additive Perturbations Theorem 2.2 opens a possibility of expressing a larger class of perturbations of A in the model space .H . Namely, consider operators in K of the form L = A +

.

α α , 2

dom(L ) = dom(A),

(13)

where . is a bounded operator in E. The family .{L | : E → E} includes A for = 0, the dissipative operator .L || for . = iIE , its adjoint .L−|| for . = −iIE , as well as self-adjoint and non-self-adjoint operators corresponding to other values of the “parameter” . . In particular, the non-dissipative operator .L = A + iV = A + i αJ2 α of (8) is recovered by putting . = iJ . Representations of the resolvent .(L −zI )−1 , .z ∈ ρ(L ) in the model space .H are obtained using the properties of .F± given in Theorem 2.2 and resolvent identities for .(L || −zI )−1 , .(L−|| −zI )−1 , and .(L −zI )−1 . The key component of the proofs is the representation of .F± (L − zI )−1 u in terms of .F± u for .u ∈ K. For instance, it can be shown that there exist two analytic operator-functions .Θ , Θ : E → E, bounded in .C− , .C+ respectively, such that for .z0 ∈ ρ(L ), .Im z0 < 0, and all .u ∈ K

.

F+ (L − z0 I )−1 u = − .

F− (L − z0 I )

−1

1 (F+ u)(k − i0) k − z0 1 Θ (k − i0)[Θ (z0 )]−1 (F+ u)(z0 ) k − z0

1 u= (F− u)(k + i0) k − z0 −

(14)

1 Θ (k + i0)[Θ (z0 )]−1 (F+ u)(z0 ) k − z0

Here .F± u ∈ H2∓ (E) since .u ∈ K and .(F+ u)(z0 ) = (g˜ + S ∗ g)(z0 ) is the analytic continuation of the function .(g˜ + S ∗ g) to the point .z0 in the lower half-plane. The possibility to express .F± (L − z0 I )−1 u using the spectral mappings .F± applied to .u ∈ K found on the right hand side of (14) is the key ingredient of calculations leading to the main theorem.   Theorem 2.3 (Model Theorem, [20]) If .z0 ∈ C− ∩ ρ(L ) and . gg˜ ∈ K , then Φ(L − z0 I )

.

−1

  g˜ 1 = PK Φ g k − z0 g − ∗

1+i 2

g˜   Θ z0 )]−1 (g˜ + S ∗ g)(z0 )



  If .z0 ∈ C+ ∩ ρ(L ) and . gg˜ ∈ K, then Φ(L − z0 I )−1 Φ ∗

.

  g˜ 1 g˜ − = PK k − z0 g

1−i 2

[Θ ( z0 )]−1 (S g˜ + g)(z0 ) g



34

A. V. Kiselev and V. Ryzhov

2.3 Smooth Vectors and the Absolutely Continuous Subspace In [20, 22] Sergey Naboko introduced absolutely continuous subspaces of the family L . He always admired Mark Kre˘ın, and in particular liked to quote him as saying: “the major instruments of self-adjoint spectral analysis arise from the Hilbert space geometry, whereas in the non-self-adjoint setup the modern complex analysis has to take the role of the main tool”. It is therefore not surprising that his definition of spectral subspaces is formulated in the language of complex analysis. In the functional model space .H consider two subspaces .N± defined as follows:

.

.N±

   +   g I ± i ∗ − . := ∈ H : P± χ ( g + S g) + χ (S g + g) = 0 , χ ± := 2 g

These subspaces are then characterised in terms of the resolvent of the operator L . This, again, can be seen as a consequence of a much more general argument (see, e.g., [37, 38]). Consider the counterparts of .N± in the original Hilbert space .K :

.

± := Φ ∗ PK N± , N

.

± . N± := clos N

+ ∩ N − of so-called smooth vectors and its closure e := N Now introduce the set .N e ). .Ne (L ) := clos(N The next assertion has been always singled out by S. Naboko in his lectures on functional models as “the main result of the whole lecture course”. In particular, e and opens up a it motivates the term “the set of smooth vectors” used for .N possibility to construct a rich functional calculus of the absolutely continuous “part” of the operator, leading in particular to the scattering theory (see details in the next Section). ± are described as follows: Theorem 2.4 The sets .N ± = {u ∈ H : α(L − zI )−1 u ∈ H±2 (E)}. N

.

Moreover, for the functional model image of .N˜ e the following representation holds:     g  g  ∈H : ∈H .Φ Ne = PK g g    g  g 1  −1 ∗ satisfies Φ(L − zI ) Φ PK ∀ z ∈ C− ∪ C+ . = PK ·−z g g

(15)

The above Theorem together with Theorem 2.6 motivated generalising the notion of the absolutely continuous subspace .Hac (L ) to the case of non-self-adjoint operators .L by identifying it with the set .Ne . Definition 2.5 For a non-self-adjoint .L the absolutely continuous subspace .Hac (L ) is defined by the formula .Hac (L ) = Ne (L ).

Functional Models of Non-Self-Adjoint Operators

35

In the case of a self-adjoint operator .L , .Hac (L ) is to be understood in the sense of the classical definition of the absolutely continuous subspace of a self-adjoint operator. Theorem 2.6 Assume that . = ∗ and let .α(L − zI )−1 be a Hilbert-Schmidt operator for at least one point .z ∈ ρ(L ). Then the definition .Hac (L ) = Ne is equivalent to the classical definition of the absolutely continuous subspace of a self-adjoint operator, i.e., .Ne = Hac (L ). Remark 2.7 Alternative conditions, which are even less restrictive in general, that guarantee the validity of the assertion of Theorem 2.6 were obtained in [22]. The absolutely continuous subspace of a non-self-adjoint operator also admits different definitions [37], which in generic case can be not equivalent to the one given above. This question is treated in full details by Romanov in [35].

2.4 Scattering Theory The intrinsic relationship between the scattering theory and the theory of dilations and functional models is due to [16]. The fact that the characteristic function of an arbitrary dissipative operator L can be realised as the scattering matrix of its dilation .L was observed by Adamyan and Arov in [1]. This fact, as was reiterated by Sergey on many occasions, together with Birman’s seminal works on the mathematical scattering theory, motivated his work on the construction of wave and scattering operators in the functional model representation. With the introduction of smooth vector sets which are dense in absolutely continuous subspaces of operators .L , it was natural to define (see [18, 20]) the action of exponential groups .exp (iL t) in .H as multiplication by .exp(ikt) on the smooth vectors. In view of the classical definition of the wave operator of a pair of self-adjoint operators, W± (L0 , L ) := s-lim eiL t e−iL t Pac , 0

.



t→±∞

is the projection onto the absolutely continuous subspace of .L , he where .Pac g   observed that, at least formally, for .Φ ∗ PK  g ∈ Ne one has

   g −(I + S)−1 (I + S ∗ )g ∗ = Φ PK , .W− (L , L )Φ PK g g 0



∗

(16)

and similar formulae hold for .W+ (L0 , L ), .W± (L , L0 ). The need to attribute rigorous meaning to the right hand side of the latter equality, and thus to prove the existence and completeness of wave operators, motivated Sergey to investigate the boundary behaviour of operator-valued .R−functions, see [21, 22] and references therein. This research has since found numerous applications

36

A. V. Kiselev and V. Ryzhov

in as seemingly unrelated areas as, say, the theory of Anderson localisation of stochastic differential operators. In the scattering theory (see [22]) it has allowed him to prove the classical Kre˘ın–Birman–Kuroda theorem, the invariance principle and their non-self-adjoint generalisations by following the approach sketched above. It is worth mentioning that the latter effectively blends together non-stationary, stationary and smooth formulations of the self-adjoint scattering theory.

2.5 Singular Spectrum of Non-self-adjoint Operators A major thrust of Sergey’s research was towards the analysis of singular spectral subspaces of non-self-adjoint operators. In the present section, we mention some of his results obtained in this direction. The notation throughout is as in Sects. 2.3 and 2.2, with . set to be equal to iJ with an involution J (see Sect. 2.1). To simplify the notation, we therefore consistently drop the corresponding superscripts, as in .L = L . It is further assumed throughout that the non-real spectrum of L is countable, with finite multiplicity. This latter condition holds in particular when the perturbation V is in trace class, which we will assume satisfied (similar results under less restrictive conditions are also available). The singular subspace of L is defined as follows: .Ni (L) := H Ne (L∗ ). For the operator .L∗ , it is set by .Ni (L∗ ) := H Ne (L). These definitions prove to be consistent with the classical one for self-adjoint operators due to the characterisation Ni (L) = {u ∈ K : ((L − t − iε)−1 − (L − t + iε)−1 )u, v → 0

.

as ε → 0 for all v ∈ K}. Define Θ1 (z) = χ − + S(z)χ + ,

.

Θ2 (z) = χ + + S(z)χ − ,

Θ1 (z) = χ − + S ∗ (¯z)χ + ,

Θ2 (z) = χ + + S ∗ (¯z)χ − ,

(17)

so that for the characteristic function .Θ(z) one has (cf. (12)) Θ(z) = Θ1 ∗ (¯z)(Θ2 ∗ )−1 (¯z),

.

z ∈ C+ ;

Θ(z) = Θ2∗ (¯z)(Θ1∗ )−1 (¯z),

Set  −

− +i (L) = Φ ∗ PK H2 (E) Θ1 H2 (E)) , N 0  0 i ∗  N− (L) = Φ PK H2+ (E) Θ2 H2+ (E))

.

z ∈ C− .

Functional Models of Non-Self-Adjoint Operators

37

for the operator L and similarly  −

− +i (L∗ ) = Φ ∗ PK H2 (E) Θ2 H2 (E)) , N 0  0 i ∗ ∗  N− (L ) = Φ PK H2+ (E) Θ1 H2+ (E))

.

for the operator .L∗ . The respective closures of these sets .N+i (L), N−i (L), .N+i (L∗ ) and .N−i (L∗ ) are introduced in [23]. These subspaces are invariant with respect to the resolvents of .(L−z)−1 , .(L∗ −z)−1 . It is shown that .N±i (L) can be seen as spectral for L, representing the parts of the singular spectrum pertaining to the (closed) upper and lower half-planes, respectively. In particular, eigenvectors and root vectors of +i (L) (.N −i (L), the operator L, corresponding to .z ∈ C+ (.z ∈ C− ), belong to .N respectively). The paper [23] discusses the conditions of separability of spectral subspaces under the additional condition .

sup max{χ + S(z)χ − , χ − S(z)χ + } < 1,

(18)

Im z>0

which guarantees that the “interaction” of the positive and negative “parts” of the perturbation V is “small”. This is to say that it restricts the class of operators considered to those which are not too far from an orthogonal sum of a dissipative and an anti-dissipative (.Im L ≤ 0) operators. In particular, [23] provides non-restrictive additional conditions such that Ni (L) ∩ Ne (L) = {0},

.

Ni (L) ∨ Ne (L) = K

and sharp estimates for the angle between .Ni (L) and .Ne (L). What’s more, N−i (L) ∩ N+i (L) = {0},

.

N−i (L) ∨ N+i (L) = Ni (L)

with an explicit estimate for the angle between .N−i (L) and .N+i (L). Further, .L|N i + (.L|N i ) is similar to a dissipative (anti-dissipative, respectively) operator with purely − singular spectrum. Dropping the separability condition (18) makes the spectral analysis of L much more involved. The corresponding problems were posed by S. Naboko in [24]. Most of them are still awaiting resolution, including the problem of a general spectral resolution of identity for a non-self-adjoint operator of the class considered here, but some were successfully tackled in [41] by S. Naboko and his student V. Veselov as well as in subsequent papers of V. Veselov. In particular, the named paper concerns with an in-depth study of the spectral subspace .N0i (L), introduced in [24]. The main result is formulated for .V ∈ S1 as follows: .

det Θ(z) = det ΘL|N+ (L)∨N− (L) (z),

N+ (L) ∨ N− (L) = N−i (L) ∨ Ne (L) ∨ N+i (L),

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generalising the corresponding result of Gohberg and Kre˘ın. It shows that the determinant of the characteristic function of L contains no information on the spectral subspace N0i (L) := K {N− (L∗ ) ∨ N+ (L∗ )} ⊂ Ni (L),

.

i.e., .det ΘL|

N0i (L)

(19)

(z) ≡ 1. Here in notation of Sect. 2.3 .N± (L∗ ) = N±−iJ .

The subspace .N0i is precisely the “additional” spectral subspace corresponding to the real part of the spectrum of L (in particular, it contains the eigenvectors and root vectors corresponding to real values of the spectral parameter), the analytic structure of which has no parallels in the case of dissipative operators. In a nutshell, it appears due to the interaction of the “incoming” and “outgoing” energy channels in the non-conservative system modelled by L. The rôle of .N0i for the spectral analysis of non-dissipative operators is further revealed by the following assertion: Ni (L) ∩ Ne (L) ⊂ N0i (L),

.

i.e., if the absolutely continuous and singular subspaces intersect, the intersection must lie in .N0i . It is therefore the presence of .N0i that ensures that .Ne (L∗ )∨Ni (L∗ ) = K, which prevents a spectral decomposition for the operator .L∗ . Sergey had mentioned to us, that he had seven to eight papers worth of further material on the functional model and spectral analysis of non-dissipative operators. Unfortunately, he had never published these results.

2.6 A Functional Model Based on the Strauss Characteristic Function In contrast to the model theory for contractions associated with the names of Sz.Nagy-Foia¸s and de Branges-Rovnyak, the models of unbounded non-selfadjoint operators are usually concerned with “concrete” operators arising in applications. In particular, the functional model for non-self-adjoint additive perturbations discussed above was motivated by the spectral analysis of the Schrödinger operator with a complex potential, see, e.g., [22, 23, 25, 33]. In fact, Sergey Naboko had reiterated to us on a number of occasions, that his primary concern was the spectral theory of the Schrödinger operator, rather than the development of abstract mathematical concepts: the functional model in his view was simply the tool of choice in this area. More precisely, the Schrödinger operator .−Δ + p(x) + iq(x) in .L2 (R3 ), where .p(x), .q(x) are real-valued bounded functions of .x ∈ R3 , can be written in the form (8) with the operator .α defined as .α : f → |q(x)|1/2 f , where .f ∈ L2 (R3 ). It is important to note that all the building blocks of the model construction are explicitly given in terms of the problem at hand. Indeed, both the characteristic function .S(z) and the “spectral maps” .F± are expressed via the non-real part of the

Functional Models of Non-Self-Adjoint Operators

39

complex potential (and the operator itself). The true nature of the problem’s “nonselfadjointess”, i.e., the non-triviality of the imaginary part of the potential, is thus faithfully preserved in the model representation. The same observation is valid for other model constructions of non-selfadjoint operators available in the literature, see, e.g., [6] in the present volume for the case of non-self-adjoint extensions of symmetric operators. Therefore it becomes increasingly important to express the non-selfadjointness of the problem not in abstract terms (as it is commonly done in the operator theory), but rather in terms of the concrete operator present in the problem statement. The standard way to calculate the characteristic function of a non-self-adoint operator is based on the definition given by A. Strauss in [39]. For a dissipative operator it reads as follows Definition 2.8 ([39]) Let L be a closed maximal densely defined dissipative operator on a Hilbert space K. The characteristic function of L is a bounded operator-valued analytic function .S(z) : E → E∗ , .z ∈ ρ(L∗ ), such that S(z)Γf = Γ∗ (L∗ − zI )−1 (L − zI )f,

.

f ∈ dom(L),

where the boundary operators .Γ , .Γ∗ are defined for .u, v ∈ dom(L), .u , v ∈ dom(L∗ ) by the equalities (Au, v) − (u, Av) = i(Γ u, Γ v)E ,

.

(u , A∗ v ) − (A∗ u , v ) = i(Γ∗ u , Γ∗ v )E∗

and .E := clos ran(Γ ), .E∗ := clos ran(Γ∗ ) are Hilbert spaces. According to this definition, the concrete form of the characteristic function of L depends on the choice of boundary operators .Γ , .Γ∗ . It is easy to see that for any Hilbert space isometries .π : E → E , .π∗ : E∗ → E∗ , the maps .π Γ and .π∗ Γ∗ are also boundary operators with the corresponding characteristic function .π∗ S(z)π ∗ : E → E∗ . In applications, a suitable definition of the boundary operators is determined according to the problem statement itself. For example, the operator .α of (10) (the root cause of the operator’s non-selfadointness) admits the rôle of both .Γ and .Γ∗ . Convenient boundary operators appear “naturally” in the analysis of non-self-adjoint extensions of symmetric operators as well. Once the triple .{Γ, Γ∗ , S(z)} is explicitly defined, the construction of the functional model follows the blueprint of S. Naboko [20]. A further important contribution is contained in the two recent papers [4, 5] by B.M. Brown, M. Marletta, S. Naboko, and I. Wood. The authors offer a model construction carried out in the abstract setting of Strauss’ boundary operators .Γ , .Γ∗ , resorting to no specific realisation of them. This work therefore makes all the steps of the model construction explicit, regardless of any particular form of the characteristic function, the latter to be set based on the requirements imposed by a concrete application at hand. In particular, this makes it possible to construct a functional model in the case where both the differential expression itself and the boundary conditions are non-self-adjoint, which in our view is especially relevant for topical problems of materials science.

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2.7 Applications of the Functional Model Technique Here we list some notable applications of the functional model technique, in which Sergey Naboko was involved, in addition to his work on the spectral analysis of nonself-adjoint Schrödinger operators mentioned earlier, see, e.g., [23] and references therein. 1. In [14, 15] Sergey, together with Yu. Kuperin and R. Romanov, studied the nonself-adjoint single-velocity Boltzmann transport operator. Using the functional model techniques, the absolute continuity of this operator’s continuous spectrum was proved; the similarity problem of the absolutely continuous “part” of the operator to a self-adjoint one was fully settled, and the existence of a spectral singularity at zero ascertained for a singular set of multiplication coefficients. 2. In [26, 27], together with R. Romanov, Sergey Naboko analysed the impact of spectral singularities on the asymtotic behaviour of the group of exponentials, generated by a maximal dissipative operator L. It was shown that this asymptotics allows one to recover the orders and locations of spectral singularities in the case, where their number is finite and they are of a finite power order. 3. In [11, 12], for a non-dissipative trace class perturbation L of a self-adjoint operator on K such that .N0i (L) coincides with the Hilbert space K, a generalisation of the Caley identity was obtained in the following form: there exists an outer in the upper half-plane .C+ uniformly bounded scalar analytic function .γ (λ) such that .w − limε↓0 γ (L + iε) = 0. A generalisation of this result was further obtained to the case of relative trace class perturbations. 4. In [13], the so-called matrix model was introduced and studied in some detail, i.e., a rank two non-dissipative additive perturbation L in K of a self-adjoint operator under the assumption that .K = N0i (L). This model represents the simplest possible case of a non-dissipative operator which exhibits the properties not found in any dissipative one; despite its seeming simplicity, it already includes the main analytic obstacles found in the general case. It has to be noted that this model was the favourite sandbox of Sergey; unfortunately, many results obtained by him, up to and including a von Neumann type estimate in BMO classes for functions of the operator L, have never been published.

References 1. V.M. Adamjan, D.Z. Arov, Unitary couplings of semi-unitary operators. (Russian) Mat. Issled. 1(2), 3–64 (1966); English translation in Amer. Math Soc. Transl. Ser. 2 95 (1970) 2. D.Z. Arov, Passive linear steady-state dynamical systems. Siberian Math. J. 20(1), 149–162 (1979) 3. T.Y. Azizov, I.S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric (Wiley, London, 1989) 4. B.M. Brown, M. Marletta, S. Naboko, I. Wood, The functional model for maximal dissipative operators: an approach in the spirit of operator knots. Trans. Am. Math. Soc. 373, 4145–4187 (2020)

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5. B.M. Brown, M. Marletta, S. Naboko, I. Wood, The spectral form of the functional model for maximally dissipative operators: an approach in the spirit of operator knots. Preprint 6. K.D. Cherednichenko, Y.Y. Ershova, A.. Kiselev, V.A. Ryzhov, L.O. Silva, Asymptotic analysis of operator families and applications to resonant media. In this volume 7. L. de Branges, J. Rovnyak, Square Summable Power Series (Holt, Rinehart and Winston, New York, 1966) 8. L. de Branges, J. Rovnyak, Canonical models in quantum scattering theory, in Perturbation Theory and its Applications in Quantum Mechanics, ed. by C.H. Wilcox (Wiley, New York, 1966) 9. H. Dym, H. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem (Academic Press, New York, 1976) 10. P. Fuhrmann, Linear Systems and Operators in Hilbert Space (McGraw-Hill, New York, 1981) 11. A.V. Kiselev, S.N. Naboko, Nonselfadjoint operators with an almost Hermitian spectrum: weak annihilators. (Russian); translated from Funktsional. Anal. i Prilozhen. 38(3), 39–51 (2004). Funct. Anal. Appl. 38(3), 192–201 (2004) 12. A.V. Kiselev, S.N. Naboko. Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure. Ark. Mat. 47(1), 91–125 (2009) 13. A.V. Kiselev, S.N. Naboko, Non-self-adjoint operators with almost Hermitian spectrum: matrix model. I. J. Comput. Appl. Math. 194(1), 115–130 (2006) 14. Y. Kuperin, S. Naboko, R. Romanov, Spectral analysis of the transport operator: a functional model approach. Indiana Univ. Math. J. 51(6), 1389–1425 (2002) 15. Y.A. Kuperin, S.N. Naboko, R.V. Romanov, Spectral analysis of a one-velocity transport operator, and a functional model. (Russian); translated from Funktsional. Anal. i Prilozhen. 33(3), 47–58, 96 (1999). Funct. Anal. Appl. 33(3), 199–207 (2000) 16. P.D. Lax, R.S. Phillips, Scattering theory, in Pure and Applied Mathematics, vol. 26 (Academic Press, New York-London, 1967) 17. M.S. Livšic, On spectral decomposition of linear non-self-adjoint operators. Mat. Sbornik N.S. 34(76), 145–199 (1954), MR 16:48f. In Russian; English translation in Amer. Math. Soc. Transl. (2) 5, 67–114 (1957) 18. S.N. Naboko, Absolutely continuous spectrum of a nondissipative operator, and a functional model. I. Zap. Nauˇcn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI) 65, 90–102 (1976). Investigations on linear operators and the theory of functions, VII 19. S.N. Naboko, Absolutely continuous spectrum of a nondissipative operator, and a functional model. II. Zap. Nauˇcn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI) 73, 118–135 (1977). Investigations on linear operators and the theory of functions, VIII 20. S.N. Naboko, Functional model of perturbation theory and its applications to scattering theory. Trudy Mat. Inst. Steklov. 147, 86–114, 203 (1980). Boundary Value Problems of Mathematical Physics, 10 21. S.N. Naboko, Nontangential boundary values of operator R-functions in a half-plane. Algebra i Analiz 1(5), 197–222 (1989) 22. S.N. Naboko, On the conditions for existence of wave operators in the nonselfadjoint case, in Wave propagation. Scattering theory, American Mathematical Society Translations, Series 2, vol. 157 (American Mathematical Society, Providence, RI, 1993), pp. 127–149 23. S.N. Naboko, On the singular spectrum of a nonselfadjoint operator. (Russian) Investigations on linear operators and the theory of functions, XI. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113, 149–177, 266 (1981). English translation: J. Math. Sci. 22(6), 1793– 1813 (1983) 24. S.N. Naboko, Similarity problem and the structure of the singular spectrum of non-dissipative operators. Lect. Notes Math. 1043, 147–151 (1984) 25. S.N. Naboko, On the separation of spectral subspaces of a nonselfadjoint operator. Dokl. Akad. Nauk SSSR 239(5), 1052–1055 (1978)

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26. S. Naboko, R. Romanov, Spectral singularities, Szkefalvi-Nagy-Foias functional model and the spectral analysis of the Boltzmann operator. Recent advances in operator theory and related topics (Szeged, 1999), in Operator Theory: Advances and Applications, vol. 127 (Birkhuser, Basel, 2001), pp. 473–490 27. S. Naboko, R. Romanov, Spectral singularities and asymptotics of contractive semigroups. I. Acta Sci. Math. (Szeged) 70(1–2), 379–403 (2004) 28. M. Neumark, Positive definite operator functions on a commutative group. (Russian) Bull. Acad. Sci. URSS Ser. Math. [Izvestia Akad. Nauk SSSR] 7, 237–244 (1943) 29. N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading, vol. 1, 2 (Mathematical Surveys and Monographs, AMS, 2002) 30. N.K. Nikol’skii, S.V. Khrushchev, A functional model and some problems of the spectral theory of functions. Proc. Steklov Inst. Math. 176, 101–214 (1988) 31. N.K. Nikol’skii, V.I. Vasyunin, A unified approach to function models, and the transcription problem, in The Gohberg Anniversary Collection (Calgary, AB, 1988), ed. by H. Dym et al., vol. 2. Operator Theory: Advances and Applications, vol. 41 (Birkhäuser, Basel, 1989) 32. N.K. Nikol’skii, V.I. Vasyunin, Elements of spectral theory in terms of the free function model, in Holomorphic Spaces, ed. by S. Axler et al., Mathematical Sciences Research Institute Publications, vol. 33 (Cambridge University Press, Cambridge, 1998), pp. 211–302 33. B.S. Pavlov. Selfadjoint dilation of a dissipative Schrödinger operator, and expansion in its eigenfunction. (Russian) Mat. Sb. (N.S.) 102(144), 511–536, 631 (1977) 34. B.S. Pavlov, Diation theory and the spectral analysis of non-selfadjoint differential operators, in Proceedings of the 7th Winter School, Drogobych, 1974, TsEMI, Moscow (1976), pp. 2–69. English translation: Transl., II Ser., Am. Math. Soc 115, 103–142 (1981) 35. R. Romanov, On the concept of absolutely continuous subspace for nonselfadjoint operators. J. Operator Theory 63(2), 375–388 (2010) 36. M. Rosenblum, J. Rovnyak, Hardy classes and operator theory, in Oxford Mathematical Monographs (The Clarendon Press Oxford University Press, New York, 1985). Oxford Science Publications 37. V. Ryzhov, Absolutely continuous and singular subspaces of a nonselfadjoint operator. J. Math. Sci. (New York) 87(5), 3886–3911 (1997) 38. V. Ryzhov, Functional model of a closed non-selfadjoint operator. Integr. Equ. Oper. Theory 60(4), 539–571 (2008) 39. A.V. Strauss, Characteristic functions of linear operators (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 24(1), 43–74 (1960) 40. B. Sz.-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Second enlarged edition. Universitext (Springer, New York, 2010) 41. V.F. Veselov, S.N. Naboko, The determinant of the characteristic function and the singular spectrum of a nonselfadjoint operator. Sb. Math. 57(1), 21–41 (1987)

On Crossroads of Spectral Theory with Sergey Naboko Pavel Kurasov

To the memory of Sergey, my older friend who taught me math and life, whom I miss more every day

Abstract Several recent achievements of Sergey Naboko in spectral theory of singular differential operators and metric graphs are described. The impact of Sergey’s work on my own research career is underlined. Keywords Hain-Lüst operator · Metric graphs · Wigner-von Neumann potentials · Embedded eigenvalues

The purpose of this note is not only to describe Sergey Naboko’s contribution to modern spectral theory with the emphasis on our joint publications, but also try to convey his style to do mathematics and describe what a wonderful friend and colleague we miss. Only our three common projects will be described leaving many of Sergey’s excellent contributions to spectral theory aside. Sergey was not only a deep mathematician, but also a devoted tourist. Working with him on different projects can be compared with walking along a trail in the Spectral Theory (International) Park. Sergey was always hungry trying to find new mathematical problems to solve. I have a feeling that he collected these problems throughout his life.

P. Kurasov () Department of Mathematics, Stockholm University, Stockholm, Sweden e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_4

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1 Embedded Eigenvalues Eigenvalues embedded into the continuous spectrum go back to the classical von Neumann-Wigner example [20]. Such eigenvalues appear due to a certain interplay between the frequency .β of the potential decaying like . sinxβx and the momentum .k of the free wave .e±ikx . One may probably argue that existence of such eigenvalues contradicts physical intuition, therefore Sergey’s construction of embedded dense point spectrum is a remarkable result [14, 15]. It was brought to the attention of the wider mathematical community by Barry Simon about 10 years later [18] making it one of the most cited of Sergey’s papers. Barry Simon even stated that his paper on infinite imbedded spectrum is a landmark [19]. Adding a von Neumann-Wigner potential to a certain background periodic potential, it is not a priori clear whether the embedded eigenvalues appear at the points where the frequency of the perturbation resonates with the frequency of the free waves or with the quasi-momentum determined by the periodic potential. After I posed this question it returned back to me the next day by Yuri Safarov, who mentioned it as an important problem in spectral theory of one-dimensional Schrödinger operators. Therefore in our studies with Sergey we always called it Safarov’s problem. It appeared that methods developed originally for periodic Jacobi matrices can effectively be applied to this problem showing that the position of the embedded eigenvalues is determined by the frequency of the perturbation and the quasi-momentum, not by the frequency of the free waves [8, 12]. Specific for the periodic background is that embedded eigenvalues may appear in different spectral bands simultaneously despite the perturbation of von Neumann-Wigner type containing only one frequency. I learned from Sergey not to be afraid of attacking problems without having any idea how and where to start. It was the first time when I realised the power of spectral theory of Jacobi matrices developed by Sergey. Our main focus was on discovering a new phenomenon and describing it in a clear and transparent way. I have appreciated Sergey’s ability to present difficult mathematics so that it was understandable since he lectured Functional analysis for me as a 3-rd year student at the Dept. of Mathematical Physics of Leningrad (St. Petersburg) University.

2 Magnetohydrodynamics Sergey’s interest in problems related to magnetohydrodynamics was attracted by R. Mennicken who hosted him in Regensburg for several months [3, 13]. These problems are described by matrix singular differential operators. The essential spectrum is not always obtained as a limit of the essential spectra of certain regular operators. This phenomenon appears due to a sophisticated interplay between the differential operator and the singularities of the coefficients. Sergey’s goal was to

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prepare the simplest model where such behaviour of the essential spectrum can be observed [4, 6, 7]. The model operator that we investigated was defined by the following formally symmetric matrix differential operator on the interval .[0, 1] ⎛ .

⎜ ⎜ ⎝

−

d d d β⎞ ρ(x) + q(x) dx dx dx x ⎟ ⎟. ⎠ m(x) β d − x dx x2

In the formal determinant which controls the spectrum of the whole operator the differential order of the formal product of the diagonal elements  .

−

m(x) d d ρ(x) + q(x) dx dx x2

coincides with that of the formal product of the antidiagonal elements .

d β β d  − . dx x x dx

The same holds for the singularities of the coefficients at the origin. Under certain additional conditions, the essential spectrum of any realisation of the operator cannot be obtained as the limit of the essential spectra of the restrictions of the differential operator to the interval .[, 1],  → 0. An additional branch of the essential spectrum appears due to the singularity of the differential expression at the origin. These studies affirmed my conviction that in order to understand mathematical phenomena one should look for the simplest models exhibiting these phenomena. Analysing any problem it is important to separate the difficulties and eliminate them one-by-one.

3 Operators on Metric Graphs During several years Sergey followed my studies of differential operators on metric graphs and often came with very insightful comments but he did not enter the area probably considering it too explicit. His first paper in this direction [16] was initiated by Boris Solomyak, who specially invited Sergey to study a certain partial differential equation suggested a few years earlier by Uzy Smilansky to model irreversible graphs. The interaction in this operator is determined by a confining harmonic potential and a delta interaction supported by a line. It appeared that the absolutely continuous spectrum drastically depends on the value of the coupling parameter: it is semibounded for small values of the parameter and fills in the

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real line for large values. One may refer to this phenomenon as a spectral phase transition. Spectral theory of Jacobi matrices played a decisive role in these studies. The most interesting results connected with operators on metric graphs concern relations between the spectrum and topology/geometry of the underlying graph. This was the subject of our studies with Gabriela Malenová which Sergey joined with enthusiasm [5]. I remember our long sessions calculating spectra of explicit graphs trying to understand what happens to the spectrum when an edge is attached. It transpired that, when adding a second edge between two connected vertices, the spectral gap (the difference between the two lowest eigenvalues) may go up or down depending on the length of the edge to be attached. But adding a third edge between the same vertices the spectral gap always goes down. Presenting this result, I could see physicists taking notes—a sign of the highest appreciation from their point of view. These studies led to what is now known as surgery of graphs well described in [1]. A complete answer to the question what happens when two graphs are glued together is given in [10]. Our journey through spectral theory of metric graphs continued by proving a universal estimate for the spectral gap [9]. We analysed dozens of examples and tried several elementary techniques until we realised that the estimate can be easily proven if the graph is Eulerian, i.e. contains a closed path visiting each edge precisely once. By doubling every edge any graph can be turned into an Eulerian graph with the same spectral gap. Sergey was extremely happy to understand the explicit connection between metric graphs and the problem of Seven Bridges of Königsberg. This trick gives the following universal estimate for the spectral gap of the standard Laplacian on a metric graph .Γ  π 2 λ2 (Γ ) − λ1 (Γ ) ≥ ,   L

.

=0

where .L is the total length of the graph. It appeared that the estimate was proven much earlier by Serge Nicaise [17] and later on by Leonid Friedlander [2]. We learned about that only after our manuscript was submitted for publication. Following Sergey’s suggestion we decided to write a letter to the editor admitting our oversight and explaining that our second estimate valid for Eulerian graphs  π 2 λ2 (Γ ) − λ1 (Γ ) ≥ 4   L

.

=0

is an improvement which does not follow from [2, 17] and probably would never have been discovered had we known the original proof. The paper was published opening the path for several other researchers to come up with their own proofs of the Nicaise-Friedlander-Kurasov-Naboko inequality. Our last project [11] was aimed at studying dissipative operators on graphs. During these studies Jacob Muller and I learned the theory of dissipative operators from probably the best possible teacher: Sergey was generous to give us hour-long

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lectures despite not feeling well. A completely new world suddenly opened to us. All material was presented in a clear and deep manner and all our questions were getting insightful answers. It was like returning back to the lectures given by Sergey back in 1983 in Leningrad. It was important for us to focus on connections between the spectral theory and topology/geometry illustrating how different areas of mathematics may come together. Doing mathematics with Sergey was like playing a serious game—this attitude helped him to overcome disappointments and temporary difficulties. Passing away Sergey left hundreds of pages containing records of our calculations—he did not like to throw away papers. It was important for him to be able to get back and check calculations. We tried to preserve and complete all his manuscripts.

References 1. G. Berkolaiko, J.B. Kennedy, P. Kurasov, D. Mugnolo, Surgery principles for the spectral analysis of quantum graphs. Trans. Am. Math. Soc 372(7), 5153–5197 (2019) 2. L. Friedlander, Extremal properties of eigenvalues for a metric graph. Ann. Inst. Fourier (Grenoble) 55(1), 199–211 (2005) 3. V. Hardt, R. Mennicken, S. Naboko, Systems of singular differential operators of mixed order and applications to 1-dimensional MHD problems. Math. Nachr. 205, 19–68 (1999) 4. P. Kurasov, I. Lelyavin, S. Naboko, On the essential spectrum of a class of singular matrix differential operators. II. Weyl’s limit circles for the Hain-Lüst operator whenever quasiregularity conditions are not satisfied. Proc. Roy. Soc. Edinburgh Sect. A 138(1), 109–138 (2008) 5. P. Kurasov, Pavel, G. Malenová, S. Naboko, Spectral gap for quantum graphs and their edge connectivity. J. Phys. A 46(27), 275309, 16 (2013) 6. P. Kurasov, Pavel, S. Naboko, On the essential spectrum of a class of singular matrix differential operators. I. Quasiregularity conditions and essential self-adjointness. Math. Phys. Anal. Geom. 5(3), 243–286 (2002) 7. P. Kurasov, Pavel, S. Naboko, Essential spectrum due to singularity. J. Nonlinear Math. Phys. 10(suppl. 1), 93–106 (2003) 8. P. Kurasov, Pavel, S. Naboko, Wigner-von Neumann perturbations of a periodic potential: spectral singularities in bands. Math. Proc. Cambridge Philos. Soc. 142(1), 161–183 (2007) 9. P. Kurasov, Pavel, S. Naboko, Rayleigh estimates for differential operators on graphs. J. Spectr. Theory 4(2), 211–219 (2014) 10. P. Kurasov, Pavel, S. Naboko, Gluing graphs and the spectral gap: a Titchmarsh-Weyl matrixvalued function approach. Studia Math. 255(3), 303–326 (2020) 11. P. Kurasov, Pavel, S. Naboko, J. Muller, Maximal Dissipative Operators on Metric Graphs (2023) (manuscript) 12. P. Kurasov, S. Simonov, Weyl–Titchmarsh-type formula for periodic Schrödinger operator with Wigner–von Neumann potential. Proc. Roy. Soc. Edinburgh Sect. A 143(2), 401–425 (2013) 13. R. Mennicken, S. Naboko, C. Tretter, Essential spectrum of a system of singular differential operators and the asymptotic Hain-Lüst operator. Proc. Am. Math. Soc. 130(6), 1699–1710 (2002) 14. S. Naboko, Schrödinger operators with decreasing potential and with dense point spectrum. Dokl. Akad. Nauk SSSR 276(6), 1312–1315 (1984) 15. S. Naboko, On the dense point spectrum of Schrödinger and Dirac operators. Teoret. Mat. Fiz. 68(1), 18–28 (1986)

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16. S. Naboko, M. Solomyak, On the absolutely continuous spectrum in a model of an irreversible quantum graph. Proc. London Math. Soc. (3) 92(1), 251–272 (2006) 17. S. NIcaise, Spectre des réseaux topologiques finis. Bull. Sci. Math. (2) 111(4), 401–413 (1987) 18. B. Simon, Some Schrödinger operators with dense point spectrum. Proc. Am. Math. Soc. 125(1), 203–208 (1997) 19. B. Simon, Private Communication 20. J. von Neumann, E.P. Wigner, Über merkwürdige diskrete Eigenwerte. Physikalische Zeitschrift 30, 465–467 (1929)

Sergey Naboko’s Legacy on the Spectral Theory of Jacobi Operators Luis O. Silva and Sergey Simonov

Abstract Sergey Naboko authored a large amount of papers on the spectral theory of Jacobi operators. The main themes of his work are the existence of eigenvalues embedded into the absolutely continuous spectrum; spectral phase transitions for specifically-chosen Jacobi matrices, where new asymptotic methods for spectral analysis were established; construction of Jacobi matrices with gaps in the essential spectrum; estimates of Green matrix for Jacobi operators, and inverse resonance problems. In his later years, Sergey Naboko worked on block Jacobi matrices where he found realizations having both similar and dissimilar properties to their scalar counterparts.

Sergey Naboko showed a keen interest in Jacobi operators, which led him to coauthor a considerable amount of papers on the subject. These papers are inescapable references in the spectral theory of Jacobi operators. His ideas and insights on the matter played a crucial role in the development of the know-how for revealing the spectral properties and finding examples of interesting spectral phenomena for Jacobi operators. Naboko’s approach to the spectral analysis of these operators is mainly based on the asymptotic analysis of the generalized eigenvectors. He developed multiple techniques and filigree methods for the analysis of the asymptotic behavior of such eigenvectors. Within this subject all papers (except for [1]) were written in collaboration with (in chronological order) S. Yakovlev, J. Janas,

L. O. Silva () Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico City, México e-mail: [email protected] S. Simonov St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St Petersburg, Russia St. Petersburg State University, Universitetskaya nab. 7–9, St. Petersburg, Russia Alferov University, St. Petersburg, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_5

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A. Laptev, O. Safronov, A. Boutet de Monvel, L. Silva, G. Stolz, M. Brown, R. Weikard, D. Damanik, E. Sheronova, I. Pchelintseva, S. Simonov, M. Marletta, R. Shterenberg, S. Kupin, I. Wood, and E. Judge. The process of doing mathematics with Sergey Naboko was always a process of friendship, sharing, and learning for us. This note is intended to cover exhaustively the literature authored or coauthored by S. Naboko on Jacobi operators. We know for sure that there are papers to which S. Naboko contributed with important ideas, but did not agree to appear as a coauthor. S. Naboko was always open and eager to discuss questions concerning these operators and due to his recognized leadership in the area was constantly addressed to as an expert. For brevity, the list of references presented in this note includes only papers in which S. Naboko appeared as an author and it is ordered chronologically. Our account is also mostly chronological. The Jacobi operator J is the operator in .l2 (N) whose matrix representation with respect to the canonical basis .{δn }n∈N is ⎛ b1 ⎜ ⎜a1 ⎜ ⎜ .⎜0 ⎜ ⎜0 ⎝ .. .

a1 0 0 · · ·

⎞

⎟ b2 a2 0 · · ·⎟ ⎟ ⎟ a2 b3 a3 ⎟, .. ⎟ ⎟ . 0 a3 b4 ⎠ .. .. .. . . .

(1)

where .an > 0 and .bn ∈ R for .n ∈ N = {1, 2, . . . }. The sequences .{an }∞ n=1 and .{bn }∞ n=1 are referred to as weights and the potential of (1). According to the definition of the matrix representation of a possibly unbounded symmetric operator,  J is the minimal closed operator such that .δn ∈ dom J for all .n ∈ N and . δj , J δk for .j, k ∈ N are the entries of the matrix (1). A Jacobi operator is either symmetric nonselfadjoint with deficiency indices .n+ (J ) = n− (J ) = 1 or it is selfadjoint (i.e. .n+ (J ) = n− (J ) = 0). According to the Weyl’s alternative, the former case is called the limit circle case, while the latter corresponds to the limit point case. In the works ∞ referred to here, with the exception of [2], the sequences .{an }∞ n=1 and .{bn }n=1 are chosen in such a way that J turns out to be selfadjoint. The sequence .u = {un }∞ n=1 being a solution to the equation an−1 un−1 + bn un + an un+1 = zun ,

.

n > 1,

(2)

is called a generalized eigenvector of J . The recurrence relation (2) can be written as a dynamical system un+1 = Bn (z)un ,

.

n > 1,

(3)

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where

Bn (z) :=

.

0

1

an−1 z−bn an an

 and

un :=



un−1 , un

(4)

The matrix .Bn (z) is called the transfer matrix of (3). The first results by S. Naboko on the spectral properties of Jacobi operators are concerned with the discrete Schrödinger operators with decaying potentials, i.e. when in (1) .an = 1 for all .n ∈ N and .bn → 0 as .n → ∞. Under this assumption J is a bounded Jacobi operator and therefore it is selfadjoint. In the papers [3, 4], sufficient conditions on the sequence .{an }∞ n=1 are provided so that the elements of a certain given sequence of numbers inside the essential spectrum are eigenvalues of the corresponding operator. This result is obtained by ingenious transformations of the product of transfer matrices .Bn (z) which lead to establishing estimates for norms of generalized eigenvectors. These estimates and the results of [5] allow one to obtain the mentioned sufficient conditions. In [3, 4] there are also conditions on the potential guaranteeing either dense pure point spectrum or absence of eigenvalues inside the essential spectrum. These results have a continuation in [6] where the approach of [3, 4] was applied to perturbed periodic Jacobi matrices to produce a possibly infinite number of eigenvalues embedded into the absolutely continuous spectrum at prescribed locations which are subject to certain conditions of rational independence. In [7], by a different approach to the dynamical system (3) and without relying on [5], criteria for absence of eigenvalues of J when .an → 1 and .bn → 0 as .n → ∞ are provided. In this case, J is a compact perturbation of the discrete free Schrödinger operator. The case .bn ≡ 0 is considered separately. In [7], the sharpness of these criteria is dealt with by providing examples and counterexamples showing the subtlety of the results. This kind of approach for illustrating results was a trademark of S. Naboko’s work. S. Naboko was constantly concocting and implementing new methods for both the asymptotic analysis of generalized eigenvectors and the ways to translate this information into the spectral properties of the corresponding operator. The paper [8] considers the absolutely continuous spectrum of the operator J with .bn ≡ 0 which S. Naboko used to refer to as the string operator (although it differs from Krein’s string operator). It is required in [8] that .an → 1 as .n → ∞ which implies that J is bounded and therefore selfadjoint. In this work, a Harris-Lutz type transform is used in the asymptotic analysis of the products of transfer matrices .Bn (z) on the one hand and the Gilbert-Pearson-Khan subordinacy theory for characterizing the spectrum from the asymptotic behavior of generalized eigenvectors on the other. ∞ A solution .{u− n }n=1 of the Eq. (2) is called subordinate, if for any other solution ∞ of this equation one has .{un } n=1 n − 2 k=1 |uk | . n → 0, n → ∞. 2 k=1 |uk |

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The existence of a subordinate solution at each point of an interval of the real line implies that the spectrum on this interval is purely point, absence of such a solution at each point of an interval implies that the interval is covered with purely absolutely continuous spectrum. The discreteness of the point spectrum cannot be determined by subordinacy theory and is established by other methods: estimating quadratic forms, controlling the uniformity of asymptotics of generalized eigenvectors, etc. A further development of the results obtained in [8] is found in [9], where a discrete Schrödinger operator perturbed by a bounded Jacobi operator is studied. The results of [9] are stronger than the ones of [8] and the method used to obtain them is completely different; it is a “natural” generalization of the Deift-Killip approach to absolutely continuous spectrum of the one dimensional Schrödinger operator with square summable potential. This approach involves the use of trace formulae and is related to [10], which is also a generalization of results by Killip and Simon. On the basis of the techniques used in [8], the paper [11] tackles the question of when the spectrum of the unbounded string operator fills the whole real line. This again makes use of a Harris-Lutz type transform and subordinacy theory with the additional innovative idea of arranging the product of transfer matrices by blocks of length determined by the particularities of the sequence .{an }∞ n=1 . It turns out that the asymptotic behavior of the dynamical system (3) can be easier to analyze when considering its collective behavior over some periods. Later, a generalization of this method with blocks of variable length was used for studying the spectral properties of the so-called Mirzoev class of Jacobi operators. In [12], an unbounded Jacobi operator J is seen as an additive perturbation of a string operator by a coupled diagonal operator. The weights of the string operator are growing and “modulated” by a periodic sequence .{cn }∞ n=1 . Thus, the resulting Jacobi operator has: an := cn μn ,

.

bn := dνn ,

n ∈ N,

(5)

∞ where .{μn }∞ n=1 and .{νn }n=1 are certain sequences going to .+∞. The growth of weights satisfies the Carleman condition so the operator is selfadjoint. Using the Stolz smoothness classes .Dk,r and .Dk and the generalized Behncke-Stolz lemma proved in [11], it is shown how the nature of the spectrum depends on the coupling constant d. Indeed, if d is in the absolutely continuous spectrum of the periodic operator .Jper with weights .{cn }∞ n=1 and zero diagonal, then the spectrum of J is purely absolutely continuous, otherwise the spectrum is discrete. Here one has examples of multithreshold spectral phase transitions. Regarding the methodology, the generalized Behncke-Stolz lemma establishes that the absolute continuity of √ spectrum follows from the asymptotic estimate .un  = O(1/ an ), .n → ∞, for generalized eigenvectors. Speaking roughly, it is shown that in the case of dominating weights, the Behncke-Stolz sufficient condition holds, whereas in the case of dominance of the main diagonal the spectrum is discrete. This fact is proven by establishing a WKB asymptotics on the basis of a Levinson type theorem for discrete linear systems and then showing that the resolvent is compact.

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A similar model to the one of [12] is studied in [13], where one has the same expression for the weights as in (5), but the potential is given by .bn := dn νn , where ∞ is a periodic sequence. Milder conditions on the sequences .{μ }∞ and .{dn } n n=1 n=1 ∞ ∞ .{νn } are considered. If M and N denote the least periods of sequences .{cn } n=1 n=1 ∞ M+N and .{dn }n=1 , then, in the space of parameters .(c, d) ∈ R , there are domains in which the spectrum of the operator is purely absolutely continuous and occupies the whole real line. The exterior of their union in the parameter space consists of domains in which the spectrum of the operator is discrete. Thus a spectral phase transition occurs at the boundaries of domains: the spectrum abruptly changes its structure. A separate article [14] is devoted to the question of semiboundedness for this class of operators. In [14], simple criteria for upper and lower semiboundedness of operator J are obtained. Whether J is upper or lower semibounded depends again on the spectrum of operator .Jper . Differential and difference equations have many similarities. One important analogy has to do with the asymptotic analysis of solutions to such equations. Indeed, every so often, the discrete linear system (3) allows one to use the socalled semiclassical analysis, i.e. it has a WKB asymptotics just as in the differential equations setting. To obtain asymptotic formulae for the behavior of solutions to (2) as .n → ∞, one may recur to transformations of the system (3) reducing it to a form for which Levinson’s discrete asymptotic theorem is applicable in one of its many variants. In [12] and [13], it is possible to apply discrete Levinson type theorems to (3), although in [13], the grouping in blocks technique has to be used before applying the Levinson theorem. Another successful use of these techniques can be found in [15] which sets precedents for [16], later referred to on this account. As a rule, the phase transition occurs in the so-called double root case of (3). This refers to the coincidence of roots of the characteristic equation or, from another point of view, the case of a multiple eigenvalue of the limit of transfer matrices (if this limit exists). This means that the limit matrix is similar to a Jordan block. In these cases, discrete Levinson type theorems cannot be used straightforwardly. Due to this, new methods were developed in [17–21] for finding the asymptotic behavior of generalized eigenvectors when the double root case takes place. The results resemble the ones obtained by the semiclassical method for ordinary differential equations; the behavior of solutions for large n is determined by lower terms in the expansion of the transfer matrix and can be “hyperbolic” or “elliptic”. Here one recurs to a chain of successive transformations of the dynamical system so that a Levinson type theorem is applicable to the transformed system. These transformations use analogies between difference and differential equations to find a correct Ansatz with the help of which one of the main steps of transforming the system to a simple form is carried out. This method is used in [17] for the matrix (1) having weights .an = n + a and potential .bn = −2an for .n ∈ N (where √.a ∈ R is a −1/4 exp(±2 (λ + 1)n), parameter), and as a result the asymptotic formula .u± ∼ n n α .n → ∞, was proved. The paper [19] deals with the case of weights .an = n (1+μn ) α α/2 α/2 and main diagonal .bn = −2n (1 + νn ), where .n μn , n νn ∈ l 1 (N) and

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α ∈ (1/2, 2/3). For the generalized eigenvectors, one has the following asymptotic formula ⎛ ⎛√ ⎞⎞ 3α α 3 1− α2 2 2 n1− 2 λn n λ ± −α/4  ⎠⎠ , n → ∞.  +√ − .un ∼ n exp ⎝± ⎝ 1 − α2 λ 24 1 − 3α

.

2

A more general case, i.e. .α ∈ (0, 1), was studied later in [22] where the Kooman method was used to prove a formula expressing the spectral density of J in terms of some coefficients in the asymptotics of orthogonal polynomials (the WeylTitchmarsh formulae). The papers [18] and [21] study the spectral properties of J with weights .an = nα for .n ∈ N and periodically modulated diagonal entries so that .bn = bnα (.b > 0) for odd n and .bn = 0 for even n. This is yet another example of a critical situation. The difficulty with the non-smoothness of coefficients caused by the periodic modulation is overcome by grouping transfer matrices into pairs (cf. [11]), after which the analogy with a second-order differential equation and the WKB method becomes less obvious. For each case, the critical elliptic [18] and the critical hyperbolic [21], a progression of transformations is found so that the asymptotic analysis of the solutions to the system (3) is reduced to the asymptotic analysis á la Levinson. In [20], another approach to finding the asymptotics of solutions to the recurrence relation (2) in the double root case was developed: the Kelly method. With its help, it is possible to obtain an asymptotic expansion which is locally uniform with respect to the spectral parameter for a matrix with the main diagonal .bn = nα and periodically modulated weights .an = cn nα , where .{cn }∞ n=1 is a sequence with period 2 and .|c1 − c2 | = 1, .c1 , c2 > 0. The generalized eigenvectors have asymptotics for .λ > 0

  α λ n1− 2 ± 1−α .un ∼ exp ± + O(n ) , n → ∞, 2c1 c2 1 − α2 locally uniform in .λ. From this result, it is deduced that the spectrum is discrete on the positive half-line. S. Naboko also worked on the exact asymptotics of eigenvalues of Jacobi operators used for modeling phenomena in quantum and classical physics. This asymptotic analysis is relevant in physics to understand the role of physical parameters in the asymptotic expansions of large eigenvalues. A completely new method, somehow related to the transformation operator used in differential equations, is developed in [23] to obtain arbitrarily many terms in the asymptotic decomposition of large eigenvalues. This technique, which Naboko called the successive diagonalization technique, is quite general and is used in [24, 25] where for treating periodic modulations of a modified Jaynes-Cummings model the corresponding Jacobi matrix is seen as a block Jacobi matrix.

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An inverse resonance problem for bounded Jacobi operators is solved in [26]. Strictly speaking, the operators considered in that paper constitute a wider class than the one of bounded Jacobi operators. Indeed, [26] studies the operator .J with ∞ and .{b }∞ being bounded complex sequences such that .a = 0 and .{an } n n=1 n n=1 .Re an ≥ 0 for any .n ∈ N. A similarity transformation is defined for the operator .J and it is shown that the Weyl m-function is the same for all operators obtained from one another by this similarity transformation. The main result of [26] establishes that the eigenvalues and resonances of .J, under some complementary conditions ∞ on the sequences .{an }∞ n=1 and .{bn }n=1 , determine uniquely these sequences. It is also shown that superexponentially decaying perturbations of the free discrete Schrödinger operator fall into the class of operators for which the main result holds. In [34], a similar resonance inverse problem is solved for unbounded Jacobi operators appearing in the context of quantum physics, namely, the sum of the creation and the annihilation operators perturbed by a compact potential. An inverse resonance problem focusing on stability and uniqueness is treated in [29]. S. Naboko was enthusiastic about constructing examples of selfadjoint Jacobi operators having predetermined spectral properties. Since any simple selfadjoint operator is unitarily equivalent to some Jacobi operator, any closed subset of .R can be realized as the spectrum of a Jacobi operator. However, finding out the weights and the potential sequences from the spectrum is not an easy task. The paper [29] gives examples of unbounded Jacobi operators having finitely many gaps in the essential spectrum. One should note that prior to [29], only few explicit examples were known of unbounded Jacobi operators having such spectral properties whereas typically the absolutely continuous spectrum of an unbounded Jacobi operator covers the whole line (cf. [12, 13]). The methodology in [29] is based on subordinacy theory and a grouping in blocks approach to (3) for the asymptotic analysis of generalized eigenvectors. Other examples of the same spectral structure are provided in [30] which is loosely motivated by a set of ideas by Last and Simon. A further development of the technique used in [30] can be found in [31]. The results in [31] can be seen as a generalization of Last and Simon approach to the essential spectrum of some operators. In [32] and [33], the problem of embedded eigenvalues is considered in a setting similar to that of [6], but with a different approach. In contrast to [6], in which, starting from a given set of points inside the absolutely continuous spectrum of the unperturbed operator, a potential is constructed having a decay slightly slower than .1/n, for which these points are eigenvalues, the approach of [32, 33] is based upon a particular kind of potentials, namely, potentials of Wigner-von Neumann type, .qn := c sin(2ωn + δ)/n, and sums of such terms. Each such term can produce eigenvalues at points .λ such that .ωN ± 2θ (λ) ∈ 2π Z, where N is the least common period ∞ of the sequences .{an }∞ n=1 and .{bn }n=1 and .θ is the quasi-momentum corresponding to the periodic Jacobi matrix (which means that .exp(±iθ (λ)) are eigenvalues of the monodromy matrix .MN (λ) := BN (λ) · · · B1 (λ)). In the general case, critical points are not eigenvalues, but at each of them a subordinate generalized eigenvector exists. In [32], it is proven that an arbitrary point (with few exceptions) inside the absolutely continuous spectrum of the unperturbed operator can be made an

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eigenvalue by adding a potential of Wigner-von Neumann type to the sequence {an }∞ n=1 . In [33] it is shown that at countably many given points inside the absolutely continuous spectrum one can construct subordinate solutions by adding potential being an infinite sum of Wigner-von Neumann terms to .{an }∞ n=1 . A similar technique was employed in [34] which deals with a differential Schrödinger operator on the half-line, but where the analysis is reduced to a discrete dynamical system resembling (3). Another line of research of S. Naboko was obtaining estimates for the entries of the Green matrix of selfadjoint Jacobi operators in gaps of the essential spectrum and non-real values of the spectral parameter. This was done for scalar Jacobi matrices in [16] and [35] and for block Jacobi matrices in [36, 37] and [38]. These works use a refinement of the Combes-Thomas method and contain sharp decay estimates which are expressed in terms of the weights .an and depend on the behavior of the √ sums . nk=1 1/ak or . nk=1 1/ ak (and .Ak  instead of .ak in the block matrix case, see below). The difference is determined by the finiteness of gaps: estimates of the first type hold for bounded gaps in the essential spectrum, while estimates of the second type hold for unbounded gaps. Moreover, in the case of an unbounded gap .(−∞, d), it is additionally required that the operator is semibounded from below. In this case, the semiboundedness of J is crucial in this analysis as is illustrated by Naboko and Simonov [21] in which it is established for a nonsemibounded Jacobi operator that the asymptotic behavior of the generalized eigenvectors depends on the parameter b from the main diagonal and the estimate from [35] is violated for some values of .b > 0. The results of [16] and [35] have applications in the study of random Jacobi operators and spectral phase transitions. The technique developed in [35] is used in [36, 37] to obtain decay estimates for the block entries of Green matrix in the case of selfadjoint block Jacobi operators when the blocks are bounded operators. A block Jacobi operator .J is defined by a sequence .{An }∞ n=1 of bounded and boundedly invertible operators in a separable Hilbert space .H which usually is finite dimensional and a sequence .{Bn }∞ n=1 of bounded selfadjoint operators in .H. Thus the block Jacobi operator .J is an operator in the space .

l2 (N) =

∞ 

.

Hn ,

n=1

where for all .n ∈ N .Hn = H. Its matrix representation is ⎛

B1 A1 0 0 · · ·

⎜ ∗ ⎜A1 ⎜ ⎜ .⎜ 0 ⎜ ⎜0 ⎝ .. .

⎞

⎟ B2 A2 0 · · ·⎟ ⎟ ⎟ A∗2 B3 A3 ⎟. .. ⎟ ∗ 0 A3 B4 . ⎟ ⎠ .. .. .. . . .

(6)

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For a block Jacobi operator .J, the deficiency indices satisfy .0 ≤ n+ (J), n− (J) ≤ dim H. The transfer matrix .B(z) of the dynamical system A∗n−1 un−1 + Bn un + An un+1 = zun ,

.

n > 1,

analogous to (3) is Bn (z) :=

.

0 I ∗ −1 (zI − B ) −A−1 A A n n n−1 n

,

where I is the identity in .H. Note that any block Jacobi operator with N -dimensional blocks can be seen as a block Jacobi operator of M-dimensional blocks where .M = kN, .k ∈ N. This trick was actually used in [24, 25] for dealing with the modified Jaynes-Cummings model. The greater complexity in the structure of block Jacobi operators means that the results of [36–38] can be used to obtain new examples of operators exhibiting spectral phase transitions. The broader class of block Jacobi operators attracted Naboko’s attention in his later works. Apart from [36–38], the papers [2, 39] also deal with block Jacobi operators. The point spectrum of periodic block Jacobi ∞ operators is studied in [39]. For the sequences of matrices .{An }∞ n=1 and .{Bn }n=1 being N -periodic, define the monodromy matrix .M(z) := BN (z) · · · B1 (z) and denote by .P− (z) the Riesz projection corresponding to the eigenvalues of .M(z) in the interior of the unit circle. If .λ is in the point spectrum of .J and .rank P− (λ) = 1, then .λ is in the spectrum of the finite matrix ⎞ B1 A1 0 · · · 0 ⎜A∗ B2 A2 · · · 0 ⎟ ⎟ ⎜ 1 ⎜ .. ⎟ . . ∗ ⎜  := 0 A B3 . .J . ⎟ ⎟. ⎜ 2 ⎟ ⎜ . . . .. .. A ⎠ ⎝ .. N −2 ∗ 0 · · · 0 AN −2 BN −1 ⎛

(7)

It is also established in [39] that when the set of matrices .{An , Bn }N n=1 is commuta tive, then all eigenvalues of .J are also eigenvalues of .J irrespectively of .rank P− (λ). If .{An , Bn }N n=1 is not commutative and .rank P− (λ) > 1, then the eigenvalue .λ of  which depends on the concrete value of .J is an eigenvalue of a modification of .J .rank P− (λ). The work [2] is concerned with the conditions for discreteness of the spectrum of block Jacobi matrices in terms of its main diagonal entries. It turns out that sometimes the interplay between odd and even elements of the main diagonal sequence can be used to draw conclusions on discreteness independently of the weight sequence. Let .n+ (J) = n− (J). If, on the one hand, for some c and large n one has .B2n ≤ cI and, on the other, .B2n−1 → +∞ (in the sense that for every .M > 0 there exists .N ∈ N such that for every .n > N .B2n−1 ≥ MI ),

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then any selfadjoint extension of .J has discrete spectrum on the interval .(c, +∞) accumulating only at .+∞. This is directly applicable to the example considered in [18] and [21]. The result remains true if one permutes the roles of even and odd entries. The paper also contains an estimate of the counting function on subintervals of .(c, +∞) as well as a refinement of the condition for discreteness of the spectrum on the positive half-line which involves the weight sequence. A simple condition of discreteness of the whole spectrum should also be mentioned: .B2n−1 → +∞ and .B2n → −∞ as .n → ∞ guarantees this. The results of [2] extend previous results in [12, 13, 17, 18].

References 1. S.N. Naboko, On the singular spectrum of discrete Schrödinger operator, in Séminaire sur les Équations aux Dérivées Partielles, 1993–1994 (École Polytech., Palaiseau, 1994), pp. Exp. No. XII, 11 2. S. Kupin, S. Naboko, On the instability of the essential spectrum for block Jacobi matrices. Constr. Approx. 48(3), 473–500 (2018) 3. S.N. Naboko, S.I. Yakovlev, The discrete Schrödinger operator. A point spectrum lying in the continuous spectrum. Algebra i Analiz 4(3), 183–195 (1992) 4. S.N. Naboko, S.I. Yakovlev, The point spectrum of a discrete Schrödinger operator. Funktsional. Anal. i Prilozhen. 26(2), 85–88 (1992) 5. S.N. Naboko, On the dense point spectrum of Schrödinger and Dirac operators. Teoret. Mat. Fiz. 68(1), 18–28 (1986) 6. E. Judge, S. Naboko, I. Wood, Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach. J. Differ. Equations Appl. 24(8), 1247–1272 (2018) 7. J. Janas, S.N. Naboko, On the point spectrum of some Jacobi matrices. J. Operator Theory 40(1), 113–132 (1998) 8. J. Janas, S.N. Naboko, Jacobi matrices with absolutely continuous spectrum. Proc. Am. Math. Soc. 127(3), 791–800 (1999) 9. A. Laptev, S. Naboko, O. Safronov, Absolutely continuous spectrum of Jacobi matrices, in Mathematical Results in Quantum Mechanics (Taxco, 2001). Contemporary Mathematics, vol. 307 (American Mathematical Society, Providence, RI, 2002), pp. 215–223 10. A. Laptev, S. Naboko, O. Safronov, On new relations between spectral properties of Jacobi matrices and their coefficients. Commun. Math. Phys. 241(1), 91–110 (2003) 11. J. Janas, S.N. Naboko, Jacobi matrices with power-like weights—grouping in blocks approach. J. Funct. Anal. 166(2), 218–243 (1999) 12. J. Janas, S.N Naboko, Multithreshold spectral phase transitions for a class of Jacobi matrices, in Recent advances in operator theory (Groningen, 1998). Operator Theory: Advances and Applications, vol. 124 (Birkhäuser, Basel, 2001), pp. 267–285 13. J. Janas, S.N Naboko, Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries. J. Funct. Anal. 191(2), 318–342 (2002) 14. S.N. Naboko, J. Janas, Criteria for semiboundedness in a class of unbounded Jacobi operators. Algebra i Analiz 14(3), 158–168 (2002) 15. J. Janas, S. Naboko, G. Stolz, Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods. J. Comput. Appl. Math. 171(1–2), 265–276 (2004) 16. J. Janas, S. Naboko, G. Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not. IMRN 2009(4), 736–764 (2009)

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17. J. Janas, S.N Naboko, Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes, in Recent Advances in Operator Theory and Related Topics (Szeged, 1999). Operator Theory: Advances and Applications, vol. 127 (Birkhäuser, Basel, 2001), pp. 387–397 18. D. Damanik, S. Naboko, Unbounded Jacobi matrices at critical coupling. J. Approx. Theory 145(2), 221–236 (2007) 19. J. Janas, S. Naboko, E. Sheronova, Asymptotic behavior of generalized eigenvectors of Jacobi matrices in the critical (“double root”) case. Z. Anal. Anwend. 28(4), 411–430 (2009) 20. S.N. Naboko, I. Pchelintseva, L.O. Silva, Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis. J. Approx. Theory 161(1), 314–336 (2009) 21. S. Naboko, S. Simonov, Spectral analysis of a class of Hermitian Jacobi matrices in a critical (double root) hyperbolic case. Proc. Edinb. Math. Soc. (2) 53(1), 239–254 (2010) 22. S. Naboko, S. Simonov, Titchmarsh–Weyl formula for the spectral density of a class of Jacobi matrices in the critical case. Funct. Anal. Appl. 55(2), 94–112 (2021) 23. J. Janas, S.N. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach. SIAM J. Math. Anal. 36(2), 643–658 (2004) 24. A. Boutet de Monvel, S.N. Naboko, L.O. Silva, Eigenvalue asymptotics of a modified JaynesCummings model with periodic modulations. C. R. Math. Acad. Sci. Paris 338(1), 103–107 (2004) 25. A. Boutet de Monvel, S.N. Naboko, L.O. Silva, The asymptotic behavior of eigenvalues of a modified Jaynes-Cummings model. Asymptot. Anal. 47(3–4), 291–315 (2006) 26. B.M. Brown, S. Naboko, R. Weikard, The inverse resonance problem for Jacobi operators. Bull. Lond. Math. Soc. 37(5), 727–737 (2005) 27. B. Brown, S. Naboko, R. Weikard, The inverse resonance problem for Hermite operators. Constr. Approx. 30(2), 155–174 (2009) 28. M. Marletta, S. Naboko, R. Shterenberg, R. Weikard, On the inverse resonance problem for Jacobi operators – uniqueness and stability. J. Anal. Math 117, 221–247 (2012) 29. A. Boutet de Monvel, J. Janas, S. Naboko, Unbounded Jacobi matrices with a few gaps in the essential spectrum: constructive examples. Integr. Equ. Oper. Theory 69(2), 151–170 (2011) 30. A. Boutet de Monvel, J. Janas, S. Naboko, Elementary models of unbounded Jacobi matrices with a few bounded gaps in the essential spectrum. Oper. Matrices 6(3), 543–565 (2012) 31. A. Boutet de Monvel, J. Janas, S. Naboko, The essential spectrum of some unbounded Jacobi matrices: a generalization of the Last-Simon approach. J. Approx. Theory 227, 51–69 (2018) 32. E. Judge, S. Naboko, I. Wood, Eigenvalues for perturbed periodic Jacobi matrices by the Wigner–von Neumann approach. Integr. Equ. Oper. Theory 85(3), 427–450 (2016) 33. E. Judge, S. Naboko, I. Wood, Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique. Studia Math. 242(2), 179–215 (2018) 34. S. Naboko, S. Simonov, Zeroes of the spectral density of the periodic Schrödinger operator with Wigner-von Neumann potential. Math. Proc. Cambridge Philos. Soc. 153(1), 33–58 (2012) 35. J. Janas, S. Naboko, Estimates of generalized eigenvectors of Hermitian Jacobi matrices with a gap in the essential spectrum. Mathematika 59(1), 191–212 (2013) 36. J. Janas, S.N. Naboko, L.O. Silva, Green matrix estimates of block Jacobi matrices I: Unbounded gap in the essential spectrum. Integr. Equ. Oper. Theory 90(4), Paper No. 49, 24 (2018) 37. J. Janas, S.N. Naboko, L.O. Silva, Green matrix estimates of block Jacobi matrices II: Bounded gap in the essential spectrum. Integr. Equ. Oper. Theory 92(3), Paper No. 21, 30 (2020) 38. S. Naboko, S. Simonov, Estimates of Green matrix entries of self-adjoint unbounded block Jacobi matrices. St. Petersburg Mathematical Journal 35(1), 243–261 (2022) 39. J. Janas, S. Naboko, On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach. J. Differ. Equations Appl. 21(11), 1103–1118 (2015)

On the Work by Serguei Naboko on the Similarity to Unitary and Selfadjoint Operators Dmitry Yakubovich

Dedicated to the memory of Sergey Nikolaevich Naboko (1950–2020)

Abstract We review Serguei Naboko’s criteria for similarity of an operator to unitary or selfadjoint one, their relation to his functional model, applications and some related results. Keywords Similarity · Resolvent estimates · Nonselfadjoint operators · Spectral operators

Serguei Naboko was not only an great mathematician, he also will be remembered as a wonderful person. His untimely death is a great loss for all of us.

1 The Results and a Discussion Unitary and selfadjoint operators on a Hilbert space are the best understood ones. So the criteria for similarity to an operator in one of these classes are very important in the operator theory. It is a difficult topic. Serguei Naboko wrote several papers on this subject. Probably, the work which has had most impact is his paper [26], written in 1984. There he proves the following integral similarity criterion, which was also proved in the same time (in 1983) by Van Casteren [37]. Theorem 1.1 Suppose the spectrum of an operator T on a Hilbert space H is contained in the unit circle .T. Then T is similar to a unitary operator if and only if D. Yakubovich () Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_6

61

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it satisfies the estimates ˆ

2π

sup(r − 1) 2

r>1

ˆ

.

sup(1 − r

−2

(T − reiθ )−1 u2 dθ ≤ Cu2 ,

0 2π

)

(1) ∗

(T − r

−1 −iθ −1

e

)

u dθ ≤ Cu , 2

2

0

r>1

u ∈ H.

.

We always assume all operators to be linear and to act on Hilbert spaces. Naboko also proved an analogue of his result for the context of selfadjoint operators. Theorem 1.2 (Naboko [26]) Suppose the spectrum of a (possibly unbounded) operator L on H is contained in the real line. Then L is similar to a selfadjoint operator if and only if ˆ (L − k − iε)−1 u2 dk ≤ Cu2 , . sup ε R ε>0 ˆ (L∗ − k − iε)−1 u2 dk ≤ Cu2 , u ∈ H. sup ε R ε>0 Van Casteren obtained in [37] several criteria, related to Theorem 1.1. In particular, he showed the following. Theorem 1.3 The following conditions on an invertible operator T are equivalent. (i) T is similar to a unitary operator; (ii) For any .u ∈ H , .

n  sup (n + 1)−1 T −k u2 < ∞; n∈N k=0

n  sup (n + 1)−1 T ∗k u2 < ∞; n∈N k=0

(iii) For .|λ| < 1, the inverses .(I −λT )−1 , .(I −λT −1 )−1 exist, and for any .u, v ∈ H , ˆ 2π    2 . sup (1 − r )  (I − re−iθ T )−1 (I − reiθ T −1 )−1 u, v  dθ < ∞. 0≤r 1). Next Gohberg and Krein [6] wrote down this condition without recurring to the characteristic function: (T − λ)−1  ≤ C(1 − |λ|)−1 ,

.

|λ| < 1.

(3)

The proof still was relying on the methods of the functional model. In 1969, Sakhnovich [32] showed that the condition .ΘT (λ) ≤ C, .|λ| = 1 is sufficient for an arbitrary T to be similar to a unitary operator. An analogous result holds for selfadjoint operators. However, examples show that this condition is no longer necessary. Let .L = A+iJ V be a nonselfadjoint operator, where .A = Re L and .Im L = J V is the polar decomposition of .Im L, .J = J ∗ = J −1 , .V ≥ 0. In this context, the characteristic function of L is given by ΘL (λ) = I + 2iJ V 1/2 (L∗ − λ)−1 V 1/2 .

.

This is the main ingredient of Naboko’s functional model for L [24], which was one of his main objects of study, beginning from 1976. This model was based on Pavlov’s transcription of the Nagy–Foias functional model of dissipative operators. In [25], Naboko introduced the condition of “small interaction” between

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the dissipative and anti-dissipative part of L, which has the form .

sup P± ΘB (λ)P∓  < 1,

(4)

λ∈Π+

where .P± = (I ± J )/2, .Π± = {λ ∈ C : ± Im λ > 0} and .ΘB is the characteristic function of the associated dissipative operator .B = A + iV . Using his functional model, he proves that for operators satisfying (4), the following conditions are equivalent. (i) (ii) (iii) (iv)

L is similar to a selfadjoint operator; (L − λ)−1  ≤ C| Im λ|−1 , .λ ∈ C \ R; 1/2 (L − λ)−1 h ∈ H 2 (Π , Range V ) for all .h ∈ H ; .V ±

.ΘL (λ) ≤ C , .λ ∈ C \ R, where .ΘL is the characteristic function of L. .

In particular, for operators of this class, Sakhnovich’s sufficient condition for similarity of L to a selfadjoint operator is also necessary. Naboko also gave an analogous list of conditions, which are equivalent to the similarity of L to a dissipative operator. This result extends the Davis-Foias theorem [2], which says that (iv) . ⇒ (i), without assuming (4) (the context of [2] is that of contractions). More equivalent conditions for similarity to a selfadjoint or a dissipative operator under the assumption (4) were given by Malamud [21] in 2000, with proofs that do not use the functional model. An operator is called almost unitary if it is a trace class perturbation of a unitary operator and its spectrum does not cover the unit disc. In [9], Kapustin used the functional model close to Naboko’s one to study the following question: given an operator T , close to unitary, when is T similar to a direct sum of an absolutely continuous unitary .Vac and a singular operator, close to unitary? He gives a concrete criterion, expressed in terms of the parameters of the functional model of T . The first order growth of a resolvent does not guarantee the similarity to a unitary operator, as a counterexample by Marcus [22] shows. Stampfli [33] asked whether the conditions .supn≥0 T n  < ∞ and the uniform estimate (3) are sufficient for the similarity to a unitary operator (the necessity is obvious). Van Casteren [36] and Naboko [26] proved that it is the case. This generalizes the above-cited results by Nagy and Foias and by Gohberg and Krein and converts them into a complete criterion for similarity. Van Casteren’s and Naboko’s tools were not using the functional model. We mention that the paper [20] by Malamud (1985) gives alternative proofs of Van Casteren’s and Naboko’s results and some new information. It is proved there, in particular, that a maximal dissipative operator is similar to a selfadjoint one if and only if it is similar to its adjoint. We would also like to mention papers by Gubreev and Tarasenko [7] and by Kantorovitz [8], where different methods are applied to obtain similarity to a selfadjoint operator of certain perturbations of Volterra operators.

On the Similarity to Unitary and Selfadjoint Operators

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The conditions in Theorem 1.2 are also equivalent to the fact that L generates a bounded .C0 group of operators .{G(t) : t ∈ R}. The conditions for L to generate a group with polynomial growth and their exactness were discussed in [11, 31]; see also the references in these papers. Naboko and his pupils and collaborators devoted much effort to investigate the spectral structure of a general nonseladjoint operator L, whose characteristic function .ΘL has a scalar multiple, and found that the condition (4) plays an important role in this study. Naboko defines the following spaces:  N+ = clos{u ∈ H : V 1/2 (L − ·I )−1 uΠ ∈ H2 (Π+ , E)}, +  1/2 −1  N− = clos{u ∈ H : V (L − ·I ) u Π ∈ H2 (Π− , E)}, −  Ne = clos{u ∈ H : V 1/2 (L − ·I )−1 uΠ ∈ H2 (Π± , E)};

.

±

here .E = clos V H . All these spaces are L-invariant (that is, invariant under the resolvent of L). The space .Ne = Ne (L) is the absolutely continuous space of L. In the same way, the .L∗ -invariant subspaces .N±∗ and .Ne∗ are defined. Next, the singular subspace .Ni = H Ne∗ of L is defined. It is known [23] that .L|N− is quasisimilar to a dissipative operator and .L|N+ is quasisimilar to the adjoint of a dissipative operator. One of the main (and probably, most enigmatic) objects of study was the singular L-invariant space Ni0 = H (N+∗ ∨ N+∗ ).

.

The restriction of L to this space has real spectrum. All eigenvectors and root vectors of L corresponding to real eigenvalues lie in .Ni0 . Under suitable assumptions, one has .det ΘL|N 0 ≡ 1 and .det ΘL ≡ det ΘL|N+ ∨N− on .C \ σ (L). In the case when i

L is similar to a dissipative operator B, .Ni0 has a simple description. Namely, let −1 , where .W ∈ L(H ) is a linear isomorphism, let .H be the maximal .L = W BW sa reducing subspace of B, on which B is selfadjoint, and let .Hs ⊂ Hsa be the singular subspace of .B|Hsa . Then .Ni0 = W Hs . These results are contained in the paper [39]. If (4) holds, then .Ni0 = 0, see Naboko [25]. In this paper, Naboko also shows that under condition (4), the singular space .Ni is a direct sum of certain L-invariant subspaces .Ni+ and .Ni− , which are responsible for certain parts of spectrum in .Π¯ + and .Π¯ − , respectively. In particular, the angle between these subspaces is nonzero. He also considers in this paper a Schrödinger operator on .L2 (R3 ) and writes down concrete inequalities involving its complex potential, that guarantee (4). Operators L such that .H = Ni0 (L) = Ni0 (L∗ ) have been considered by Kiselev and Naboko in several works, see [16] and references therein. These operators are called operators with almost Hermitian spectrum. It has been discovered, in particular, that these operators always have an annihilator, which is scalar outer

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function in half-plane .Π+ (or in .Π− ). In particular, it is true for any singular selfadjoint operator L. Even in this case, this result is nontrivial. More information on spectral subspaces of nondissipative operators can be found in [27], where several problems are posed and a review of this topic is given.

2 Applications In his paper [26], Naboko already applied his result to a nonselfadjoint perturbation of rank one of a selfadjoint operator Lu(x) = x · u(x) + u, ϕψ,

.

on .L2 (R), where .ϕ, ψ ∈ L2 (R). He derived necessary and sufficient conditions for similarity of this operator to a selfadjoint one, expressed in terms of uniform boundedness of certain operators, involving Riesz projections and operators of multiplication by functions .ϕ, ¯ .ψ¯ and the perturbation determinant ˆ D(λ) = 1 +

.

R

ψ(x)ϕ(x)(x ¯ − λ)−1 dx.

In the case when .ϕ, .ψ belong to classes .Lip α for .α > 1/2 and decay at infinity, this leads to a more tractable necessary and sufficient integral condition [26, Theorem 7]. It shows that the zeros of .D on the real line should be at the same time zeros of .ϕ and .ψ. See also [28] for the study of a finite rank Friedrichs model. Later, in [29], Naboko and Tretter applied the same abstract criterion to the operator of multiplication in .L2 (0, 1), perturbed by a Volterra operator: ˆ (Lu)(x) = xu(x) +

x

ϕ(x)ψ(s)u(s) ds,

.

u ∈ L2 (0, 1), x ∈ [0, 1],

0

where .ϕ, ψ are two given functions from .L2 [0, 1]. Such operator always has a real spectrum. The resolvent of L can be written down explicitly. On this way, Naboko and Tretter show that L is similar to a selfadjoint operator whenever ´ δ −1 .ϕ ∈ Lip(ω1 ), .ψ ∈ Lip(ω2 ), . 0 t ω1 (t)ω2 (t) dt < ∞ and .ϕψ = 0 on .[0, 1]. Here .ωj : [0, ∞) → R are moduli of continuity. Naboko and Tretter also show the exactness of this condition: if .ωj (t) = | ln t|−δj for .0 < t ≤ 1/2 and .δ1 + δ2 < 1 ´δ (in which case the integral . 0 t −1 ω1 (t)ω2 (t) dt diverges), then the above assertion is no longer valid. The paper [12] by Kiselev is devoted to the same Friedrichs model in an unbounded case under the hypotheses that .ϕψ = 0 a.e. and .ϕψ ≥ 0 a.e. The absolute continuous and the singular subspaces of L are discussed and various examples are given.

On the Similarity to Unitary and Selfadjoint Operators

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In [14], Kiselev considers the case when L has purely absolutely continuous spectrum. Given a Borel set .δ ⊂ R, the corresponding spectral projection can be expressed as ˆ

1 ε→+0 2π i

χδ (L)u = lim

.

[(L − k − iε)−1 − (L − k + iε)−1 ]u dk,

δ

for a dense set of vectors u. Kiselev gives a limit local condition for the restriction of L to its spectral invariant subspace .χδ (L)H to be similar to a selfadjoint operator. In particular, he showed that the conditions ˆ .

(I ± J )V 1/2 (L − k − iε)−1 u2 dk ≤ Cu2 ,

u∈H

δ

imply such similarity. Earlier this form of conditions appeared in a paper by Kiselev and Faddeev [15] and in [13], for the case of nonselfadjoint extensions of symmetric operators. We should remark that Volterra perturbations of selfadjoint operators of the form ˆ Lu(x) = α(x)u(x) + 2i

1

k(x, s)u(s) dμ(s)

.

(5)

x

on .L2 ([0, 1], μ; Cr ) had been considered already in the old paper by Gohberg and Krein [6] under the assumptions that .k(x, s) is a nonnegative .r × r matrix-valued kernel on .[0, 1] and .dμ is the Lebesgue measure. This operator is dissipative. Gohberg and Krein show that this operator is similar to a selfadjoint one if and only if the function ˆ σ (s) =

tr k(x, x) dx

.

α(x)≤s

satisfies the Lipschitz condition. Malamud [21] generalized this criterion to the case of an arbitrary continuous measure on .[0, 1], .r = ∞ and non-trace-class perturbations. Naboko’s criterion has also been applied to several classes of differential operators. In [4], Faddeev and Shterenberg use this criterion to study the operator d 2 f (x) .L1 f (x) = −(sign x) + p(x)f (x) on the Sobolev space .W22 (R). Assuming p dx 2 to be a real valued potential with finite second moment, they show that L is similar to selfadjoint operator if and only if it has no nonreal eigenvalues. This is done by comparing the resolvent of L with the resolvents of certain auxiliary selfadjoint operators, for which Naboko’s criterion holds trivially. In a related work [5], these two authors make the same analysis of the operator L2 = −

.

signx d 2 , |x|α p(x) dx 2

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acting on .L2 (p(x)|x|α , R) (where .α > −1). They observe that Sakhnovich’s sufficient condition for similarity to a selfadjoint operator cannot be applied to this operator. These two papers were motivated by an earlier work by A. Fleige and B. Najman, where the operator .L2 (for the case .p = 1) was treated by means of the spectral  theory of selfadjoint operators in the Krein spece setting. The operator .(sign x) −  d2 + q(x) was studied in [10] for the case of finite-zone potentials. Its similarity dx 2 to a normal operator (with possibly non-real eigenvalues) and of its essential part to a selfadjoint operator were investigated. In [9], Kapustin showed an abstract approach for passing from his results on almost unitary operators to the study of non-selfadjoint extensions of abstract symmetric operators and their similarity to selfadjoint ones.

3 Similarity to Normal Operators A natural question related with Naboko’s work is to find criteria for similarity of an operator T to a normal operator. This context is more difficult, because the techniques of the characteristic function no longer work. An obvious necessary condition is the first order growth of the resolvent: (T − λ)−1  ≤

.

C , dist(λ, σ (T ))

(6)

but in most cases it is not sufficient. An easy way to construct a counterexample is to define T by .Tfk = λk fk , where .λk → 0 are real and distinct and .{fk } is a Schäuder basis in our Hilbert space, not equivalent to an orthonormal one. In 1972, Stampfli [33] considered the question of similarity to general normal operators in terms of resolvent criteria. His earlier result of 1969 asserts that T is normal whenever .σ (T ) is contained in a .C 2 smooth Jordan curve .Γ and the resolvent has first order growth with constant one: (T − λ)−1  ≤

.

1 dist(λ, Γ )

for .λ in some open neighborhood of .Γ . In this paper he showed that if the curve is closed, the closure of its interior S is a spectral set for T and  −1 (T − λ)−1  ≤ C dist(λ, Γ )

.

(7)

for .λ ∈ S, then T is similar to a normal operator. This paper by Stampfli contains many other results, counterexamples and questions. In particular, he asks whether an estimate like (7), with a general constant

On the Similarity to Unitary and Selfadjoint Operators

69

C on one side of a curve .Γ and with the constant 1 on the other side will be sufficient for similarity of T to a normal operator. This was shown to be the case in the paper [3] by Dritschel, Estévez and the author. This paper also contains generalizations of Naboko’s and Van Casteren’s integral criteria to operators with spectra on smooth curves. There are several works concerning similarity to normal operators of contractions and operators similar to contractions. Benamara and Nikolski [1] proved that for weak contractions with defects of finite rank, the condition (6) is sufficient. Kupin and Treil showed [19] that for trace class defects this is no longer true, and Kupin [17] established the following necessary and sufficient condition: a weak contraction is similar to a normal operator if and only if it satisfies (6) and .

  sup tr I − bμ (T )∗ bμ (T ) < ∞,

|μ| 0 }.

2 Preliminaries 2.1 Linear Relations Let .H be a Hilbert space. A linear subspace T of .H × H is called a linear relation in H. For a linear relation T in .H the symbols .dom T , .ker T , .ran T , and .mul T denote the domain, kernel, range, and the multi-valued part, respectively. The adjoint .T ∗ is the closed linear relation in .H defined by (see [10])

.

T∗ =

.

     h f ∈ H ⊕ H : (k, f )H = (h, g)H , ∈T . k g

(2.1)

The null spaces of .T − zI , .z ∈ C, are denoted by .Nz (T ) = ker(T − zI ). Moreover, ρ(T ) (resp. .ρ (T )) stands for the set of regular (resp. regular type) points of T . The closure of a linear relation T will be denoted by .clos T . If T is closed, then also the eigenspace .Nz (T ) is closed for every .z ∈ C. Recall that a linear relation T in .H is called symmetric if .Im (h , h) = 0 for all .{h, h } ∈ T . These properties remain invariant under closures. By polarization it follows that a linear relation T in .H is symmetric if and only if .T ⊂ T ∗ . A linear relation T in .H is called selfadjoint if .T = T ∗ . If T is symmetric and closed, then ∗ ⊥ .mul T ⊂ mul T . In this case the orthogonal decomposition .H = (mul T ) ⊕ mul T induces an orthogonal decomposition of T as .

T = graph A ⊕ T∞ ,

.

T∞ = {0} × mul T ,

(2.2)

where A is a closed symmetric operator in .H  mul T . A symmetric linear relation T in .H is called simple if there is no nontrivial reducing subspace .H0 ⊂ H such that .T0 := T |H0 is a selfadjoint relation in .H0 . Decomposition (2.2) for .T = graph A ⊕ T∞ shows that a simple closed symmetric relation is necessarily the graph of a simple symmetric operator. Recall

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(cf. e.g. [41]), that a closed symmetric relation T in .H is simple if and only if H = span { Nz (T ∗ ) : z ∈ C \ R }.

.

Importance of the concept of simplicity of a symmetric operator A is demonstrated by Proposition 2.7. We also recall, (see [25, Exercise 6.23]) that every linear relation . in .H with a nonempty resolvent set admits a representation

=

.

   u0 ∈ H2 : B0 u0 + B1 u1 = 0 , u1

(2.3)

with .B0 , B1 ∈ B(H ) such that 0 ∈ ρ(B0 B0∗ + B1 B1∗ )

.

(2.4)

2.2 Ordinary Boundary Triples and Weyl Functions Definition 2.1 ([37]) A triple  = {H, 0 , 1 } is called boundary   an ordinary g = gg ∈ A∗ and the triple for A∗ , if the Green identity (1) holds for all f = ff ,  T  mapping  = 0 1 is surjective. In this case A∗ = A∗ . During the last four decades there appeared numerous papers devoted to the theory of boundary triples and the corresponding Weyl functions as well as its applications to different aspects of extension theory and boundary value problems. We refer to the monographs [33] and [25] for a detailed discussion of such applications, as well as for historical background. Definition 2.1 immediately leads to a parametrization of the set of all selfadjoint

of A by means of abstract boundary conditions via extensions A

= A := {f ∈ A∗ :  f ∈ }, A

.

where A ranges over the set of all selfadjoint extensions of A and ranges over the set of all selfadjoint relations in H (see [33] and [25]). This correspondence is

Another description is obtained in [55]. bijective and in this case := (A). Let A be a symmetric operator in a Hilbert space H and let {H, 0 , 1 } be an ordinary boundary triple for A∗ . If a linear relation is given by (2.3) then the proper extension A of A corresponding to takes the form  f + B1 1 f=0 . A =  f ∈ A∗ : B0 0

.

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In the following statement the characterization of the spectrum of the extension A and a formula for the resolvent of A in terms of the boundary relation and the Weyl function M are presented. Proposition 2.2 ([24]) Let {H, 0 , 1 } be an ordinary boundary triple for A∗ , let M(·) be the corresponding Weyl function, let be a linear relation in H given by (2.3)–(2.4), and let z ∈ ρ(A0 ). Then: (i) z ∈ ρ(A ) if and only if 0 ∈ ρ(B0 + B1 M(z)). (ii) The resolvent of A is given by .

(A − zI )−1 = (A0 − zI )−1 − γ (z)(B0 + B1 M(z))−1 B1 γ (¯z)∗ ,

z ∈ ρ(A ). (2.5)

2.3 B-Generalized Boundary Triples Here we briefly discuss B-generalized boundary triples introduced in Definition 1.1 and the corresponding Weyl functions. Such triples turns out to be important in the spectral theory (see [9] and [21]). By definition, .A0 ⊂ dom  = A∗ , which implies that .A0 = A∗0 is an extension of A. The next result collects some further properties of B-generalized boundary triples. Proposition 2.3 ([24]) Let .{H, 0 , 1 } be a B-generalized boundary triple for .A∗ and let .M(·) and .γ (·) be the corresponding Weyl and .γ (·)-field given by (1.5). Then: (i) (ii) (iii) (iv)

 z ∩ A∗ .  z (A∗ ) := N  z (A∗ ) for every .z ∈ ρ(A0 ), where .N N A∗ = A0 +  .0 (Nz (A∗ )) = H, for all .z ∈ ρ(A0 ). 2 .1 (A0 ) = H and .ran  = H . Equalities (1.4) define a .B(H, Nz )-valued function .γ (·), and a .B(H )-valued function .M(·) holomorphic on .ρ(A0 ) and satisfying (1.5). .

It follows from (1.5) that .M(·) belongs to the class .Rs [H]. But in contrast with the case of ordinary boundary triple, if .A∗ = A∗ , then .γ (z) has no bounded inverse, and, thus, the Weyl function M, corresponding to the B-generalized boundary triple belongs to the subset .Rs [H] of .R[H]-functions satisfying .0 ∈ σp (Im M(i)). The following statement gives a criterion that the transposed triple .{H, 1 , −0 } to a B-generalized boundary triple .{H, 0 , 1 } is also B-generalized. In this case the Krein’s formula for the resolvent remains in force. Proposition 2.4 ([24]) Let . = {H, 0 , 1 } be a B-generalized boundary triple for .A∗ and let .M(·) and .γ (·) be the corresponding Weyl function and the .γ -field. Then: (i) .0 ∈ ρ(Im M(i)) .⇐⇒ ran  = H2 .

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(ii) .0 ∈ ρ(M(i)) .⇐⇒ ran 1 = H and .A1 = A∗1 , i.e. . is a double B-generalized boundary triple for .A∗ . (iii) If (ii) holds, then the resolvent of .A1 admits a representation (A1 − z)−1 h = (A0 − z)−1 − γ (z)M(z)−1 γ (¯z)∗ ,

.

z ∈ C \ R.

(2.6)

Proof (i)–(ii) are contained in [24, Prop. 6.2], and (iii) is implied by Derkach et al. [22, Prop. 4.15].   Formula (2.5) is a version of the Krein type formula for B-generalized boundary triples. It was proved by Krein in [38] for operators with .n± (A) < ∞ and by Saakyan [57] for operators with .n± (A) = ∞, see also [30]. A connection with boundary triples and historical details can be found in the monograph [25]. Lemma 2.5 Let .+ = {H, 0+ , 1+ } be a double B-generalized boundary triple for .A∗ and let .M+ (·) and .γ+ (·) be the corresponding Weyl function and .γ -field. Then .− = {H, 0− , 1− } = {H, −1+ , 0+ } is also a B-generalized boundary triple for .A∗ and the corresponding Weyl function .M− (·) and .γ -field .γ− (·) are given by .M− (z) = −M+ (z)−1 and .γ− (z) = −γ+ (z)M+ (z)−1 . Proof It follows from Definition 1.2 that −1 −1    −1 γ− (z) = 0−  Nz = − 1+  Nz = − M+ (z)(0+  Nz ) −1  = − 0+  Nz M+ (z)−1 = −γ+ (z)M+ (z)−1 .

.

 

This proves the result. (1) .A

(2.7)

(2) .A

Definition 2.6 Let and be two symmetric operators in Hilbert spaces (k) ∗ H(1) and .H(2) , respectively, and let .A(k) ∗ be linear relations dense in .(A ) , and let .(k) = {H; 0k , 1k } be B-generalized boundary triples for .(A(k) )∗ , .k ∈ {1, 2}. Then the B-generalized boundary triples .(1) and .(2) are called unitary equivalent if there exists a unitary operator .U : H(1) → H(2) such that

.

A(1) = U −1 A(2) U,

.

and

(1)

(2)

0 = 0 U,

(1)

(2)

1 = 1 U.

(2.8)

Proposition 2.7 ([23, 24]) Let .A(k) be as in Definition 2.6, let .(k) = {H; 0k , 1k } be B-generalized boundary triples for .(A(k) )∗ , .k ∈ {1, 2} and let .Mk (·) be the corresponding Weyl functions. If, in addition, .A(1) and .A(2) are simple, then the triples .(1) and .(2) are unitary equivalent if and only if .M1 (z) = M2 (z), .z ∈ C+ . (1) (2) Moreover, in this case the extensions .Aj of .A(1) and extensions .Aj of .A(2) , .j ∈ {0, 1}, defined by (1.2) are also unitarily equivalent by means of U , i.e. A(1) = U −1 A(2) U,

.

−1 (2) A(1) 0 = U A0 U,

−1 (2) A(1) 1 = U A1 U.

(2.9)

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85

The corresponding .γ -fields .γ (1) and .γ (2) are connected by .γ (2) (z) = U γ (1) (z).

2.4 Weyl Function and Spectral Multiplicity It is shown in [47] that the multiplicity function .NAac (·) of the ac-part of the 0 self-adjoint extension .A0 := A∗  ker(0 ) can be computed by means of the Weyl function .M(·). To state the corresponding result choose any Hilbert-Schmidt operator D (.D ∈ S2 (H )) and introduce the sandwiched Weyl function .M D (·), (M D )(z) := D ∗ M(z)D,

.

z ∈ C+ .

(2.10)

It is well known that the strong limit .(M D )(t) := s − limy↓0 M D (t + iy) exists for a.e. .t ∈ R. Therefore the following function is well-defined dM D (t) := rank (Im (M D (t))) = dim (ran (Im (M D (t))))

.

for a.e.

t ∈ R.

Proposition 2.8 ([47, Proposition 3.2]) Let A be a simple densely defined closed symmetric operator, let . = {H, 0 , 1 } be a B-generalized boundary triple for .A∗ and let .M(·) be the corresponding Weyl function. Further, let .EA0 (·) be the spectral measure of .A0 = A∗  ker(0 ). If .D ∈ S2 (H ) and .ker(D) = ker(D ∗ ) = {0}, then NAac (t) = dM D (t) for a.e. 0

.

t ∈ R,

σac (A0 ) = clac (supp (dM D )),

and

where .supp (dM D ) = {λ ∈ R : dM D (λ) > 0} and .clac (K) is the ac-closure of K, see definition in [47]. If, in addition, the limit .M(t) := s − limy→+0 M(t + iy) exists for a.e. .t ∈ R, (t) = dM (t) for a.e. .t ∈ R. and .σac (A0 ) = clac (supp (dM )). then .NAac 0

3 Functional Models in L2 (, H ) 3.1 Space L2 (, H ) Following the book [11] we recall the definition of the space .L2 ( , H ). Let .C00 (H ) denote the set of strongly continuous compactly supported functions f with values in a finite-dimensional subspace of .H depending on f . On the set .C00 (H ) one defines the semiscalar product  (f, g)L2 ( ,H ) =

.

R

(d (t)f (t), g(t))H =

lim

d(πn )→0

n

( (k )f (ξk ), g(ξk )),

k=1

(3.1)

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where, as usual, .πn = {a = t0 < t1 < . . . < tn = b} is a partition of a segment .[a; b]; this segment is chosen so as to include the supports of f and g. Next, . (k ) := (tk ) − (tk−1 ), ξk ∈ [tk−1 , tk ], and .d(πn ) is the diameter of the partition .πn . The limit in (3.1) is understood in the same sense as in the definition of the Riemann–Stieltjes integral, i.e., the particular choice of .πn with a given diameter and of .ξk ∈ [tk−1 , tk ] is irrelevant. The completion of .C00 (H ) with respect to the seminorm 1/2

p(f ) := (f, f )L

2 ( ,H )

.

(3.2)

2 ( , H ) (i.e., a complete space with a seminorm in place is a semi-Hilbert space .L

2 ( , H ) :

2 ( , H ) over the kernel .ker p := {f ∈ L of a norm). The quotient of .L 2 p(f ) = 0} of the seminorm is the Hilbert space .L ( , H ). The problem of an intrinsic functional description of .L2 ( , H ) was posed by Krein [38] and, in the case where .dim H = n < ∞, it was completely solved by Kats [35] (see also [1]). To state his result we set . = (σij )ni,j =1 and let .σ :=  σii . Since the measure . is absolutely continuous with respect to .σ , the Radon– Nikodym theorem ensures the existence of a .σ -measurable matrix density . = (ψij )ni,j =1 , satisfying  () =

(t)dσ (t),

.



 n (t) := (ψij )ni,j =1 = dσij /dσ i,j =1 .

(3.3)

2 ( , Cn ) is isometrically identified In accordance with the Kac theorem the space .L with the space of .σ -measurable vector-valued functions .f : R → Cn satisfying  .

R

((t)f (t), f (t))dσ (t) (= f 2L

2 ( ,Cn ) ) < ∞.

(3.4)

2 ( , Cn )/N0 respectively, where .N0 = {f ∈ Moreover, .L2 ( , Cn ) := L n 2 L ( , C ) : f L2 ( ,Cn ) = 0} is the kernel of the seminorm. Passing to the infinite-dimensional case we introduce a scalar measure .τ chosen to be equivalent to the measure . , .τ ∼ . In this case provided that the measure . is of locally bounded variation instead of (3.3) one has the following identities  (δ) =

(t)dσ (t), δ ∈ Bb (R)

.

δ

where

(t) := w −

d (t) (τ − a.e.), dτ (3.5)

(see [11, Theorem V.1.1]). Here the density .(t) is defined to be the weak .τ -a.e. limit in .H and the integral in (3.5) is a Bochner integral convergent in .B(H ). Besides, .Varδ ( ) = δ (t)dρ(t). Note that in the case .dim H = ∞ the structure of the space .L2 ( , H ) is much more complicated even if . ∈ BVloc (R). The corresponding description is obtained in [46] and reads as follows.

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87

Theorem 3.1 Let . ∈ BVloc (R), .τ be a scalar measure equivalent to . , .τ ∼ , and let . be the density of . given by (3.5). Then

2 ( , H ) and .L2 ( , H ) the following expansions hold (i) For the spaces .L

2 ( , H ) = L



.

R

t dτ (t) =:

⊕H H,

 L2 ( , H ) =

R

⊕Ht dτ (t) =: H. (3.6)

t (⊃ H ) is the semi–Hilbert space obtained by completion of .H with Here .H  1/2 f  , and .H := respect to the seminorm .p p(f ) := f H

t := (t) t

t / ker p. H (ii) If f is a bounded .H–valued function with compact support that is continuous a.e. with respect to . , then the Riemann–Stieltjes integral (3.1) exists,

2 ( , H ) f ∈L

.

 and

R

(d (t)f (t), f (t)) = f 2L2 ( ,H ) .

Theorem 3.1 is no longer true if . ∈ / BVloc (R; H ). Simple examples show that in this case there exist continuous .H-valued functions f with compact support not

2 ( , H ) (see [46]). In particular, . is of locally bounded belonging to the space .L variation whenever it has a locally finite trace: .Varδ ( ) ≤ tr (δ). In this case Theorem 3.1 holds with .τ (δ) = tr (δ), .δ ∈ Bb (R). Note that the orthogonal measure (resolution of identity) E is of locally bounded variation if and only if its support consists of finitely many points. In the case of an arbitrary operator measure . ∈ / BVloc (R; H ) a procedure of differentiating the measure . was first proposed by Gelfand and Kostjuchenko [27] and was substantially improved by Berezanskii [11]. Namely, let K be a HilbertSchmidt operator, .K ∈ S2 (H ) satisfying .ker K = ker K ∗ = {0}, and let .τ be a scalar measure, .τ ∼ . For instance, one can take .τ (δ) := tr K ∗ (δ)K, .δ ∈ Bb (R). Then there exists the weak limit d(K ∗ K) K ∗ (n )K =: (t) =: K (t) = (t) n→∞ τ (n ) dτ

w − lim

.

(τ − a.e.)

where .n = (t − 1/n, t + 1/n). Now instead of (3.5) one has  ( (δ)f, g) =

.

δ

((t)K −1 f, K −1 g)H dτ (t),

f, g ∈ dom K −1 ,

t be the completion of .dom K −1 with respect to the seminorm Let .H 1/2 −1 hH K hH

t = (t)

.

h ∈ dom K −1 .

δ ∈ Bb (R).

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2 ( , H ) is isometrically In accordance with [46, Theorem 2.14], the space .L

t . In particular, the space identified with the direct integral (3.6) of the spaces .H

t (⊃ H ). An

2 ( , H ) contains functions .f (·) taking values in wider spaces .H .L

2 ( , H ) is given by inner product in .L  (f, g)L

2 ( ,H ) =

.

R

(f (t), g(t))H

t dτ (t),

2 ( , H ). f, g ∈ L

(3.7)

In connection with Theorem 3.1 we mention another functional space close to L2 ( , H ) investigated in [28].

.

3.2 The Lebesgue-Stieltjes Integral I with Respect to the Operator Measure Let .Q := Q be the multiplication operator in .L2 ( , H ) with the domain  dom Q = {f ∈ L2 ( , H ) :

.

R

(1 + t 2 )(d (t)f (t), f (t))H < ∞}.

For a continuous vector valued function (vvf) .f ∈ C([a, b]; H ) with compact support the definition of the Riemann-Stieltjes integral  IRS ( , f ) =

.

R

d (t)f (t)

(3.8)

is standard. Note however, that if . ∈ / BVloc (R) there exists a vvf .f ∈ C([a, b]; H ) such that the integral (3.8) diverges (see [46]). Next we recall the definition of the Lebesgue-Stieltjes integral by an operator measure, following [46]. Consider the linear space .S of step functions f =

n

.

1j uj ,

uj ∈ H,

{1 ≤ j ≤ n},

(3.9)

j =1

where .j = [αj , βj ) are disjoint intervals on .R such that . (·) is continuous at the points .αj , .βj .(1 ≤ j ≤ n). Introduce the operator .I : S → H by I (f ) =

n

.

(j )uj .

(3.10)

j =1

The following lemma plays an important role in our construction of functional model.

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89

Lemma 3.2 Let . be a .B(H )-valued measure on .R such that  .K = (1 + t 2 )−1 d (t) ∈ B(H ). R

(3.11)

Then the operator .I admits a unique continuation .I to the set .dom Q in the graph norm of Q. Moreover, the following estimate holds for all .f ∈ dom Q   I f 2H ≤ K  f 2L2 ( ,H ) + Qf 2L2 ( ,H ) .

.

3.3

(3.12)

Functional Model of a Symmetric Operator in H(M)

Let M be an .R[H]-function. Then it admits an integral representation   M(z) = Dz + C +

.

R

1 t − t − z 1 + t2

 d (t),

(3.13)

with .D = D ∗ , C = C ∗ ∈ B(H ), .D ≥ 0 and . is a .B(H )-valued measure on .R, that meets condition (3.11). Introduce the Hilbert space H(M) := ran D ⊕ L2 ( , H )

.

(3.14)

equipped with the norm .f2H(M) = u2H + f 2L2 ( ,H ) , where  f=

.

u f

 ∈ H(M),

u ∈ ran D,

f ∈ L2 ( , H ).

Denote by .PD the orthogonal projection in .H onto .ran D. Lemma 3.3 Let M be an .R[H]-function of the form (3.13) and let .H(M) be the Hilbert space given by (3.14). Then: (i) The linear relation .A0 (M)  A0 (M) =

.

0 f

    u , : f ∈ dom Q, u ∈ ran D) Qf

is a selfadjoint linear relation in .H(M).

(3.15)

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(ii) The operator  A(M)

.

0 f



 =

 −D [−1/2] I (f ) , Qf

f ∈ dom Q, PD I (f ) ∈ ran D 1/2 , (3.16)

is a closed symmetric operator in .H(M). Proof (1) The linear relation .A0 (M) is a selfadjoint linear relation in .H(M) = ran D ⊕ L2 ( , H ) because it admits an orthogonal decomposition A0 (M) = ({0} × ran D) ⊕ gr Q

.

of the graph of the selfadjoint operator Q in .L2 ( , H ) and the purely multivalued linear relation .{0} × ran D in .ran D. (2) Since .A0 (M) is an extension of .A(M) the operator .A(M) is symmetric. Next, assume that .fn ∈ dom Q and fn → f,

.

Qfn → g in L2 ( , H ), D [−1/2] I (fn ) → u in H as n → ∞.

Then .f ∈ dom Q and .g = Qf . Moreover, in accordance with Lemma 3.2 I (fn ) → I (f ) as .n → ∞. Since .PD I (fn ) → D 1/2 u, one has

.

PD I (f ) = D 1/2 u ∈ ran D 1/2 .

.

This proves that  .

0 f





0 ∈ dom A(M), and A(M) f



 =

−u g



 =

 −D [−1/2] I (f ) . Qf

Thus, the operator .A(M) is closed.   Theorem 3.4 Let .M(·) be an .Rs [H]-function of the form (1.6) and let .A(M) be defined by (3.16). Then: (i) The defect subspace .Nz (A(M)∗ ) of .A(M) is the closure of the set  ∗ .Nz

= fz =



D 1/2 h h t−z



 : h∈H

(z ∈ C \ R).

(ii) .A(M) is a simple symmetric operator in .H(M).

(3.17)

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91

(iii) The adjoint linear relation .A(M)∗ is the closure of the set    f= .A∗ (M) =

D 1/2 h fQ + t 2th +1

  ,

v QfQ −

 h t 2 +1

fQ ∈ dom Q, : h ∈ H, v ∈ ran D

 .

(3.18) (iv) The triple .A(M) = {H, 0A(M) , 1A(M) }, where A(M)

A(M)

f = h,

0

.

1

f = D 1/2 v + Ch + I (fQ ),

(3.19)

is a B-generalized boundary triple for .A∗ (M). (v) The Weyl function corresponding to the B-generalized boundary triple .A(M) , coincides with M and the corresponding .γ -field takes the form .(z ∈ C \ R)  (γ

.

A(M)

(z)h)(t) =

D 1/2 h



h t−z

.

(3.20)

Proof

  u (i) Assume that a vector . , where .u ∈ ran D, .g ∈ L2 ( , H ), belongs to g ∗ ∗ .Nz = Nz (A(M) ) and is orthogonal to .Nz for some .z ∈ C \ R. Then (D

.

1/2

   g(t) d (t) , h = 0, u, h)H + t − z¯ R

h ∈ H.

(3.21)

This implies, D 1/2 u + I (f ) = 0 for f (t) =

.

g(t) . t − z¯

Therefore, .I (f ) ∈ ran D 1/2 , and in accordance with (3.16)  .

u g(t)



 =

−D [−1/2] I (f ) (t − z¯ )f (t)

 ∈ ran (A(M) − z¯ ).

  u Hence, . ∈ Nz ∩ ran (A(M) − z¯ ), and thus .u = 0, .g = 0 and .N∗z is dense g in .Nz .

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  u (ii) Assume now, that a vector . is orthogonal to .N∗z for all .z ∈ C \ R. Then in g accordance with direct decomposition (3.6) equality (3.21) takes the form (D 1/2 u, h)H +

.

 (g(t), h) dτ (t)

t H t − z¯

R

=0

The uniqueness theorem for Cauchy-Stieltjes transform implies (D 1/2 u, h)H = 0,

(g(t), h)H

t = 0 (mod τ ),

.

h ∈ H,

and hence .u = 0, .g = 0. (iii) Denote   ∗z =  hz = {hz , zhz } : hz ∈ N∗z . N

.

 ∗z ⊆ A∗ (M), .z ∈ C \ R, is implied by the formula The inclusion .N   .hz =

D 1/2 h h t−z

  ,

zD 1/2 h



 =

zh t−z

D 1/2 h fQ + t 2th +1

  ,

zD 1/2 h QfQ − t 2h+1

 , (3.22)

h where .fQ = 1+zt t−z 1+t 2 ∈ dom Q.  ∗ . The inclusion .A0 (M) ⊆ A∗ (M) is Let us show that .A∗ (M) = A0 (M)  N i ∗  clear and hence .A0 (M)  Ni ⊆ A∗ (M). Conversely, for

  .f =

D 1/2 h fQ + t 2th +1

  ,

v QfQ −

 h

t 2 +1

∈ A∗ (M),

h ∈ H, v ∈ ran D, fQ ∈ dom Q (3.23)

f0 +  we obtain  .f =  hi , where   .f0 =

0 fQ − t 2ih +1 

 .hi =

  ,

D 1/2 h h t−i

v − iD 1/2 h QfQ − t 2ith +1

  ,

iD 1/2 h ih t−i

 ∈ A0 (M),

 ∗. ∈N i

 ∗ . Since .N∗ is dense in .Ni the linear relation Therefore, .A∗ (M) = A0 (M)  N i i A∗ (M) is dense in .A∗ (M).

.

Functional Models of Symmetric and Selfadjoint Operators

93

(iv) Let  .f be given by (3.23), where .f ∈ dom Q, .h ∈ H, .u ∈ ran D. Let us show that the mapping f  A(M) :  f ∈ A∗ (M) →  A(M)

.

is single-valued. Indeed, if  .f = 0 then .v = 0 and g := fQ +

.

th ∈ ker p 1 + t2

and

Qg−h = QfQ −

h ∈ ker p. 1 + t2

(3.24)

 Hence .h ∈ ker p, and therefore, . R d( (t)h, h)H = 0. Since .M ∈ Rs [H], this implies .h = 0. By (3.24), .fQ , QfQ ∈ ker p and hence .I (fQ ) = 0. Thus, the mapping . A(M) is single-valued. Let  .f be given by (3.23), where .fQ ∈ dom Q, .h ∈ H, .v ∈ ran D. Then the lefthand part of (1.1) takes the form (f , f)H(M) − (f, f )H(M) = 2iIm (v, D 1/2 h)H      th h + 2iIm tfQ (t), 2 + fQ (t), 2 t + 1 L2 ( ,H ) t + 1 L2 ( ,H ) .

(3.25)

For step functions .fQ ∈ S equality (3.25) yields  (f , f)H(M) − (f, f )H(M) = 2iIm (D 1/2 v, h)H + 2iIm I (fQ ), h

.

A(M)

= (1

A(M)

f, 0

A(M)

f)H − (0

A(M)

f, 1

f)H .

(3.26)

Since the set .S is dense in .dom Q, in accordance with Lemma 3.2 equality (3.26) can be extended to the set .A∗ (M). Now (1.1) follows from (3.26) by polarization. ∗ (v) Let .fz = {fz , fz } ∈ Nz , where .fz has the form (3.17). Hence taking into account (3.19), we get A(M) fz . 0

= h,

A(M) fz 1 

 = zDh + Ch + I

h th − t − z 1 + t2

 (3.27)

A(M)

and thus the mapping .0 is surjective. Moreover, the statement (iv) is implied by combining (3.27) with (3.17), (3.13) and definition (1.4).   Remark 3.5 Theorem 3.4 for .M ∈ Ru [H] was proved in [24] without precise treatment of the operator integral .I , which was done later in [46], see Lemma 3.2. Next we specify Theorem 3.4 assuming additionally that either .M ∈ Ru [H] or the symmetric operator A is densely defined.

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Corollary 3.6 (cf. [24]) Let .M ∈ Ru [H] and let .A(M) be defined by (3.15). Then: (i) The defect subspace .Nz of .A(M) is given by (3.17), i.e. .Nz = N∗z . (ii) The adjoint linear relation .A(M)∗ is given by (3.18), i.e. .A(M)∗ = A∗ (M). A(M) A(M) (iii) The triple .A(M) = {H, 0 , 1 } given by (3.19) is an ordinary T  ∗ boundary triplet for .A (M), i.e. the mapping . = 0A(M) 1A(M) is surjective. (iv) The Weyl function corresponding to .A(M) , coincides with M. Proof It follows from the equality (3.20)  γ

.

A(M)

= (Dh, h)H +

(z)h2H(M)

R

(d (t)h, h)H = |t − z|2



Im M(z) h, h Im z

 H

that .0 ∈ ρ(Im M(z)) (.z ∈ C \ R) if and only if .ran γ A(M) (z) is closed. The latter   implies statements (1)–(3) of the corollary. In the case of ovf M given by (3.13) with .D = 0 the functional model in Theorem 3.4 is substantially simplified. The following statement was proved in [24]. Corollary 3.7 Let M be an .Rs [H]-function of the form (1.6), such that BM = 0

.

lim y(M(iy)h, h) = ∞ for all

and

y↑∞

h ∈ H \ {0}

and let .A(M) be defined by A(M) := Q  dom A(M),

.

dom A(M) = {fQ ∈ dom Q : I (fQ ) = 0}. (3.28)

Then: (i) .A(M) is a closed densely defined symmetric operator in .L2 ( , H ). (ii) The adjoint linear operator .A(M)∗ is the closure of the operator .A∗ (M):  .A∗ (M) fQ (t) +

th t2 + 1

 = tfQ (t) −

h , t2 + 1

fQ ∈ dom Q, h ∈ H. (3.29) A(M)

(iii) A B-generalized boundary triple .A(M) = {H, 0 given by A(M)

0

.

f = h,

A(M)

1

A(M)

, 1

f = Ch + I (fQ ).

} for .A∗ (M) is

(3.30)

(iv) The linear operator .A0 (M) = A∗ (M)  ker 0A(M) coincides with .Q = Q∗ .

Functional Models of Symmetric and Selfadjoint Operators

95

(v) The Weyl function corresponding to the B-generalized boundary triple .A(M) , coincides with M and the corresponding .γ -field takes the form (γ

.

A(M)

h (z)h)(t) = , t −z

h∈H

and

  ∗  f (t) A(M) (¯z) f = I γ . t −z

Remark 3.8 The linear relation .A∗ (M) in Theorem 3.4 can be rewritten in an equivalent form    1/2    D h h ∈ H, u ∈ ran D u , A∗ (M) =  f= , : f, Qf − h ∈ L2 ( , H ) f Qf − h (3.31)

.

where .Q : f (t) → tf (t) is the formal multiplication by t. In this case the boundary A(M) A(M) , 1 } for .A∗ (M) takes the form triple .{H; 0 A(M) . f= 0

h,

A(M) 1 f=

D

1/2

 v + Ch + I f −

 th . t2 + 1

(3.32)

3.4 Functional Models for Proper Extensions of a Symmetric Operator We recall [25, Exercise 6.23] that every linear relation . with a nonempty resolvent set admits the representation (2.3), where .B0 , B1 ∈ B(H ) and satisfy (2.4). In the following statement we use the representation (3.31) for .A∗ (M) and write a functional model for a proper extension .A (M) of .A(M) corresponding to a linear relation . ∈

C(H ) via the equality   A(M) A(M) f + B1  1 f=0 . A (M) =  f ∈ A∗ (M) : B0 0

.

(3.33)

Here we assume (for simplicity) that the ovf M has no linear term .(D = 0). Theorem 3.9 Let . = {H; 0A(M) , 1A(M) } be the boundary triple for .A∗ (M) of the form (3.32) and let .M(·) be the corresponding Weyl function of the form (1.6), satisfying the conditions (1.9), let .B0 , B1 ∈ B(H ) and let the linear relation . be given by (2.3)–(2.4). Then:

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(i) The linear relation .A (M) takes the form 

f, Qf − h ∈ L2 ( , H ), h ∈ H, .A (M) = {f, Qf − h} : th (B0 + B1 C)h + B1 I (f − 1+t 2) = 0

 . (3.34)

(ii) The multi-valued part of .A (M) is given by mul A (M)     th = 0 . = h ∈ H : h ∈ L2 ( , H ), (B0 + B1 C)h − B1 I 1 + t2

.

(iii) .A0 (M) = ker 0A(M) coincides with the graph of .Q . (iv) The linear relation .A1 (M) is defined by 

f, Qf − h ∈ L2 ( , H ), h ∈ H, .A1 (M) = {f, Qf − h} : th Ch + I (f − 1+t 2 ) = 0;

 .

(3.35)

Proof (1) Formula (3.34) is implied by combining (3.31), (3.32) and (3.33). th (2) If .f = 0 in (3.34), then .h ∈ L2 ( , H ) and hence . 1+t 2 ∈ dom Q and the second relation in (3.34) becomes   (B0 + B1 C)h − B1 I th(1 + t 2 )−1 = 0.

.

(3) The proper extension .A0 (M) corresponds to the linear relation . in (2.3), where .B0 = I , .B1 = 0. Thus (iii) is implied by (3.34). (4) .A1 (M) is also determined by (3.34) with .B0 = 0 and .B1 = I . Therefore, h can be found from the equality Ch + I (fQ ) = 0,

.

where

fQ = f −

th . 1 + t2

This proves (3.35).  

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4 Functional Models in Reproducing Kernel Hilbert Spaces 4.1 Reproducing Kernel Hilbert Space L(M) Let .M ∈ R[H] and let M have the integral representation (1.6). Consider the kernel  Kω (z) =

.

M(z)−M(ω)∗ , z−ω¯ M (z),

z, ω ∈ C \ R; z = ω. ¯

(4.1)

Due to Aronszajn [7] there exists a Hilbert space .(L(M), ·, ·L(M) ) of .H-valued functions defined on .C \ R such that: (1) for every .ω ∈ C \ R and every .u ∈ H the vvf .Kω (·)u belongs to .L(M); (2) for every .f ∈ L(M), .ω ∈ C \ R and .u ∈ H the following identity holds .

f, Kω uL(M) = (f (ω), u)H .

(4.2)

The space .L(M) is uniquely defined by the properties (1), (2) and is called a reproducing kernel Hilbert space with the Hermitian kernel .Kω (z). Moreover, it follows from (1) and (2) that the set SM := span {Kω (·)u : ω ∈ C \ R, u ∈ H}

.

(4.3)

is dense in .L(M) and the evaluation operator E(ω) : f ∈ L(M) → f (ω) ∈ H

.

(4.4)

is continuous in .L(M). In the following theorem we show that the space .L(M) is a “Cauchy transform” of the space .H(M) defined by (3.14) and discussed in the previous section. In the case of a finite dimensional .H, .dim H < ∞, this result goes back to de Branges [14, Lemma 2], see also [5, Lemma 6.5]. Theorem 4.1 Let .M ∈ R[H] and let M admit the integral representation (1.6). u Then for any pair . ∈ H(M) the function f  .

F (z) = D

1/2

u + I

 f (t) , t −z

u ∈ ran D,

f ∈ L2 ( , H ),

z ∈ C \ R, (4.5)

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is well defined and “the Cauchy transform” .CM : H(M) → L(M),  CM :

.

u f



 → F (z) = D 1/2 u + I

f (t) t −z

 (4.6)

  u is unitary from .H(M) onto .L(M). In particular, for any . ∈ H(M) the following f “Parseval” identity holds:  .

F, F L(M) = (u, u)H +

R

(4.7)

(d (t)f (t), f (t))H .

−1 f (·) ∈ dom Q for any .f ∈ L2 ( , H ) and .z ∈ C , Lemma 3.2 Proof Since .(·−z) ±   f (t) ensures that .I t−z exists and estimate (3.12) holds. Due to (3.12) the mapping .C : H(M) → L(M) is well defined and bounded. Let us establish its unitarity. Let .G ∈ SM where the linear set .SM is given by (4.3). Combining definition (4.3) with integral representation (1.6) of .M(·) one arrives at the following representation

F (z) =

n

.

Kωj (z)uj =

j =1

n

M(z) − M(ωj ) j =1

z − ω¯ j

 uj = D

1/2

u + I

 f (t) , t −z (4.8)

where u=

n

.

D 1/2 uj ∈ ran D,

and

f (t) =

n

j =1

j =1

uj ∈ L2 ( , H ). t − ω¯ j

(4.9)

By definition (4.6) of the operator .CM the vvf F admits the representation F = CM f,

.

where

f=

  u ∈ C−1 M SM ⊂ H(M). f

  u By Theorem 3.4(2) the set .C−1 S of vvf’s . of the form (4.9) is dense in .H(M). M M f Combining formulas (4.8) and (4.9) with the reproducing kernel property (2) yields

.

F, F L(M) =

n n

  Kωj (z)uj , Kωk (z)uk L(M) = Kωj (ωk )uj , uk H j,k=1

=

n

j,k=1

j,k=1

  D 1/2 uj , D 1/2 uk + H

n

j,k=1

= u2H + f 2L2 ( ,H ) = f2H(M) .



uj uk , t − ωj t − ωk

 L2 ( ,H )

(4.10)

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Hence, the mapping .CM : H(M) → L(M) is isometric on the subspace .C−1 M SM of .H(M). Further, in accordance with Theorem 3.4(ii) and formulas (4.9) and (3.17), the set .C−1 M SM is dense in .H(M). Combining this fact with the density of .SM in .L(M) yields the unitarity of the operator .CM .   Corollary 4.2 The operator  + :

.

u f



 → D

1/2

u + I

 f (t) , t −z

u ∈ ran D, f ∈ L2 ( , H )

(4.11)

is a unitary operator from .H(M) onto .L(M). Lemma 4.3 Let .M ∈ R[H] and let .0 ∈ ρ(M(λ)) for some .λ ∈ C \ R. Then M− := −M −1 ∈ R[H] and the multiplication operator

.

F ∈ L(M) → M− F ∈ L(M− )

.

isometrically maps .L(M) onto .L(M− ). For a pair .(A0 , A1 ) of two selfadjoint operators satisfying the Kuroda hypothesis this statement goes back to [14], in the case .dim H < ∞ its proof is contained in [6, Lemma 2.6] and can be easily extended to the case .dim H = ∞. The ovf .M− = −M −1 also admits an integral representation   M− (z) = D− z + C− +

.

R

1 t − t − z 1 + t2

 d − (t),

(4.12)

∗,C ∗ with operators .D− = D− − = C− ∈ B(H ), .D− ≥ 0 and a .B(H )-valued measure . − on .R, which satisfies (3.11).

Definition 4.4 ([13]) Let .M ∈ R[H] and let .0 ∈ ρ(M(i)). Let .CM+ : H(M+ ) → L(M+ ) and .CM− : H(M− ) → L(M− ) be Cauchy transforms defined in (4.6). Then HM := C−1 M− MM− CM+

.

(4.13)

is a unitary operator from .H(M+ ) onto .H(M− ). If the spaces .H(M+ ) and .H(M− ) are reduced to .L2 ( + , H ) and .L2 ( − , H ), then .HM is called the generalized Hilbert transform.

4.2 Functional Model in L(M) A functional model of a symmetric operator .S(M) in .L(M) with the Weyl function M can be derived from the analogous functional model in .H(M) using the unitary mapping .CM .

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Let us define the operator .Rω in .L(M) for every .ω ∈ C \ R by (Rω F )(z) =

.

F (z) − F (ω) , z−ω

(Rω F )(ω) = F (ω),

F ∈ L(M).

(4.14)

Next we will show that .Rω is the resolvent of a selfadjoint linear relation in .L(M). Lemma 4.5 (cf. [4]) Let .M ∈ R[H] and let .S(M) be the multiplication operator in the reproducing kernel Hilbert space .L(M) with the domain dom S(M) = {F ∈ L(M) : F, S(M)F ∈ L(M)} = {F ∈ L(M) : F, λF ∈ L(M)}.

.

Then: (i) .Rω is the resolvent of a selfadjoint linear relation .S0 (M) given by     for some u1 ∈ H F 2

 .S0 (M) = F =

∈ L(M) : F (z) − zF (z) ≡ u1 and all z ∈ C \ R . F (ii) .S(M) is a closed simple symmetric operator in .L(M). (iii) The defect subspace .Nω of .S(M) is the closure in .L(M) of the set {Kω¯ u : u ∈ H},

ω ∈ C \ R.

.

(4.15)

Proof (1) Let a vvf .G ∈ L(M) have a representation (4.5), i.e. G = CM g

.

  v with g = , g

v ∈ ran D,

g ∈ L2 ( , H ).

It follows from (4.14) that  (Rω G)(z) = I

.

g(t) (t − ω)(t − z)

On the other hand .(A0 (M) − ωI )−1 g =

(CM (A0 (M) − ωI )

.

−1

 g ∈ L2 ( , H ).  0 and hence (Q − ωI )−1 g



 g)(z) = I

 g(t) . (t − ω)(t − z)

It follows from (4.16) and (4.17) that Rω = CM (A0 (M) − ωI )−1 C−1 M

.

(4.16)

(4.17)

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and hence .Rω is the resolvent of the selfadjoint operator .

S0 := CM A0 (M)C−1 M in .L(M). The proof of the equality .

S0 = S0 (M) is straightforward. (2) Assume that .G ∈ L(M) and G is orthogonal to .Kω¯ u for some .ω ∈ C \ R and all .u ∈ H. Then (G(ω), ¯ u)H = (G, Kω¯ u)L(M) = 0

.

and hence, .G(ω) ¯ = 0. Therefore, the vvf F (z) :=

.

G(z) , z − ω¯

z ∈ C \ R,

belongs to .L(M) and .G = (S(M)− ω)F ¯ ∈ ran (S(M)− ω). ¯ Assuming .G ∈ Nω one obtains .G = 0. This proves that the set (4.15) is dense in .Nω . Next, assume that .G ∈ L(M) and G is orthogonal to .Kω u for all .ω ∈ C \ R, .u ∈ H. Then, as is shown above, .G ≡ 0 and, thus, the operator .S(M) is simple.     = F ∈ Theorem 4.6 Let .M ∈ Rs [H] and let .S∗ (M) consists of the vvf’s .F

F 2 L(M) such that for some .u0 , u1 ∈ H the following relation holds

(z) − zF (z) = u1 − M(z)u0 F

.

Then: (i) .S∗ (M) is dense

in .S(M)∗

z ∈ C \ R.

for all

(4.18)

  F  and for all .F = ∈ S∗ (M), satisfying (4.5) the F

following equality holds

, F L(M) − F, F

L(M) = (u1 , u0 )H − (u0 , u1 )H . F

.

S(M)

(ii) The triple .S(M) = {H, 0

S(M) 

0

.

S(M)

, 1

F = u0 ,

(4.19)

} defined by S(M) 

1

F = u1 ,

(4.20)

is a B-generalized boundary triple for .S(M)∗ . (iii) The Weyl function corresponding to the boundary triple .S(M) coincides with .M(·) and the corresponding .γ -field takes the form γ S(M) (ω)u = Kω¯ u,

.

u ∈ H, ω ∈ C \ R.

(4.21)

The adjoint mapping is the evaluation operator γ S(M) (ω)∗ F = F (ω), ¯

.

F ∈ L(M), ω ∈ C \ R.

(4.22)

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Remark 4.7 Lemma 4.5 and Theorem 4.6 were presented in another form in [24, Theorem 6.1]. In the case .dim H < ∞ Lemma 4.5 was proved in [4]. Proposition 4.8 Let under the assumptions of Theorem 4.6 .0 ∈ ρ(M(i)). Then S1 (M) is a selfadjoint linear relation and its resolvent is given by

.

(S1 (M) − ωI )−1 G = Rω G − (Rω M)M(ω)−1 G(ω),

.

ω ∈ C \ R, G ∈ L(H ). (4.23)

Proof The first statement is implied by Proposition 2.4 and Theorem 4.6. Assume

} ∈ S1 (M) satisfies the equality that .G ∈ L(H ) and the pair .{F, F

(z) − ωF (z), G(z) = F

.

z ∈ C \ R.

  F Since . ∈ S1 (M) and satisfies the equality (4.18) for some .u0 , u1 ∈ H, we have F (z − ω)F (z) = G(z) + M(z)u0 .

.

Setting .z = ω here one obtains .u0 = −M(ω)−1 G(ω) and F (z) =

.

G(z) − M(z)M(ω)−1 G(ω) . z−ω

It remains to notice that the right hand parts of (4.23) and (4.24) coincide.

(4.24)  

4.3 Unitary Equivalence In accordance with Proposition 2.7 a B-generalized boundary triple . for .A∗ with the corresponding Weyl function M is unitary equivalent to the model triple .S(M) defined in Theorem 4.6. Next result clarifies this statement by expressing the unitary operator U in terms of .. Theorem 4.9 Let A be a simple symmetric operator in a Hilbert space .H, let . = {H, 0 , 1 } be a B-generalized boundary triple for .A∗ , let M be the corresponding S(M) S(M) Weyl function. Let also .S(M) = {H, 0 , 1 } be a B-generalized boundary ∗ triple for .S(M) defined in Theorem 4.6 and let .Rω be given by (4.14). Then: (i) The operator .U+ : h ∈ H → γ (¯z)∗ h, .z ∈ C \ R, isometrically maps .H onto .L(M). In particular, the following identity holds U+ γ (ω)u = Kω¯ u,

.

ω ∈ C \ R.

(4.25)

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(ii) The resolvents of linear relations .A0 and .S0 (M) are unitary equivalent by means of the operator .U+ , i.e. the following identity holds U+ (A0 − ωI )−1 U+−1 = Rω ,

.

ω ∈ C \ R.

(4.26)

(iii) The boundary triples . and .(M) are unitary equivalent by means of .U+ , i.e. U+ A = S(M)U+ ,

.

S(M)

0 = 0

U+ ,

S(M)

1 = 1

(4.27)

U+ .

(iv) If, additionally, .0 ∈ ρ(M(i)), then the resolvents of linear relations .A1 and .S1 (M) are related by the equality U+ (A1 −ω)−1 U+−1 = (S1 (M)−ω)−1 = Rω −Kω¯ M(ω)−1 E(ω),

.

ω ∈ C \ R,

where .E(ω) is the evaluation operator (4.4). Proof (1) Consider a restriction .U0 of the operator .U+ to the linear set H0 = span{Nω : ω ∈ C \ R} = span{γ (ω)u : ω ∈ C \ R, u ∈ H}

.

Since the operator A is simple, the linear set .H0 is dense in .H. By definition, any .h ∈ H0 admits a representation h=

n

.

γ (ωj )uj

with ωj ∈ C \ R and

uj ∈ H.

j =1

Taking identity (1.5) into account and using definition (4.1) one gets

(U+

n

.

γ (ωj )uj )(z) := γ (¯z)∗

j =1

n

γ (ωj )uj =

j =1

=

n

j =1

Kω¯ j (z)uj .

n

M(ωj ) − M(z) j =1

ωj − z

uj

(4.28)

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Clearly, for .n = 1 we arrive at equality (4.25). In turn, combining this identity with property (4.2), definition (4.1), and identity (1.5) implies  2   n

   . U+ γ (ω )u j j    j =1

L(M)

=

 n

Kω¯ j uj ,

j =1

n

 Kω¯ k uk

k=1

L(M)

 2  n     = (Kω¯ j (ω¯ k )uj , uk )H =  γ (ωj )uj   . j =1  j,k=1 n

H

(4.29) Combining this equality with (4.28) shows that the operator .U+ isometrically maps .H0 onto the set .SM (⊂ L(M)) given by (4.3). Since both sets .SM and .H0 are dense in .L(M) and .H, respectively, the isometric map .U0 admits a continuation to a unitary operator from .H onto .L(M). Since .H0 is dense in .H, for any .h ∈ H there is a sequence .hn ∈ H0 converging to h in .H. Then in accordance with (4.29) the sequence Fn (ω) := γ (ω) ¯ ∗ hn

.

ω ∈ C \ R,

for all

n ∈ N,

is a Cauchy sequence in .L(M). Hence there exists .F ∈ L(M) such that .Fn converges to F in .L(M). This implies the pointwise convergence γ (ω) ¯ ∗ hn → F (ω) for all

ω ∈ C \ R.

.

On the other hand, .γ (ω) ¯ ∗ hn → γ (ω) ¯ ∗h equality

for all

(U+ h)(ω) = γ (ω) ¯ ∗ h = F (ω),

.

ω ∈ C \ R. This proves the ω ∈ C \ R,

and hence .U+ is a unitary continuation of .U0 . (2) To prove (4.26) it suffices to check that U+ (A0 − ωI )−1 h = Rω U+ h

.

for all

h ∈ H, ω ∈ C \ R.

(4.30)

Indeed, it follows from (4.1), the definition of the operators .U+ and .Rω and identity (1.5) that for all .h = γ (λ¯ )u ∈ H0 , .u ∈ H, and .λ, z ∈ C \ R: Rω U + h =

.

M(z)(ω−λ¯ )+M(λ¯ )(z−ω)+M(ω)(λ¯ −z) Kλ (z)u−Kλ (ω)u = u. z−ω (ω−z)(ω−λ¯ )(λ¯ −z)

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On the other hand, U+ (A0 − ωI )−1 h = U+

.

γ (ω)u − γ (λ¯ )u Kω¯ (z)u − Kλ (z)u = ω − λ¯ ω − λ¯

¯ + M(λ)(z ¯ M(z)(ω − λ) − ω) + M(ω)(λ¯ − z) u. (ω − z)(ω − λ¯ )(λ¯ − z)

=

(4.31)

Combining this relation with the previous one leads to equality (4.30) for all h ∈ H0 . Since .H0 is dense in .H, equality (4.30) is easily extended to all .h ∈ H. But (4.26) coincides with (4.30). (3) The first equality in (4.27) follows from (4.26). Next the equality (4.25) yields .

S(M)

0

.

U+ | N  ω (A∗ ) = 0 |N  ω (A∗ ) .

 ω (A∗ ), we get . S(M) U+ = 0 . Since .A∗ = A0  N 0 Next, for every .h ∈ H one obtains S(M)

1

.

U+ (A0 − ωI )−1 h = 1

S(M)

(A0 (M) − ωI )−1 U+ h = γ+ (ω) ¯ ∗ U+ h

= E(ω)U+ h = γ (ω) ¯ ∗ h = 1 (A0 − ωI )−1 h Since S(M)

1

.

U+ 1 γ (ω)u = M(ω)u = 1 γ (ω)u S(M)

for all .u ∈ H, .ω ∈ C \ R this proves the equality .1 (4) It follows from (2.6) that for all .ω ∈ C \ R, h ∈ H

U+ =  1 .

U+ (A1 − ω)−1 h = U+ (A0 − ωI )−1 h − U+ γ (ω)M(ω)−1 γ (ω) ¯ ∗h

.

= Rω U+ h − Kω¯ M(ω)−1 U+ h = (S1 (M) − ω)−1 U+ h. This proves (4.26).   Remark 4.10 The proof of Theorem 4.9 can be extracted from Theorem 4.1 where H = H(M) and the unitary mapping .U+ turns out to be the Cauchy transform:

.

U+ : H(M) → L(M),

.

U+ : h ∈ H → CM h = γ (¯z)∗ h,

h ∈ H(M).

Here .γ (¯z)∗ is defined in Theorem (3.4) (see formula (3.20)). However we present independent proof because certain its details could be of interest.

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Remark 4.11 Note that results on functional models were preceded by results on triangular models constructed first by Livšic in [44] for bounded operators with trace class imaginary part using the concept of characteristic functions introduced in the same paper (see also the survey of Brodskii and Livšic [17]). Generalizations to wider classes of operators with compact imaginary part as well as the theory of triangular representation of certain classes of bounded operators originally created by Brodskii can be found in monographs [32] and [16]. Generalizations to the case of unbounded operators can be traced in the monograph [8]. A functional model for a contraction on a Hilbert space was discovered by Sz.Nagy and Foias (see [59] and references therein). They have indicated numerous applications of the model in spectral theory of non-selfadjoint operators (innerouter factorization, similarity, unicellularity criteria, etc.). Many other applications including completeness and Riesz basis property, interpolation problems, etc., are discussed in [53]. A version of a functional model for a dissipative operator was proposed by Pavlov [54]. In the framework of this model eigenfunction expansions of m-dissipative Sturm-Liouville operator T , as well as eigenfunctions of the ac-spectrum of T , and the scattering matrix were investigated. An interesting approach to the spectral theory of non-dissipative operators was firstly proposed by Naboko [49, 50]. Namely, he treated a non-dissipative operator .T = A + iB as a perturbation of its “dissipative part” .Tdis := A + i|B| in the framework of Pavlov’s model of .Tdis . In the framework of this model he introduced the absolutely continuous, singular, and singular continuous subspaces [49, 50]. Using this technique Naboko obtained non-selfadjoint versions of the Kato-Rosenblum theorem, conditions for similarity to selfadjoint and dissipative operators, investigated the scattering matrix and obtained its factorization [49–51], see also [56] for extension to the case of almost solvable extensions of a symmetric operator A. The reproducing kernel space .L(M) was first introduced by de Branges in [14] for a scalar M, the functional model of a symmetric operator in .L(M) in the matrix case was build in [4]. Another two-component functional model for a pair of selfadjoint operators was introduced by de Branges and Rovnyak [15] in connection with the scattering problem for this pair, see further development and applications to interpolation problems in [5, 6]. Notice also, that functional models for symmetric operators in the space .B(E) based on a Hermite-Biehler function .E were introduced and applyed to canonical systems by de Branges [13, 14], see also further development to Pontryagin-de Branges spaces in [36], and to matrix case in [20, 26].

5 Perturbation Theory and Functional Models Functional models for finite-dimensional perturbations of selfadjoint operators have been studied in [6]. In the present section we consider functional models for pairs of

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107

selfadjoint operators in a general setting. First, we apply the functional model from Theorem 3.4 to the case when .A− is an additive perturbation of .A+ . Next, a model for .H−1 -perturbation of a selfadjoint operator is presented.

5.1 The Case of Additive Perturbation Let .A+ and .A− be selfadjoint operators on a Hilbert space .H such that .dom A+ = dom A− and (A− − A+ )f = GJ G∗ f

.

for f ∈ dom A+ ,

(5.1)

where G is a bounded linear operator from an auxiliary Hilbert space .H to .H, such that .ker G = {0}, and J is a signature operator in .H, i.e. .J = J ∗ = J −1 . Consider a symmetric linear operator .A = A+  ker G∗ . Clearly, .n+ (A) = n− (A), because .A+ is its selfadjoint extension. Since .A+ and .A− are disjoint extensions of A there is a double B-generalized boundary triple .+ = {H, 0+ , 1+ } such that A+ = ker 0 ,

.

A− = ker 1 .

(5.2)

Here we construct a B-generalized boundary triple .+ = {H, 0+ , 1+ } with the properties (5.2) explicitly. Proposition 5.1 Let .A± be selfadjoint operators in a Hilbert space .H satisfying (5.1) and let .A∗ be a linear relation in .H given by   f : f ∈ dom A+ , u ∈ H . A+ f − Gu

 A∗ =

.

(5.3)

Then: (i) .A∗ is dense in .A∗ . (ii) The triple .+ = {H, 0+ , 1+ } with 0+ f = u,

.

1+ f = J u + G∗ f,

(5.4)

is a B-generalized boundary triple for .A∗ such that (5.2) holds. (iii) The .γ -field .γ+ (·) and the Weyl function .M+ (·) corresponding to the Bgeneralized boundary triple .+ are given by γ+ (z) = (A+ − z)−1 G,

.

M+ (z) = J + G∗ (A+ − z)−1 G,

z ∈ ρ(A+ ). (5.5)

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Proof (1) Clearly, .A∗ ⊂ A∗ . In view of (5.3), the set .Nz (A∗ ) = ker(A∗ − zI ) admits the representation Nz (A∗ ) = {fz = (A+ − z)−1 Gu :

.

u ∈ H}.

(5.6)

Clearly, .Nz (A∗ ) is contained in .Nz := Nz (A∗ ) and is dense in .Nz , since the equality ((A+ − z)−1 Gu, f ) = 0 for all u ∈ H

.

implies .(Gu, (A+ − z¯ )−1 f ) = 0 for all .u ∈ H, and hence .g = (A+ − z¯ )−1 f ∈ dom A or .f = (A − z¯ )g ∈ ran (A − z¯ ) ∩ Nz = {0}. Since .Nz (A∗ ) is dense in .Nz , one obtains  ∗z , A∗ = A+  N

.

z A∗ = A+  N

∗ and thus .A   ∗ is dense in.A .  g f  ∈ A∗ . Then (2) Let .f = , . g= A+ g − Gv A+ f − Gu

(A+ f − Gu, g)H − (f, A+ g − Gv)H = (G∗ f, v)H − (u, G∗ g)H

.

= (G∗ f + J u, v)H − (u, G∗ g + J v)H and, hence, the Green identity (1.1) holds for .0+ , .1+ defined by (5.4). The equalities (5.2) are implied by (5.1), (5.3) and (5.4). Since .ker 0+ = A+ and is surjective, the triple . is a B-generalized boundary triple for .A∗ . For every .u ∈ H and .z ∈ C \ R one obtains

+ . 0

z = .f



fz zfz





 (A+ − z)−1 Gu = . A+ (A+ − z)−1 Gu − Gu

By (5.6) and (5.4), one has 0+ fz = u,

.

1+ fz = J u + G∗ (A+ − z)−1 Gu.

Combining (5.7) with (5.6) yields (5.5).

(5.7)  

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Proposition 5.2 Let the assumptions of Proposition 5.1 be satisfied. Then: (i) The linear relation .A∗ admits the dual representation    .A∗ = f =

  f : f ∈ dom A− , v ∈ H . A− f − Gv

(5.8)

(ii) The triple .− = {H, 0− , 1− } with 0− f = J v,

.

1− f = −v + J G∗ g,

(5.9)

is a B-generalized boundary triple for .A∗ , such that .A+ = ker 1− , .A− = ker 0− . (iii) The .γ -field .γ− and the Weyl function .M− of A corresponding to the Bgeneralized boundary triple .− are given by γ− (z) = (A− − z)−1 GJ,

.

z ∈ ρ(A− ),

M− (z) = −M+ (z)−1 = −J + J G∗ (A− − z)−1 GJ,

.

(5.10)

z ∈ ρ(A− ).

(5.11)

Proof In view of (5.1) f = {f, A+ f − Gu} = {f, A− f − Gv},

.

where v = u + J G∗ u.

Hence the formulas (5.4) take the form 0+ f = u = v − J G∗ f,

.

1+ f = J u + G∗ h = J v,

Then the transposed B-generalized boundary triple .− = {H, 0− , 1− } := {H, 1+ , −0+ } coincides with (5.9). Setting fz =



.

fz zfz



 =

 (A− − z)−1 GJ v , A− (A− − z)−1 GJ v − GJ v

v ∈ H,

z ∈ C \ R,

one obtains 0− fz = v,

.

1− fz = −J v + J G∗ (A− − z)−1 GJ v.

Since the B-generalized boundary triple .− is transposed to the triple ., one gets −1 .M− = −M+ . Besides, one easily obtains formula (5.11) for .M− using (5.1).   Proposition 5.3 Let in the assumptions of Proposition 5.1 the symmetric operator A is simple and let .M+ and .M− be given by (5.5) and (5.11), respectively. Then:

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(i) There exist operator valued finite measures . ± (·) such that .M± (·) admit the following representations   d − (t) d + (t) , M− (z) = −J + . (5.12) .M+ (z) = J + t − z R t −z R (ii) There is a unitary operator .V+ : H → L2 ( + ) such that (V+ A+ V+−1 f )(t) = tf (t),

(V+ A− V+−1 f )(t) = tf (t) + J I + (f ). (5.13)

.

(iii) There is a unitary operator .V− : H → L2 ( − ) such that (V− A+ V−−1 f )(t) = tf (t) − J I − (f ),

.

(V− A− V−−1 f )(t) = tf (t). (5.14)

Proof (1) Let .E± (t) be resolutions of the identity for the operators .A± . Let us set + (t) = G∗ E+ (t)G,

.

− (t) = J G∗ E− (t)GJ.

Then representations (5.5) and (5.11) for .M± take the form (5.12). Note also that . + (R) = G∗ G and . − (R) = J G∗ GJ , and hence both measures . ± are finite. (2) Consider the B-generalized boundary triple .+ = {H, 0+ , 1+ } for .A∗ defined by formulas (5.4). By Proposition 5.1, the corresponding Weyl function M is given by (5.12), .M = M+ . Consider also the minimal multiplication operator 2 .A(M) in .L ( + , H ) defined by (1.10) and the B-generalized boundary A(M) A(M) A(M) triple . = {H, 0 , 1 } for .A(M)∗ defined in Theorem 3.4. Since the corresponding Weyl function is also M and the operator .A(M) is simple, Proposition 2.7 ensures unitary equivalence of the triples .+ and .A(M) . In particular, it means (see (2.9)) that there exists a unitary mapping .V+ : H → L2 ( + , H ) that intertwines the triples .{A, A+ , A− } and .{A(M), A0 (M), A1 (M)}, i.e. V+ AV+−1 = A(M),

V+ A+ V+−1 = A0 (M),

.

V+ A− V+−1 = A1 (M). (5.15)

Further, comparing representations (1.6) and (5.12) for .M = M+ and noting that the measure . + (·) is finite we obtain  CM −

.

R

t d + (t) = J. 1 + t2

(5.16)

By Theorem 3.9 the domain .dom A1 (M) is given by (3.32). Inserting the last th identity into (3.32) and noting that . 1+t 2 ∈ dom I + , because the measure . +

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is finite, we obtain  0 = CM h + I + f −

.

th 1 + t2



 = CM h − I +

th 1 + t2

 + I + f

= J h + I + f

(5.17)

for any .f ∈ dom A1 (M) and .h ∈ H. This identity implies .h = −J I + f . In turn, inserting this expression into formula (3.35) for .A1 (M) gives A1 (M)f = Qf − h = Qf + J I + f.

.

Combining this formula with (5.15) proves the second formula in (5.13). (3) To prove the third statement let us consider the unitary operator V− = HM V+ : H → L2 ( − , H ),

.

where .HM is the generalized Hilbert transform introduced in (4.13). It follows from (5.15) that .V− A+ V−−1 = A1 (M− ), .V− A− V−−1 = A0 (M− ).  

5.2 Applications to de Branges-Rovnyak and Carey Perturbation Results Theorem 5.4 Let A+ and A− be selfadjoint operators in H and let condition (5.1) be satisfied with G ∈ B(H, H), i.e. A− − A+ = GJ G∗ and ker G = {0}. Let also the symmetric operator A := A+  ker G∗ be simple. Then there exists a Nevanlinna function M(·) ∈ Rs [H] with −M(·)−1 ∈ Rs [H] and such that the following hold: (i) There exist unitary transformations U+ : H → L(M) and

.

U− : H → L(−M −1 )

satisfying the identities .

U+ (A+ − ωI )−1 = Rω U+ ,

U− (A− − ωI )−1 = Rω U− ,

ω ∈ C \ R. (5.18)

Here Rω is the resolvent of the model operator S0 (M) (see (4.14)). (ii) The following identity holds: −M −1 (z)U+ h = U− h for every h ∈ H.

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Proof (1) In accordance with Propositions 5.1 and 5.2 there exist boundary triples ± = {H, 0± , 1± } for A∗ such that A+ = ker 0+ = ker 1− and A− = ker 1+ = ker 0− . Let also M± and γ± (·) be the corresponding Weyl functions and γ fields. As a desired Rs [H]-function M we choose the Weyl function .

M(z) = M+ (z) = J + G∗ (A+ − z)−1 G

(5.19)

and note that M ∈ Rs [H] because ker G = {0}. Next we introduce the reproducing kernel Hilbert space L(M) with M = M+ and define the operators U± : H → L(M+ ) by formulas (4.28), i.e. we set U+ : h → γ+ (¯z)∗ h

and

.

U− : h → γ− (¯z)∗ h,

h ∈ H,

z ∈ C± , (5.20)

(cf. (4.28)). Then, by Theorem 4.9, the operators U± unitarily map H onto L(M) and intertwines the operators (A± − ωI )−1 and Rω , i.e. satisfy the identities (5.18). (2) Let us prove the second statement. Using definitions (5.20) of the operators U± and taking adjoints in identity (2.7) one gets for h ∈ H ∗  U− h = γ− (¯z)∗ h = −M(z)−1 γ+ (¯z)∗ h

.

= −M(z)−1 γ+ (¯z)∗ h = −M(z)−1 U+ h. This proves statement (ii).   Proposition 5.5 Let A± be selfadjoint operators in H with A− − A+ = GJ G∗ ∈ B(H). Let also the symmetric operator A := A+  ker G∗ be simple. Then: (i) The operator valued function (5.19) is the complete unitary invariant of the triple {A, A+ , A− }, i.e. it defines this triple uniquely up to unitary equivalence. (ii) Let also G ∈ S2 (H ). Then the spectral multiplicity function NAac (t) for the + of the operator A is given by ac-part Aac + +  NAac (t) = rank +

.

 d ∗ G EA+ (t)G dt

for a.e.

t ∈ R.

(5.21)

Here the derivative exists in S1 -norm for a.e. t ∈ R. Proof (i) Consider the double B-generalized boundary triple for A∗ defined by (5.4). The corresponding Weyl function is given by (5.19). Now the required statement is immediate from Proposition 2.7.

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(ii) It easily follows from (5.19) that  Im M+ (t + iy) =

.

R

 y d G∗ EA+ (s)G (t − s)2 + y 2

In accordance with Fatou theorem the strong limit Im M+ (t + i0) = s − lim Im M+ (t + iy) exists and the following equality holds y↓0

 Im M+ (t + i0) = π · d G∗ EA+ (t)G /dt

.

for a.e.

t ∈ R,

(5.22)

where the derivative on the right hand side exists in S1 (H)-norm. Combining this equality with Proposition 2.8 applied to the Weyl function (5.19) yields   NAac (t) = rank Im M+ (t + i0) = rank d G∗ EA+ (t)G /dt for a.e. t ∈ R. +

.

This proves the required relation (5.21). Besides, Proposition 2.8 ensures that   σac (A+ ) = clac (supp (dM )). Remark 5.6 (i) Theorem 5.4 generalizes the result by de Branges and Rovnyak obtained in [15] for the case of trace class difference A− − A+ ∈ S1 (H). From another point of view the case of finite dimensional difference GJ G∗ (the case of matrix function M+ ) was treated in details by Alpay and Gohberg in [6]. In fact, boundary triple approach allows to treat general pairs {A+ , A− } with A− being a singular (i.e. non-additive) perturbation of A+ . Besides, in the framework of this approach we indicate the unitary operators U± = γ± (¯z)∗ satisfying identities (5.18). As in (5.20) they are expressed explicitly in terms of B-generalized triple. (ii) For J = I Proposition 5.5 was proved by Carey [19, Section 5]. Regarding the statement (ii) we note that according to the Naboko result [52] the normal limit M+ (x +i0) exists in Sp (H)-norm for any p > 1, while the limit Im M+ (x +i0) in (5.22) exists in S1 (H)-norm (see [14] and also [52]). By applying Proposition 2.8 the statement (ii) can easily be extended to the case of any bounded difference GJ G∗ ∈ B(H) where the Weyl function M+ (·) of the form (5.19) is called in [19] “the determining function”. Operator valued functions of the form (5.19) with J = I appeared in [29] in connection with spectral shift functions.

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5.3 H−1 -perturbations of Selfadjoint Operators Let .A+ be a selfadjoint operator in a Hilbert space .H, let .{Hs }s∈R be the scale generated by .A+ :  Hs := dom |A+ |s = {f ∈ H :

.

R

(1 + |t|)s d(E+ (t)f, f ) < ∞},

(5.23)

where .E+ (·) = EA+ (·) is the spectral measure of .A+ . In particular, .H2 = dom A+

+ the and .H1 = dom |A+ |1/2 . Clearly, .H−s is the dual space to .Hs . Denote by .A .B(H1 , H−1 )-continuation of .A+ , see [11]. Let G be an injective bounded operator from .H into .H−1 , .G ∈ B(H, H−1 ), and let .G∗ denote its adjoint, .G∗ ∈ B(H1 , H ). Consider the operator

+ + V := A

+ + GJ G∗ , A− = A

.

+ + V )f ∈ H}. dom A− = {f ∈ H : (A (5.24)

Then .A− is also a self-adjoint operator in .H. It is called the .H−1 -perturbation of .A+ . Since .G∗ ∈ B(H1 , H ) and embedding .B(H2 ) → B(H1 ) is continuous, then also .G∗  H2 ∈ B(H2 , H ). Hence, the linear space .ker G∗ is closed in .H2 = dom A+ and, therefore, the operator A defined as the restriction of .A+ to the domain dom A = dom A+ ∩ ker G∗

(5.25)

.

is closed and symmetric in .H. Let us set     f

 .A∗ = f =

+ f − Gu : f ∈ H1 , A+ f − Gu ∈ H, u ∈ H . A

(5.26)

Proposition 5.7 Let .A+ be a selfadjoint operator in a Hilbert space .H, .G ∈ B(H, H−1 ), .J ∈ B(H ) and let A, .A∗ be defined by (5.25), (5.26). Then: (i) The linear relation .A∗ is dense in .A∗ . (ii) The triple . = {H, 0 , 1 } with 0 f = u,

.

1 f = J u + G∗ f,

(5.27)

is a B-generalized boundary triple for .A∗ , such that .A+ = ker 0 . (iii) The corresponding .γ -field .γ and the Weyl function M of A corresponding to the B-generalized boundary triple . take the form

+ − z)−1 G, γ (z) = (A

.

+ − z)−1 G, M(z) = J + G∗ (A

z ∈ ρ(A+ ). (5.28)

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(iv) The extension .A1 = ker 1 is a selfadjoint extension of A, which coincides with the extension .A− defined in (5.24). Proof The proof of items (i)–(iii) is similar to that in Proposition 5.1.

+ − z)−1 G (iv) The coincidence .A1 = A− follows from (5.26), (5.27). Since .G∗ (A converges uniformly to 0 as .|Im z| → ∞, the ovf M has the property .0 ∈ ρ(M(z)) for .|Im z| big enough. Therefore, by Proposition 2.3 the extension .A1 is selfadjoint.   To state the next result we recall that .Q denotes the formal multiplication operator defined on .L2 ( , H ), .Q : f → tf (t). If the measure . is not finite, then .Qf ∈ / L2 ( , H ) for .f ∈ L2 ( , H ) \ dom Q ± . In the case of .H−1 -perturbations (.G ∈ B(H, H−1 )) the integral .I ± can be extended to a wider set {f ∈ L2 ( ± , H ) : Qf − h ∈ L2 ( ± , H )

.

for some h ∈ H}( dom Q ± )

as follows  I ± f := I ± f −

.

th 1 + t2



 +

R

td ± (t) h. 1 + t2

(5.29)

Due to Lemma 3.2, this mapping is an extension of .I ± defined originally on dom Q ± .

.

Theorem 5.8 Let .A± be the same as in Proposition 5.7, let .M+ be the corre−1 sponding .B(H )-valued Weyl function given by (5.5), and .M− = −M+ . Let also . ± (·) be the corresponding representative measures from representation (1.6) of .M± . Then:  ± (t) (i) There are nondecreasing ovf’s . ± (t) such that . R d 1+|t| ∈ B(H ) and  M+ (z) = J +

.

R

d + (t) , t −z

 M− (z) = −J +

R

d − (t) . t −z

(5.30)

(ii) There is a unitary operator .V+ : H → L2 ( + , H ) such that (V+ A+ V+−1 f ) = Q + f,

.

(V+ A− V+−1 f ) = Qf + J I + f,

(5.31)

where .dom (V+ A− V+−1 ) = {f ∈ L2 ( + , H ) : Qf + J I + f ∈ L2 ( + , H )}. (iii) There is a unitary operator .V− : H → L2 ( − , H ) such that (V− A− V−−1 f ) = Q − f,

.

(V− A+ V−−1 f ) = Qf − J I − f,

(5.32)

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where dom (V− A+ V−−1 ) = {f ∈ L2 ( − , H ) : Q − f − J I − f ∈ L2 ( − , H )}.

.

Proof (i) Let .E± (t) be resolutions of the identity for the operators .A± . As is known (see, for instance [58]), for every finite interval . the operator .E± () admits a ± (), such that .(−, +)-bounded continuation .E  .

R

± (dt)f, f ) < ∞ for all (1 + |t|)−1 (E

f ∈ H− .

(5.33)

Let us set + ()G, + () := G∗ E

.

− ()GJ. − () := J G∗ E

Combining (5.33) with the inclusion .G ∈ B(H, H−1 ) yields  .

R

d( ± (t)h, h) = 1 + |t|

 R

± (dt)Gh, Gh) (E < ∞, 1 + |t|

h ∈ H.

(5.34)

 Thus, . R (1 + |t|)−1 d ± (t) ∈ B(H ) and the first identity in (5.30) is implied by (5.28). (ii)–(iii) It follows from (5.34) that identity (5.16) remains valid in this case. Now   combining this fact with Theorem 3.9 (iv) yields identities (5.31) and (5.32). Remark 5.9 For additive finite dimensional perturbations identities (5.31)–(5.32) were obtained by another method in [6] (see also [15]). Note that our method based on the functional model in Theorem 3.4 allows to extend this result to the case of arbitrary pair .{A+ , A− } with slightly different operator .V+ A− V+−1 . Next we prove the existence of a spectral shift function (SSF) for a pair

+ + GJ G∗ , given by (5.24) with trace class singular .H−1 (A+ , A− ), .A− = A ∗ perturbation .GJ G . To this end we systematically use the form (5.28) of the Weyl function

.

+ − z)−1 G, M(z) = J + G∗ (A

.

z ∈ ρ(A+ ).

(5.35)

+ + GJ G∗ be given by (5.24) and let .G ∈ Proposition 5.10 Let .A+ and .A− = A S2 (H, H−1 ) be an injective Hilbert Schmidt operator. Then: (i) The resolvent difference is a trace class operator, i.e. (A+ − z)−1 − (A− − z)−1 ∈ S1 (H).

.

(5.36)

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(ii) There exists a spectral shift function .ξ for the pair .(A+ , A− ), i.e. .ξ ∈ L1 (R; (1 + t 2 )−1 ), .ξ(t) ∈ R and the following trace formula holds   tr (A− − z)−1 − (A+ − z)−1  1 =− ξ(t) dt, z ∈ ρ(A+ ) ∩ ρ(A− ). 2 R (t − z)

.

(5.37)

Proof (i) The proof is easily extracted from the identity    

+ −z)−1 −(A− −z)−1 = (A− −z)−1 G J G∗ (A

+ −z)−1 ∈ S1 (H−1 , H1 ). (A

.

However, this also follows directly from the Krein-type formula with account of formulas (5.28) for the .γ -field and the Weyl function. (ii) Consider the operator .A∗ defined by (5.26) and a B-generalized boundary triple . = {H, 0 , 1 } for .A∗ constructed in Proposition 5.1 (see (5.27)) and let .M(·) be the corresponding Weyl function given by (5.35). Passing in (1.5) to the limit as .ω¯ → z one gets γ (z)∗ γ (z) =

.

d M(z), dz

z ∈ C+ .

(5.38)

We also need the classical formula (see [27, Chapter 4.1]) d  tr log(M(z)) = tr . dz

  −1 d (M(z)) M(z) , dz

z ∈ C+ .

(5.39)

Inserting formula (5.38) in the Krein type formula for resolvents (2.6), applying formula (5.39) and using identity .tr T1 T2 = tr T2 T1 one gets that for .z ∈ C+    

+ − z)−1 = −tr (M(z))−1 γ (z)∗ γ (z) tr (A− − z)−1 − (A

.

=−

d  tr log(J M(z)) . dz

(5.40)

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To compute the trace of the right hand side in (5.40) we set .J = P+ − P− , V± := GP± G∗ ,

.

V := V+ − V− ,

H0 := A+ ,

and

H− := H0 − GP− G∗ .

Next one easily checks that the following factorization holds I + V (H0 − z)−1 = (I + V+ (H− − z)−1 )(I + V− (H− − z)−1 )−1 .

.

(5.41)

Note that for any dissipate operator K the implication .K ∈ S1 (H ) ⇒ log(I + K) ∈ S1 (H ) takes place and the following identity holds .

det(I + K) = exp (tr log(I + K))

(5.42)

Taking logarithm of both sides in (5.41), applying identities (5.42) and .det(I + T1 T2 ) = det(I + T2 T1 ) and using formula (5.35) for .M(·), we derive

.

d  d  tr log(J M(z)) = tr log(I + J G∗ (H0 − z)−1 G) dz dz  d  d tr log(I + V (H0 − z)−1 ) = log det(I + V (H0 − z)−1 ) = dz dz   d d log det(I + V+ (H− − z)−1 ) − log det(I + V− (H− − z)−1 ) . = dz dz (5.43)

Further, in accordance with [29, Theorem 2.10] there exist bounded selfadjoint ∗ ∈ B(H ) and .B(H )-valued measurable functions .t →  (t), operators .C± = C± ± ± (t) = ± (t)∗

.

0 ≤ ± (t) ≤ IH ,

and

t ∈ R,

such that the following representations hold for all .z ∈ C+ .

log(I + P± G∗ (H− − z)−1 G) = C± +

 R

  ± (t) (t − z)−1 − t (1 + t 2 )−1 dt. (5.44)

Since the inclusions .T (z) := P± G∗ (H− − z)−1 G ∈ S1 (H) imply .log(I + T (z)) ∈ S1 (H), the operator functions .± (·) in (5.44) satisfy .± (t) ∈ S1 (H) for a.e. .t ∈ R (see [29]). Therefore setting .ξ± (t) := tr (± (t)) and using the identity .det(I +

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T1 T2 ) = det(I + T2 T1 ) one gets   tr log I + V± (H− − z)−1 ) = tr log(I + P± G∗ (H− − z)−1 G)    = tr (C± ) + ξ± (t) (t − z)−1 − t (1 + t 2 )−1 dt, z ∈ C+ .

.

R

(5.45)

Further, setting .C := C+ − C− and .ξ := ξ+ − ξ− , taking the difference of two identities in (5.45), and inserting the result in (5.43) and differentiating it yields d  d  . tr log(M(z)) = tr log(J M(z)) = dz dz

 R

(t − z)−2 ξ(t) dt,

z ∈ C+ .

Finally, combining this identity with (5.40) we arrive at the required trace formula (5.37). Besides, taking the imaginary part in both sides of identity (5.45), setting .(λ) := + (λ) − − (λ) and applying the Stieltjes inversion formula we derive .

ξ(λ) := tr ((λ)) = lim ε↓0

1  Im tr (log(J M(λ + iε))) π

for a.e. λ ∈ R. (5.46)

The integrability condition .ξ ∈ L1 (R; (1 + t 2 )−1 ) is also implied by Gesztesy et al. [29, Theorem 2.10]. This completes the proof of assertion (ii).   Remark 5.11 The notion of the spectral shift function (SSF) was first introduced by physicist Lifshits in [42] for a pair of selfadjoint operators .(A+ , A− ) with .A1 being a finite rank additive perturbation of .A0 . Later Krein [39] proposed another definition of SSF and extended it to the case of self-adjoint operators .A0 and .A1 with trace class resolvent difference. To prove the existence of the spectral shift function he introduced the concept of perturbation determinant .A1 /A0 and proved the inversion formula similar to that of (5.46). Further development of this concept is discussed by Birman and Pushnitski in the survey [12], by Alpay, Gohberg [6], and in [48] (see also the lists of references therein). A concept of spectral shift operator was introduced by Carey [19] for a special class of .R[H]-functions of the form (5.35) with .J = I and .G ∈ S2 (H ). Another approach to the most general situation of arbitrary .R[H]-functions was proposed by Gesztesy, Makarov, and Naboko [29]. They extracted the existence of SSF from [29, Theorem 2.10] in the case of additive (non-singular) perturbations, i.e. for pairs .(A+ , A− ) with .A− = A+ + GJ G∗ , .G ∈ S2 (H ). Note however, that in the proof of Proposition 5.10 we used another idea based on the factorization identity (5.41) to overcome the difficulty with appearance of the signature operator .J = I in (5.35). This approach is somehow close to that used in [48]. Acknowledgments The present research was supported by DFG under project 436 UKR 113/85/0-1. V. D. gratefully acknowledges also financial support by Ministry of Education and Science of Ukraine (project 0121U109525).

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Schrödinger Operators with δ-potentials Supported on Unbounded Lipschitz Hypersurfaces .

Jussi Behrndt, Vladimir Lotoreichik, and Peter Schlosser

Dedicated to the memory of our friend and colleague Sergey Naboko

Abstract In this note we consider the self-adjoint Schrödinger operator .Aα in L2 (Rd ), .d ≥ 2, with a .δ-potential supported on a Lipschitz hypersurface . ⊆ Rd of strength .α ∈ Lp () + L∞ (). We show the uniqueness of the ground state and, under some additional conditions on the coefficient .α and the hypersurface ., we determine the essential spectrum of .Aα . In the special case that . is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schrödinger operator as Birman-Schwinger operator. As an application we prove an optimization result for the bottom of the spectrum of .Aα .

.

Keywords Schrödinger operator · Singular potential · Essential spectrum · Ground state · Birman-Schwinger operator · Eigenvalue optimization

J. Behrndt () · P. Schlosser Institute of Applied Mathematics, Graz University of Technology, Graz, Austria e-mail: [email protected]; [email protected] V. Lotoreichik ˇ Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, Rež, Czech Republic e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_8

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1 Introduction In this paper we are interested in spectral properties of a class of self-adjoint operators .Aα with singular .δ-potentials in the Hilbert space .L2 (Rd ), .d ≥ 2, which correspond to the formal differential expression −  − α δ(x − ),

.

(1.1)

where . ⊂ Rd is the graph of a Lipschitz function .ξ : Rd−1 → R and the function .α :  → R is the strength of the .δ-potential; cf. [8, 13], the monograph [25] and the references therein. Note that the unbounded Lipschitz surface . splits .Rd into two unbounded disjoint parts and that the special choice .ξ = 0 corresponds to the situation where . is the hyperplane in .Rd . Assuming .α ∈ Lp () + L∞ () for some .1 < p < ∞ in .d = 2 and for .d − 1 ≤ p < ∞ in .d ≥ 3 dimensions we define 2 d .Aα as the semibounded self-adjoint operator in .L (R ) associated with the densely defined, symmetric, semibounded, and closed form ˆ aα [u, v] := (∇u, ∇v)L2 (Rd ;Cd ) − .

α τD u τD v dx, 

(1.2)

dom aα := H 1 (Rd ), where .τD : H 1 (Rd ) → H 1/2 () is the Dirichlet trace operator. Let us denote the bottom of the spectrum of .Aα by λ1 (α) := inf σ (Aα ).

.

(1.3)

The first issue we discuss in this paper is the essential spectrum of the self-adjoint operator .Aα . In the present situation one always has the inclusion .[0, ∞) ⊂ σess (Aα ) and in Theorem 2.3 we prove that if . is a local deformation of the hyperplane d−1 × {0} and .α is close to a constant .α ∈ R outside of sets of finite measure (that .R 0 is, the set .{x ∈ ||α(x) − α0 | > ε} is of finite measure for every .ε > 0), then  σess (Aα ) =

.

α2

[− 40 , ∞), if α0 ≥ 0, [0, ∞), if α0 ≤ 0;

see [42] and also [6, 16–18] for related results on Schrödinger operators with singular interactions supported on hyperplanes, deformed hyperplanes and other unbounded surfaces. Next we investigate the uniqueness of the ground state of .Aα , which is a typical property for Schrödinger operators .− + V with regular potentials. More precisely, if .λ1 (α) in (1.3) is a discrete eigenvalue then it will be shown in Sect. 2.3 that .λ1 (α) is simple and the corresponding eigenfunction can be chosen strictly positive on .Rd \ ; this observation is based on a standard argument using Harnack’s inequality.

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In Sect. 3 we focus on the special case that . is the hyperplane and we obtain a Birman-Schwinger principle, where the Birman-Schwinger operator is a relativistic Schrödinger operator in .L2 (Rd−1 ). The operators appearing in this context can also be viewed as (extensions of) the .γ -field and Weyl function corresponding to a certain quasi boundary triple; cf. [9, Section 8] for more details. Under the additional assumption that .α is close to a constant .α0 outside of sets of finite measure, we then provide a more detailed analysis of the spectrum of the Birman-Schwinger operator and link these spectral properties to those of .Aα . As an interesting application we prove an optimization result for the bottom of the spectrum of .Aα which is formulated in terms of the so-called symmetric decreasing rearrangement: Consider again a real-valued .α ∈ L∞ (Rd−1 ) + Lp (Rd−1 ) for some .1 < p < ∞ in .d = 2 and for .d − 1 ≤ p < ∞ in .d ≥ 3 dimensions, which is close to a constant .α0 outside of sets of finite measure. Furthermore, let in the following .α1 := α − α0 and ∗ .(α1 )+ = max{α1 , 0}, and let .(α1 )+ be the symmetric decreasing rearrangement of .(α1 )+ defined in (3.23). Then we have the inequality λ1 (α0 + (α1 )∗+ ) ≤ λ1 (α0 + α1 ).

.

(1.4)

Our proof of (1.4) relies on the fact that the symmetric decreasing rearrangement decreases the kinetic energy term corresponding to the relativistic Schrödinger operator. This property of the kinetic energy can be viewed as an analogue of the Pólya-Szeg˝o inequality. We note that a different argument for (1.4) based on Steiner symmetrization was communicated to us; cf. Remark 3.11 for more details. We wish to mention that eigenvalue optimization is a trademark topic in spectral theory; see the monographs [33, 34] and the references therein. In particular, optimization of eigenvalues induced by .δ-potentials supported on hypersurfaces is a topic of growing interest [22, 23, 26, 39]. There are also closely related works on eigenvalue optimization for .δ-potentials supported on sets of higher co-dimension [7, 24], for the Robin Laplacian [3, 12, 14, 19, 28, 29, 31, 36, 37], for .δ  -interactions [40] and for Dirac operators with surface interactions [2, 4].

2 The Schrödinger Operator with δ-potential Supported on a Lipschitz Graph In this section let .d ≥ 2 and  := {(x, ξ(x))|x ∈ Rd−1 } ⊂ Rd

.

(2.1)

be the graph of a Lipschitz function .ξ : Rd−1 → R. Furthermore, let α ∈ Lp () + L∞ ()

.

(2.2)

be a real-valued function with .1 < p < ∞ in .d = 2 and .d − 1 ≤ p < ∞ in .d ≥ 3 dimensions. In this setting we will define the self-adjoint operator .Aα associated to

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the form (1.2) and study its essential spectrum. In particular, if the support . is a local deformation of a hyperplane and if the coefficient .α is close to a constant in the sense of (2.6), we explicitly compute .σess (Aα ). Furthermore, we verify that the ground state .λ1 (α) (if it is a discrete eigenvalue) is simple.

2.1 The Form aα and the Operator Aα In this subsection we will prove that the form (1.2), which models a .δ-potential of strength .α supported on ., is well defined and gives rise to a self-adjoint operator 2 d .Aα in .L (R ); cf. [13, 27] and [10, Proposition 3.8]. In the following the Dirichlet trace operator .τD in (1.2) is viewed for . 12 < s < 32 as a bounded operator 1

τD : H s (Rd ) → H s− 2 ();

(2.3)

.

cf. [41, Proof of Theorem 3.38]. Proposition 2.1 The form .aα in (1.2) is densely defined, symmetric, semibounded, and closed in .L2 (Rd ). Proof It is clear, that .dom aα = H 1 (Rd ) is dense in .L2 (Rd ). Furthermore, we split .aα into a0 [u, v] := (∇u, ∇v)L2 (Rd ;Cd ) , ˆ a1 [u, v] := − α τD u τD v dx,

.

with

dom a0 := H 1 (Rd ),

with

dom a1 := H 1 (Rd ).



Observe that .a0 is densely defined, nonnegative, and closed in .L2 (Rd ). Furthermore, since .α is real-valued it is clear that .a1 is symmetric. The estimate (A.3) shows that for every .ε > 0 there exists some .cε ≥ 0, such that   a1 [u, u] ≤ ε2 τD u 2

.

1

H 2 ()

+ cε2 τD u 2L2 () ,

u ∈ H 1 (Rd ).

Using the boundedness (2.3) of the trace operator, the absolute value of .a1 [u, u] can be further estimated by   a1 [u, u] ≤ ε2 d 2 u 2

.

1

H 1 (Rd )

+ cε2 ds2 u 2H s (Rd ) ,

u ∈ H 1 (Rd ),

where .d1 and .ds are the operator norms of (2.3) with .s = 1 and some fixed .s ∈ ( 12 , 1), respectively. Since .s < 1, we can use [32, Theorem 3.30] to find a constant .c ˜ε ≥ 0 with   a1 [u, u] ≤ ε2 (d 2 + 1) u 2 1 d + c˜2 u 2 2 d , ε 1 H (R ) L (R )

.

u ∈ H 1 (Rd ).

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That is, the form .a1 is .a0 -bounded with form bound 0. The semiboundedness and closedness of .aα = a0 + a1 now follow from [35, Chapter VI, Theorem 1.33].  Proposition 2.1 combined with the First Representation Theorem [35, Chapter VI, Theorem 2.1] implies that there is a unique self-adjoint operator .Aα in 2 d .L (R ) representing the form .aα in the sense that .dom Aα ⊂ dom aα and (Aα f, g)L2 (Rd ) = aα [f, g],

.

f ∈ dom Aα , g ∈ dom aα .

(2.4)

2.2 Essential Spectrum of Aα In this subsection we investigate the essential spectrum of .Aα . The following preparatory lemma shows that in the present situation the essential spectrum of .Aα always covers the nonnegative real axis. Lemma 2.2 For any .α of the form (2.2) we have [0, ∞) ⊆ σess (Aα ).

.

(2.5)

Proof In a similar way as in the proof of [20, Chapter 10, Theorem 6.5] one d constructs for .λ ∈ (0, ∞) an orthonormal sequence .( n )n ∈ C∞ 0 (R ) with support d in .R \  and n→∞

(− − λ) n L2 (Rd ) −→ 0.

.

From .supp n ⊆ Rd \  we have .τD n = 0 and hence it follows from (1.2) that .Aα n = − n . This implies n→∞

(Aα − λ) n L2 (Rd ) −→ 0,

.

so that .( n )n is a singular sequence and we conclude .λ ∈ σess (Aα ). This proves that (0, ∞) ⊆ σess (Aα ) and since the essential spectrum is closed we obtain (2.5). 

.

For a subclass of hypersurfaces ., which are local deformations of a hyperplane, and interaction strengths that are close to a constant in the sense of (2.6), we are able to determine the essential spectrum explicitly. Theorem 2.3 If the function .ξ : Rd−1 → R in (2.1) is compactly supported and if for some .α0 ∈ R {x ∈ ||α(x) − α0 | > ε} has finite measure for every ε > 0,

.

(2.6)

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then the essential spectrum of the corresponding Schrödinger operator .Aα is given by  σess (Aα ) =

.

α2

[− 40 , ∞), if α0 ≥ 0, [0, ∞), if α0 ≤ 0.

(2.7)

Proof Step 1. First, we consider the hyperplane . = Rd−1 × {0} ∼ = Rd−1 and the constant potential .α(x) = α0 . We introduce two auxiliary closed forms d[φ, ψ] := (∇φ, ∇ψ)L2 (Rd−1 ;Cd−1 ) ,

.

tα0 [f, g] := (f  , g  )L2 (R) − α0 f (0)g(0),

with

dom d := H 1 (Rd−1 ),

with

dom tα0 := H 1 (R),

with the corresponding self-adjoint operators .− and .Tα0 in the Hilbert spaces L2 (Rd−1 ) and .L2 (R), respectively. The spectra of these operators are explicitly given by

.

 σ (−) = [0, ∞)

.

and

σ (Tα0 ) =

α2

{− 40 } ∪ [0, ∞), if α0 ≥ 0, [0, ∞), if α0 ≤ 0,

where the proof of the latter one can be found in [1, Theorem 3.1.4]. The Schrödinger operator . Aα0 with .δ-potential supported on a hyperplane of constant strength .α0 can be decomposed as  Aα0 = (−) ⊗ IR + IRd−1 ⊗ Tα0

.

with respect to .L2 (Rd ) = L2 (Rd−1 ) ⊗ L2 (R); here .IR and .IRd−1 denote the identity operators in .L2 (R) and .L2 (Rd−1 ), respectively. Hence, it follows from [45, Eq. (4.44)] that  .σ ( Aα0 ) =

α2

[− 40 , ∞), if α0 ≥ 0, [0, ∞), if α0 ≤ 0.

(2.8)

Step 2. Let .Aα0 be the Schrödinger operator with .δ-potential of constant strength α0 supported on the hypersurface .. Since the Lipschitz mapping .ξ is compactly supported, the surface . is a local deformation of the hyperplane .Rd−1 × {0} in the sense that . \ B = (Rd−1 × {0}) \ B for a ball .B ⊂ Rd of sufficiently large radius. Hence it follows from (2.8) using [6, Theorem 4.7] that

.

 .σess (Aα0 ) = σess ( Aα0 ) =

α2

[− 40 , ∞), if α0 ≥ 0, [0, ∞), if α0 ≤ 0.

(2.9)

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Step 3. With .α0 from (2.6) we define .α1 := α − α0 , such that .{x ∈ ||α1 (x)| > ε} has finite measure for every .ε > 0. The self-adjoint operators .Aα0 and .Aα are both semibounded since they correspond to semibounded forms. Hence, we can fix some .λ < inf(σ (Aα0 ) ∪ σ (Aα )) and consider the resolvent difference W := (Aα0 − λ)−1 − (Aα − λ)−1 .

(2.10)

.

Our aim is to show that .W is a compact operator in .L2 (Rd ). For this consider some 2 d .f, g ∈ L (R ) and set u := (Aα0 − λ)−1 f

.

and

v := (Aα − λ)−1 g.

(2.11)

Using (2.11) and the definition of the operator .W in (2.10) we obtain     (Wf, g)L2 (Rd ) = (Aα0 − λ)−1 f, g L2 (Rd ) − (Aα − λ)−1 f, g L2 (Rd ) = (u, g)L2 (Rd ) − (f, v)L2 (Rd )     = u, (Aα − λ)v L2 (Rd ) − (Aα0 − λ)u, v L2 (Rd )

.

(2.12)

= (u, Aα v)L2 (Rd ) − (Aα0 u, v)L2 (Rd ) . We can express the above inner products via the corresponding forms (2.4) and conclude that .(Wf, g)L2 (Rd ) reduces to the surface integral ˆ (Wf, g)L2 (Rd ) = −

.



α1 τD u τD v dx = (T1 f, T2 g)L2 () ,

where .T1 , T2 : L2 (Rd ) → L2 () are defined by 1

T1 := |α1 | 2 τD (Aα0 − λ)−1

.

and

1

T2 := − sgn(α1 )|α1 | 2 τD (Aα − λ)−1 .

As .(Aα0 − λ)−1 and .(Aα − λ)−1 are bounded operators from .L2 (Rd ) into .H 1 (Rd ), it follows from (2.3) that .τD (Aα0 − λ)−1 and .τD (Aα − λ)−1 are bounded from .L2 (Rd ) 1 into .H 2 (). Consequently, both .T1 and .T2 are compact as operators from .L2 (Rd ) into .L2 () by Proposition A.3. Thus the operator .W = T∗2 T1 is compact as well and the stability of the essential spectrum under compact perturbations in resolvent sense combined with (2.9) yields the claim. 

2.3 Uniqueness of the Ground State In this subsection we assume that the bottom of the spectrum .λ1 (α) in (1.3) is a discrete eigenvalue of .Aα . The aim is to prove in Theorem 2.7 that this eigenvalue is simple and the corresponding eigenfunction can be chosen strictly positive on d .R \ .

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Lemma 2.4 Let .u ∈ H 1 (Rd ) be a real-valued eigenfunction of .Aα corresponding to .λ1 (α). Then also .|u| is an eigenfunction of .Aα corresponding to .λ1 (α). Proof From .|∇|u|| = |∇u|, cf. [38, Theorem 6.17], and .τD |u| = |τD u|, we obtain .

aα [|u|] aα [u] = = λ1 (α). 2 |u| L2 (Rd ) u 2L2 (Rd )

Since .λ1 (α) is the bottom of the spectrum it can be represented by the min-max principle [43, Theorem XIII.2] as λ1 (α) =

.

inf

0=v∈H 1 (Rd )

aα [v] . v 2L2 (Rd )

Since .λ1 (α) is assumed to be a discrete eigenvalue, it follows from [15, Chapter 10.2, Theorem 1] that .|u| is indeed an eigenfunction of .Aα corresponding to the eigenvalue .λ1 (α). 

Lemma 2.5 Let . ⊆ Rd be open and connected. Assume that .u ∈ H 1 () and .λ ∈ R satisfy (∇u, ∇v)L2 (;Cd ) = λ(u, v)L2 () ,

.

v ∈ H01 ().

Then .u ∈ C∞ () and if .u ≥ 0 and .u(x0 ) = 0 for some .x0 ∈ , then .u ≡ 0. Proof The interior regularity .u ∈ C∞ () is well known; cf. [21, §6.3, Theorem 3]. Assume now .u ≥ 0 and .u(x0 ) = 0 for some .x0 ∈ . Since . is connected, for every .x ∈  there exists a path .γ connecting x and .x0 . Since . is also open, there even exists some open and bounded U with .γ ⊆ U ⊆ U ⊆ . Then it follows from Harnack’s inequality [30, Corollary 8.21], that .

sup u(y) ≤ C inf u(y), y∈U

y∈U

for some constant .C > 0. Since .u(x0 ) = 0, the right and hence also the left hand side of this inequality vanishes. Therefore, .u|U = 0 and, in particular, .u(x) = 0. Since .x ∈  was arbitrary, we conclude .u ≡ 0. 

Lemma 2.6 Let .u ∈ H 1 (Rd ) be a real-valued eigenfunction of .Aα corresponding to .λ1 (α). Then .u ∈ C∞ (Rd \ ) is either strictly positive or strictly negative on d .R \ . Proof From Lemma 2.5 we conclude .u ∈ C∞ (Rd \ ). In order to show that u has no zeros in .Rd \, we assume the converse, i.e. that .u(x0 ) = 0 for some .x0 ∈ Rd \. It is clear that . cuts the whole space .Rd into the two domains + := {(x, xd ) ∈ Rd−1 × R|xd > ξ(x)},

.

− := {(x, xd ) ∈ Rd−1 × R|xd < ξ(x)}.

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131

We will assume without loss of generality that .x0 ∈ + . Since, by Lemma 2.4, .|u| is also an eigenfunction corresponding to .λ1 (α), we have (∇|u|, ∇v)L2 (+ ;Cd ) = λ1 (α)(|u|, v)L2 (+ ) ,

.

v ∈ H01 (+ ),

and Lemma 2.5 implies .u|+ ≡ 0. In particular, we have .τD u = 0 and the eigenvalue equation for u reduces to (∇u, ∇v)L2 (− ;Cd ) = λ1 (α)(u, v)L2 (− ) ,

.

v ∈ H 1 (Rd ).

Since .λ1 (α) is a discrete eigenvalue, it is negative by Lemma 2.2, and consequently choosing .v = u, we conclude .u|− ≡ 0. But this is a contradiction to the fact that u is a (nonzero) eigenfunction; hence u has no zeros in .Rd \ . Since we already know that .u ∈ C∞ (Rd \ ) has no zeros in .Rd \ , it has to be either strictly positive or strictly negative on each of the domains .± . However, a priori the signs of u may not coincide. If, e.g. u|+ > 0 and

.

u|− < 0,

then .τD u = 0 and the eigenvalue equation for u reduces to (∇u, ∇v)L2 (Rd ;Cd ) = λ1 (α)(u, v)L2 (Rd ) ,

.

v ∈ H 1 (Rd ).

Choosing .v = u we again conclude .u ≡ 0 by the negativity of .λ1 (α); a contradiction as u is a (nonzero) eigenfunction. 

Theorem 2.7 If the bottom (1.3) of the spectrum of .Aα is a discrete eigenvalue, then it is simple and the corresponding eigenfunction can be chosen strictly positive on .Rd \ . Proof Note that there exists a real-valued basis of the eigenspace corresponding to λ1 (α) since for every eigenfunction the complex conjugate is also an eigenfunction. Now consider two orthogonal real-valued eigenfunctions .u1 and .u2 . According to Lemma 2.6 each eigenfunction is either strictly positive or strictly negative on .Rd \ . But this is a contradiction to the orthogonality condition

.

ˆ .

Rd

u1 u2 dx = 0.

Hence, the eigenspace is one-dimensional and thus .λ1 (α) is a simple eigenvalue.



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3 The Birman-Schwinger Principle and an Optimization Result for δ-potentials on a Hyperplane In this section we assume that the support of the .δ-potential is a hyperplane and we shall therefore identify . = Rd−1 × {0} ∼ = Rd−1 . Moreover, as in (2.2), everywhere in this section we consider a real-valued function α ∈ Lp (Rd−1 ) + L∞ (Rd−1 )

.

with .1 < p < ∞ if .d = 2 and .d − 1 ≤ p < ∞ if .d ≥ 3. Later we shall also assume that there exists some .α0 ∈ R such that {x ∈ Rd−1 ||α(x) − α0 | > ε} has finite measure for every ε > 0.

.

(3.1)

We first discuss the Birman-Schwinger principle for the operator .Aα in this special situation, by means of which the spectral problem can be reduced to the spectral analysis of a relativistic Schrödinger operator in .L2 (Rd−1 ). As an application and illustration we prove an optimization result for the bottom of the spectrum of .Aα in Theorem 3.7.

3.1 The Birman-Schwinger Principle for δ-potentials Supported on a Hyperplane For every .λ < 0 we consider the form   1 1 dα,λ [φ, ψ] := 2 (− − λ) 4 φ, (− − λ) 4 ψ L2 (Rd−1 ) − .

ˆ Rd−1

α φ ψ dx,

1

dom dα,λ := H 2 (Rd−1 ). (3.2) It follows from Lemma A.1 that for every .ε > 0 there exists a .cε > 0 such that  1 2 1 2 2 .|α| 2 φ  2 d−1 ≤ ε φ 1 + cε2 φ 2L2 (Rd−1 ) , φ ∈ H 2 (Rd−1 ). (3.3) L (R ) H 2 (Rd−1 )

Using this inequality it follows (see the proof of Proposition 2.1) that .dα,λ is a densely defined, symmetric, semibounded and closed form in .L2 (Rd−1 ). We denote the corresponding self-adjoint operator in .L2 (Rd−1 ) by .Dα,λ . It turns out in Proposition 3.2 below that the eigenvalue 0 of this relativistic Schrödinger operator is directly linked to the eigenvalue .λ of the Schrödinger operator .Aα .

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We first formulate and prove a preparatory lemma; here and in the following we shall denote the extension of the .L2 (Rd−1 ) scalar product onto the dual pair −1 d−1 ) × H 12 (Rd−1 ) by . · , ·  .H 2 (R . 1 −1 d−1 d−1 H

2 (R

)×H 2 (R

)

Lemma 3.1 For every fixed .λ < 0 there exists a unique bounded linear operator 1 γ (λ) : H − 2 (Rd−1 ) → H 1 (Rd ) such that the identity     . ∇γ (λ)φ, ∇v − λ γ (λ)φ, v L2 (Rd ) = φ, τD v − 1 d−1 1 L2 (Rd ;Cd ) d−1

.

H

2 (R

)×H 2 (R

)

(3.4) 1

holds for all .φ ∈ H − 2 (Rd−1 ), .v ∈ H 1 (Rd ). Moreover, the trace of .γ (λ) is given by τD γ (λ) =

.

1 1 (− − λ)− 2 , 2 1

(3.5) 1

and acts as a bounded linear operator from .H − 2 (Rd−1 ) to .H 2 (Rd−1 ). Proof Let .Fd and .Fd−1 be the unitary Fourier transforms in .L2 (Rd ) and .L2 (Rd−1 ), respectively, and consider Schwartz functions .φ ∈ S(Rd−1 ). We first define the operator .γ (λ) in Fourier space as ˜ := √(Fd−1 φ)(k) , (Fd γ (λ)φ)(k) ˜ 2 − λ) 2π (|k|

.

k˜ = (k, kd ) ∈ Rd−1 × R.

(3.6)

As .λ < 0 and .Fd−1 φ ∈ S(Rd−1 ), this is a well defined function in .L2 (Rd ). The 1 fact, that .γ (λ) is bounded from .H − 2 (Rd−1 ) to .H 1 (Rd ) follows from the estimate ˆ 2 1 2 ˜ 2 ) |(Fd−1 φ)(k)| dk˜ γ (λ)φ H 1 (Rd ) = (1 + |k| ˜ 2 − λ)2 2π Rd (|k| ˆ ˆ 1 + |k|2 + kd2 1 = dkd |(Fd−1 φ)(k)|2 dk 2π Rd−1 R (|k|2 + kd2 − λ)2 . ˆ 2|k|2 + 1 − λ 1 |(Fd−1 φ)(k)|2 dk = 4 Rd−1 (|k|2 − λ) 23 ≤

c(λ) , φ 2 − 1 4 H 2 (Rd−1 )

where .c(λ) denotes the maximum of the function k →

.

(2|k|2 + 1 − λ)(|k|2 + 1)1/2 . (|k|2 − λ)3/2 1

Since .S(Rd−1 ) is dense in .H − 2 (Rd−1 ) the operator .γ (λ) can be extended by 1 continuity onto .H − 2 (Rd−1 ).

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In order to prove the identity (3.4) for Schwartz functions .φ ∈ S(Rd−1 ) and d .v ∈ S(R ), we use the Fourier representation ˜ = i k(F ˜ d v)(k), ˜ (Fd ∇v)(k)

.

k˜ ∈ Rd ,

(3.7)

of the gradient. For .x ∈ Rd−1 the trace can be written as (τD v)(x) = (F−1 d Fd v)(x, 0) ˆ 1 ˜ ˜ k˜ = eik,(x,0) (Fd v)(k)d d d (2π ) 2 R ˆ ˆ . 1 ik,x e (Fd v)(k, kd )dkd dk = d R (2π ) 2 Rd−1 ˆ 1 −1 (Fd v)( · , kd )dkd (x) = √ Fd−1 2π R

(3.8)

and hence 1 .(Fd−1 τD v)(k) = √ 2π

ˆ R

k ∈ Rd−1 .

(Fd v)(k, kd )dkd ,

(3.9)

The definition (3.6) of .γ (λ), together with (3.7) and (3.9) leads to  .

∇γ (λ)φ, ∇v



  − λ γ (λ)φ, v L2 (Rd ) ˆ ˜ 2 − λ) (Fd γ (λ)φ)(k) ˜ (Fd v)(k) ˜ dk˜ = (|k|

L2 (Rd ;Cd )

Rd

ˆ 1 (Fd−1 φ)(k) (Fd v)(k, kd ) dkd dk =√ 2π Rd ˆ = (Fd−1 φ)(k) (Fd−1 τD v)(k) dk Rd−1

= (φ, τD v)L2 (Rd−1 ) = φ, τD v

H

− 21

1

(Rd−1 )×H 2 (Rd−1 )

,

and hence (3.4) holds for .φ ∈ S(Rd−1 ) and .v ∈ S(Rd ). By density and continuity 1 this identity extends to all .φ ∈ H − 2 (Rd−1 ) and .v ∈ H 1 (Rd ). Also note, that the identity (3.4) uniquely defines the operator .γ (λ). For the proof of (3.5) note first that the identity (3.9) and its derivation (3.8) remain valid for .v ∈ H 1 (Rd ) ∩ C(Rd ) with .Fd v ∈ L1 (Rd ). In particular, for .φ ∈ S(Rd−1 ) it is not difficult to see that .Fd γ (λ)φ ∈ L1 (Rd ) by its definition (3.6)

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and hence also that .γ (λ)φ = F−1 d Fd γ (λ)φ is continuous as the inverse Fourier 1 transform of an .L -function. This means that from (3.9) we get 1 (Fd−1 τD γ (λ)φ)(k) = √ 2π

ˆ

.

R

(Fd γ (λ)φ)(k, kd )dkd

(Fd−1 φ)(k) = 2π =

(Fd−1 φ)(k) 1

2(|k|2 − λ) 2

ˆ R

dkd ˜ |k|2 − λ

,

which is exactly Eq. (3.5) in Fourier space. Again, by continuity this identity also 1 

holds for every .φ ∈ H − 2 (Rd−1 ). With this lemma we find a connection between the eigenvalue 0 of the relativistic Schrödinger operator .Dα,λ and the eigenvalue .λ of the Schrödinger operator .Aα . Proposition 3.2 For every .λ < 0 the restriction of the Dirichlet trace operator τD : ker(Aα − λ) → ker Dα,λ

(3.10)

.

is bijective and, in particular, .dim ker(Aα − λ) = dim ker Dα,λ . Proof In order to see that the restriction of .τD onto .ker(Aα − λ) maps into .ker Dα,λ consider some .u ∈ ker(Aα − λ). By (1.2) we have .u ∈ H 1 (Rd ) and   1 1 (∇u, ∇v)L2 (Rd ;Cd ) − λ(u, v)L2 (Rd ) = sgn(α)|α| 2 τD u, |α| 2 τD v L2 (Rd−1 ) (3.11)

.

1

1

for all .v ∈ H 1 (Rd ). Since .τD u ∈ H 2 (Rd−1 ), we get .|α| 2 τD u ∈ L2 (Rd−1 ) from 1 (3.3) and hence there exist .ψn ∈ H 2 (Rd−1 ) such that 1

sgn(α)|α| 2 τD u = lim ψn

.

n→∞

in L2 (Rd−1 ).

(3.12)

1

Again, by (3.3), we have .|α| 2 ψn ∈ L2 (Rd−1 ) and inserting these into (3.4) leads to  .

1

∇γ (λ)|α| 2 ψn , ∇v

 L2 (Rd ;Cd )

    1 1 − λ γ (λ)|α| 2 ψn , v L2 (Rd ) = ψn , |α| 2 τD v L2 (Rd−1 )

for all .v ∈ H 1 (Rd ). Combining this with (3.11) and (3.12) implies the convergence 1

γ (λ)|α| 2 ψn  u

.

weakly in H 1 (Rd ).

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Applying the bounded operator .(− − λ) 4 τD : H 1 (Rd ) → L2 (Rd−1 ) on this weak convergence and using (3.5) gives .

1 1 1 1 1 1 (− − λ)− 4 |α| 2 ψn = (− − λ) 4 τD γ (λ)|α| 2 ψn  (− − λ) 4 τD u 2 1

weakly in .L2 (Rd−1 ). Hence, for every .ψ ∈ H 2 (Rd−1 ) we get   1 1 1 dα,λ [τD u, ψ] = lim (− − λ)− 4 |α| 2 ψn , (− − λ) 4 ψ L2 (Rd−1 ) n→∞ ˆ − α τD u ψ dx

.

Rd−1

  1 = lim ψn , |α| 2 ψ L2 (Rd−1 ) − n→∞

ˆ Rd−1

α τD u ψ dx

= 0, where (3.12) was used in the last step. Thus, we conclude .τD u ∈ ker Dα,λ . Next we show that (3.10) is injective. In fact, assume that .τD u = 0 for some .u ∈ ker(Aα − λ). Then (1.2) leads to (∇u, ∇v)L2 (Rd ;Cd ) = λ(u, v)L2 (Rd−1 ) ,

.

v ∈ H 1 (Rd ).

Since .λ < 0 we can choose .v = u and conclude .u = 0. For the surjectivity of (3.10) we choose .φ ∈ ker Dα,λ . By (3.2) we then have 1 d−1 ) and .φ ∈ H 2 (R ˆ   1 1 1 .2 (− − λ) 4 φ, (− − λ) 4 ψ = α φ ψ dx, ψ ∈ H 2 (Rd−1 ). d−1 L2 (R ) Rd−1

(3.13)

1

Now define .uφ := 2γ (λ)(− − λ) 2 φ. Then .τD uφ = φ by (3.5) and using (3.4) with 1 1 d .φ replaced by .2(− − λ) 2 φ, gives for any .v ∈ H (R )   ∇uφ , ∇v L2 (Rd ;Cd ) − λ (uφ , v)L2 (Rd ) 1

.

= 2(− − λ) 2 φ, τD v − 1 d−1 1 H 2 (R )×H 2 (Rd−1 ) ˆ = α φ τD v dx Rd−1

ˆ =

Rd−1

α τD uφ τD v dx,

where in the second step we used (3.13) with .ψ = τD v. Summing up, for every φ ∈ ker Dα,λ we found .uφ ∈ ker(Aα − λ) such that .τD uφ = φ, which is the surjectivity of (3.10). 

.

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137

Next we analyse how the bottom of the spectrum .σ (Dα,λ ) behaves as a function of .λ < 0. Lemma 3.3 For .λ < 0 the mapping λ → μα (λ) := inf σ (Dα,λ ) =

.

inf

0=φ∈H 1/2 (Rd−1 )

dα,λ [φ] φ 2L2 (Rd−1 )

(3.14)

is nonincreasing, continuous and admits the limit lim μα (λ) = ∞.

.

(3.15)

λ→−∞

Proof With the help of the Fourier transform in .L2 (Rd−1 ) we see that the form .dα,λ admits the representation ˆ dα,λ [φ] = 2

.

Rd−1

ˆ

1 2

(|k| − λ) |(Fd−1 φ)(k)| dk − 2

2

Rd−1

α |φ|2 dx,

(3.16)

1

for any .φ ∈ H 2 (Rd−1 ), which shows that .dα,λ [φ] is nonincreasing in .λ. Hence the same is true for .μα in (3.14). For the continuity of the function .μα consider .λ1 ≤ λ2 < 0. Then for every 1 d−1 ) we can estimate the difference .φ ∈ H 2 (R ˆ  2 1 1 (|k| − λ1 ) 2 − (|k|2 − λ2 ) 2 |(Fd−1 φ)(k)|2 dk .dα,λ1 [φ] − dα,λ2 [φ] = 2 Rd−1

 

≤ 2 −λ1 − −λ2 φ 2L2 (Rd−1 ) , and via (3.14) we also conclude

 

−λ1 − −λ2 ,

μα (λ1 ) − μα (λ2 ) ≤ 2

.

which proves the continuity of .λ → μα (λ). It remains to verify (3.15). For this we use the estimate ˆ  

.

Rd−1

  α |φ|2 dx  ≤ φ 2

1

H 2 (Rd−1 )

+ c12 φ 2L2 (Rd−1 ) ,

1

φ ∈ H 2 (Rd−1 ),

from (3.3). Plugging this in (3.16) gives ˆ dα,λ [φ] ≥



.

Rd−1

1 1 2(|k|2 − λ) 2 − (1 + |k|2 ) 2 |(Fd−1 φ)(k)|2 dk − c12 φ 2L2 (Rd−1 )

≥ (c(λ) − c12 ) φ 2L2 (Rd−1 ) ,

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1

where .c(λ) ∈ R is the minimum of .k → 2(|k|2 − λ) 2 − (1 + |k|2 ) 2 . From (3.14) we then conclude λ→−∞

μα (λ) ≥ c(λ) − c12 −→ ∞.

.

 

Next, we compute the essential spectrum of .Dα,λ under the additional assumption that .α is close to a constant in the sense of (3.1). Proposition 3.4 Assume that .α satisfies (3.1) with some .α0 ∈ R. Then for every λ < 0 the essential spectrum of .Dα,λ is given by

.

 √  σess (Dα,λ ) = 2 −λ − α0 , ∞ .

.

(3.17)

Furthermore, the mapping .λ → μα (λ) from (3.14) is strictly decreasing on (−∞, 0).

.

Proof It is clear that for constant .α(x) = α0 ∈ R the relativistic Schrödinger 1 operator is given by .Dα0 ,λ = 2(− − λ) 2 − α0 with .dom Dα0 ,λ = H 1 (Rd−1 ). Hence we have  √  σ (Dα0 ,λ ) = σess (Dα0 ,λ ) = 2 −λ − α0 , ∞ .

.

(3.18)

For a nonconstant function .α we define the function .α1 (x) := α(x) − α0 . Then the set .{x ∈ Rd−1 | |α1 (x)| > ε} has finite measure for every .ε > 0 by the property (3.1). To prove (3.17) we proceed in the same way as in Step 3 of the proof of Theorem 2.3 and check that for some .μ < inf(σ (Dα0 ,λ ) ∪ σ (Dα,λ )) the resolvent difference W := (Dα0 ,λ − μ)−1 − (Dα,λ − μ)−1

.

is a compact operator in .L2 (Rd−1 ). For this let .φ, ψ ∈ L2 (Rd−1 ) and set φμ := (Dα0 ,λ − μ)−1 φ

.

and

ψμ := (Dα,λ − μ)−1 ψ.

In the same way as in (2.12) one verifies     (Wφ, ψ)L2 (Rd−1 ) = φμ , Dα,λ ψμ L2 (Rd−1 ) − Dα0 ,λ φμ , ψμ L2 (Rd−1 ) ˆ =− α1 φμ ψμ dx

.

Rd−1

= (T1 φ, T2 ψ)L2 (Rd−1 ) ,

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139

where 1

T1 := |α1 | 2 (Dα0 ,λ − μ)−1

.

and

1

T2 := − sgn(α1 )|α1 | 2 (Dα,λ − μ)−1 .

As .(Dα0 ,λ − μ)−1 and .(Dα,λ − μ)−1 are bounded operators from .L2 (Rd−1 ) into 1 d−1 ) it follows from Proposition A.3 that both .T and .T are compact .H 2 (R 1 2 operators in .L2 (Rd−1 ). Thus the resolvent difference .W = T∗2 T1 is compact as well, which implies .σess (Dα0 ,λ ) = σess (Dα,λ ) and (3.17) follows from (3.18). For the proof of the strict monotonicity of .λ → μα (λ), let .λ1 < λ2 < 0. Then

μα (λj ) ≤ 2 −λj − α0 ,

.

j = 1, 2,

(3.19)

√ √ by (3.17). If .μα (λ1 ) = 2 −λ1 − α0√we conclude from .μα (λ2 ) ≤ 2 −λ2 − α0 that .μα (λ2 ) < μα (λ1 ). If .μα (λ1 ) < 2 −λ1 − α0 we know from (3.17) that .μα (λ1 ) is a discrete eigenvalue of .Dα,λ1 and hence there is a corresponding eigenfunction 1 d−1 ). Since, in particular, .φ = 0 we conclude from (3.16) .φ ∈ dom Dα,λ1 ⊂ H 2 (R that .λ → dα,λ [φ] is strictly decreasing, and hence .

μα (λ1 ) =

dα,λ1 [φ] dα,λ2 [φ] > ≥ μα (λ2 ). φ 2L2 (Rd−1 ) φ 2L2 (Rd−1 )

 

Lemma 3.5 Assume that .α satisfies (3.1) with some .α0 ∈ R. For the lowest spectral point .λ1 (α) of .Aα in (1.3) and the lowest spectral point .μα (λ) of .Dα,λ in (3.14) the following are equivalent: (i) .λ1 (α) ∈ σd (Aα )



α2

0 (ii) .μα admits a zero strictly below . − 4 , if α0 ≥ 0 0, if α0 ≤ 0.

In this situation the zero of .μα coincides with .λ1 (α). Proof For an easier notation we write .λ1 := λ1 (α). For the implication .(i) ⇒ (ii) let .λ1 ∈ σd (Aα ) and note that due to the explicit form of the essential spectrum (2.7) we have  α2 − 40 , if α0 ≥ 0, .λ1 < (3.20) 0, if α0 ≤ 0. It follows from Proposition 3.2 that zero is an eigenvalue of .Dα,λ1 . Assume now μα (λ1 ) = 0.

.

• The case .μα (λ1 ) = inf σ (Dα,λ1 ) > 0 is a contradiction to the fact that zero is an eigenvalue of .Dα,λ1 .

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˜ = 0 for some .λ˜ < λ1 by Lemma 3.3. Also note, that • If .μα (λ1 ) < 0, then .μα (λ)

.

inf σess (Dα,λ˜ ) = 2 −λ˜ − α0 ≥ 2 −λ1 − α0 > 0

by Proposition 3.4 and the estimate (3.20). But then the bottom of the spectrum ˜ = inf σ (D ˜ ) 0 = μα (λ) α,λ

.

is a point in the discrete spectrum and hence an eigenvalue of .Dα,λ˜ . Consequently, Proposition 3.2 implies that .λ˜ < λ1 is an eigenvalue of .Aα ; a contradiction as .λ1 is the smallest spectral point of .Aα . Hence our assumption is wrong and we conclude .μα (λ1 ) = 0. Due to the strict monotonicity in Proposition 3.4, this is also the only zero of .μα . For the implication .(ii) ⇒ (i) assume that .μα admits a zero  λ˜
0 by (3.21) we conclude from α,λ (3.17) that zero belongs to the discrete spectrum of .Dα,λ˜ , and hence Proposition 3.2 implies that .λ˜ is an eigenvalue of .Aα . Hence, also the bottom of the spectrum  ˜ < .λ1 = inf σ (Aα ) ≤ λ

α2

− 40 , if α0 ≥ 0, 0, if α0 ≤ 0,

belongs to the discrete spectrum of .Aα by (2.7).



3.2 Optimization of λ1 (α) and the Symmetric Decreasing Rearrangement In this subsection we prove an optimization result for the bottom of the spectrum of .Aα , which will be formulated in terms of the so-called symmetric decreasing rearrangement of the positive part of the function .α1 (x) := α(x) − α0 , with .α0 ∈ R from (3.1). We first briefly recall the definition and some basic properties of the symmetric decreasing rearrangement and formulate our main result in Theorem 3.7 below. Further details on symmetric decreasing rearrangements can be found in the monographs [5, 38]. Let .A ⊆ Rd−1 , .d ≥ 2, be a measurable set of finite volume. Then its symmetric rearrangement .A∗ is defined as the open ball centered at the origin and having

Schrödinger Operators with .δ-potentials

141

the same volume. Let .u : Rd−1 → R be a nonnegative measurable function, that vanishes at infinity in the sense that {x ∈ Rd−1 |u(x) > t} has finite measure for every t > 0.

(3.22)

.

We define the symmetric decreasing rearrangement .u∗ of u by symmetrizing its level sets as ˆ ∞ ∗ .u (x) := χ{u>t}∗ (x) dt. (3.23) 0

Here .χA : Rd−1 → R denotes the characteristic function. The rearrangement .u∗ has a number of straightforward properties, which will be needed below in the proofs of Theorem 3.7 and Lemma 3.9; cf. [38, Section 3.3 (iv) and Theorem 3.4]. Lemma 3.6 Let .u, v : Rd−1 → R be nonnegative measurable functions satisfying (3.22). Then the following holds: (i) .u∗ is nonnegative; (ii) .u∗ is radially symmetric and nonincreasing; (iii) u and .u∗ are equi-measurable, i.e.,     {x ∈ Rd−1 |u(x) > t} = {x ∈ Rd−1 |u∗ (x) > t},

.

t > 0;

(iv) .(u∗ )2 = (u2 )∗ . (v) . u Lp (Rd−1 ) = u∗ Lp (Rd−1 ) , .p ≥ 1 (Conservation of .Lp -norm); ´ ´ (vi) . Rd−1 u v dx ≤ Rd−1 u∗ v ∗ dx (Hardy-Littlewood inequality). Next we formulate our optimization result for the bottom of the spectrum of .Aα . Theorem 3.7 Assume that .α satisfies (3.1) with some .α0 ∈ R. For the function α1 (x) := α(x) − α0 we then have the inequality

.

λ1 (α0 + (α1 )∗+ ) ≤ λ1 (α0 + α1 ),

.

where .(α1 )+ := max{α1 , 0} is the positive part and .(α1 )∗+ its symmetric decreasing rearrangement defined in (3.23). Corollary 3.8 Let .ω ⊂ Rd−1 be a set of finite measure and .ω∗ ⊂ Rd−1 be a ball with the same volume as .ω, and let .χω and .χω∗ be the characteristic functions of .ω and .ω∗ , respectively. Then for .β ≥ 0 we have the inequality λ1 (βχω∗ ) ≤ λ1 (βχω ).

.

The proof of Theorem 3.7 relies on the Birman-Schwinger principle for the operator .Aα , by means of which the problem is reduced to an eigenvalue inequality for the relativistic Schrödinger operator in .L2 (Rd−1 ). The latter is proven with the

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help of the fact that the symmetric decreasing rearrangement decreases the kinetic energy term corresponding to the relativistic Schrödinger operator; cf. Lemma 3.9. This property of the kinetic energy can be viewed as an analogue of the Pólya-Szeg˝o inequality. 1

Lemma 3.9 For every .λ < 0 and nonnegative .φ ∈ H 2 (Rd−1 ) the rearrangements ∗ ∗ .(α1 )+ , .φ in (3.23) and the form (3.2) satisfy dα0 +(α1 )∗+ ,λ [φ ∗ ] ≤ dα0 +α1 ,λ [φ].

(3.24)

.

Proof First, in view of Lemma 3.6 (iv), (v) and (vi) we have ˆ .

Rd−1

ˆ (α0 + α1 )φ 2 dx ≤

ˆ Rd−1

(α0 + (α1 )+ )φ 2 dx ≤

Rd−1

(α0 + (α1 )∗+ )(φ ∗ )2 dx. (3.25)

Moreover, it is proven in [38, Section 7.11 (5), Section 7.17 (2) and the remark afterwards] that   (− − λ) 41 φ ∗ 2 2

.

L (Rd−1 )

 1  ≤ (− − λ) 4 φ L2 (Rd−1 ) .

Combining (3.25) and (3.26) then proves the stated inequality (3.24).

(3.26) 

Proof of Theorem 3.7 Observe that by Theorem 2.3 and Lemma 3.6 (v) the essential spectra of the Schrödinger operators .Aα0 +α1 and .Aα0 +(α1 )∗+ are given by  σess (Aα0 +α1 ) = σess (Aα0 +(α1 )∗+ ) =

.

α2

[− 40 , ∞), if α0 ≥ 0, [0, ∞), if α0 ≤ 0.

We assume that .α1 is such that  λ1 := λ1 (α0 + α1 )
0 and .q ∈ [1, ∞] φ ∈ H s () ⇔ φ ◦  ∈ H s (Rd−1 ) and φ H s () := φ ◦  H s (Rd−1 ) , .

φ ∈ Lq () ⇔ φ ◦  ∈ Lq (Rd−1 ) and φ Lq () := φ ◦  Lq (Rd−1 ) ,

where .(x) := (x, ξ(x)) is a bijective map from .Rd−1 onto ..

(A.2)

Schrödinger Operators with .δ-potentials

145

Lemma A.1 For every .ε > 0 there exists some .cε ≥ 0, depending on .α, such that Mα φ 2L2 () ≤ ε2 φ 2

.

1

H 2 ()

1

+ cε2 φ 2L2 () ,

φ ∈ H 2 ().

(A.3)

Proof We decompose .α ∈ Lp () + L∞ () into β ∈ Lp (), γ ∈ L∞ ().

α = β + γ,

.

Fix .ε > 0. Then the integrability condition .β ∈ Lp () ensures the existence of some .Cε ≥ 0 such that .β = β1 + β2 , where

β1 (x) :=

.

0, |β(x)| ≤ Cε , β(x), |β(x)| > Cε ,

and

β2 (x) :=

β(x), |β(x)| ≤ Cε , 0, |β(x)| > Cε ,

and β1 Lp () ≤ ε2 .

(A.4)

.

We now split .α = β1 + (β2 + γ ) into a bounded part .β2 + γ and an unbounded remainder .β1 and estimate both parts separately. For .β1 we use Hölder’s inequality and the estimate (A.4) to get   |β1 | 21 φ 2 2

.

L ()

≤ β1 Lp () φ 2

2p L p−1 ()

≤ ε2 φ 2

2p L p−1 ()

2 ≤ ε2 cE φ 2

1

,

H 2 ()

(A.5) where in the last inequality we additionally used the Sobolev embedding ·

.

2p

L p−1 ()

≤ cE ·

1

H 2 ()

on the surface, which follows from the classical Sobolev embedding [11, Theorem 8.12.6] on .Rd−1 and the definition of the Sobolev and Lebesgue norms in (A.2). On the other hand, .β2 + γ can simply be estimated by   |β2 + γ | 21 φ 2 2

.

L ()

  ≤ Cε + γ L∞ () φ 2L2 () .

Now the estimate (A.3) follows from (A.5) and (A.6). 1

(A.6) 

The next lemma treats the transition from weak .H 2 -convergence on . to strong on subsets of finite measure of .; this observation is preparatory for the compactness result in Proposition A.3.

2 .L -convergence

146

J. Behrndt et al. 1

Lemma A.2 For every .φ0 , (φn )n ∈ H 2 (), the convergence 1

weakly in H 2 (),

φn  φ0

.

(A.7)

implies for any Borel set .A ⊆  with finite measure, the convergence φ n → φ0

strongly in L2 (A).

.

(A.8)

Proof In Step 1 we consider the hyperplane case . = Rd−1 × {0} ∼ = Rd−1 . For every .t > 0, we define the mollifier ϕt (x) :=

1

.

(4π t)

d−1 2

e−

|x|2 4t

x ∈ Rd−1 .

,

(A.9)

Then by the weak convergence (A.7), we conclude the pointwise convergence of the convolution   lim (ϕt ∗ φn )(x) = lim ϕt (x − · ), φn − 1 d−1 1 d−1 n→∞

n→∞



= ϕt (x − · ), φ0

.

H

 H

− 21

2 (R

)×H 2 (R

)

(A.10)

1

(Rd−1 )×H 2 (Rd−1 )

= (ϕt ∗ φ0 )(x). As the weakly convergent sequence .(φn )n is bounded, i.e. . φn

1

H 2 (Rd−1 )

≤ M for

some .M ≥ 0, we also conclude the uniform boundedness of the convolution |(ϕt ∗ φn )(x)| ≤ ϕt L2 (Rd−1 ) φn L2 (Rd−1 ) ≤ M ϕt L2 (Rd−1 )

(A.11)

.

for every .x ∈ Rd−1 , .n ∈ N. Since A is a set of finite measure, (A.10) and (A.11) are sufficient to apply the dominated convergence theorem, which leads to the norm convergence .

lim ϕt ∗ (φn − φ0 ) L2 (A) = 0.

(A.12)

n→∞

For the Fourier transform of the mollifier (A.9) we have (Fϕt )(k) =

.

=

ˆ

1 (2π )

d−1 2

1 (8π 2 t)

Rd−1

e d−1 2

e

−ikx

−t|k|2

ϕt (x)dx =

ˆ Rd−1

e−

ˆ

1 (8π 2 t)

(x+2itk)2 4t

dx =

d−1 2

Rd−1

1 (2π )

d−1 2

e−ikx e−

e−t|k|

2

|x|2 4t

dx

Schrödinger Operators with .δ-potentials

147

for .k ∈ Rd−1 and we use the estimate   1 1 1 2 1 − (2π ) d−1 2 (Fϕt )(k) = 1 − e −t|k| ≤ c(t|k|2 ) 4 ≤ ct 4 (1 + |k|2 ) 4 ,

k ∈ Rd−1 ,

.

1

where .c := supy>0 (1 − e−y )y − 4 . Since the Fourier transform of the convolution can be written as the product .F(ϕt ∗ φn ) = (2π )

d−1 2

(Fϕt )(Fφn ), we can estimate

   d−1 φn − ϕt ∗ φn L2 (Rd−1 ) =  1 − (2π ) 2 Fϕt Fφn L2 (Rd−1 )  1 1 ≤ ct 4 (1 + | · |2 ) 4 Fφn L2 (Rd−1 ) . 1

= ct 4 φn

1

H 2 (Rd−1 )

(A.13)

.

The inequality (A.13) of course also holds with .φn replaced by .φ0 , which leads to the estimate 1

1

φn − φ0 L2 (A) ≤ ct 4 M + ϕt ∗ (φn − φ0 ) L2 (A) + ct 4 φ0

.

1

H 2 (Rd−1 )

,

for every .n ∈ N and .t > 0. The first and third term can be made arbitrary small by the choice of .t > 0 and the second term converges by (A.12). This proves the statement of the lemma for . ∼ = Rd−1 × {0}. In Step 2 we consider the general case of a Lipschitz graph .. By the definition of the boundary spaces (A.2), it follows immediately from the weak convergence (A.7), that also φn ◦   φ0 ◦ 

.

1

weakly in H 2 (Rd−1 ).

Since A has finite measure, the preimage .−1 (A) = {x ∈ Rd−1 | (x) ∈ A} has finite measure as well, and we conclude from the first step φn ◦  → φ0 ◦ 

.

strongly in L2 (−1 (A)).

By the definition of the boundary spaces (A.2) this implies (A.8).



Next we prove the compactness of the multiplication operator .Mα for functions α which are close to a constant in the sense of (A.14). Note that, although stated for ∞ p d−1 ) satisfies .α, this property only affects the .L -part of .α. Any function in .L (R (A.14) automatically. .

Proposition A.3 Assume that the function .α satisfies {x ∈ ||α(x)| > ε} has finite measure for every ε > 0.

.

Then the multiplication operator .Mα in (A.1) is compact.

(A.14)

148

J. Behrndt et al.

Proof From Lemma A.1 we conclude that .Mα in (A.1) is an everywhere defined and bounded operator. In order to prove that .Mα is compact, we verify that for any 1 sequence .φn  φ0 weakly in .H 2 (), the sequence .Mα φn → Mα φ0 converges 2 strongly in .L (). As in the proof of Lemma A.1, let .ε > 0 and decompose the potential into α = β1 + β2 + γ .

.

Next, we define the set Aε := {x ∈ ||β2 (x)| > ε2 } ∪ {x ∈ ||γ (x)| > ε2 }.

(A.15)

.

The integrability condition .β2 ∈ Lp () implies that the set .{|β2 | > ε2 } has finite 2 2 measure. Furthermore, since .{|γ | > ε2 } ⊆ {|β| > ε2 } ∪ {|α| > ε2 } it follows from the integrability condition .β ∈ Lp () and from (A.14) that .{|γ | > ε2 } also has finite measure. Then Lemma A.2 shows .

lim φn − φ0 L2 (Aε ) = 0.

n→∞

This convergence in particular gives an index .Nε ∈ N, such that φn − φ0 2L2 (A ) ≤

.

ε

ε2 , Cε + γ L∞ ()

n ≥ Nε ,

(A.16)

with .Cε the cut-off from (A.4). Then the Eqs. (A.5) and (A.16), as well as the fact that .|β2 + γ | ≤ Cε + γ L∞ () on . and .|β2 + γ | ≤ 2ε2 on . \ Aε , allows us to estimate  1  |α| 2 (φn − φ0 )2 2

.

L ()

 2  2 1 1 ≤ |β1 | 2 (φn − φ0 )L2 () + |β2 + γ | 2 (φn − φ0 )L2 (A ) ε  2 1 + |β2 + γ | 2 (φn − φ0 )L2 (\A ) ε

≤

2 ε2 cE φn



− φ0

2

+ ε + 2ε2 φn − φ0 2L2 (\A 2

1

H 2 ()

ε)

2 ≤ ε2 (cE + 2) φn − φ0 2

1

H 2 ()

 +1 ,

n ≥ Nε .

Since . φn − φ0 H 1/2 () on the right hand side is bounded as a consequence of the weak .H 1/2 -convergence, this inequality implies the norm convergence .

 1 2 lim |α| 2 (φn − φ0 )L2 () = 0,

n→∞

and hence the compactness of the operator .Mα .



Schrödinger Operators with .δ-potentials

149

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Improved Lieb–Thirring Type Inequalities for Non-selfadjoint Schrödinger Operators Sabine Bögli

Dedicated to the memory of Sergey Naboko

Abstract We improve the Lieb–Thirring type inequalities by Demuth, Hansmann and Katriel (J. Funct. Anal. 2009) for Schrödinger operators with complex-valued potentials. Our result involves a positive, integrable function. We show that in the one-dimensional case the result is sharp in the sense that if we take a non-integrable function, then an analogous inequality cannot hold. Keywords Lieb–Thirring inequalities .· Schrödinger operator with complex potential

1 Introduction Lieb–Thirring inequalities appeared first in the work of Lieb and Thirring in the proof of stability of matter, see [12, 13]. Since then, the involved constants have been improved, and various attempts have been made to generalise the inequalities to allow for non-selfadjoint Schrödinger operators. The classical Lieb–Thirring inequality for a Schrödinger operator .− + V in 2 d .L (R ) reads  .

p

|λ|p−d/2 ≤ Cd,p V Lp ,

(1.1)

λ∈σd (−+V )

S. Bögli () Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Durham, United Kingdom e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_9

151

152

S. Bögli

for real-valued potentials .V ∈ Lp (Rd ), where the range for p depends on the dimension d as follows: .

p ≥ 1,

if d = 1,

p > 1,

if d = 2,

p ≥ d2 ,

if d ≥ 3.

(1.2)

Here .σd (− + V ) denotes the set of discrete eigenvalues, outside the essential spectrum .σe (− + V ) = [0, ∞). The inequality (1.1) cannot be true for complexvalued .V ∈ Lp (Rd ) with .p > (d + 1)/2 since then .σd (− + V ) can accumulate anywhere in the essential spectrum, see [1, 2]. It was proved by Demuth, Hansmann and Katriel in [4] that if .p ≥ d/2 + 1, then for any .τ ∈ (0, 1) we have the inequality  .

λ∈σd (−+V )

(dist(λ, [0, ∞)))p+τ p ≤ Cd,p,τ V Lp |λ|d/2+τ

(1.3)

for any (complex-valued) .V ∈ Lp (Rd ). Note that in the selfadjoint case (V realvalued), the inequality reduces to (1.1) since all discrete eigenvalues are in .(−∞, 0) and hence .dist(λ, [0, ∞)) = |λ|. In [5], it was published as an open problem to prove whether (1.3) remains true for .τ = 0, i.e. whether  .

λ∈σd (H )

(dist(λ, [0, ∞)))p p ≤ Cp,d V Lp . d/2 |λ|

(1.4)

For .d = 1, a counterexample was found in [3]; it is still an open problem whether the inequality can hold in higher dimensions .d ≥ 2. The main ingredient in the proof of (1.3) is the following Lieb–Thirring type inequality by Frank, Laptev, Lieb and Seiringer in [7], which sums only over all eigenvalues outside a sector: If .p ≥ d/2 + 1 then, for any .t > 0,  . λ∈σd (−+V ), | Im λ|≥t Re λ

|λ|

p−d/2

  2 p p ≤ Cd,p 1 + V Lp . t

(1.5)

Now (1.3) follows by taking a weighted integral over the parameter t of the sector. However, the weight of the integral wasn’t chosen to optimise the inequality. In this paper we improve the result by taking an optimal weight function. In Theorem 2.1 we prove a Lieb-Thirring type inequality where the left hand side of (1.4) is multiplied by .f (− log (dist(λ, [0, ∞))/|λ|)) where f is a positive function that can decay quite slowly (slower´than in the result (1.3) which corre∞ sponds to .f (s) = e−τ s ) but is still integrable, . 0 f (s) ds < ∞. In Theorem 2.2 we show that in dimension .d = 1 the inequality is sharp in the sense that if we take a

Improved Lieb–Thirring Type Inequalities for Non-selfadjoint Schrödinger Operators

153

´∞ non-integrable function, . 0 f (s) ds = ∞, then such an inequality cannot hold for all .V ∈ Lp (R). This suggests that the t-dependence (asymptically .t −p as .t → 0) on the right hand side of (1.5) is optimal. We prove this sharpness in Theorem 2.4. In Sect. 3 we discuss a few classes of integrable functions f . Each class can be combined with Theorem 2.1 to give inequalities that are better than (1.3). We mention that in [6, 8–11], different Lieb-Thirring type inequalities were proved. They don’t reduce to (1.1) in the selfadjoint case, and are therefore difficult to compare with the equalities proved here.

2 New Lieb–Thirring Type Inequalities In this section we prove new Lieb-Thirring type inequalities and discuss their sharpness. Theorem 2.1 Let .d ∈ N and .p ≥ d/2 ´+ 1. Let .f : [0, ∞) → (0, ∞) be ∞ a continuous, non-increasing function. If . 0 f (s) ds < ∞, then there exists p d .Cd,p,f > 0 such that, for any .V ∈ L (R ),  .

λ∈σd (−+V )

dist(λ, [0, ∞))p f |λ|d/2

  dist(λ, [0, ∞))  p ≤ Cd,p,f V Lp . − log |λ| (2.1)

The dependence of .Cd,p,f on f is as follows: ˆ Cd,p,f = Cd,p

.

∞

 f (s) ds + f (0)

0

with a constant .Cd,p > 0 that is independent of f . Proof First we show that it suffices to prove the theorem for a continuous, nonincreasing, integrable, piecewise .C 1 -function f for which there exists .c > 0 with  .(log f ) ≥ −c almost everywhere. To this end, first note that since f is continuous and non-increasing, we find a non-increasing .C 1 -function .f1 : [0, ∞) → (0, ∞) with .f (s) ≤ f1 (s) ≤ 2f (s) for all .s ∈ [0, ∞). Let .c > 0. Define a function .a : [0, ∞) → [0, ∞) by    a(x) := max {0} ∪ y ≤ x : (log f1 ) (y) ≥ −c and f1 (x) ≤ f1 (y)e−c(x−y) .

.

Then a is a non-decreasing and piecewise .C 1 -function. For each .x ∈ [0, ∞) we have .a(x) = x or .a(x) < x; in the latter case the derivative is .a  (x) = 0. Now

154

S. Bögli

define another non-increasing function .f2 : [0, ∞) → (0, ∞) by −c(x−a(x))

f2 (x) := f1 (a(x))e

.

= f1 (x)

if a(x) = x,

≥ f1 (x)

if a(x) < x.

Note that a may have discontinuities, namely it may happen that .

lim a(x) < a(x0 ) = x0 ;

x x0

in this case, for all .x < x0 sufficiently close to .x0 , f1 (x) < f1 (a0 )e−c(x−a0 ) ,

a(x) =: a0 is constant,

.

but .f1 (x0 ) = f1 (a0 )e−c(x0 −a0 ) , which implies lim f2 (x) = lim f1 (a0 )e−c(x−a0 ) = f1 (a0 )e−c(x0 −a0 ) = f1 (x0 ) = f2 (x0 ).

.

x x0

x x0

This proves that the function .f2 is continuous. In addition, .f2 is piecewise .C 1 and, for almost every .x ∈ [0, ∞), (log f2 ) (x) = (log f1 ) (a(x))a  (x) − c(1 − a  (x)) (log f1 ) (x) ≥ −c if a(x) = x, = −c if a(x) < x.

.

Hence .(log f2 ) ≥ −c almost everywhere. In addition, using .f1 ≤ 2f , ˆ .

∞

ˆ f2 (x) dx =

0

ˆ ≤

ˆ

{x: a(x)=x}

{x: a(x)=x}

ˆ

∞

≤

f2 (x) dx + ˆ f1 (x) dx + ˆ

f1 (x) dx +

0

∞

{x: a(x) 0 and .dist(λ, [0, ∞)) = | Im λ| ≥ t|λ|. Then, multiplying both sides by .g(t) := t p−1 f (− log t) and integrating over .t ∈ (0, 1), we get ˆ



1

|λ|

.

0

p−d/2

g(t) dt ≤

p Cd,p V Lp

λ∈σd (−+V ), Re λ>0, dist(λ,[0,∞))≥t|λ|

ˆ

1

2 1+ t

0

p g(t) dt. (2.2)

Using that .1 ≤ 1/t for .t ∈ (0, 1) and substituting .s = − log t, the integral on the right hand side of (2.2) side becomes ˆ

1

1+

.

0

2 t

p

ˆ

1

g(t) dt ≤

3p t −1 f (− log t) dt = 3p

ˆ

0

∞

f (s) ds < ∞.

0

The left hand side of (2.2) becomes, using the substitution .s = − log t, ˆ



1

0

.

|λ|p−d/2 g(t) dt

λ∈σd (−+V ), Re λ>0, dist(λ,[0,∞))≥t|λ|

ˆ



=

|λ|p−d/2



g(t) dt 0

λ∈σd (−+V ), Re λ>0

=

dist(λ,[0,∞))/|λ|

ˆ |λ|p−d/2

λ∈σd (−+V ), Re λ>0

∞

− log(dist(λ,[0,∞))/|λ|)

(2.3)

e−ps f (s) ds.

Recall that by assumption or construction, .f = f2 satisfies .(log f ) ≥ −c almost everywhere. This implies .f  (s) ≥ −cf (s) and thus integration by parts yields ˆ .

∞

e−ps f (s) ds ≥

a

  ˆ ∞ 1 −pa e f (a) − c e−ps f (s) ds , p a

whence ˆ .

a

∞

e−ps f (s) ds ≥

1 −pa e f (a). p+c

156

S. Bögli

Thus the last line in (2.3) can be estimated from below by .

1 p+c =



|λ|p−d/2 e−pa f (a)

 λ∈σd (−+V ), Re λ>0

1 p+c



|λ|p−d/2

λ∈σd (−+V ), Re λ>0

 a=− log

dist(λ,[0,∞)) |λ|



 dist(λ, [0, ∞)) p   dist(λ, [0, ∞))  . f − log |λ| |λ|

Combining the estimates of the left and right hand sides of (2.2), we obtain (2.1) with the left hand side restricted to .Re λ > 0. For the sum over the eigenvalues with .Re λ ≤ 0, we use (1.5) with .t = 1 (in fact, any .t > 0 would work here). Note that then all .λ satisfying .Re λ ≤ 0 also satisfy .| Im λ| ≥ t Re λ. Thus we get  .

p

|λ|p−d/2 ≤ Cd,p 3p V Lp .

λ∈σd (−+V ), Re λ≤0

This implies, with .dist(λ, [0, ∞)) = |λ| if .Re λ ≤ 0,  . λ∈σd (−+V ), Re λ≤0

=

dist(λ, [0, ∞))p f |λ|d/2



  dist(λ, [0, ∞))  − log |λ| p

|λ|p−d/2 f (0) ≤ Cd,p 3p f (0) V Lp ,

λ∈σd (−+V ), Re λ≤0

which concludes the proof of (2.1). The stated dependence of .Cd,p,f on f follows immediately from the derived inequalities.

In dimension .d = 1, Theorem 2.1 is optimal in the sense that if f is no longer integrable, then the corresponding Lieb–Thirring inequality is false. Theorem 2.2 Let .d = 1 and´.p ≥ 1. Let .f : [0, ∞) → (0, ∞) be a continuous, ∞ non-increasing function with . 0 f (s) ds = ∞. Then  .

sup

V ∈Lp (R)

dist(λ,[0,∞))p f λ∈σd (−+V ) |λ|1/2 p V Lp



− log



dist(λ,[0,∞)) |λ|

 = ∞.

(2.4)

Remark 2.3 Note that Theorem 2.2 holds for .p ≥ 1 but Theorem 2.1 is proved for p ≥ 3/2 in dimension .d = 1. It is still an open question how to optimise the LiebThirring type inequalities in .d = 1 for .p ∈ (1, 3/2), and in higher dimensions. In particular, it is not known whether Theorem 2.1 is optimal for .d ≥ 2.

.

Proof (Proof of Theorem 2.2) We use a slight modification of the example in [3, Proposition 10] with potential .Vh (x) = ihχ[−1,1] (x) where .h > 0 with .h → ∞.

Improved Lieb–Thirring Type Inequalities for Non-selfadjoint Schrödinger Operators

157

In the proof of [3, Proposition 10], asymptotic formulas were derived for all eigenvalues .λj = μ2j + ih with .α log h ≤ Im μj ≤ β log h, .Re μj ≤ −hγ | Im μ| where .α, β, γ > 0 are chosen to be h-independent constants with .γ < 2α < 2β < 1. We now want to allow .α, γ to be h-dependent (.β ∈ (0, 1/2) still constant), with .0 < γ (h) < 2α(h) → 0 as .h → ∞, but sufficiently slowly so that the asymptotics still hold. It turns out that if we still have .h−γ (h) → 0, then   π(8j − 7) π + O(h−γ (h) ) (7 − 8j ) + i log .μj = √ 4 2 h uniformly for all integers j with .hα(h)+1/2 ≤ j ≤ hβ+1/2 . Hence we set .ε(h) = h−γ (h) (converging to zero arbitrarily slowly) and .α(h) = γ (h). The latter implies α(h)+1/2 = h1/2 . Analogously we then obtain that there exists .h > 0 such that for .h 0 ε(h) 1/2

h all .h > h0 and . ε(h) ≤ j ≤ hβ+1/2 , we have the estimates

dist(λj , [0, ∞)) = Im λj >

.

h , 2

πj ≤ |λj |1/2 ≤ 2πj.

(2.5)

This implies, for all .h > h0 ,  .

λ∈σd (−+Vh )

dist(λ, [0, ∞))p f |λ|1/2



≥

h1/2 β+1/2 ε(h) ≤j ≤h

≥

 p 1 h 2 2π

  dist(λ, [0, ∞))  − log |λ|

dist(λj , [0, ∞))p f |λj |1/2 

h1/2 β+1/2 ε(h) ≤j ≤h

1 f j

  dist(λ , [0, ∞))  j − log |λj |

  − log

 h . 2(2πj )2

  2  j h 2 = log Note that .− log 2(2πj 2 h + log(8π ). Since f is non-increasing, with ) .f (s) ≤ f (0) for .s ≥ 0, we see that  . h1/2 β+1/2 ε(h) ≤j ≤h

ˆ ≥ ˆ ≥

hβ+1/2 h1/2 ε(h)

1 f j

1 f x

2β log(h)

−2 log(ε(h))

  2  j 2 log + log(8π ) h

  2  x log + log(8π 2 ) dx − 2f (0) h f (s + log(8π 2 )) ds − 2f (0)

158

S. Bögli

where we have used the substitution .s = log



x2 h

 . The appearance of .−2f (0) 1/2

h , .x = hβ+1/2 may not be compensates for the fact that the two endpoints .x = ε(h) integers j . Note that .2β log(h) → ∞. Since f is not integrable, we can choose ´ 2β log(h) .ε(h) → 0 so slowly (i.e. .−2 log(ε(h)) → ∞ so slowly) that . −2 log(ε(h)) f (s + p 2 p log(8π )) ds → ∞. Now, with .Vh Lp = 2h , we arrive at



dist(λ,[0,∞))p f |λ|1/2 p Vh Lp

λ∈σd (−+Vh ) .



− log



dist(λ,[0,∞)) |λ|



ˆ  p 2β log(h) 1 1 2 f (s + log(8π )) ds − 2f (0) → ∞, ≥ 2 4π −2 log(ε(h))

h → ∞.



This completes the proof.

Finally we also prove that in one dimension, the t-dependence in (1.5) is optimal. Theorem 2.4 Let .d = 1 and .p ≥ 1. Let .ϕ : (0, ∞) → (0, ∞) be a continuous function with .ϕ(t) = o(t −p ) as .t → 0. Then  .

lim sup t→0

sup V ∈Lp (R)

|λ|p−1/2

λ∈σd (−+V ), | Im λ|≥t Re λ

(2.6)

= ∞.

p

ϕ(t)V Lp

Proof We use the same potentials .Vh as in the proof of Theorem 2.2. Recall that 1/2 p Vh Lp = 2hp . Again we use the index .j ∈ Z with . hε(h) ≤ j ≤ hβ+1/2 to enumerate the eigenvalues .λj with uniform estimates in (2.5); we take .ε(h) → 0 with .1/ε(h) = −2β o(hβ ). We take t to be h-dependent as .t (h) = h8π 2 . Then one can check that each of these .λj satisfies .| Im λj | ≥ t (h)|λj | ≥ t (h) Re λj . Therefore,

.

 λ∈σd (−+Vh ), | Im λ|≥t (h) Re λ .



|λ|p−1/2 p

ϕ(t (h))Vh Lp

≥

(πj )2(p−1/2)

h1/2 β+1/2 ε(h) ≤j ≤h

.

ϕ(t (h))2hp

In the limit .h → ∞, the sum on the right hand side is of the same order as ˆ .

hβ+1/2 h1/2 ε(h)

j

2p−1

dj =

h2βp+p − 2p

hp ε(h)2p

.

(2.7)

Improved Lieb–Thirring Type Inequalities for Non-selfadjoint Schrödinger Operators

159

Thus, up to a multiplicative constant, the right hand side of (2.7) is asymptotically .

t (h)−p h2βp 1 = → ∞, ϕ(t (h)) (8π 2 )p ϕ(t (h))

where we used the assumption .ϕ(t) = o(t −p ) as .t → 0. This proves the claim.



3 Examples In this section we verify the assumptions of Theorem 2.1 for a few examples. If we combine the below part (i) with Theorem 2.1, we recover [4, Corollary 3], which was the hitherto best known result. Parts (ii)–(v) yield improvements of this result. We use the notation .g ◦n for the n-th iterated function g, i.e. .g ◦0 (s) = s, .g ◦1 (s) = g(s), .g ◦2 (s) = g(g(s)) etc. We also use the super-logarithm (to the basis .e) defined for an .s ∈ R by .slog(s) := min{n ∈ N0 : log◦n (s) ≤ 1}. Proposition 3.1 ´ ∞The following continuous, non-increasing functions .f : [0, ∞) → (0, ∞) satisfy . 0 f (s) ds < ∞ if .ε > 0: (i) .f (s) = e−εs ; 1 (ii) .f (s) = (1+s) 1+ε ; 1/e (iii) .f (s) = 1 1

s ≤ e,

s log(s)1+ε , s > e; ⎧ 1 ⎨n , ◦j j =1 (iv) .f (s) = n−1exp 1(1)  ⎩ ◦j

slog(s) ≤ n (i.e. s ≤ exp◦n (1)),

slog(s) ≥ n + 1 (i.e. s > exp◦n (1)), for .n ∈ N ⎧ (where .n = 1 corresponds to the case (iii)); ⎨1/e, s ≤ e, (v) .f (s) = slog(s)−1 1  1 ⎩ j =0 s > e. ◦j ◦slog(s) 1+ε , j =0 log (s)

1 , log◦n (s)1+ε

log (s)

(slog(s)−1+log

´∞

In each case (i)–(v), if we set .ε = 0 then .

0

(s))

f (s) ds = ∞.

Remark 3.2 The functions in (i)–(iv) are the piecewise derivatives of the following respective functions: (i) .F (s) = − 1ε e−εs ; 1 (ii) .F (s) = − 1ε (1+s) ε; 1 (iii) .F (s) = − 1ε log(s) ε for .s > e;

(iv) .F (s) = − 1ε (log◦n1(s))ε for .s > exp◦n (1); (v) .F (s) = − 1ε

1 (slog(s)−1+log◦slog(s) (s))ε

for .s > e.

160

S. Bögli

In (v), note that .log◦slog(s) (s) ∈ (0, 1], so asymptotically we have .F (s) ∼ 1 − 1ε (slog(s)) ε as .s → ∞. More examples can be generated by taking F to be a piecewise continuous function with .F (s) 0 very slowly as .s → ∞, and then take .f = F  piecewise. There exists no function with the slowest convergence, as if .F (s) = −1/g(s) with .g(s) ∞ slowly, then .g(g(s)) ∞ even slower and hence .−1/g(g(s)) 0 even slower than .F (s). ´∞ Proof (Proof of Proposition 3.1) By Remark 3.2, we can write . 0 f (s) ds = ´a 0 f (s) ds + F (a) for any .a´ ∈ (0, ∞). This proves that f is integrable. It is left ∞ to show that if .ε = 0, then . 0 f (s) ds = ∞. The corresponding antiderivatives (modulo additive constants) are the following functions: (i) (ii) (iii) (iv) (v)

F (s) = s; F (s) = log(1 + s); ◦2 .F (s) = log (s) for .s > e; ◦(n+1) .F (s) = log (s) for .s > exp◦n (1); ◦slog(s) .F (s) = log(slog(s) − 1 + log (s)) for .s > e. . .

Since each of these functions satisfies .lims→∞ F (s) = ∞, the corresponding .f = F  is not integrable.

Acknowledgments The author is grateful for the comments of the anonymous referee, and thanks Jean-Claude Cuenin, Rupert L. Frank, František Štampach and Alexei Stepanenko for helpful discussions.

References 1. S. Bögli, Schrödinger operator with non-zero accumulation points of complex eigenvalues. Commun. Math. Phys. 352(2), 629–639 (2017) 2. S. Bögli, J.-C. Cuenin, Counterexample to the Laptev–Safronov conjecture. Commun. Math. Phys. 398, 1349–1370 (2023) 3. S. Bögli, F. Štampach, On Lieb–Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schrödinger operators. J. Spectr. Theory 11(3), 1391–1413 (2021) 4. M. Demuth, M. Hansmann, G. Katriel, On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal. 257(9), 2742–2759 (2009) 5. M. Demuth, M. Hansmann, G. Katriel, Lieb-Thirring type inequalities for Schrödinger operators with a complex-valued potential. Integr. Equ. Oper. Theory 75(1), 1–5 (2013) 6. R.L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Am. Math. Soc. 370(1), 219–240 (2018) 7. R.L. Frank, A. Laptev, E.H. Lieb, R. Seiringer, Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77(3), 309–316 (2006) 8. R.L. Frank, J. Sabin, Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Am. J. Math. 139(6), 1649–1691 (2017) 9. L. Golinskii, A. Stepanenko, Lieb–Thirring and Jensen sums for non-self-adjoint Schrödinger operators on the half-line. arXiv:2111.09629 10. M. Hansmann, An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators. Lett. Math. Phys. 98(1), 79–95 (2011)

Improved Lieb–Thirring Type Inequalities for Non-selfadjoint Schrödinger Operators

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11. A. Laptev, O. Safronov, Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys. 292(1), 29–54 (2009) 12. E.H. Lieb, W.E. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975) 13. E.H. Lieb, W.E. Thirring, Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities (Springer, Berlin, 1991), pp. 135–169

Ballistic Transport in Periodic and Random Media Anne Boutet de Monvel and Mostafa Sabri

Dedicated to the memory of our friend Sergey Naboko

Abstract We prove ballistic transport of all orders, that is, .x m e−itH ψ  t m , for the following models: the adjacency matrix on .Zd , the Laplace operator on .Rd , periodic Schrödinger operators on .Rd , and discrete periodic Schrödinger operators on periodic graphs. In all cases we give the exact expression of the limit of m −itH ψ/t m as .t → +∞. We then move to universal covers of finite graphs .x e (these are infinite trees) and prove ballistic transport in mean when the potential is lifted naturally, giving a periodic model, and when the tree is endowed with random i.i.d. potential, giving an Anderson model. The limiting distributions are then discussed, enriching the transport theory. Some general upper bounds are detailed in the appendix. Keywords Ballistic transport · Delocalization · Periodic Schrödinger operators · Periodic graphs · Trees

1 Introduction 1.1 Background Ballistic motion is a dynamical property found in certain delocalized Schrödinger operators. A. Boutet de Monvel () Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, Paris, France e-mail: [email protected] M. Sabri Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_10

163

164

A. Boutet de Monvel and M. Sabri

Broadly speaking, in considering a Schrödinger operator, say .H = − + V on L2 (Rd ), we can understand localization in some spectral interval I in three senses: (a) Spectral localization means that .σ (H ) ∩ I is pure point. (b) Exponential localization means that moreover the eigenfunctions decay exponentially. (c) Dynamical localization means that states initially localized in a bounded domain will not leave this domain much as time goes on. The opposite regime of delocalization is similarly understood in three senses: (a’) spectrally, one expects absolutely continuous (AC) spectrum, (b’) spatially, the (generalized) eigenfunctions may ideally be equidistributed in some sense, (c’) dynamically, one expects the wave packets to spread on the space as time goes on. The RAGE theorem establishes some links between spectral and dynamical (de)localization. Recall that if .ψ0 ∈ L2 (Rd ) is some initial state of the system, then .e−itH ψ0 describes the state of the system at time t. Now define the following subspaces of .H := L2 (Rd ):

.

Hp := span{eigenvectors of H }

.

Hc := Hp⊥ .

and

The RAGE theorem (see [33, Section 5.2]) asserts that f ∈ Hp ⇐⇒ ∀ε > 0 ∃K ⊂ Rd compact s.t. sup χK c e−itH f  ≤ ε, .

.

(1.1)

t∈R

f ∈ Hc ⇐⇒ ∀K ⊂ Rd compact :

1 T →+∞ T lim

ˆ

T

−T

χK e−itH f 2 dt = 0. (1.2)

Hence: (i) A state is in the pure point space .Hp iff at all times, most of its mass lies within a fixed compact set. (ii) A state is in the continuous space .Hc iff its time evolution escapes (in average) from any compact set, after sufficient time has passed. Exploring further the dynamical aspects of delocalization, a common object of study is the mean square displacement .rf2 (t) := xe−itH f 2 . In presence of AC spectrum, i.e., if the absolutely continuous subspace .Hac is nontrivial, one can deduce from the RAGE theorem1 that . lim x m e−itH f  = ∞ for any .f ∈ Hac , t→∞ .f = 0. In this article we are interested in the rate of this divergence. It is known that 2 2 .r (t)  t , see Theorem A.4 for a more general result. The operator exhibits f ballistic transport if we also have .rf2 (t)  t 2 , at least in a time-averaged sense.

is known [33] that for .f ∈ Hac , (1.2) strengthens to the fact that .limt→+∞ χK e−itH f  = 0. This implies that .limt→+∞ χK c e−itH f  = limt→+∞ e−itH f  = f . Taking .K = Λr := {|x| ≤ r}, we thus get .lim inft→+∞ x m e−itH f  ≥ lim inft→+∞ r m χΛcr e−itH f  = r m f . As r is arbitrary, this shows .limt→∞ x m e−itH f  = ∞. 1 It

Ballistic Transport in Periodic and Random Media

165

Here we discuss several models that exhibit the exact ballistic rate, that is, x m e−itH f   t m for .f ∈ Hac , .f = 0 and all .m ≥ 1.

.

1.2 Contents Our aim is to provide a comprehensive outlook of various techniques to prove ballistic motion for different models. In Sect. 2 we establish ballistic transport for the adjacency matrix on .Zd (Sect. 2.1), followed by the Laplace operator on .Rd (Sect. 2.2). This is of course folklore, however we are not aware of references computing the exact limit of m −itH ψ2 /t 2m as .t → +∞, apart from [9] where a proof was given for Jacobi .x e matrices (.d = 1). Then, in Sect. 3.1, we move on to Schrödinger operators with periodic potential on .Rd , extending the result of [5] to all m. After that, in Sect. 3.2, we proceed to the less studied discrete periodic Schrödinger operators on periodic graphs. This includes the hexagonal lattice, face-centered and body-centered cubic lattices, and a lot more. For these models we establish ballistic motion, assuming .ψ has a nontrivial AC component, i.e., its spectral projection onto .Hac is nontrivial. In Sect. 4 we move to infinite trees that enjoy some form of periodicity. More specifically, we consider a finite graph G endowed with some potential W , then  endowed with the naturally lifted potential consider the universal cover .T = G in Sect. 4.2. We proceed afterwards to replace such lifted potentials by random i.i.d. potentials in Sect. 4.3, which gives an Anderson model. For both models (periodic and random) we establish ballistic transport in mean, that is, we consider ´ ∞ −2ηt .2η e |x|β/2 e−itH ψ2 dt instead, with .η ↓ 0. The periodic result seems new, 0 the random result appeared before in the special case of the d-regular tree in [1, 20]. In Sect. 5 we refine our results for the periodic models of Sect. 3 by studying the limiting distribution of .Xt /t, when .Xt has distribution .|e−itH ψ(x)|2 and .ψ ψ is normalized. Note that if .νt is the distribution of .Xt /t, then ballistic motion is merely the statement that the second moment . lim Eν ψ (x 2 ) is .> 0. Here we t→+∞

ψ

t

ψ

actually compute the nontrivial limit .ν∞ of the measures .νt as .t → +∞. This gives a richer understanding of the transport theory; in particular, this gives an expression for the limit . lim Eν ψ (f ) = Eν ψ (f ) for any bounded continuous f . t→+∞

t

∞

Appendix A contains complete proofs of the ballistic upper bounds, which is well-known [27] for .m = 1, 2. We also prove some differentiability results for the moments in the Heisenberg picture, i.e., for the maps .t → eitH x m e−itH , which are needed in the text. To conclude this introduction we refer to the beautiful paper [25] for finer interplay between the spectral measures and quantum dynamics. We also mention the recent papers [10, 11, 13, 17] which prove ballistic transport for limit-periodic and quasiperiodic Schrödinger operators, in various degrees of generality and strength.

166

A. Boutet de Monvel and M. Sabri

2 The Free Laplacian 2.1 Discrete Case We consider the integer lattice .Zd and the space .2 (Zd ) of square summable d sequences .ψ : Z → C equipped with the scalar product .φ, ψ :=  d d d n∈Zd φ(n)ψ(n). On the other hand we consider the torus .T = R /(2π Z) = d 2 d d [0, 2π ) and the space .L (T ) of square ´ integrable functions .f : T → C equipped with the scalar product .f, g := Td f (θ )g(θ )dθ . The Fourier transform 2 d 2 d .F : L (T ) →  (Z ) is defined by ˆ 1 .(F f )(n) ≡ fˆ(n) := √ f (θ )e−in·θ dθ = en , f , ( 2π )d Td in·θ √e . In particular, .F en = δn where .{δn }n∈Zd is the standard ( 2π)d 2 d of . (Z ). The inverse Fourier transform .F −1 : 2 (Zd ) → L2 (Td ) is

where .en (θ ) := Hilbert basis given by

  1 ˆ )= √ [F −1 (ψ)](θ ) = ψ(θ ψ(n)ein·θ = ψ(n)en (θ ). ( 2π )d d d

.

n∈Z

n∈Z

In this section we consider the adjacency matrix .A acting in .2 (Zd ), that is, (Aψ)(n) =



.

w∼n

ψ(w) =

d    ψ(n − ej ) + ψ(n + ej ) j =1

where .{ej }dj =1 is the standard basis of .Zd and .w ∼ v means that the vertices d are nearest neighbors. Under the action of the Fourier transform, .v, w ∈ Z iθ −iθj )ψ(θ ˆ := F −1 AF is the ˆ ). So .A .ψ(n − ej ) + ψ(n + ej ) becomes .(e j + e d 2 operator of multiplication by .φ(θ) := j =1 2 cos θj in .L (Td ) and the spectrum of .A is absolutely continuous: .σ (A) = σac (A) = [−2d, 2d]. For .n ∈ Zd , .m ∈ N, and .ψ : Zd → C, let .nm ψ : Zd → Cd denote the map m m m defined by .nm ψ(n) := (nm 1 ψ(n), . . . , nd ψ(n)) and let .n ψ = n ψ2 (Zd ,Cd ) be its norm. Theorem 2.1 For any .ψ ∈ 2 (Zd ) with .nm ψ < ∞, .m ∈ N we have .

  m−1  2m nm eitA ψ2 2m 2 (−1)q+m ψ, S2m−2q ψ, = d ψ + t→+∞ q m t 2m lim

q=0

(2.1) where .(Sk ψ)(n) :=

d

j =1

  ψ(n − kej ) + ψ(n + kej ) is a k-step adjacency matrix.

Ballistic Transport in Periodic and Random Media

167

Moreover, this limit is .> 0 if .ψ = 0 and we have the upper bound .

nm eitA ψ2 ≤ 4m dψ2 . t→+∞ t 2m lim

(2.2)

Proof Denote .ψt := eitA ψ. We first prove that .nm ψ < ∞ implies .nm ψt  < ∞ mψ = Dmψ ˆ where for any .t ≥ 0. By applying the Fourier transform, we get .n m m m m m m d ˆ .D := (−i) (∂θ1 , . . . , ∂θd ). So .n ψ < ∞ means .ψ ∈ H (T ). Moreover,  ˆ is the operator of multiplication by .φ(θ) = dj =1 2 cos θj it leaves invariant since .A ˆ m d it A itA ψ mψ = ˆ ∈ H m (Td ) and .n .H (T ) and similarly for .eitA = e . So, .ψˆ t = e t m 2 d m 2 d ˆ D ψt ∈ L (T ) which means that .n ψt ∈  (Z ). The limit in (2.1) is strictly positive if .ψ = 0 as a consequence of (2.6) below. We then show how formula (2.1) implies the upper bound (2.2) which is a slight improvement over the general bound (A.8) of Appendix A . By using .Sk  ≤ 2d to estimate the right-hand side of (2.1) we get

.

   m−1  2m nm eitA ψ2 2m ψ2 ≤ d + 2d t→+∞ q m t 2m lim

q=0

   2m  2m ψ2 = (1 + 1)2m dψ2 . = d q q=0

itA ψ be as above. Since .A ˆ Now we give the proof of formula (2.1). Let t := e .ψ d is the operator of multiplication by .φ(θ ) := 2 j =1 cos θj , we have mψ = Dmψ ˆ t = D m (eit Aˆ ψ) ˆ = D m (eitφ ψ). ˆ n t

.

In particular, ˆ L2 (Td ,Cd ) . nm eitA ψ2 (Zd ,Cd ) = D m (eitφ ψ)

.

(2.3)

By Leibniz, m .∂θ j

m    itφ(θ)   m r itφ(θ) m−r ˆ ˆ ). ∂ e e ∂θj ψ(θ ψ(θ ) = r θj

(2.4)

r=0

Clearly the leading term in t is for .r = m and we have ∂θmj eitφ(θ) = (−2it sin θj )m eitφ(θ) + O(t m−1 )

.

(2.5)

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A. Boutet de Monvel and M. Sabri

for some .O(t m−1 ) which can be made explicit. Since .ψˆ ∈ H m (Td ), we deduce from (2.3)–(2.5) that nm eitA ψ2 = 4m t 2m

d 

.

ˆ 2 + O(t 2m−1 ). (sin θj )m ψ

(2.6)

j =1

Now we observe that (sin θ )2m =

.

 2m  (eiθ − e−iθ )2m (−1)m  2m iqθ −iθ(2m−q) e e = (−1)2m−q . q 4m (2i)2m q=0

 2m  2m = q , we thus have Using .(−1)2m−q = (−1)q and . 2m−q  4m (sin θ )2m =

.

 m−1  2m 2m (−1)q+m (2 cos(2m − 2q)θ ) . + q m q=0

Recall that .Sk : 2 (Zd ) → 2 (Zd ) is defined by (Sk ψ)(n) :=

.

d    ψ(n − kej ) + ψ(n + kej ) . j =1

Under the action of the Fourier transform, .ψ(n−kej )+ψ(n+kej ) becomes .(eikθj + ˆ So .Sˆk := F −1 Sk F is the operator of multiplication by the function .φk in e−ikθj )ψ. d 2 d .L (T ), with .φk (θ ) := j =1 2 cos kθj . It follows that 4m

d 

.

d 

 ˆ 2 = ψ, ˆ 4m (sin θj )m ψ (sin θj )2m ψˆ

j =1

j =1

 =d

 m−1  2m 2m ˆ ˆ 2+ ˆ φ2m−2q ψ (−1)q+m ψ, ψ q m q=0

 m−1  2m 2m 2 ˆ ˆ S ˆ (−1)q+m ψ, =d ψ + 2m−2q ψ q m 

q=0

 m−1  2m 2m 2 (−1)q+m ψ, S2m−2q ψ. =d ψ + q m 

q=0

Recalling (2.6), dividing by .t 2m and taking .t → +∞, we obtain the statement.

 

Ballistic Transport in Periodic and Random Media

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Remark 2.2 For .m = 1 we get .limt→+∞ ne t 2 ψ = 2dψ2 − ψ, S2 ψ. In particular a bound like .neitA ψ ≥ Ctψ cannot be valid for a positive constant C independent of .ψ. In fact, let .d = 1. Consider .ψ = (ψj ) given by .ψ1 = · · · =  ψn = √1n and .ψj = 0 otherwise. Then .ψ2 = 1 and . j ψj ψj +2 = ψ1 ψ3 + · · · + itA

2

ψn−2 ψn + ψn−1 ψn+1 + ψn ψn+2 = n1 + · · · + n1 + 0 + 0 = n−2 n . So in this case the 4 ) = , which can get arbitrarily small while conserving .ψ = 1. limit is .2(1 − n−2 n n

2.2 Continuous Case Consider now the case of the Laplacian . on Euclidean space .Rd . The space .L2 (Rd ) ´ is equipped with the scalar product .φ, ψ := Rd φ(x)ψ(x)dx and the Fourier transform .F : L2 (Rd ) → L2 (Rd ) is defined by 1 ˆ .F (ψ)(y) ≡ ψ(y) := √ ( 2π )d

ˆ Rd

ψ(x)e−iy·x dx.

For .x = (x1 , . . . , xd ) ∈ Rd we denote .x k := (x1k , . . . , xdk ) and .x k ψ(x) := (x1k ψ(x), . . . , xdk ψ(x)). Let also .D k := (−i)k (∂xk1 , . . . , ∂xkd ) and . := ∂x21 + · · · + ∂x2d .  m−k D k ψ2 < ∞. Then Theorem 2.3 Assume .ψ ∈ L2 (Rd ) satisfies . m k=0 x .

x m eit ψ2 = 4m D m ψ2 . t→+∞ t 2m lim

Let .m = (m1 , . . . , md ), .m := max{m1 , . . . , md }, and .x m := (x1m1 , . . . , xdmd ). The proof shows more generally that .

 x m eit ψ2 = 4m ∂xmj ψ2 . 2m t→+∞ t lim

1≤j ≤d mj =m

Proof Using the Fourier transform one sees that eit ψ(x) =

.

eix·x/4t (4π it)d/2

ˆ Rd

eiy·y/4t ψ(y)e−ix·y/2t dy,

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A. Boutet de Monvel and M. Sabri

see, e.g., [33, Section 7.4, (7.43)]. Denoting .φt (y) := eiy·y/4t ψ(y), this can be written as eit ψ(x) =

.

eix·x/4t x . φˆ t 2t (2it)d/2

(2.7)

Thus, x m eit ψ =

.

  1  1   mˆ ·   x mˆ ·  m = (2t) φ φ   x  t t 2t 2t (2t)d/2 (2t)d/2 2t

m φ  = (2t)m D m φ .  = (2t)m x m φˆ t  = (2t)m D t t

By Leibniz formula, ∂ymj φt (y) =

m    m

.

k=0

k

∂ym−k eiy·y/4t ∂ykj ψ(y) = eiy·y/4t ∂ymj ψ(y) + Oψ (t −1 ). j

(2.8)

The error term .Oψ (t −1 ) depends on .ψ and its derivatives. It can be made explicit as follows. The Faà di Bruno formula implies that p iy·y/4t .∂yj e

=e

iy·y/4t

p/2  r=0

ip−r p−2r p! y . (p − 2r)!r! 2p t p−r j

Thus, Oψ (t −1 ) =

.

eiy·y/4t

m−1  k=0

m k

 (m−k)/2  r=0

im−k−r (m − k)! y m−k−2r ∂ykj ψ(y). m−k (m − k − 2r)!r! 2 t m−k−r j

By (2.8) and for any x, .|∂xmj φt (x)| → |∂xmj ψ(x)| as .t → +∞. The explicit form  m−k D k ψ2 < ∞ allow to use dominated of .Oψ (t −1 ) and our assumption . m k=0 x   m 2 convergence to conclude that .D φt  = dj =1 ∂xmj φt 2 → dj =1 ∂ymj ψ2 =   D m ψ2 . Remark 2.4 The regularity of .ψ above is important. In fact, as shown in [27, √ p. 294] for .d = 1, the function .ψ(x) = 2|x|e−|x| satisfies .x 2 eit ψ2 = eit ψ, x 4 eit ψ = +∞ for any .t > 0. Note that here .D 2 ψ does not exist.

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3 Periodic Operators in Euclidean Space 3.1 Continuous Case Consider .H = H0 + V = − + V on .L2 (Rd ) with V a periodic potential. We assume .V ∈ C m−1 (Rd ) and its partial derivatives of order .< m are bounded. Denote as above D := −i∇x = −i(∂x1 , . . . , ∂xd ).

.

Recall the direct integral notations. If .(M, dm) is some measure space and .h a Hilbert space, its direct integral over M is the Hilbert space .H defined by ˆ H =

⊕

.

ˆ h dm := L2 (M, dm; h) ,

f, gH :=

M

fm , gm h dm ,

(3.1)

M

where .f = (fm )m∈M and .g = (gm )m∈M with .fm , gm ∈ h. Moreover, a measurable family .(A(m))m∈M of operators in .h defines an operator A in .H by ˆ A=

⊕

(Af )m = A(m)fm .

A(m)dm,

.

(3.2)

M

For definiteness we assume the potential is .(2π Z)d -periodic. Let Td := Rd /(2π Z)d = [0, 2π )d and Td∗ := Rd /Zd = [0, 1)d .

.

We define an operator ˆ U : L2 (Rd ) →

⊕

.

Td∗

L2 (Td )dθ,

  ψ → (U ψ)θ θ∈Td

∗

by setting (U ψ)θ (x) :=



.

e−iθ·(x+2π n) ψ(x + 2π n)

(3.3)

n∈Zd

for each .θ ∈ Td∗ and .x ∈ Td ⊂ Rd . So we indeed have .(U ψ)θ ∈ L2 (Td ). This operator U is unitary and we have (see proof below) U H U −1 =

ˆ

⊕

.

Td∗

H (θ )dθ

(3.4)

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A. Boutet de Monvel and M. Sabri

where .H (θ ) = (D V on .L2 (Td ) with form domain .H 1 (Td ). This d+ θ ) · (D + θ ) + 2 means .H (θ ) = j =1 (−i∂xj + θj ) + V . We prefer to denote .D · D rather than .D 2 to avoid confusing with the notation .D m = (−i)m (∂xm1 , . . . , ∂xmd ) from the previous section. Here .H (θ ) has compact resolvent and H (θ ) =

∞ 

.

En (θ )Pn (θ ),

n=1

where .En (θ ) are its eigenvalues in non-decreasing order and .Pn (θ ) are the corresponding eigenprojections. Moreover, for every n, the following set has full Lebesgue measure in .Td∗ : Sn := {θ ∈ Td∗ : Pn (θ ) and En (θ ) are smooth at θ, and ∇θ En (θ ) = 0}.

.

(3.5)

These facts are standard, see [34, Theorems 1 and 2] or [6, Proposition 10.11]. For the last assertion, note that .En (θ ) is non-constant and analytic outside a nullset [34], hence .∇θ En (θ ) = 0 a.e. [24, Lemma 5.22]. Proof (of (3.4)) For future use we prove (3.4). We have for .φ ∈ S (Rd ), ˆ



⊕

.

Td∗

 (x) := D(U φ)θ (x)

Ddθ U φ θ

=



{De−iθ·(x+2π n) φ(x + 2π n) + e−iθ·(x+2π n) Dφ(x + 2π n)}

n∈Zd

= −θ (U φ)θ (x) + (U Dφ)θ (x),

(3.6)

Relation (3.6) means (U Dφ)θ = (D + θ )(U φ)θ ,

.

(3.7)

that is, .−i(U ∂xj φ)θ = (−i∂xj + θj )(U φ)θ for .j = 1, . . . , d. So by iterating this relation we get .−(U ∂x2j φ)θ = (−i∂xj + θj )2 (U φ)θ . We thus find for .H = − + V , using periodicity of V , that [U H φ]θ = [U (−φ) + U (V φ)]θ

.

=

d 

(−i∂xj + θj )2 (U φ)θ + V (U φ)θ = H (θ )(U φ)θ ,

(3.8)

j =1

with .H (θ ) =

d

j =1 (−i∂xj

+ θj )2 + V , which is (3.4).

 

Ballistic Transport in Periodic and Random Media

173

In the following, for .ψ ∈ L2 (Rd ), .x ∈ Rd and .f ∈ L2 (Td∗ ), .θ ∈ Td∗ , x m ψ(x) := (x1m ψ(x), . . . , xdm ψ(x)) and (∇θ f )m := ((∂θ1 f )m , . . . , (∂θd f )m ).

.

Theorem 3.1 Assume .ψ ∈ H 2m (Rd ), .ψ = 0 satisfies .x m ψ < ∞. Then .

x m e−itH ψ2 = t→+∞ t 2m

ˆ

lim

where .|(∇θ En (θ ))m |2 =

∞  |(∇θ En (θ ))m |2 Pn (θ )(U ψ)θ 2L2 (Td ) dθ > 0,

Td∗ n=1

d

j =1 (∂θj En (θ ))

2m .

This theorem was previously obtained for .m = 1 by J. Asch and A. Knauf in [5]. In contrast to [5] we will not consider temporal means of moment derivatives in the proof, as their rigorous manipulation becomes overly complicated for higher m. The extension to potentials periodic with respect to a different lattice than d .(2π Z) is quite immediate once the lattice vocabulary has been settled, we prefer to postpone this to the next subsection where it is a less standard material. Proof The proof is divided into three steps. Step 2 is in some sense the main argument, which is quite simple, while Steps 1 and 3 are technical justifications. In Step 1 we show it suffices to prove relation (3.10) below. In Step 2 we prove this relation for .ψ in some spectral subspace, then we consider general .ψ in Step 3. Step 1 For a linear operator O, define the propagator sandwich O(t) := eitH Oe−itH

(3.9)

.

suggested by the Heisenberg picture. Below we take for O the operator of multiplication by .x m . Since .x m ψ < ∞, we have .x m e−itH ψ ∈ L2 (Rd ) for all .t ≥ 0 by Theorem A.4, so we may apply U to it. We claim Theorem 3.1 will follow if we prove the strong convergence ∞ ˆ ⊕ 

U x m (t)ψ m . lim = (∇ E (θ )) P (θ )dθ U ψ, (3.10) θ n n t→+∞ tm Td∗ n=1

for since .eitH and U are unitary, we then get by definitions (3.1) and (3.2) that  ˆ ⊕  2 ∞

  x m e−itH ψ2 U x m (t)ψ2 m  . = −→  (∇θ En (θ )) Pn (θ )dθ U ψ   2m 2m d t t T∗ n=1

ˆ =

Td∗

ˆ =

∞  2   (∇θ En (θ ))m Pn (θ )(U ψ)θ  2  n=1 ∞ 

Td∗ n=1

since the .Pn (θ ) are orthogonal projections.

L (Td )

dθ

|(∇θ En (θ ))m |2 Pn (θ )(U ψ)θ 2L2 (Td )dθ

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A. Boutet de Monvel and M. Sabri

This is indeed .> 0: if .S ⊂ Td∗ is the intersection over n of all sets .Sn considered in (3.5), each of them being of full measure, S also has full measure, and restricting the integral over S, we see that it is zero iff ∞  .

|(∇θ En (θ ))m |2 Pn (θ )(U ψ)θ 2L2 (Td ) = 0 for a.e. θ ∈ Td∗ ,

n=1

which occurs on S iff .Pn (θ )(U ψ)θ 2L2 (Td ) = 0 for all n, a.e. .θ ∈ Td∗ , and so .U ψ = 0 and .ψ = 0, in contradiction with our assumption on .ψ.   Step 2 Now by (3.3), 

(U x m φ)θ (x) =

.

e−iθ·(x+2π n) (x + 2π n)m φ(x + 2π n) = im ∇θm (U φ)θ (x),

n∈Zd

where .∇θm f = (∂θm1 f, . . . , ∂θmd )f . Recalling (3.8), we thus have (U x m (t)ψ)θ = im eitH (θ) ∇θm e−itH (θ) (U ψ)θ ,

.

Assume for now that .ψ satisfies .(U ψ)θ = independent of .θ , i.e.,

N

n=1 Pn (θ )(U ψ)θ

for some N

Pn (θ )(U ψ)θ = 0 for all n > N.

(3.11)

.

p

q

p

q

p

q

Then, denoting .∇θ f ∇θ g = (∂θ1 f ∂θ1 g, . . . , ∂θd f ∂θd g), we have ∇θm e−itH (θ) (U ψ)θ =

N 

.

∇θm [e−itEn (θ) Pn (θ )(U ψ)θ ]

n=1

=

m   N   m n=1 p=0

=

N 

p

(∇θ e−itEn (θ) )∇θ p

m−p

[Pn (θ )(U ψ)θ ]

e−itEn (θ) (−it∇θ En (θ ))m Pn (θ )(U ψ)θ + On,θ (t m−1 )

n=1

for some error term .On,θ (t m−1 ) which can be made explicit. By dominated convergence,

.

ˆ  N 2  1 m itH (θ)   i e On,θ (t m−1 ) dθ = 0.  m t→+∞ Td t ∗ lim

n=1

(3.12)

Ballistic Transport in Periodic and Random Media

175

Indeed, the error term consists of derivatives of .En (θ ), .Pn (θ ) and .(U ψ)θ of order ≤ m, which are all bounded. In fact, .∇θk (U ψ)θ (x) = (−i)k (U x k ψ)θ and the .En (θ ) and .Pn (θ ) are analytic outside nullsets .XN , .ZN of lower dimension [34], so their potential singularities can be ignored in the integrals by avoiding an .ε-neighborhood of .XN ∪ ZN and taking .ε to zero at the end. On the other hand,

.

im eitH (θ)

.

N  (−it∇θ En (θ ))m e−itEn (θ) Pn (θ )(U ψ)θ n=1

= tm

∞ 

eitEk (θ) Pk (θ )

k=1

= tm

∞ 

N  (∇θ En (θ ))m e−itEn (θ) Pn (θ )(U ψ)θ n=1

(∇θ Ek (θ ))m Pk (θ )(U ψ)θ

k=1

since .Pk (θ )Pn (θ ) = δk,n Pk (θ ) and .Pk (θ )(U ψ)θ = 0 for .k > N. We thus showed that .

ˆ  ∞ 2 m   (U x (t)ψ)θ  m − (∇ E (θ )) P (θ )(U ψ)  θ k k θ  dθ = 0 , m t→+∞ Td t ∗ lim

(3.13)

k=1

completing the proof in the case where .ψ satisfies (3.11) for some N .

 

Step 3 It remains to justify why we can reduce the general case to the particular case where .ψ satisfies (3.11), i.e., .Pk (θ )(U ψ)θ = 0 for all .k > N. Let ψN := U −1

ˆ

N ⊕

.

Aψ := U

−1

Td∗ n=1

ˆ

∞ ⊕

Td∗ n=1

 Pn (θ )dθ U ψ,  (∇θ En (θ )) Pn (θ )dθ U ψ. m

Then   x m (t)ψ x m (t)ψ   x m (t)ψ   x m (t)ψ N N      ≤ −Aψ −Aψ −  +  +A(ψN −ψ).   N tm tm tm tm

.

We showed in Step 2 that, N being fixed, the first norm vanishes as .t → +∞. If we show that the two other norms vanish as .t → +∞ followed by .N → ∞, then the proof will be complete.

176

A. Boutet de Monvel and M. Sabri

By Theorem A.4 we know that  x m (ψN − ψ) x m (t)(ψN − ψ) ≤ c + C H k (ψN − ψ). m m tm tm m

.

k=0

N being fixed, the first term in the right-hand side vanishes as .t → +∞. In fact for ψ, by assumption, we indeed have .x m ψ ∈ L2 (Rd ), and for .ψN :

.

ˆ  N 2  m  .x ψN  = Pn (θ )(U ψ)θ  dθ ∇θ 2

m

Td∗

n=1

ˆ  m   N  2 m   m−p = (∇θ Pn (θ ))(U x p ψ)θ  dθ,  d p T∗ n=1 p=0

which is finite for any N . Next, by definition, (U ψN )θ − (U ψ)θ =

N 

.

∞ 

Pn (θ )(U ψ)θ − (U ψ)θ = −

Pn (θ )(U ψ)θ .

n=N +1

n=1

Using (3.8), i.e., .(U H k φ)θ = H k (θ )(U φ)θ , we thus have ˆ  ∞ 2    k .H (ψN −ψ) = U H (ψN −ψ) = Pn (θ )(U ψ)θ  2 H (θ ) 2

k

k

2

Td∗

n=N +1

L (Td )

dθ.

But, for any .k ≤ m, ∞  .

Pn (θ )H (θ )k (U ψ)θ 2L2 (Td ) = H (θ )k (U ψ)θ 2L2 (Td )

n=1

≤ c"" (U ψ)θ 2H 2r (Td ) < ∞. Hence, .H k (ψN − ψ)2 vanishes as .N → ∞ by dominated convergence. This x m (t)(ψ−ψN )    completes the proof that the second norm . vanishes as .t → +∞ tm followed by .N → ∞. Finally ˆ Aφ = 2

.

ˆ =

∞    (∇θ En (θ ))m Pn (θ )(U φ)θ 2 2

Td∗ n=1

L (Td )

dθ

∞    (∇θ En (θ ))m Pn (θ )(H (θ ) + i)−m/2 Pn (θ )(U (H + i)m/2 φ)θ 2 2 d dθ, L (T )

Td∗ n=1

Ballistic Transport in Periodic and Random Media

177

where we used (3.8) and the fact that .H (θ ) commutes with .Pn (θ ). This can now be bounded by ˆ

∞ 

.

Td∗ n=1

(∇θ En (θ ))m Pn (θ )(H (θ ) + i)−m/2 2L2 →L2

× Pn (θ )(U (H + i)m/2 φ)θ 2L2 (Td ) dθ .

(3.14)

Now (∇θ En (θ ))m Pn (θ )(H (θ ) + i)−m/2 = (∇θ En (θ ))m Pn (θ )

.

∞  (Ek (θ ) + i)−m/2 Pk (θ ) k=1

=

(θ ))m

(∇θ En Pn (θ ). (En (θ ) + i)m/2

So for .g ∈ L2 (Td ) we have  (∇ E (θ )) m θ n   (∇θ En (θ ))m Pn (θ )(H (θ ) + i)−m/2 g =   Pn (θ )g. (En (θ ) + i)1/2

.

(3.15)

It thus suffices to estimate this norm for .m = 1. For this, we observe that (∇θ En (θ ))Pn (θ ) = 2Pn (θ )(D + θ )Pn (θ ).

.

(3.16)

This is based on .En (θ )Pn (θ ) = H (θ )Pn (θ ) and .∇θ H (θ ) = ∇θ ((D + θ )2 + V ) = 2(D + θ ). More precisely, on the one hand, we have ∇θ (Pn H (θ )Pn ) = (∇θ Pn )H (θ )Pn + Pn (∇θ H (θ ))Pn + Pn H (θ )∇θ Pn

.

= Pn (∇θ H (θ ))Pn + En ∇θ (Pn Pn ) = Pn (∇θ H (θ ))Pn + En ∇θ Pn . On the other hand, ∇θ (Pn H (θ )Pn ) = ∇θ (En Pn ) = (∇θ En )Pn + En ∇θ Pn .

.

Thus, .(∇θ En )Pn = Pn (∇θ H (θ ))Pn = 2Pn (D + θ )Pn as stated2 in (3.16).

fact that .∇θ H (θ)g = 2(D + θ)g for any .g ∈ L2 (Td ) is clear by definition of the derivative. Computing .∇θ e−itH (θ ) however is less clear. This is why we used the spectral decomposition of .H (θ) in Step 2 to estimate it. 2 The

178

A. Boutet de Monvel and M. Sabri

We next note that, since .V ≥ −b for .b = V ∞ , then (D + θ )f 2 = f, (D + θ )2 f  = f, H (θ )f  − f, Vf  ≤ (H (θ ) + b)1/2 f 2 .

.

We thus have, using (3.16), (∇θ En (θ ))Pn (θ )(H (θ ) + i)−1/2 g = 2Pn (θ )(D + θ )Pn (θ )(H (θ ) + i)−1/2 g

.

≤ 2(H (θ ) + b)1/2 (H (θ ) + i)−1/2 Pn (θ )g ≤ cPn (θ )gL2   θ En (θ)  with c independent of .θ . Recalling (3.15), this says that . (E∇(θ)+i) 1/2 ≤ c. By (3.15), n

this implies .(∇θ En (θ ))m Pn (θ )(H (θ ) + i)−m/2 L2 →L2 ≤ cm . Back to (3.14), we thus showed that ˆ Aφ ≤ c

.

2

∞ 

2m

Td∗ n=1

Pn (θ )(U (H + i)m/2 φ)θ 2L2 (Td ) dθ

ˆ = c2m

Td∗

(U (H + i)m/2 φ)θ 2L2 (Td ) dθ

= c2m (H + i)m/2 φ2L2 (Rd ) . Thus, .A(ψN − ψ) ≤ cm ψ − ψN , (H + i)m (ψ − ψN )1/2 . Since we previously showed that .H k (ψN − ψ) → 0 for any .k ≤ m, we get that the third norm .A(ψN − ψ) vanishes as .N → ∞. This completes the proof.   Example 3.2 (The Laplacian) For .V = 0, i.e., .H = − we have .Ek (θ ) = d (k + θj )2 , corresponding to the eigenvectors .ek (x) = (2π1)d/2 eik·x , .k ∈ Zd , j j =1 and .Pk (θ )f = ek , f ek . So the limit reduces to ˆ



.

Td∗

ˆ |2 (k + θ ) | |ek , (U ψ)θ | dθ = 4 m

m 2

2

m

k∈Zd

Td∗

ˆ = 4m

Td∗



|ek , (D + θ )m (U ψ)θ |2 dθ

k∈Zd

(U D m ψ)θ 2L2 (Td ) dθ

= 4mD m ψ2 . This agrees with Theorem 2.3 where this special case was already considered.

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179

3.2 Periodic Discrete Graphs We now consider periodic Schrödinger operators on discrete graphs . ⊂ Rd .

3.2.1

Preliminaries

The spectral theory has been considered in great generality in [21] (and earlier in [23] in the special case of .Zd ). The infinite graph . = (V , E) is assumed to be locally finite and invariant under translations by some fixed basis .{a1 , . . . , ad } in d .R . This includes the hexagonal lattice, the triangular lattice, the body-centered and face-centered cubic lattices, and much more. The authors in [21] take the convention of writing the vertices in this coordinate system. This shortens the notation compared to say [29, pp. 303–309] but it can also be quite confusing and nonstandard. As a compromise we introduce notations as follows. • Given .x = (x1 , . . . , xd ) ∈ Rd , we let .xa = x1 a1 + · · · + xd ad . • Let .Zda := {n1 a1 + · · · + nd ad : nj ∈ Z} = {na : n ∈ Zd }. • Let .b1 , . . . , bd be the dual basis, that is, .ai · bj = 2π δi,j . We similarly denote .xb = x1 b1 + · · · + xd bd . • Let .Ca = {x1 a1 + · · · + xd ad : xj ∈ [0, 1)} and .Cb = {x1 b1 + · · · + xd bd : xj ∈ [0, 1)} be the unit cells in the respective bases. d • Let .Td∗ := [0, 1)d . Consider .ρ : Td∗ → Cb , .ρ(θ ) = θb = i=1 θi bi . We endow .Cb with the image measure .dρ of the Lebesgue measure. In particular, ´ ´ . Cb f dρ = Td∗ f (θb ) dθ . For example, if .ai = ei is the standard basis, then .bi = 2π ei , .xa =x, .θb = 2π θ , dθ Ca = [0, 1)d , .Cb = [0, 2π )d and .dρ = (2π . In general, if .bi = dj =1 βj (i)ej , )d

.

dθ then .ρ(θ ) = Bθ , where B is the matrix .(βi (j ))di,j =1 and we get .dρ = | det B| , see e.g. [33, Appendix A.7]. We let .Vf = V ∩ Ca be the unit crystal, .Vf = {v1 , . . . , vν }. Since . + ak = , we have

V = Zda + Vf .

.

(3.17)

At the level of edges, such additive invariance implies that .Nna +vn = na + Nvn , i.e., .{u : u ∼ na + vn } = {na + v : v ∼ vn }. Recall (3.17). If .v ∈ V , we shall denote v = va + {v}a ,

.

where .va ∈ Zda and .{v}a ∈ Vf are the integer and fractional parts of v, respectively.

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3.2.2

A. Boutet de Monvel and M. Sabri

The Schrödinger Operator

We now consider the Schrödinger operator H =A+Q

.

on ., where Q is a periodic potential satisfying .Q(v + ai ) = Q(v). In particular, Q has at most .ν distinct values; those on .Vf . One could also consider . + Q, where . = d − A and d is the degree matrix; this is in fact the framework in [21] and d can just be regarded as a periodic ´ ⊕ potential. We define .U : 2 () → Cb 2 (Vf ) dρ by (U ψ)θb (vn ) =



.

e−iθb ·(vn +ka ) ψ(vn + ka ) .

(3.18)

ka ∈Zda

Then U is unitary: ˆ U ψ2 =

.

=

Cb

(U ψ)θb 22 (V ) dρ



f

ˆ



ψ(vn + ka )ψ(vn + na )

vn ∈Vf ka ,na ∈Zda

=

 

Td∗

eiθb ·(na −ka ) dθ

|ψ(ka + vn )|2 = ψ2 ,

vn ∈Vf ka ∈Zda

 where we used that .θb · xa = i,j θi bi · xj aj = 2π θ · x. This shows isometry. ´⊕ 2 ´ Given .g ∈ Cb  (Vf ) dρ , the function .ψ(vn + ka ) = Td∗ eiφb ·vn gφb (vn )eika ·φb dφ satisfies .(U ψ)θb (vn ) = gθb (vn ). This can be seen by expanding .θ → eiθb ·vn gθb (vn ) in the orthonormal basis .{e−iθb ·ka }ka ∈Zda of .L2 (Td∗ ). Thus, U is unitary as asserted. ´⊕ We claim that .U H U −1 = Cb H (θb )dρ , where .H (θb ) acts on .2 (Vf ) by3 [H (θb )f ](vn ) =



.

eiθb ·(u−vn ) f ({u}a ) + Q(vn )f (vn ),

(3.19)

u∼vn

where the sum runs over the neighbors u of .vn in the full graph . (not just in .Vf ).

3 Our

operators U and .H (θb ) differ slightly from those of [21]. Namely, they consider (θb ) = = eiθb ·vn (U ψ)θb (vn ), so they obtain instead the fiber operator .H (θb ) are unitarily eiθb · H (θb )e−iθb · , where .(e±iθb · f )(vk ) = e±iθb ·vk f (vk ). Since .H (θb ) and .H n (θb ) = eiθb · Pn (θb )e−iθb · . We equivalent, they share the same eigenvalues .En (θb ), moreover .P have avoided the introduction of “bridges” and quotient graphs and used fractional parts instead, which we think is more transparent for our purposes.

ψ)θb (vn ) .(U

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181

In fact, given .ψ ∈ 2 (), (U Aψ)θb (vn ) =



.

e−iθb ·(vn +ka )





eiθb ·(u−vn )

u∼vn

=



ψ(u) =

u∼ka +vn

ka ∈Zda

=





ka ∈Zda

e−iθb ·(vn +ka )



ψ(ka + u)

u∼vn

e−iθb ·({u}a +ua +ka ) ψ({u}a + ua + ka )

ka ∈Zda

eiθb ·(u−vn ) (U ψ)θb ({u}a ).

u∼vn

Since Q is .Zda -periodic, if follows that .(U H ψ)θb (vn ) = H (θb )(U ψ)θb (vn ) as asserted. We assume . is not disconnected as a direct sum of infinitely many copies of .(Vf , E(Vf )). That is, in (3.19), at least one .vn has some .u ∼ vn such that .u ∈ / Vf (and thus .ua = 0). For the purposes of ballistic transport, we regard .H (θb ) as a complex .ν × ν matrix (indeed .2 (Vf ) ≡ Cν ) and observe  that .θj → H (θb ) is analytic for ν any fixed .(θk )k=j . We may write .H (θb ) = n=1 En (θb )Pn (θb ), where .En (θb ) are the eigenvalues (in increasing order, counting multiplicities) and .Pn (θb ) the eigenprojections. The authors in [21] then use Floquet theory to deduce that the spectrum of H consists of at most .ν bands (intervals), .σ (H ) = σac (H ) ∪ σfb (H ), where .σac (H ) are the bands of AC spectrum, while .σfb (H ) is the set of flat bands, i.e., intervals degenerate to a single point, which can occur in general. For example, for the Laplacian there are no flat bands in the case of the triangular lattice, while there is a flat band in the case of the face-centered cubic lattice [21, Proposition 8.3]. Theorem 3.3 Let .H = A + Q, with Q periodic on the periodic discrete infinite graph .. Assume .ψ ∈ 2 () satisfies .x m ψ < ∞. Then .

x m e−itH ψ2 = t→+∞ t 2m lim

ˆ

ν  2

  ∇θa En (θb ) m  Pn (θb )(U ψ)θb  ν dθ,  C 2π Td∗ n=1

(3.20)  where .∇θa := di=1 ai ∂θi . The limit is nonzero if .ψ has a nontrivial AC component. The limit is zero if .ψ is an eigenvector. dThe quantity .2π did not appear in Theorem 3.1 because we considered .∇θ = i=1 ei ∂θi , while in that result .ai = 2π ei , so .∇θa = 2π ∇θ . There, .θb = θ . It can indeed happen that .ψ is an eigenvector (corresponding to a flat band, as in face-centered cubic lattices). On the other hand, H always has at least one band of AC spectrum, see [22, Theorem 2.1].

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Proof We have .x m (t)ψ ∈ 2 () by Theorem A.1. As in Theorem 3.1, it suffices to show that ν ˆ ⊕ 

∇θa En (θb ) m U x m (t)ψ = P (θ )dθ U ψ. . lim n b t→+∞ tm 2π Td∗

(3.21)

n=1

 Recall (3.18). We note that if .vn = dj =1 sj aj , .sj ∈ [0, 1), then .θb · (ka + vn ) = d −iθb ·(ka +vn ) = −2π i(k + i i,j =1 θi bi · (kj + sj )aj = 2π θ · (k + s). Hence, .∂θi e −iθ ·(k +v ) −iθ ·(k +v ) −iθ a n a n si )e b and .∇θa e b = −2π i(ka + vn )e b ·(ka +vn ) . We thus get (U x m φ)θb (vn ) =



.

e−iθb ·(ka +vn ) (ka + vn )m φ(ka + vn )

ka ∈Zda

=

im ∇ m (U φ)θb (vn ) . (2π )m θa

(3.22)

Hence, (U x m (t)ψ)θb =

.

im itH (θb ) m −itH (θb ) e ∇θa e (U ψ)θb . (2π )m

On the other hand, ∇θma e−itH (θb ) (U ψ)θb =

ν 

.

∇θma [e−itEn (θb ) Pn (θb )(U ψ)θ ]

n=1

=

ν 

e−itEn (θb ) (−it∇θa En (θb ))m Pn (θb )(U ψ)θ + On,θ (t m−1 )

n=1

(3.23) for some error term .On,θ (t m−1 ) which can be made explicit. From here we conclude the proof of (3.20) as in (3.12)–(3.13). Here it is even easier because the technical Step 3 is no longer needed, as the sums are already finite. It remains to prove the last two claims. Suppose first that .ψ has a nontrivial AC component. Since .σ (H ) = ∪νn=1 σn , where .σn = En (Cb ) = [En− , En+ ] are the spectral bands, then we have .1B ψ = 0 for some .B ⊂ σn of positive measure, where .σn is non-degenerate. This implies that .En is a non-constant piecewise analytic function [21], so .∇θa En (θb ) = 0 a.e. If B lies in the intersection of several bands .σk , we ignore the degenerate ones as they are just finitely many points in B, as for the non-degenerate ones, they also have .∇θa Ek (θb ) = 0 a.e.

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183

If the RHS of (3.20) is zero, then 2      ∇θa En (θb ) m   Pn (θb )(U ψ)θb    ν dθ = 0, 2π

ˆ .

Td∗ E (θ )∈B n b

C

 2

  ∇ E (θ ) m  so . En (θb )∈B  θa 2πn b Pn (θb )(U ψ)θb  ν = 0 a.e., so C



Pn (θb )(U ψ)θb 2Cν = 0 a.e.

.

En (θb )∈B

by the previous paragraph, and so .1B (H (θb ))(U ψ)θb 2 = 0 a.e. But .1B (H )ψ = 0 ´⊕ by hypothesis. If .H " = Cb H (θb ) dρ , we have 1B (H )ψ2 = U 1B (H )ψ2 = 1B (H " )U ψ2 ˆ = (1B (H " )U ψ)(θb )2 dρ

.

Cb

ˆ =

Cb

1B (H (θb ))(U ψ)θb 2 dρ .

We see that .1B (H (θb ))(U ψ)θb 2 = 0 a.e. would imply .1B (H )ψ2 = 0, a contradiction. Thus, the RHS of (3.20) is nonzero. This checks the claim for .ψ with a nontrivial AC component. Now let .ψ be an eigenvector. Clearly then .

x m e−itH ψ2 x m e−itλ ψ2 x m ψ2 = = → 0 as t → +∞. 2m 2m t t t 2m

In that case the RHS of (3.20) is also zero because such an eigenvector corresponds to some flat band .σfb (H ) = Er (Cb ), i.e. .Er (θb ) = λ is constant in .θ , so .∇θa Er (θb ) is identically zero. On the other hand (U ψ)θb = (U 1{λ} (H )ψ)θb = [1{λ} (H " )(U ψ)]θb

.

= 1{Er (θb )} (H (θb ))(U ψ)θb = Pr (θb )(U ψ)θb . Since .Pn (θb )Pr (θb ) = δn,r Pr (θb ), the RHS of (3.20) reduces to 2  ˆ   ∇θa Er (θb ) m   Pr (θb )(U ψ)θb  .   ν dθ = 0. 2π Td∗ C  

184

A. Boutet de Monvel and M. Sabri

Example 3.4 (The Integer Lattice) In the case of the adjacency matrix .A on .Zd , we  iθ ·u have .Vf = {0}, .H (θb ) = u∼0 e b = 2 dj =1 cos 2π θj , only one eigenvalue d .E1 (θb ) = 2 j =1 cos 2π θj , which traces the spectrum .[−2d, 2d] as .θ varies over d .T∗ , and .P1 (θb ) = Id. So Theorem 3.3 asserts that nm e−itH ψ2 . lim = t→+∞ t 2m

2    ˆ   ∇E1 (θb ) m 2   −ik·θ  b    e ψ(k) dθ    2π Td∗ d

= 4m

k∈Z

ˆ  d Td

 2m ˆ )|2 dθ. sin θj |ψ(θ

j =1

This expression is the same as (2.6), so we recover Theorem 2.1. Example 3.5 (The Triangular Lattice) The triangular lattice has .V = Z2 , but with two additional links: Aψ(n) = ψ(n − e1 ) + ψ(n + e1 ) + ψ(n − e2 ) + ψ(n + e2 ) + ψ(n − e1 − e2 )

.

+ ψ(n + e1 + e2 ), where .(e1,2 ) is the standard basis,4 see [21, Fig. 3]. This is a 6-regular graph. Here  iθb ·u , there is one eigenvalue .E (2π θ ) = 2 cos 2π θ + .Vf = {0}, .H (θb ) = 1 1 u∼0 e 2 cos 2π θ2 +2 cos(2π(θ1 +θ2 )) which traces the interval .[−3, 6] = σ (A) = σac (A) as .θ varies over .T2∗ . Again .P1 (2π θ ) = Id. So Theorem 3.3 asserts that .

x m e−itH ψ2 = t→+∞ t 2m ˆ   m ˆ )|2 dθ. (sin θ1 + sin(θ1 + θ2 ))2m + (sin θ2 + sin(θ1 + θ2 ))2m |ψ(θ 4 lim

T2

Example 3.6 (The Hexagonal Lattice) The hexagonal (honeycomb/graphene) lat√ tice is 3-regular, see [21, Fig. 7]. Here .a1 = a(1, 0), .a2 = a2 (1, 3), where a is the distance between a vertex and its second nearest neighbor. We have .Vf = {0, v}, 2 2 where .v = a2 (1, √1 ) = a1 +a 3 , .V = Za + Vf , and 3

Aψ(ka ) = ψ(ka + v) + ψ(ka + v − a1 ) + ψ(ka + v − a2 ), .

Aψ(ka + v) = ψ(ka ) + ψ(ka + a1 ) + ψ(ka + a2 ).

4 Some authors replace the last two terms by .ψ(n − e

(3.24)

1 + e2 ) + ψ(n + e1 − e2 ), this is just a different shearing convention and slightly changes the eigenvalue .E1 (θ).

Ballistic Transport in Periodic and Random Media

185

 iθb ·u f ({u} ). Using (3.24), this yields By (3.19), we have .H (θb )f (0) = a u∼0 e iθ ·v iθ ·(v−a ) 1 f (v) + eiθb ·(v−a2 ) f (v). Similarly, we have .H (θb )f (0) = e b f (v) + e b H (θb )f (v) =



.

eiθb ·(u−v) f ({u}a ) = e−iθb ·v f (0)

u∼v

+ eiθb ·(a1 −v) f (0) + eiθb ·(a2 −v) f (0). This shows that   0 eiθb ·v ξ(θb ) , where ξ(θb ) := 1 + e−iθb ·a1 + e−iθb ·a2 . .H (θb ) = 0 e−iθb ·v ξ(θb ) −2π iθ1 + e−2π iθ2 . The eigenvalues Since .θb · ka = 2π θ · k, this √gives .ξ(θb ) = 1 + e are given by .±|ξ(θb )| = ± 3 + 2 cos 2π θ1 + 2 cos 2π θ2 + 2 cos 2π(θ1 − θ2 ). So

.

∓(sin 2π θ1 + sin 2π(θ1 − θ2 )) ∇θa E± (θb ) = a1 2π |ξ(θb )| +

∓(sin 2π θ2 + sin 2π(θ2 − θ1 )) a2 . |ξ(θb )|

Since we have .|(∇θa E+ (θb ))m |2 = |(∇θa E− (θb ))m |2 , the definitions of .P± (θb ) do not matter as we use .P1 (θb )(U ψ)θb 2 + P2 (θb )(U ψ)θb 2 = (U ψ)θb 2 . We conclude that .

x m e−itH ψ2 = t→+∞ t 2m √ ˆ 3 2m 1 −θ2 ) 2m ) + ( a 2m [(sin θ1 + sin θ2 +sin(θ 2 2 (sin θ2 + sin(θ2 − θ1 ))) ] |1 + eiθ1 + eiθ2 |2m T2    2    2 dθ    −ika ·θb −i(ka +v)·θb ×  e ψ(ka ) +  e ψ ka + v)  . (2π )2 2 2 lim

ka ∈Za

ka ∈Za

Remark 3.7 While revising the paper, we discovered the recent work [12] which considers ballistic transport for periodic Jacobi matrices. This corresponds to . = Zd , .m = 1 and adjacency matrices carrying periodic weights (besides the periodic potential).

186

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4 Universal Covers 4.1 Background Let .T be a tree, that is, a connected graph with no cycles. Consider a Schrödinger operator H =A+W

.

on .T, where .A is the adjacency matrix of the tree and W a potential on .T. We start by recalling some properties of the Green function of H . Given .γ ∈ C+ := {z ∈ C : Im z > 0} and .(v, w) a directed edge in .T, let Gγ (v, w) := δv , (H − γ )−1 δw 

.

denote the Green function of H and ζvγ (w) :=

.

Gγ (v, w) . Gγ (v, v)

The following identities are then classical (see, e.g., [3, Section 2]): .

 1 = W (v) + ζvγ (u) − γ , Gγ (v, v) u∼v

 −1 = W (v) + ζvγ (u) − γ , . γ ζw (v) u∈N \{w} v

(4.1) 1 γ ζw (v)

− ζvγ (w) =

−1 Gγ (v, v)

ζwγ (v) =

,

Gγ (v, v) Gγ (w, w)

ζvγ (w), . (4.2)

G (v0 , vk ) = G γ

γ

(v0 , v0 )ζvγ0 (v1 ) · · · ζvγk−1 (vk ),

G (v, w) = G (w, v) γ

γ

(4.3) for any non-backtracking path .(v0 ; vk ) in a tree .T. γ γ The quantity .−ζv (w) can also be expressed as the Green function .G (w|v) (w, w) T γ γ of a subtree of .T. By the Herglotz property, we thus have .− Im ζv (w) = | Im ζv (w)| for any .(v, w). In particular, (4.1) implies the important relation  .

u∈Nv \{w}

γ

| Im ζvγ (u)| =

| Im ζw (u)| − Im γ . γ |ζw (v)|2

(4.4)

Ballistic Transport in Periodic and Random Media

187

4.2 Universal Covers Suppose .(G, W ) is any finite graph endowed with a potential W and let .(T, W) =  W  ) be its universal cover, where the potential W is lifted naturally by .W  (v) = (G, W (π v), if .π : T → G is the covering projection. For example, if G is d-regular, then .T is the infinite d-regular tree. We assume that the minimal degree of G is .≥ 2. The spectral theory of such trees has a rich history, see [4] and references therein. It is known in particular that the spectrum consists of bands of absolutely continuous spectrum, possibly with infinitely degenerate eigenvalues between the bands. Moreover, in the interior of the bands, all Green functions limits .Gλ (v, w) := λ+iη limη↓0 Gλ+iη (v, w) and .ζvλ (w) := limη↓0 ζv (w) exist, .Im Gλ (v, v) > 0 for any λ v, while .Im ζv (w) is strictly negative, that is .|Im ζvλ (w)| = − Im ζvλ (w) > 0 within the bands. We henceforth denote γ := λ + iη

.

with η > 0.

As in [26], we define the spectral quantities zγ :=

inf

.

(v,w)∈B(T)

|Im ζvγ (w)| and

zλ :=

inf

(v,w)∈B(T)

|Im ζvλ (w)|, γ

where the infimum is over the set .B(T) of directed edges of .T. Actually .ζv (w) = γ ζv " (w " ) if .π(v, w) = π(v " , w " ), so .zλ is just a minimum over the (lifts of) directed edges of G. In particular, .zλ > 0 within the bands. Back to the topic of ballistic transport, fix an origin .o ∈ T, let .|x| := d(x, o) and .(xψ)(x) := |x|ψ(x). In this section we shall consider averaged moments instead: ˆ x β ψ,η := 2η

.

∞

e−2ηt x β/2 e−itH ψ2 dt .

0

Let .Gz := (H − z)−1 . A well-known application of the Plancherel identity yields that ˆ η ∞ β/2 λ+iη 2 β .x ψ,η = x G ψ dt. (4.5) π −∞ Our aim is to find lower bounds for .lim inf ηβ x β ψ,η . Using Theorem A.1 and the η↓0

Tauberian theorem [32, Theorem 10.3], such a lower bound implies .

lim inf

1

a→+∞ a β+1

ˆ 0

a

x β/2 e−itH ψ2 dt > 0

and

lim sup t→+∞

x β/2 e−itH ψ2 > 0. tβ

188

A. Boutet de Monvel and M. Sabri

Theorem 4.1 We have .

lim inf η x δo ,η ≥ β

β

η↓0

ˆ

1 2β+1 π

β

σ (H )

zλ Im Gλ (o, o)dλ > 0.

Note For .β = 0 it follows from (4.5) that η0 x 0 ψ,η =

.

η π

ˆ

∞

−∞

Gλ+iη ψ2 dt =

1 π

ˆ

∞

−∞

Imψ, Gλ+iη ψ dt.

So the presence of .Im Gλ (o, o) above is quite natural, while the moment effect is β captured by .zλ . Proof We have ηβ+1



.



|v|β |Gλ+iη (o, v)|2 ≥ ηβ+1

|v|β |Gλ+iη (o, v)|2

|v|>bη−1

v

≥ ηβ+1 bβ η−β



|v|>bη−1

 = ηbβ Gλ+iη δo 2 − 

|Gλ+iη (o, v)|2 

|Gλ+iη (o, v)|2

|v|≤bη−1

= bβ Im Gλ+iη (o, o) − η





 |Gλ+iη (o, v)|2 ,

|v|≤bη−1

(4.6) ´ 1 where we used the spectral theorem .Gz (o, o) = σ (A) x−z dμo,o (x) in the last equality. Note that  .η |Gλ+iη (o, v)|2 ≤ ηGλ+iη δo 2 = Im Gλ+iη (o, o). |v|≤bη−1

This trivial estimate makes the lower bound (4.6) useless however, which is natural since it doesn’t use any delocalization of H . The argument below can be summarized as showing that b can be chosen so that η



.

|v|≤bη−1

|Gλ+iη (o, v)|2 ≤

1 Im Gλ+iη (o, o). 2

Ballistic Transport in Periodic and Random Media

189

Denoting .v0 := o, we have 

|G

λ+iη

.

(v0 , v)| = |G 2

λ+iη

(v0 , v0 )| + 2

M   

|Gλ+iη (v0 , vr )|2

r=1 v1 ∼v0 (v2 ;vr )

|v|≤M

where the last sum is over all nonbacktracking paths .(v2 ; vM ) of length .r − 2 “outgoing” from the directed edge .(v0 , v1 ), i.e., .v2 ∼ v1 and .v2 = v0 . Now, for .γ = λ + iη,   . |Gγ (v0 , vr )|2 = |Gγ (o, o)|2 |ζvγ0 (v1 ) · · · ζvγr−1 (vr )|2 (v2 ;vr )

(v2 ;vr ) γ |ζv (w)|2  γ |ζv0 (v1 ) · · · ζvγr−2 (vr−1 )|2 | Im ζvγr−1 (vr )| |Gγ (o, o)|2 max γ (v,w) | Im ζv (w)| (v2 ;vr )

≤

1

γ

≤ |Gγ (o, o)|2 | Im ζo (v1 )| max

| Im ζ γ 1(w) |

(v,w)

(4.7)

,

v

where we applied (4.4) .r − 1 times in the last step.  λ+iη By (4.1), .Im Gλ+iη (o, o) = ( u∼o | Im ζo (u)| + η)|Gλ+iη (o, o)|2 . Thus,  .

(v2 ;vr )

u∼o

λ+iη

| Im ζo (v1 )| λ+iη zλ+iη | Im ζo (u)| + η

Im Gλ+iη (o, o)

|Gλ+iη (v0 , vr )|2 ≤ 

where we estimated .Im ζ γ 1(w) using (4.1) and the definition of .zλ+iη . Thus, v



|Gλ+iη (o, v)|2 ≤ |Gλ+iη (o, o)|2 + M

.

|v|≤M

Im Gλ+iη (o, o) . zλ+iη

(4.8)

Take .M = bη−1 . It follows from (4.6) that  β+1 .η |v|β |Gλ+iη (o, v)|2 ≥ v

Choose .b =

η

.

β+1

 v

  Im Gλ+iη (o, o) bβ Im Gλ+iη (o, o) − b − η|Gλ+iη (o, o)|2 . zλ+iη zλ+iη 2 .

Then zλ+iη  β

|v| |G β

λ+iη

(o, v)| ≥ 2

2β+1

 Im Gλ+iη (o, o) − 2η|Gλ+iη (o, o)|2 . (4.9)

β

zλ

As .η ↓ 0, we get the lower bound . 2β+1

Im Gλ (o, o)

> 0.

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This estimate is true for any .λ in the union of intervals .∪j I˚j of AC spectrum, which is a set of positive measure. Using Fatou’s lemma, we thus get ˆ .

lim inf η

β+1

η↓0

∞



−∞ v

ˆ |v| |G β

λ+iη

(o, v)| dλ ≥ 2

∪j ˚ Ij

β

zλ

2β+1

Im Gλ (o, o) dλ > 0 .  

The claim follows by (4.5).

Remark 4.2 We may extend the result to functions .ψ of compact support, but the bound we get is not very good. Namely, if .ψ is supported in .Λ ⊂ T, we obtain .

lim inf ηβ x β ψ,η ≥ η↓0

1 2β+1 2 |Λ|β

ˆ σ (H )

β

zλ (Imψ, Gλ ψ)β+1 dλ, ψ, Im Gλ (·, ·)ψβ

where .[Im Gλ (·, ·)ψ](w) = Im Gλ (w, w)ψ(w). In particular, the bound becomes useless as the size of the support becomes infinite, which does not seem natural in view of our previous ballistic estimates. We thus omit the proof and think it a worthwhile question to obtain estimates for general .ψ, without .η-time averaging, and to find the exact limit.

4.3 The Anderson Model on Universal Covers We now study weak random perturbations of universal covering trees. The simplest example is the Anderson model on the .(q + 1)-regular tree, for which ballistic transport was previously established in the time-averaged sense by A. Klein [20], and by M. Aizenman and S. Warzel [1]. Here we study the general case. Assumptions Compared to the previous subsection, we need to assume the tree  has a bit nicer geometry. If .b ∈ B(T) we let .o(b) and .t (b) denote the origin T=G and terminus of b, respectively.

.

• We now assume the minimal degree is .≥ 3, • Since G is finite, there are finitely many isomorphism classes of .(T, b) as b runs over the directed edges of .T and .(T, b) is considered as a tree with “root” b. If .Nb := {b+ : o(b+ ) = t (b)} is the set of edges outgoing from b, then we assume that for each b, there is at least one .b+ ∈ Nb such that .(T, b) is isomorphic to .(T, b+ ).

Ballistic Transport in Periodic and Random Media

191

The second condition is perhaps better visualized using the language of cone types, see [19, condition (M1*)] in which it was introduced. Explicit examples of such trees can be found in [4]. Concerning the random potential, we assume • Each vertex .v ∈ T is endowed with a random variable .W(v), the .(W(v))v∈T are i.i.d. with common distribution .ν of compact support. We also impose some regularity as in [1, assumption A2] for comfort, but perhaps this can be avoided. It is known [19] that under these conditions the random Schrödinger operator H = A + W inherits the purely absolutely continuous spectrum of .A almost surely if . is small enough—one of the few results of Anderson delocalization. The result of [19] can also be used to derive some inverse moments bounds on the imaginary parts of the Green functions, see [2, Theorem 5.2] for details (in a different model). More precisely, it can be shown that within the stable intervals I of pure AC spectrum, we also have

.

.

sup E(| Im ζo(b) (tb )|−s ) < ∞ , λ+iη

sup sup

(4.10)

λ∈I η∈(0,1) b∈B(T)

for .0 ≤ s ≤ 5, where .supb runs over the set .B(T) of directed edges of .T and it is in fact a maximum as it can be equivalently taken over the directed edges of G. Theorem 4.3 Under the previous assumptions, .

 lim inf ηβ E x β 1I (H )δo ,η ) > 0. η↓0

The proof also works for states .f (H )δo instead of .1I (H )δo , if f is piecewise continuous and supported in I . This result appeared before in [1, 20] in the special case .T = Tq , the .(q + 1)-regular tree. Proof The main argument is the same as in Theorem 4.1, but there are two technical difficulties. First, we are dealing with the state .1I (H )δo instead of .δo . Second, we cannot estimate the expectation of a maximum as in (4.7). Let us begin the proof. Let .f (H ) = 1I (H ). By (4.5), we have ηβ E(x β f (H )δo ,η ) =

.

ηβ+1 π

ˆ

∞



−∞ v∈T

|v|β E(|(f (H )Gλ+iη δo )(v)|2 ) dλ.

Essentially the same proof as [1, Lemma 2.1] shows that .f (H ) can be replaced by f (λ). More precisely, we get

.

ηβ E(x β 1I (H )δo ,η ) =

.

ηβ+1 π

ˆ 

|v|β E(|Gλ+iη (o, v)|2 ) dλ + o(η).

I v∈T

This settles the first difficulty mentioned above.

(4.11)

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Note We mention in passing that for the proof of [1, Lemma 2.1], one doesn’t really need a Wegner bound. Inequality [1, (2.7)] holds deterministically since η . π

ˆ

∞

−∞

|f (λ)| G 2

λ+iη

f 2∞ ϕ dλ ≤ π

ˆ

=

f 2∞ π

ˆ

∞

2

−∞

ˆ

σ (H )

( σ (H )

η dμϕ (x) dλ (x − λ)2 + η2

−π π − ) dμϕ (x) = f 2∞ ϕ2 . 2 2

Similarly, one can avoid the Wegner bound ´ ∞ in [1, (2.10)] by writing .Q(η) = ´ E σ (H ) Fη (x) dμϕ (x), where .Fη (x) = π1 −∞ |f (x) − f (λ)|2 (λ−x)η2 +η2 dλ, then use dominated convergence. In fact .Fη (x) → 0 by a well-known calculation, cf. [16, p. 6], for continuous f . The extension to piecewise continuous f is straighforward as long as .E(μϕ ({a})) = 0 for any a, which is weaker than a Wegner bound. Using Fatou’s lemma and (4.6), we see as before that the proof is now reduced to showing that . (v2 ;vr ) E(|Gγ (v0 , vr )|2 ) stays bounded independently of r and .η ∈ (0, 1). Let .γ := λ + iη. We have E(|Gγ (v0 ; vr )|2 ) = E(|ζvγ1 (v0 ) · · · ζvγr (vr−1 )|2 |Gγ (vr , vr )|2 )

.

but |Gγ (vr , vr )| ≤ 

.

u∼vr γ

γ

1 1 ≤ , γ γ | Im ζvr (u)| + η | Im ζvr (vr+1 )| γ

and .ζvr (vr+1 ) is independent of .ζv1 (v0 ) · · · ζvr (vr−1 ). Indeed, ζvγr (vr+1 ) = −GT(vr+1 |vr ) (vr+1 , vr+1 )

.

is the Green function of the connected component .T(vr+1 |vr ) containing .vr+1 obtained by removing the directed edge .(vr , vr+1 ) from .T, while γ γ .ζv1 (v0 ) · · · ζvr (vr−1 ) = −G (vr−1 |vr ) (v0 , vr−1 ) lives on a disjoint subtree (see [3, T Eq. (2.7)]). Thus, E(|Gγ (v0 ; vr )|2 ) ≤ E(|ζvγ1 (v0 ) · · · ζvγr (vr−1 )|2 )E(| Im ζvγr (vr+1 )|−1 ).

.

γ

E(| Im ζ (v

)|)

Again by independence, we may insert . E(| Im ζvγr (vr+1 )|) and get vr

r+1

E(| Im ζvr (vr+1 )|−1 ) . γ E(| Im ζvr (vr+1 )|) (4.12) γ

E(|Gγ (v0 ; vr )|2 ) ≤ E(|ζvγ1 (v0 ) · · · ζvγr (vr−1 )|2 | Im ζvγr (vr+1 )|)

.

Ballistic Transport in Periodic and Random Media E(| Im ζv0 (v1 )|−1 ) γ E(| Im ζv0 (v1 )|) γ

The second fraction is .

193

≤ E(| Im ζv0 (v1 )|−1 )2 ≤ c by (4.10). For γ

γ

the first term, now that we gained the .| Im ζvr (vr+1 )|, we would like to put back the .|Gγ (vr , vr )|2 we removed. More precisely, we would like to find some C independent of r such that E(|ζvγ1 (v0 ) · · · ζvγr (vr−1 )|2 | Im ζvγr (vr+1 )|)

.

≤ CE(|ζvγ1 (v0 ) · · · ζvγr (vr−1 )|2 | Im ζvγr (vr+1 )||Gγ (vr , vr )|2 ) . γ

γ

γ

E(|ζv1 (v0 )···ζvr (vr−1 )|2 | Im ζvr (vr+1 )|f ) . We should γ γ γ E(|ζv1 (v0 )···ζvr (vr−1 )|2 | Im ζvr (vr+1 )|) bound .E2 (|Gγ (vr , vr )|2 ) ≥ C− for some .C− independent of

Define .E2 (f ) :=

(4.13)

thus establish a

lower r and .γ . This is nontrivial, but can be done exactly like [1, pp. 8-10], using (4.10). γ γ Since .|ζv1 (v0 ) · · · ζvr (vr−1 )Gγ (vr , vr )|2 = |Gγ (v0 , vr )|2 , we have finally shown that E(|Gγ (v0 , vr )|2 ) ≤ cCE(|Gγ (v0 , vr )|2 | Im ζvγr (vr+1 )|) .

.

We may now apply (4.4) multiple times to get  .

E(|Gγ (v0 , vr )|2 | Im ζvγr (vr+1 )|) ≤ E(|Gγ (v0 , v0 )|2 | Im ζvγ0 (v1 )|)

(v2 ;vr )

 =E

Taking .b = ηβ+1



.

cC 2D ,

γ

Im Gγ (o, o)| Im ζo (v1 )|  γ u∼o | Im ζo (u)| + η

 ≤ E(Im Gγ (o, o)).

where .D is the maximal degree, we deduce as in (4.9) that

|v|β E(|Gλ+iη (o, v)|2 ) ≥

v

1 cC β E[Im Gλ+iη (o, o) − 2η|Gλ+iη (o, o)|2 ]. 2 2D

By (4.11), we thus have .

ˆ 1 cC β lim inf E[Im Gλ+iη (o, o)] dλ η↓0 2 2D I

π cC β π cC β = E[μδo (I )] = E(1I (H )δo 2 ) > 0. 2 2D 2 2D

lim inf ηβ E(x β 1I (H )δo ,η )≥ η↓0

194

A. Boutet de Monvel and M. Sabri

The last quantity is strictly positive by (4.10) and classical identities. Note indeed that γ

E(Im Gγ (o, o)) ≥ E(|Gγ (o, o)|2 | Im ζo (u)|) for any u ∼ o.

.

By Cauchy–Schwarz this is .≥

|Gγ (o, o)|−2 ≤ 2|ζo (u)|2 + γ

.

γ

Also, .E(| Im ζo (u)|1/2 ) ≥ E(Im Gγ (o, o)) ≥

.

γ

E(| Im ζo (u)|1/2 ) . E(|Gγ (o,o)|−2 )

But, by (4.1)–(4.2),

2 γ γ ≤ 2(| Im ζu (u" )|−2 + | Im ζu (o)|−2 ). γ |ζu (o)|2

1 . γ E(| Im ζo (u)|−1/2 )

Thus, using (4.10),

1 γ γ min E(| Im ζo (u)|−1/2 )−1 E(| Im ζu (u" )|−2 )−1 > 0. 4 u,u" ∼o

The proof is completed.

 

5 Epilogue: Limiting Distributions As the paper was being finalized, we came upon some articles on the topic of quantum walks and found interesting connections. The results we derive in this section are inspired by Grimmett et al. [14], Saigo and Sako [30], and Sako [31]. Quantum walks are discrete in nature, traditionally consisting of simple shift operators on .2 (Z) with additional degrees of freedom. Schrödinger evolutions −itH are a lot more complex than shifts, however the quantum walks have been .e recently generalized in [30, 31] and the theories now seem to share a good common ground. Suppose in either continuous or discrete model that .ψ = 1. Then .e−itH ψ = 1 as the operator is unitary, so the entries .|e−itH ψ(x)|2 define a probability density on .Rd (resp. G), and .x m e−itH ψ2 is just the 2m-moment of this measure. Our results thus imply that for certain models, all the (normalized, even) moments of this probability measure converge. This naturally evokes the method of moments, see [7, Theorem 30.2]. We shall here give a fuller probabilistic/quantum mechanical picture of the transport theory by discussing the limiting behavior of this measure. If H is a Schrödinger operator on a general graph and .Xt is a random vector with distribution −itH ψ(v)|2 , we first show that . Xt is always asymptotically confined in a compact .|e t set. Afterwards we compute exactly the limiting distribution of . Xt t for the periodic models of Sect. 3.

Ballistic Transport in Periodic and Random Media

195

5.1 General Facts Our first task is to properly modify the probability measure to incorporate the division by .t m . This is easy. On .Rd , if dμt (x) := |e−itH ψ(x)|2 dx,

.

ψ

(5.1)

and if .Q(t) : Rd → Rd is defined by .Q(t) (x) := x/t then 1 . E ψ (x m ) = t m μt

ˆ ˆ ˆ  (t) m ψ x m ψ ψ   Q (x) dμt (x) = dμt (x) = y m dQ(t)  μt (y), d d d t R R R

(t) ψ

where .Q μt is the image measure, see e.g. [33, Appendix A.7]. Thus, 1 m m . m E ψ (x ) = E (t) ψ (x ) and we have a statement about the convergence of t μt Q μt moments of the measure ψ

ψ

νt := Q(t)  μt .

.

(5.2)

This measure has a very natural meaning. Suppose .Xt is a random vector in .Rd ψ ψ with distribution .|e−itH ψ(x)|2 dx. Then for .B ⊂ Rd , we have .νt (B) = μt (tB) = ψ Xt Xt P(Xt ∈ tB) = P( t ∈ B). In other words, .νt is the distribution of . t . Note that ψ ψ ψ d d .νt is also a probability measure since .νt (R ) = μt (R ) = 1. ψ Similarly, on a graph .G ⊂ Rd , if .ψ is normalized, we define .μt (v) := ψ (t) ψ |e−itH ψ(v)|2 , then .Q(t) : G → Rd by .Q(t) (v) := v/t, and .νt := Q μt . Note ψ ψ d that .νt is a measure on .R while .μt is a measure on G. ψ The map .ψ → νt enjoys a nice form of uniform continuity: Lemma 5.1 For any measurable B we have ψ

ϕ

|νt (B) − νt (B)| ≤ (ψ + ϕ)ψ − ϕ.

.

Proof We write the proof on .Rd , the same works on graphs. We have ˆ  ψ ψ ϕ ϕ |νt (B) − νt (B)| = |μt (tB) − μt (tB)| = 

.

ˆ ≤

Rd



tB



  |e−itH ψ(x)|2 − |e−itH ϕ(x)|2 dx 

|e−itH ψ(x)| + |e−itH ϕ(x)| |e−itH ψ(x) − e−itH ϕ(x)| dx

≤ (e−itH ψ2 + e−itH ϕ2 )e−itH (ψ − ϕ)2 = (ψ2 + ϕ2 )ψ − ϕ2 .  

196

A. Boutet de Monvel and M. Sabri

To understand this measure further, we state a result which is always valid, regardless of the transport being ballistic or not. Proposition 5.2 Let .H = A + V be a Schrödinger operator on a countable graph G ⊂ Rd with maximal degree .≤ D. Assume .|x − y|2 ≤ L for any .x ∼ y in G, where 2 .| · |2 is the Euclidean norm. Assume the potential V is bounded. Suppose .ψ ∈  (G) ψ satisfies .ψ = 1. Let .ΛLD := [−LD, LD]d . Then . lim νt (ΛLD ) = 1. .

t→+∞

Physically, this means that if .Xt is a random vector with distribution |e−itH ψ(v)|2 , .v ∈ G, then .Xt /t is asymptotically confined in the compact set d .ΛLD , which is independent of V , regardless of the transport behavior. If .G = Z then .L = 1 and .D = 2d. A quantum walk analog of this result previously appeared in [31, Theorem 3.10] for “smooth” .ψ, with a different proof. .

Proof This follows from the moment upper bounds, Theorem A.1. Fix some .o ∈ G and denote .|x| := d(x, o) where .d( · , · ) is the graph distance. We first assume that m .x ψ < ∞ for all m. We then know from Remark A.3 that .

lim sup t→+∞

1  k −itH |x| |e ψ(x)|2 ≤ Dk tk

(5.3)

x∈G

for any k. To pass from the graph distance to the Euclidean distance .| · |2 , we note that given .x ∈ G, if .|x| = n,then there is a path .(v0 , . . . , vn ) such that .v0 = o and .vn = x. So .|x − v0 |2 ≤ n−1 j =0 |vj +1 − vj |2 ≤ nL by our assumptions. Thus, d d 2m 2m 2 2 m m .|x − v0 |2 ≤ L|x|. Since .|x | = j =1 xj ≤ ( j =1 xj ) = |x|2 ≤ (|x − v0 |2 + 2 2 |v0 |2 )2m , we get .|x m |2 ≤ (L|x| + |o|2 )2m . It now follows from (5.3) that .

lim sup Eν ψ (|x m |22 ) = lim sup t→+∞

t

t→+∞

= lim sup t→+∞

1 t 2m

Eμψ (|x m |22 ) t

1  t 2m

|x m |22 |e−itH ψ(x)|2 ≤ (LD)2m .

x∈G

Now let .K = ΛLD , .Kδ = [−LD − δ, LD + δ]d and .B ⊂ Kδc . Then .x ∈ B implies d 2m m 2 2m .|xi | > LD + δ for some i, which implies .|x | = j =1 xj > (LD + δ) . Thus, 2 ´ ψ m 2 m 2 2m ψ .E ψ (|x | ) ≥ 2 B |x |2 dνt (x) ≥ (LD + δ) νt (B). It follows that ν t

.

ψ

lim sup νt (B) ≤ t→+∞

LD 2m for all m. LD + δ

ψ

ψ

Taking .m → ∞ yields .limt→+∞ νt (B) = 0. As .δ is arbitrary and .νt is a ψ probability measure, this shows that .limt→+∞ νt (K) = 1. This completes the proof when .x m ψ < ∞ for all m.

Ballistic Transport in Periodic and Random Media

197

For general .ψ, we argue by approximation. In fact, if .ψN → ψ, .ψN has compact ψ ψ ψ ψ support, then .|νt (K) − 1| ≤ |νt N (K) − 1| + |νt (K) − νt N (K)|. Using Lemma 5.1 ψ then taking .t → +∞, we get .lim supt→+∞ |νt (K)−1| ≤ (ψN +ψ)ψ −ψN . Since .ψN  → ψ, the claim follows by taking .N → ∞.   Remark 5.3 One may be tempted to deduce the same result in case of the continuous Schrödinger operator with smooth potential, evoking Theorem A.4. This does not work however because the constant .Cm in Theorem A.4 depends on m, in contrast to the discrete case. This is not just an artifact, the result is wrong in general in the continuous case. In fact, for the simplest case .H = − on .Rd , we have by (2.7), ˆ ˆ   1   x 2 ψ ψ it 2 .νt (B) = μt (tB) = |e ψ(x)| dx =  dx φt (2t)d tB 2t tB ˆ t (y)|2 dy = |φ B/2 2

for any measurable .B ⊂ Rd , with .φt (y) := eiy /4t ψ(y). As .t → +∞ we have 2 .φt (y) → ψ(y) pointwise, so .φt → ψ in .L by dominated convergence, and 2 ˆ ˆ 2 → 0. We thus get for  t − ψ) .φt → ψ in .L by Parseval. In particular, .1B (φ d 2 d any measurable .B ⊂ R and .ψ ∈ L (R ), ˆ ψ 2 ˆ .νt (B) → |ψ(y)| dy. (5.4) B/2

This has no reason to vanish for B outside some compact region. For example, if 2 2 ψ ˆ = e−y /2 and .limt→+∞ νt (B) > 0 for any B of positive ψ(x) = e−x /2 , then .ψ(y) measure.

.

On a different note, (5.4) also implies the following result. ψ

Lemma 5.4 Consider .H = − on .Rd . Then for any .ψ ∈ L2 (Rd ), .ψ2 = 1, .νt ψ ψ ˆ x )|2 dx. converges weakly to the AC measure .ν∞ given by .dν∞ (x) = 2−d |ψ( 2 Proof This follows immediately from the portemanteau theorem. Xt . t

 

Lemma 5.4 says that if .Xt is a random vector with distribution .|eit ψ|2 dx then ˆ x )|2 dx. converges in distribution to a random vector Y with distribution .2−d |ψ( 2

5.2 Periodic Models We start with the discrete case. Given a periodic graph . as in Sect. 3.2, and a .Zda periodic Schrödinger operator H = A + Q,

.

198

A. Boutet de Monvel and M. Sabri

let .ψ ∈ 2 () satisfy .ψ = 1. Let . := Td∗ × {1, . . . , ν} and define .μψ on . by ˆ

ˆ f (θ, n) dμψ (θ, n) =

.



ν 

Td∗

f (θ, n)Pn (θb )(U ψ)θb 2 dθ.

(5.5)

n=1

This is a probability measure since ˆ μ () =

.

ψ

ν 

Td∗ n=1

Pn (θb )(U ψ)θb 2 dθ = U ψ2 = ψ2 = 1.

Next, consider the map .h :  → Rd given by .h(θ, n) := be the image measure on .Rd given by

1 2π ∇θa En (θb )

ψ

ν∞ := h μψ .

.

ψ

and let .ν∞

(5.6)

Theorem 5.5 Let . be a periodic discrete graph endowed with a periodic Schrödinger operator H . Let .Xt be a random vector with distribution .|e−itH ψ(v)|2 , ψ Xt d .v ∈ . Then . t converges to a random vector Y on .R with distribution .ν∞ given by (5.5)–(5.6). This agrees with Theorem 3.3, as Eν ψ (|x m |22 ) =

.

∞

d  i=1

Eν ψ (xi2m ) = ∞

d 

Eμψ (h2m i (θ, n))

i=1

is the RHS of (3.20). We prove this result using our analysis in Sect. 3 and the idea from [14] to use the Cramér-Wold device. Perhaps the theory in [31] could also be adapted to prove a theorem of this kind. Proof We may assume .x m ψ < ∞ for all m, see Remark 5.10. ψ We first note that .ν∞ is supported on a compact set since . is compact and h is continuous. In one dimension this implies that Y is characterized by its moments. In higher dimensions and lattices things are a bit more complicated. d m d (c) Denote .x = where .c = i=1 αi (x)ai and let .pm (x) := i=1 ci αi (x) (c) (c) (c1 , . . . , cd ). Suppose we showed that .Eν ψ (pm (x)) → Eν ψ (pm (x)) for any ∞ t m d () .m = 1, 2, . . . and c. Denoting .x = (x1 , . . . , xd ) and .qm (x) := , j =1 j xj d since .xj = x · ej = i=1 αi (x)ai · ej , we get ⎛ () .qm (x)

=⎝

d  i,j =1

⎞m j ai · ej αi (x)⎠ =

 d  i=1

m ci αi (x)

(c) = pm (x)

Ballistic Transport in Periodic and Random Media

for .ci :=

d

j =1 j ai

199 ()

()

· ej . Consequently, we have .Eν ψ (qm (x)) → Eν ψ (qm (x)) ()

t

()

∞

for any m and ., so .E(qm ( Xt t )) → E(qm (Y )) for Y as in the statement. This   X implies that . dj =1 j jt,j converges in distribution to . dj =1 j Yj for any . by Billingsley [7, Theorem 30.2], where .Xt = (Xt,1 , . . . , Xt,d ) and .Y = (Y1 , . . . , Yd ). By the Cramér-Wold device [7, Theorem 29.4], this implies that . Xt t converges in distribution to Y . (c) 1 1 d So consider .pm (x). Since .h(θ, n) = 2π ∇θa En (θb ) = 2π i=1 ai ∂θi En (θb ), we have (c) (c) Eν ψ (pm (x)) = Eμψ (pm (h))

.

∞

ˆ =

ν d

m  1  cj ∂θj En (θb ) Pn (θb )(U ψ)θb 2 dθ 2π Td∗ j =1

n=1

ˆ = U ψ,

ν ⊕

Td∗ n=1

d

m

 1  cj ∂θj En (θb ) Pn (θb )dθ U ψ . 2π j =1

On the other hand, (c) (c) Eν ψ (pm (x)) = Eμψ (pm (x/t)) =

d  

.

t

t

cj

x∈ j =1

αj (x) m −itH |e ψ(x)|2 t

m  1 −itH  −itH e ψ, c α (x) e ψ j j tm d

=

j =1



m  1

U ψ, U eitH cj αj (x) e−itH ψ . m t d

=

j =1

It thus suffices to prove that

.

lim

U eitH (

d

j =1 cj αj (x)) tm

t→+∞

ˆ

ν ⊕

Td∗ n=1

m e−itH ψ

=

d

m

1  cj ∂θj En (θb ) Pn (θb )dθ U ψ. 2π j =1

By the multinomial theorem we have  d .

j =1

m cj αj (x)

=

 l1 +···+lr =m

m! cl1 · · · crlr α1 (x)l1 · · · αr (x)lr . l1 ! · · · lr ! 1

(5.7)

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We showed before (3.22) that .∂θi e−iθb ·(ka +vn ) = −2π i(ki + si )e−iθb ·(ka +vn ) if .ka = d d i=1 ki ai and .vn = i=1 si ai , i.e., if .αi (ka + vn ) = ki + si . We thus have .αi (ka + vn )li e−iθb ·(ka +vn ) = ( 2πi )li ∂θlii e−iθb ·(ka +vn ) . This yields (U α1 (x)l1 · · · αr (x)lr φ)θb (vn ) =  e−iθb ·(ka +vn ) α1 (ka + vn )l1 · · · αr (ka + vn )lr φ(ka + vn ) =

.

ka ∈Zda

=

im ∂ l1 · · · ∂θlrr (U φ)θb (vn ) (2π )m θ1

and so,  .

itH

Ue

 d

m cj αj (x)

−itH

e

 =e

itH (θb )

ψ θb

j =1



=

l1 +···+lr =m

  m  d −itH U cj αj (x) e ψ j =1

θb

m! im itH (θb ) l1 c1l1 · · · crlr e ∂θ1 · · · ∂θlrr e−itH (θb ) (U ψ)θb . l1 ! · · · lr ! (2π )m

As in (3.23), ∂θl11 · · · ∂θlrr e−itH (θb ) (U ψ)θb

.

ν 

=

e−itEn (θb ) (−it∂θ1 En (θb ))l1 · · · (−it∂θr En (θb ))lr Pn (θb )(U ψ)θb

n=1

+ On,θ (t l1 +···+lr −1 ) , where the error term contains as usual derivatives of .En (θb ), .Pn (θb ), and .(U ψ)θb . Since .l1 + · · · + lr = m, using the multinomial theorem again we conclude that (5.7) holds, the details are the same as (3.12)–(3.13).   Example 5.6 (The Integer Lattice) If .H = A on .Zd , this gives, for a Borel function f : [−2, 2]d → C,

.

ˆ Eν ψ (f ) = Eμψ (f (h)) =

.

∞

Td

ˆ )|2 dθ. f (−2 sin θ1 , . . . , −2 sin θd )|ψ(θ

Example 5.7 (The Triangular Lattice) In this case, we similarly get ˆ Eν ψ (f ) =

.

∞

T2

ˆ )|2 dθ. f (−2 sin θ1 − 2 sin(θ1 + θ2 ), −2 sin θ2 − 2 sin(θ1 + θ2 ))|ψ(θ

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Example 5.8 (The Hexagonal Lattice) The hexagonal distribution can be calculated similarly using the information in Example 3.6. Here we also need .P± (θb ). This is simple: if .eiθb ·v ξ(θb ) = |ξ(θb )|eiφ(θb ) , then the normalized eigenvectors of 1 −iφ(θb ) )T , so .P (θ ) = w , · w . In particular, .H (θb ) are .w± (θb ) = √ (1, ±e ± b ± ± P± (θb )(U ψ)θb

.

2

=

2 |(U ψ)θb (0)±eiφ(θb ) (U ψ)θb (v)|2 . 2

We may now tackle the continuous case. Theorem 5.9 Consider a Schrödinger operator H on .Rd with smooth periodic potential V having bounded derivatives and let .ψ ∈ L2 (Rd ). Let .Xt be a random vector with distribution .|e−itH ψ(x)|2 dx. ψ d Then . Xt t converges to a random vector distribution .ν∞ having the ν Y on .R with ∞ analogous expression (5.5)–(5.6) (so . n=1 becomes . n=1 ). Proof Here . := Td∗ × N∗ is no longer compact so we cannot use the method of moments directly. However we can by approximation. Namely, given .ψ ∈ L2 ,  ´ ⊕ argue N −1 we know that .ψN := U n=1 Pn (θ ) dθ U ψ satisfies .ψN − ψ2 → 0 Td∗ (in fact, in the proof of Theorem 3.1, Step 3 we showed the stronger fact .ψN − ψH 2m → 0 if .ψ ∈ H 2m (Rd )). For fixed N, we may take .N := Td∗ × {1, . . . , N}. Then the proof of ψ ψ Theorem 5.5 shows that .νt N converges weakly to .ν∞N as .t → +∞. Given .B ⊂ , ψ ψ ψ ψ it remains to control .|νt N (B)−νt (B)| and .|ν∞N (B)−ν∞ (B)|. The former vanishes uniformly in .t, B by Lemma 5.1. Similarly, ψ

ψ

|ν∞ (B) − ν∞N (B)| ∞ ˆ     = 1h−1 B (θ, n)(Pn (θ )(U ψN )θ 2 − Pn (θ )(U ψ)θ 2 )dθ 

.

Td∗ n=1

ˆ ≤

∞  (Pn (θ )(U ψN )θ  + Pn (θ )(U ψ)θ )Pn (θ )((U ψ)θ − (U ψN )θ )dθ

Td∗ n=1

≤ (ψN  + ψ)ψ − ψN  ´ where we used Cauchy–Schwarz for .  in the last step and the fact that .φ2 = ´  2 U φ2 = Td∗ ∞   n=1 Pn (θ )(U φ)θ  dθ . This completes the proof. Remark 5.10 The same approximation trick works for any dense subspace. In fact we did not use the explicit form of .ψN in the previous proof. Remark 5.11 In this paper we proved directly the convergence of moments x m (t)/t m for all m, then proceeded to identify the limiting distribution by generalizing the arguments. If one is only interested in moments, there is an alternative method which we now explain, and only relies on .m = 1. This works in the discrete setting.

.

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Suppose, as in Sect. 3, that we could prove the strong convergence of .Xt,j := to some operator .Qj . By combining [28, Theorems VIII.25 and VIII.20],

xj (t) t

s

we deduce that .f (Xt,j ) − → f (Qj ), for any bounded continuous f . In discrete s m − → Qm models, this guarantees that .Xt,j j for any .m > 0. To see this, write ´ s s x xj 1 t d → Qj and . tj − → 0, it follows that .Xt,j = t + t 0 ds xj (s) ds. Since .Xt,j − ´ s 1 t d .Bt,j := → Qj . Now using Corollary A.2, since .[A, xj ] ≤ LD t 0 ds xj (s) ds − is bounded, where .L = maxy∼x |yj − xj | and .D = max deg x, we see that .Bt,j  ≤ s → f (Qj ) for any continuous f (possibly unbounded), LD. It follows that .f (Bt,j ) − as we can set .g = f on .[−LD, LD], which contains .σ (Bt,j ), and extend g in a s bounded continuous fashion on .[−LD, LD]c . Then .f (Bt,j ) = g(Bt,j ) − → g(Qj ) = f (Qj ). (xj (t)−xj )m s − → Qm j . Using tm m s xj (t) that . t m − → Qm j , as asserted.

Applying this to .f (s) = s m , this shows that

.

the

binomial theorem and Theorem A.1, it follows Still, as we discussed in Theorem 5.5, to gain access to the limiting distribution, one should deal with quite complicated polynomials of .x1 (t)/t, . . . , xd (t)/t. It is not clear if strong convergence of each .xj (t)/t alone can ensure this, as these operators are unbounded. Acknowledgments We thank Christopher Shirley and Jake Fillman for pointing out references, and for an interesting discussion reflected in Remark 5.11.

Appendix A: Upper Bounds and Derivatives Here we first prove that the m-moments grow at most like .t m . This already appeared in various forms: for continuous Schrödinger operators see [27] for .m = 1, 2; for discrete Schrödinger operators, upper bounds can be deduced from [1, Appendix B] for general moments but .ψ = δx . Using the upper bounds, we then give rigorous proofs of the moment derivative formulas (A.11) and (A.22). We start with some general remarks on lower bounds.

A.1 Lower Bounds On .L2 (X), if .x0 ∈ X is fixed and .|x| := d(x, x0 ), then we have ˆ |x|j φ2 = φ, |x|2j φ =

|x|2j |φ|2 ≤

.

X

ˆ

|x|2m |φ|2

j/m ˆ

X

(m−j )/m

X

= |x| φ m

|φ|2

2j/m

φ

2(m−j )/m

,

(A.1)

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where we used Hölder’s inequality with .f = |x|2j |φ|2j/m and .g = |φ|(2m−2j )/m . It follows that .|x|j φm ≤ |x|m φj φm−j . In particular, .

lim inf t→+∞

|x|m e−itH ψ2 |x|e−itH ψ2 m ≥ lim inf ψ2−2m . t→+∞ t 2m t2

(A.2)

On .Rd we usually consider .x j φ instead of .|x|j φ, where .x j φ := j We have .xk φ2 ≤ xkm φ2j/m φ2(m−j )/m by the same argument. Using the Plancherel identity gives the Gagliardo–Nirenberg inequality

j j (x1 φ, . . . , xd φ).

j

Dk φ ≤ Dkm φj/m φ(m−j )/m .

.

If H is a Schrödinger operator and xk (t) := eitH xk e−itH ,

.

j

j

Dk (t) := eitH (−i∂xk )j e−itH , j

then this implies j

xk (t)ψ ≤ xkm (t)ψj/m ψ(m−j )/m , .

(A.3)

j

Dk (t)ψ ≤ Dkm (t)ψj/m ψ(m−j )/m .

Lastly in this connection, recall the uncertainty principle .φ2 ≤ 2xk φ∂xk φ. The above yields the generalization φ2 ≤ 2xkm φ1/m φ(m−1)/m ∂xmk φ1/m φ(m−1)/m ,

.

i.e. φ2 ≤ 2m xkm φ · Dkm φ.

(A.4)

.

Applying this to .φ = e−itH ψ, we thus get j

j

xk (t)ψ · Dk (t)ψ ≤ 2m−j xkm (t)ψ · Dkm (t)ψ.

.

(A.5)

More generally, for .j, n ≤ m, j

xk (t)ψ · Dkn (t)ψ ≤ 2

.

2m−j −n 2

xkm (t)ψ

2m+j −n 2m

Dkm (t)ψ

2m−j +n 2m

.

(A.6)

The preceding estimates provide useful lower bounds for .x m (t)ψ and .D m (t)ψ in terms of lower moments. We now consider upper bounds.

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A.2 Discrete Case In the following, given a countable graph G, we fix some vertex .o ∈ G regarded as an origin and denote .|x| := d(x, o) and .x m ψ(x) := |x|m ψ(x). Theorem A.1 Let .H = A + V be a Schrödinger operator on a countable graph G with maximal degree .≤ D. We assume the potential V is bounded. Then for any m .t ≥ 0 and .m ∈ N, if .x ψ < ∞, then m −itH

x e

.

ψ ≤

m 

pm−r (t)x r ψ,

(A.7)

r=0

where .pk (t) is a polynomial in t of degree k with .p0 (t) = 1, and the leading term of the top polynomial .pm (t) is .Dm t m . In particular, .

lim sup t→+∞

x m e−itH ψ ≤ Dm ψ. tm

(A.8)

Note that each .pk (t) also depends on m, that is, for each fixed m there is a set of polynomials .p0,m (t), . . . , pm,m (t) with .pk,m of degree k such that (A.7) is satisfied with .pk ≡ pk,m . See Remark A.3 for a further comment. Proof By induction on m. The statement is trivial for .m = 0 since .e−itH is unitary. Let .m = 1. For an operator O, recall we denote .O(t) := eitH Oe−itH . In particular, .x(t) := eitH xe−itH when O is the operator of multiplication by x. We formally have .

d x(t)ψ = iH eitH xe−itH ψ − ieitH xH e−itH ψ dt = ieitH [H, x]e−itH ψ = ieitH [A, x]e−itH ψ .

(A.9)

This calculation is formal because the first term .iH eitH xe−itH ψ in the derivative requires .xe−itH ψ ∈ 2 (G), while the second term .−ieitH xH e−itH ψ requires e−i(t+δ)H −e−itH .limδ→0 x ψ = −ixH e−itH ψ. See, e.g., [15, Lemma 10.17]. None of δ these facts is a priori clear (in fact the first point is partly what the theorem tries to prove, we only know that .xψ ∈ 2 (G), a priori). Note that this formal calculation is justified however if instead of x we multiply by a bounded function. λ So, similar to [27], given . > 0, we consider .f (λ) := 1+λ for .λ ≥ 0. d Then multiplication by .f (|x|) is a bounded operator and we have . dt [f (|x|)](t) =

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205

ieitH [A, f (|x|)]e−itH ψ. But [A, f (|x|)]φ(x) =



.

[f (|y|) − f (|x|)]φ(y) =:

y∼x



αx,y φ(y).

y∼x

Here .|αx,y | = |f" (λ)| for some .λ ∈ [|x| − 1, |x| + 1]. Hence, .|αx,y | ≤ We thus get [A, f (|x|)]φ2 ≤ D



.

1 (1+λ)2

≤ 1.

|φ(y)|2 ≤ D2 φ2 .

x y∼x

Applying this to .φ = e−itH ψ, we get .[A, f (|x|)]e−itH ψ ≤ Dψ. So using [8, Theorem 5.6.1], ˆ   .(f (|x|))(t)ψ = (f (|x|))(0)ψ + ˆ ≤ f (|x|)ψ +

0

t

t

 d  (f (|x|))(s)ψ ds  ds

[A, f (|x|)]e−isH ψ ds

0

≤ xψ + tDψ . Since . > 0 is arbitrary, taking . ↓ 0 and using Fatou’s lemma, we get xe−itH ψ2 =



.

|x|2 |(e−itH ψ)(x)|2 ≤ lim inf ↓0

x

 x

|x|2 |(e−itH ψ)(x)|2 (1 + |x|)2

= lim inf f (|x|)e−itH ψ2 ↓0

≤ (xψ + tDψ)2 ,

(A.10)

where we used .f (|x|)e−itH ψ = (f (|x|))(t)ψ since .eitH is unitary. This settles .m = 1. λm Now assume the statement holds for all .k < m. Let .f (λ) = 1+λ m . Here " m−1 .|f (λ)| ≤ m|λ| . Arguing as before, we get [A, f (|x|)]φ2 ≤ Dm2

.

  (|x| + 1)2(m−1) |φ(y)|2 x

y∼x

≤ D2 m2 (|x| + 2)m−1 φ2 .

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Hence, (f (|x|))(t)ψ ≤ f (|x|)ψ + Dm

m−1 

.

q=0

 ˆ m − 1 m−1−q t q −isH 2 x e ψ ds. q 0

Since .f (|x|) ≤ |x|m , using the induction hypothesis we get (f (|x|))(t)ψ ≤ x m ψ + Dm

m−1 

.

q=0

 q m − 1 m−1−q  2 p˜ q−r+1 (t)x r ψ . q r=0

As the RHS is independent of ., taking . ↓ 0 and using Fatou’s lemma again yields s m x m (t)ψ ≤ m s=0 pm−s (t)x ψ. The above also shows the coefficient of .x ψ is .p0 (t) = 1. The top polynomial .pm´(t) is found by taking .q = m−1 and .r = 0 and t equals .Dmp˜ m (t), where .p˜ m (t) := 0 pm−1 (s) ds. As the leading term of .pm−1 (s) ´t is .Dm−1 s m−1 by hypothesis, the leading term of .Dm 0 pm−1 (s) ds is .Dm t m .  

.

A posteriori, the formal differentiation (A.9) is actually valid. Recall the notation x(t) := eitH xe−itH .

.

Corollary A.2 Under the same assumptions, . lim x m (s)ψ = x m (t)ψ and s→t

.

d m x (t)ψ = ieitH [A, x m ]e−itH ψ . dt

(A.11)

Proof We have x m (s)ψ − x m (t)ψ = eisH x m e−isH ψ − eitH x m e−itH ψ

.

≤ eisH x m e−isH ψ − eisH x m e−itH ψ + eisH x m e−itH ψ − eitH x m e−itH ψ = x m e−isH ψ − x m e−itH ψ + eisH x m e−itH ψ − eitH x m e−itH ψ . (A.12) We know from Theorem A.1 that .φ = x m e−itH ψ ∈ D(H ) = 2 (G) for any isH x m e−itH ψ = eitH x m e−itH ψ. This settles the second term in .t ≥ 0, so .lims→t e the RHS. λm For the first term, we use Fatou’s lemma as in (A.10). Let .f (λ) = 1+λ m . We ´ t d have .f (|x|)e−itH ψ = f (|x|)e−isH ψ + s dα f (|x|)e−iαH ψ dα. Now  d     f (|x|)e−iαH ψ  = f (|x|)H e−iαH ψ dα m  ≤ x m e−iαH H ψ ≤ pm−r (α)x r H ψ

.

r=0

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for some polynomials .pk , by Theorem A.1. These are uniformly bounded by some −isH ψ − f (|x|)e−itH ψ ≤ M(t) for all  .α ∈ [t − 1, t + 1] and we get .f (|x|)e  m r |t − s| M(t) r=0 x H ψ, with .M(t) independent of .. By Fatou’s lemma, .x m e−isH ψ − x m e−itH ψ2 ≤ lim inf f (|x|)e−isH ψ − →0 m−1 r f (|x|)e−itH ψ2 . Thus, .x m e−isH ψ − x m e−itH ψ ≤ |t − s| M(t) r=0 x H ψ → 0 as .s → t. Recalling (A.12), this completes the proof of the first claim. For the derivative we first make some simplifications. Given .δ > 0,

.

 x m (t + δ)ψ − x m (t)ψ    − ieitH [H, x m ]e−itH ψ   δ  eiδH x m e−i(t+δ)H ψ − x m e−itH ψ    = − i[H, x m ]e−itH ψ  δ  eiδH − I    ≤ x m e−i(t+δ)H ψ − iH x m e−i(t+δ)H ψ  δ

.

+ H x m (e−itH ψ − e−i(t+δ)H )ψ  e−i(t+δ)H − e−itH

   ψ + iH e−itH ψ  + x m δ  eiδH − I 

  ≤ − iH x m e−itH ψ  δ  eiδH − I 

  + − iH x m (e−i(t+δ)H ψ − e−itH ψ) δ + H x m (e−itH ψ − e−i(t+δ)H )ψ  e−i(t+δ)H − e−itH

   ψ + iH e−itH ψ  . + x m δ

(A.13)

For the first term, we know from Theorem A.1 that .x m e−itH ψ ∈ D(H ) = 2 (G), so this term vanishes as .δ → 0. For the second term, we use the spectral theorem: ´ iδλ ´ eiδH −I . φ2 = | e δ−1 |2 dμφ (λ) ≤ λ2 dμφ (λ) = H φ2 . With this bound, we δ see the second and third terms vanish as .δ → 0 by the argument of (A.12) (note that H is bounded; see also Corollary A.5 for unbounded operators). So it remains to control the last term. For this we first use Fatou’s lemma to replace .x m by .f (|x|) as follows. We know that .

d f (|x|)e−isH ψ = f (|x|)(−iH e−isH )ψ, ds

d f (|x|)(−iH e−isH ψ) = −f (|x|)H 2 e−isH ψ. ds

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It follows from [8, Theorem 5.6.2] that for small .δ, 

 |δ| e−i(t+δ)H − e−itH   x m H 2 e−isH ψ, ψ + iH e−itH ψ  ≤ sup f (|x|) δ 2 s∈[t−1,t+1]

.

where we used that .f (|x|) ≤ |x|m . Using Theorem A.1 again, we may bound |δ| r 2 the RHS by . 2 M(t) m r=0 x H ψ, with .M(t) independent of .. Fatou’s lemma  implies as before that the last term in (A.13) is now bounded by |δ| m r 2 . M(t)   r=0 x H ψ. Taking .δ → 0 finally completes the proof. 2 Remark A.3 Theorem A.1 implies that .

lim sup t→+∞

1  t 2m

|x|2m |e−itH ψ(x)|2 ≤ D2m ψ2 .

x∈G

This also implies a control for odd powers. Namely, if .x m ψ < ∞, letting .ψt := e−itH ψ, we have by Cauchy–Schwarz that

1/2 

1/2   . . |x|m |ψt (x)|2 ≤ |ψt (x)|2 |x|2m |ψt (x)|2 So . t1m



1 |x|m |ψt (x)|2 ≤ ( t 2m

.

lim sup t→+∞

 2m |x| |ψt (x)|2 )1/2 ψ and thus 1  m |x| |ψt (x)|2 ≤ Dm ψ2 . tm x∈G

A.3 Continuous Case Assume now that on .Rd , we have a potential .V ∈ C m−1 such that V and its partial derivatives of order .< m are bounded. Let .H = H0 + V = − + V . Then we claim that D m φ ≤ Cm,V

m 

.

H k φ.

(A.14)

k=0

Indeed, using .Xm − Y m =

m−1 p=0

Xp (X − Y )Y m−1−p , we have

D m φ2 = φ, D 2m φ ≤ φ, H0m φ = φ, H m φ −

m−1 

.

p=0

 with the convention . −1 p=0 := 0.

p

φ, H0 V H m−1−p φ.

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Now (A.14) is clear for .m = 0. If (A.14) holds for all .p < m, then using Cauchy– Schwarz, Leibniz formula, and our assumption on V , we get p

|φ, H0 V H m−1−p φ| ≤ cd,p D p φ · D p V H m−1−p φ

.

≤ cV ,m

p 

D p φ · D q H m−1−p φ.

q=0

So by induction hypothesis, D φ ≤ φ · H

.

2

m

m

φ + cV" ,m

p  p  q m−1 

H r φ · H m−1−p+s φ.

p=0 r=0 q=0 s=0

 k 2 Using .ab ≤ 12 (a 2 + b2 ), we thus get .D m φ2 ≤ cV"" ,m m k=0 H φ , implying (A.14). This implies that for .D(t) := eitH De−itH and .ψt := e−itH ψ, we have D(t)r ψ = D r ψt  ≤ Cr,V

r 

.

" H k ψ ≤ Cr,V ψH 2r ,

(A.15)

k=0

independently of t, where we used that .H k e−itH = e−itH H k . Theorem A.4 If .V ∈ C m−1 (Rd ), if V and its partial derivatives of order .< m are bounded, and if .ψ ∈ H 2m (Rd ), then m

 x m e−itH ψ ≤ 2m−1 x m ψ + Cm t m H k ψ .

.

k=0

A sketch of an earlier result can also be found in [27, Theorem 4.1]. We first give a formal proof, then indicate how to make it rigorous. Proof (Formal) Recall that .x m = (x1m , . . . , xdm ). In this proof we denote .x 2m = x12m + · · · + xd2m instead of .|x m |22 to avoid too cumbersome formulas. Formally, . dtd x 2m (t)ψ = ieitH [−, x 2m ]e−itH ψ for .ψ ∈ D(H ). But, for F smooth on .Rd we have .[−, F ]φ = −(F )φ − 2∇F · ∇φ = −∇ · [(∇F )φ] − ∇F · ∇φ. In particular, for .F (x) = x 2m , since .∇x 2m = 2m(x12m−1 , . . . , xd2m−1 ), we get .

d m d x (t)ψ2 = ψ, x 2m (t)ψ dt dt = iψt , [−, x 2m ]ψt  = ψ, D(t) · [(∇x 2m )(t)ψ] + (∇x 2m )(t) · D(t)ψ ≤ 4m

d  j =1

xjm (t)ψ · xjm−1 (t)Dj (t)ψ,

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where .Dj := −i∂xj and .Dj (t) := eitH Dj e−itH . We have in general xjn (t)Djk (t)ψ = ψ, Djk (t)xj2n (t)Djk (t)ψ1/2

.

≤

k    k p=0

p

2n−p

(2n) · · · (2n − p)|ψ, xj

2k−p

(t)Dj

(t)ψ|1/2. (A.16)

≤

k 

2n−p−m

cp,k,n xjm (t)ψ1/2 xj

2k−p

(t)Dj

(t)ψ1/2 .

p=0

(A.17) 2n−p−m

2k−p

We may apply the same inequality to .xj (t)Dj times we see that the term with highest power is 1

Ck,n xjm (t)ψ 2

.

+ 14 +···+

1 2−1

xj2

−1 (n−m)+m

(t)Dj2

(t)ψ. Doing this . − 1

−1 k

1

(t)ψ 2−1 .

The Case .m = 2 Let .n = m − 1 = 2 − 1. Then by applying (A.17) . − 1 times, we get −1

d 2  1 d 1− 1 x(t)m ψ2 ≤ Cm cr,m xjr (t)Djr (t)ψ 2−1 . xjm (t)ψ · xjm (t)ψ 2−1 . dt j =1

r=0

(A.18) In fact, the terms have the general form 2−1 (n−m)+m−2−2 p1 −2−3 p2 −···−p−1

xj

.

2−1 k−2−2 p1 −···−p−1

(t)Dj

(t)ψ.

For .m = 2 , .n = m − 1, .k = 1, we see the powers of .xj (t) and .Dj (t) match indeed. We next apply (A.16) plus Cauchy–Schwarz to get .

d m x (t)ψ2 dt ≤ Cm

d  j =1

2− 1 xjm (t)ψ 2−1

−1 2

r=0

cr,m

r 

2r−p

cp,r xj

1

2r−p

(t)ψ 2 Dj

1

(t)ψ 2 .

p=0

(A.19)

Ballistic Transport in Periodic and Random Media

211

Recalling (A.5) and (A.15), we conclude that for .m = 2 ,  1 d m x (t)ψ2 ≤ Cm,d xjm (t)ψ2− m Djm (t)ψ1/m dt d

.

j =1

1

≤ Cm,d,V x m (t)ψ2− m

m 

H k ψ1/m .

(A.20)

k=0

Thus,  1 1 d d 1 ψ, x 2m (t)ψ 2m = ψ, x 2m (t)ψ 2m −1 ψ, x 2m (t)ψ ≤ c H k ψ1/m . dt 2m dt m

.

k=0

 k 1/m . The result We thus have .x m (t)ψ1/m ≤ x m ψ1/m + Ct m k=0 H ψ follows in this case.   The General Case For general m we let . such that .2 ≤ m < 2+1 . Say .m = 2 + q with .0 ≤ q < 2 . Then following the scheme, we apply (A.17) . − 1 times. Then 1

1

r+q

xjr (t)Djr (t)ψ 2−1 in (A.18) is replaced by .xj (t)Djr (t)ψ 2−1 . The proof must be slightly modified as now .2r + 2q ≤ 2 + 2q = m + q, i.e., the powers of .xj (t) in (A.19) can exceed m. So to the q highest terms .r = 2−1 −q +1, . . . , 2−1 , we apply 1 1  2r+2q−p−m 2r−p (A.17) once more to get . rp=0 cp,q,r xjm (t)ψ 2 xj (t)Dj (t)ψ 2 . We can now apply (A.16) plus Cauchy–Schwarz to this and the lower terms as before. Then (A.19) is replaced by

.

 d 2− 1 x(t)m ψ2 ≤ Cm . xjm (t)ψ 2−1 dt

−1 2

d

j =1

×

2r−p 

4r+4q−2p−2m−p"

+

2−1 −q r=0

cr,m

r 

1

cp,q,r xjm (t)ψ 2

r=2−1 −q+1 p=0

cp" ,r,p,q,m xj

p" =0

r 

2r+2q−p

cp,r xj

4r−2p−p"

1

(t)ψ 2+1 Dj

1

2r−p

(t) 2 Dj

1

(t)ψ 2+1

1 (t)ψ 2 .

p=0

(A.21) We may now apply (A.6) and (A.15) to get 4r+4q−2p−2m−p"

xj

.

4r−2p−p"

(t)ψ · Dj 2q

(t)ψ

≤ 22m−r−2q xjm (t)ψ m Djm (t)ψ

2m−2q m

.

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A. Boutet de Monvel and M. Sabri

2q Recall .q = m − 2 , so . 2+1 = 21 − m1 . In the first sum of (A.21), .xjm (t)ψ thus m 1 + 21 + 21 − m1 = 2 − m1 as required. For the gets elevated to the power .2 − 2−1

lower terms, the power is similarly .2 − formal proof.

1 2−1

+

2m+2q 2+1 m

= 2−

1 m.

This completes the  

Proof (Completed) To make the formal proof rigorous we consider the operator of multplication by x12m + · · · + xd2m x 2m = , 2m 1 + x 1 + (x12m + · · · + xd2m )

F (x) :=

.

 > 0.

This is a bounded operator. For .F (x)(t) := eitH F (x)e−itH we get .

d F (x)(t)ψ = ieitH [−, F (x)]e−itH ψ for ψ ∈ D(H ). dt

Again .[−, F ]φ = −(F )φ − 2∇F · ∇φ = −∇ · [(∇F )φ] − ∇F · ∇φ. On x 2m−1

x 2m−1

d 1 the other hand .∇F = 2m( (1+x 2m )2 , . . . , (1+x 2m )2 ). So

.

d ψ, F (x)(t)ψ = ψ, D(t) · [(∇F )(x)(t))ψ] + (∇F )(x)(t) · D(t)ψ dt d x m−1   xjm j (t)D (t)ψ, (t)ψ = 2m j 1 + x 2m 1 + x 2m j =1

+ ≤ 4m

d  

 

j =0



xjm 1 + x

(t)ψ, 2m



xjm−1 1 + x

(t)Dj (t)ψ 2m

  x m−1   j  (t)ψ (t)D (t)ψ   . j 1 + x 2m 1 + x 2m xjm

We have   

.

xjm 1 + x



 (t)ψ  = ψt , 2m

xj2m (1 + x 2m )

ψ 2 t

= ψ, F (x)(t)ψ1/2

1/2

≤ ψt , F (x)ψt 1/2

Ballistic Transport in Periodic and Random Media

213

where .ψt := e−itH ψ. On the other hand,   

.

xjr 1 + x



 (t)Djk (t)ψ  = ψ, Djk (t) 2m ≤

xj2r (1 + x 2m )2

(t)Djk (t)ψ

1/2

k  

1/2  xj2r k   p 2k−p D ψ   ψt , Dj t p (1 + x 2m )2 j

p=0

and xj2r

p

∂x j

.

(1 + x 2m )2

If .f (u) := ∂xnj

.

1 (1+u)2

=

p    p



=0

p−

(2r) · · · (2r −  + 1)xj2r− ∂xj

and .g(x) := x 2m then by the Faà di Bruno formula,

1 = ∂xnj f (g(x)) = (1 + x 2m )2

×

n 

1 . (1 + x 2m )2



cn,mi f(m1 +···+mn ) (g(x))

(m1 ,...,mn )  n i=1 imi =n

(∂xi j g(x))mi .

i=1

But .f (u) = (−)q (q + 1)!(1 + u)−2−q and .∂xi j g(x) = (2m) · · · (2m − i + (q)

1)xj(2m−i) . Thus, ∂xnj

.

1 = (1 + x 2m )2



c˜n,mi

(m1 ,...,mn )  n i=1 imi =n

 m1 +···+mn (2m−1)m1 (2m)xj (1 + x 2m )2+m1 +···+mn (2m−2)m2

× (2m)(2m − 1)xj =



Cn,mi

(m1 ,...,mn )  n i=1 imi =n

 m1 +···+mn 2m(m1 +···+mn )−n x , (1 + x 2m )2+m1 +···+mn j

 where we used . imi = n in the last step. But 2m(m1 +···+mn )

 m1 +···+mn xj

.

(2m−n)mn

· · · (2m) · · · (2m − n + 1)xj

≤ (1 + x 2m )m1 +···+mn .

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Thus, xj2r

p

∂x j

.

(1 + x 2m )2 p    p (2r) · · · (2r −  + 1)xj2r− ≤  =0

=

p 



−p

Cp−,mi

(m1 ,...,mp− )

xj

(1 + x 2m )2

2r−p

cp,,r

=0

xj

(1 + x 2m )2

.

It follows that   

.

xjr 1 + x ≤

  k (t)D (t)ψ  j 2m

k 



 cp,k,r  ψ,

p=0

≤

k 

  cp,k,r  

p=0

2r−p

xj

(1 + x 2m )

2k−p

(t)Dj 2

1/2  (t)ψ 

1/2  2r−p−m 1/2   xj  2k−p    . (t)ψ (t)D (t)ψ j    1 + x 2m 1 + x 2m xjm

This proves the analog of (A.16)–(A.17). From here, the proof goes as before and we get   

.

xjm 1 + x

m 1/m   m 1/m ψ ≤ x ψ + Ct H k ψ1/m ,  2m k=0

independently of .. The result follows by taking . ↓ 0, using Fatou’s lemma.

 

Corollary A.5 Under the same assumptions on V , if .ψ ∈ L2 (Rd ), then

∈

.

H 2m+4 (Rd )

d m x (t)ψ = ieitH [−, x m ]e−itH ψ. dt

and .x m ψ

(A.22)

Proof The proof is the same as that of Corollary A.2, using Theorem A.4. In more details, the fact that .x m (s)ψ → x m (t)ψ as .s → t for .ψ ∈ H 2m+2 (Rd ) m is proved the same way by considering .f (x) := √ x 2m instead. For the 1+x

derivative, to deal with the second and third terms at the end of (A.13), we use that m = x m H + [−, x m ] instead. The term .x m H is dealt with as before. For the .H x second term .[−, x m ], let .φtδ := e−itH ψ − e−i(t+δ)H ψ. We have .[−, x m ]φtδ  ≤ d m−2 δ φt +2mxim−1 ∂xi φtδ ]. The calculations (A.16)–(A.19) and i=1 [m(m−1)xi

Ballistic Transport in Periodic and Random Media

215

their later generalization to all m imply that we may bound the second term by 1 1 Cx m φtδ 1− m D m φtδ  m . The norm .x p φtδ  → 0 as .δ → 0 for .p = m − 2, m using the analog of (A.12), while .D m φtδ  is controlled using (A.15). Finally the last term in (A.13) is controlled using the same Fatou argument and we get (A.22).  

.

References 1. M. Aizenman, S. Warzel, Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs. J. Math. Phys. 53(9), 095205, 15 (2012) 2. N. Anantharaman, M. Ingremeau, M. Sabri, B. Winn, Absolutely continuous spectrum for quantum trees. Commun. Math. Phys. 383(1), 537–594 (2021) 3. N. Anantharaman, M. Sabri, Poisson kernel expansions for Schrödinger operators on trees. J. Spectr. Theory 9(1), 243–268 (2019) 4. N. Anantharaman, M. Sabri, Recent results of quantum ergodicity on graphs and further investigation. Ann. Fac. Sci. Toulouse Math. (6) 28(3), 559–592 (2019) 5. J. Asch, Joachim, A. Knauf, Motion in periodic potentials. Nonlinearity 11(1), 175–200 (1998) 6. A.-M. Berthier, Spectral theory and wave operators for the Schrödinger equation, in Research Notes in Mathematics, vol. 71 (Pitman (Advanced Publishing Program), Boston, 1982) 7. P. Billingsley, Probability and measure, in Wiley Series in Probability and Mathematical Statistics, 3rd edn. (Wiley, New York, 1995) 8. H. Cartan, Calcul différentiel (Hermann, Paris, 1967) (French) 9. D. Damanik, M. Lukic, W. Yessen, Quantum dynamics of periodic and limit-periodic Jacobi and block Jacobi matrices with applications to some quantum many body problems. Commun. Math. Phys. 337(3), 1535–1561 (2015) 10. M. Duerinckx, A. Gloria, C. Shirley, Approximate normal forms via Floquet-Bloch theory: Nehorošev stability for linear waves in quasiperiodic media. Commun. Math. Phys. 383(2), 633–683 (2021) 11. J. Fillman, Ballistic transport for limit-periodic Jacobi matrices with applications to quantum many-body problems. Commun. Math. Phys. 350(3), 1275–1297 (2017) 12. J. Fillman, Ballistic transport for periodic Jacobi operators on Zd , in From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory. Operator Theory: Advances and Applications, vol. 285 (Birkhäuser, Cham, 2021), pp. 57–68 13. L. Ge, I. Kachkovskiy, Ballistic Transport for One-dimensional Quasiperiodic Schrödinger Operators (2020). https://arxiv.org/abs/2009.02896 14. G.R. Grimmett, S. Janson, P.F. Scudo, Weak limits for quantum random walks. Phys. Rev. E 69(2), 026119 (2004) 15. B.C. Hall, Quantum theory for mathematicians, in Graduate Texts in Mathematics, vol. 267 (Springer, New York, 2013) 16. R.P. Kanwal, Generalized Functions: Theory and Applications, 3rd edn. (Birkhäuser Boston Inc., Boston, MA, 2004) 17. Y. Karpeshina, L. Parnovski, R. Shterenberg, Ballistic transport for Schrödinger operators with quasi-periodic potentials. J. Math. Phys. 62(5), Paper No. 053504, 12 (2021) 18. T. Kato, Perturbation theory for linear operators, in Classics in Mathematics (Springer, Berlin, 1995). Reprint of the 1980 edition 19. M. Keller, D. Lenz, S. Warzel, Absolutely continuous spectrum for random operators on trees of finite cone type. J. Anal. Math. 118(1), 363–396 (2012) 20. A. Klein, Spreading of wave packets in the Anderson model on the Bethe lattice. Commun. Math. Phys. 177(3), 755–773 (1996) 21. E. Korotyaev, N. Saburova, Schrödinger operators on periodic discrete graphs. J. Math. Anal. Appl. 420(1), 576–611 (2014)

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22. E. Korotyaev, N. Saburova, Spectral estimates for the Schrödinger operator on periodic discrete graphs. Algebra i Analiz 30(4), 61–106 (2018) (Russian), English transl., St. Petersburg Math. J. 30(4), 667–698 (2019) 23. H. Krüger, Periodic and Limit-periodic Discrete Schrödinger Operators (2011). https://arxiv. org/abs/1108.1584 24. P. Kuchment, An overview of periodic elliptic operators. Bull. Am. Math. Soc. (N.S.) 53(3), 343–414 (2016) 25. Y. Last, Quantum dynamics and decompositions of singular continuous spectra. J. Funct. Anal. 142(2), 406–445 (1996) 26. E. Le Masson, M. Sabri, Lp norms and support of eigenfunctions on graphs. Commun. Math. Phys. 374(1), 211–240 (2020) 27. C. Radin, B. Simon, Invariant domains for the time-dependent Schrödinger equation. J. Differ. Equ. 29(2), 289–296 (1978) 28. M. Reed, B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980) 29. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978) 30. H. Saigo, H. Sako, Space-homogeneous quantum walks on Z from the viewpoint of complex analysis. J. Math. Soc. Japan 72(4), 1201–1237 (2020) 31. H. Sako, Convergence theorems on multi-dimensional homogeneous quantum walks. Quantum Inf. Process. 20(3), Paper No. 94, 24 (2021) 32. B. Simon, Functional integration and quantum physics, in Pure and Applied Mathematics, vol. 86 (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979) 33. G. Teschl, Mathematical methods in quantum mechanics. With applications to Schrödinger operators, in Graduate Studies in Mathematics, vol. 157, 2nd edn. (American Mathematical Society, Providence, RI, 2014) 34. C.H. Wilcox, Theory of Bloch waves. J. Analyse Math. 33, 146–167 (1978)

On the Spectral Theory of Systems of First Order Equations with Periodic Distributional Coefficients Kevin Campbell and Rudi Weikard

In memory of Sergey Nikolaevich Naboko (1950–2020)

Abstract We establish a Floquet theorem for a first-order system of differential equations .u = ru where r is an .n×n-matrix whose entries are periodic distributions of order 0. Then we investigate, when .n = 1 and .n = 2, the spectral theory for the equation .J u +qu = wf on .R when J is a real, constant, invertible, skew-symmetric matrix and q and w are periodic matrices whose entries are real distributions of order 0 with q symmetric and w non-negative.

1 Introduction Periodic structures and periodic phenomena have always played a large role in the sciences and in mathematics. In 1883 Floquet [4] gave a canonical form of the solutions of an m-th order homogeneous differential equation with periodic coefficients. Later his result turned out to be instrumental in the understanding of the associated spectral theory. Such results are now classical even in the somewhat more general case of a first order system .u = Au with a periodic locally integrable matrix A. There are many excellent sources for these matters but we have benefited most from the books by Eastham [3] and Brown, Eastham, Schmidt [1]. Both have extensive lists of references to further literature on the subject.

K. Campbell · R. Weikard () Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_11

217

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K. Campbell and R. Weikard

In this paper we generalize some of these classical results by allowing the coefficients of the differential equation to be periodic distributions of order 0.1 In Sect. 2 we recall the concept of periodicity for distributions and state its most important properties. This includes the relationship of distributions of order 0 with measures. In Sect. 3 we state and prove the generalization of Floquet’s theorem. In Sect. 4 we recall some basic facts of the spectral theory for the case of distributional coefficients. These are taken from Ghatasheh and Weikard [5] where one may also find additional information on the history and background of the subject. In Sects. 5 and 6 we specialize to the cases of first and second order systems, respectively. Some simple examples are briefly considered in Sect. 7. We end this introduction with a few words on notation. The set of complexvalued functions of locally bounded variation on .R is represented by .BVloc (R). Any ± .f ∈ BVloc (R) has right- and left-hand limits denoted by .f , respectively. We also # + − use .f for the function .(f + f )/2 which we call balanced. Corresponding to # these kinds of functions we have the subspaces .BV± loc (R) and .BVloc (R) of .BVloc (R). Identity operators are denoted by .1 and .χE is the characteristic function associated with the set E. A function .R ∈ BVloc (R) generates a complex measure dR at least on compact subsets of .R. The corresponding total variation measure is denoted the by .|dR|. In particular, for Lebesgue measure we use the ´ symbol dx, regarding ´ symbol x as the identity function. We will often write . f in place of . f dx, i.e., when an integral does not explicitly specify a measure it may be taken for granted that integration is with respect to Lebesgue measure. Similarly, if a range for the integration is not specified, integration is over the whole real axis.

2 Basic Properties of Periodic Distributions A distribution r is a linear functional on the set of test functions, i.e., the set of compactly supported infinitely often differentiable functions from .R to .C, satisfying the following property: for every compact subset K of .R there are numbers .C ≥ 0 and .k ∈ N0 such that |r(φ)| ≤ C

k 

.

φ (j ) ∞

(1)

j =0

whenever .φ is a test function whose support is contained in K. For example, if .x0 ∈ R is fixed, we have the Dirac distribution defined by 1 (dx) a distribution .f is defined by .δx0 : φ → φ(x0 ). Also, for any .f ∈ L loc

1 Recall that distributions of order 0 are distributional derivatives of functions of locally bounded variation and hence may be thought of, on compact subintervals of .R, as measures. For simplicity we might use the word measure instead of distribution of order 0 below.

Periodic Distributional Coefficients

219

´ φ → f (x)φ(x)dx. Below we will often follow the ubiquitous convention to use the same symbol for .f and f . Context will serve to distinguish the two meanings. If we define .r  (φ) = −r(φ  ) it follows that .r  is again a distribution, called the derivative of r. Distributions also have antiderivatives and we have the following important lemma.

.

Lemma 2.1 (Du Bois-Reymond) Suppose the derivative of the distribution r is zero. Then´r is the constant distribution, i.e., there is a complex number C such that .r(φ) = C φ for every test function .φ. The number .ω ∈ R is called a period of the distribution r, if .r(φ) = r(φ(· + ω)) for every test function .φ. The distribution r is called periodic (or, more specifically, .ω-periodic) if it has a non-zero period .ω. Of course, 0 is a period of any distribution. Theorem 2.2 A periodic distribution r has the following properties: If .ω1 and .ω2 are periods of r, then so are .ω1 ± ω2 . If .ω is a period of r, then any integer multiple of .ω is also a period. If r is constant, then all real numbers are periods of r. If the set of periods of r has a finite limit point, then r is constant. The infimum .ω0 of the set of positive periods of r is itself a period. If .ω0 > 0, every period of r is an integer multiple of .ω0 . The period .ω0 is then called the fundamental period of r. 6. If R is an antiderivative of r, then there is a complex number .α and ´ a periodic distribution P with the same periods as r, such that .R(φ) = α xφ(x) dx + P (φ). 1. 2. 3. 4. 5.

Proof Properties (1)–(3) are trivial. To prove (4) notice first that 0 must also be a limit point of the periods of r. We show that .r  = 0, since our claim follows then from du Bois-Reymond’s lemma. Suppose .ω is a period of r with .0 < ω < 1 and let .φ be a test function. Pick a compact interval K such that the supports of both .φ and .φ(· + ω) are in K. Then   φ(· + ω) − φ r(φ(· + ω)) − r(φ) =r .0 = ω ω and hence, for appropriate numbers C and k, |r  (φ)| = |r(φ  )| = |r(ψ)| ≤ C

k 

.

j =0

where ψ=

.

φ(· + ω) − φ − φ. ω

ψ (j ) ∞

220

K. Campbell and R. Weikard

Using the mean value theorem twice, we obtain ψ (j ) (x) = φ (j +1) (c) − φ (j +1) (x) = (c − x)φ (j +2) (c) ˜

.

for some .c ∈ (x, x + ω) and some .c˜ ∈ (x, c). Hence .|r  (φ)| ≤ C(k + 1)Mω where (j +2)  .M = max{φ ∞ : 0 ≤ j ≤ k}. Since .ω may be arbitrarily small, we find   .r (φ) = 0 and since .φ was arbitrary, we get .r = 0 as promised. Property (5) is clear when the infimum .ω0 is 0, so assume it is not. Then r is not constant and the previous result shows that .ω0 is not a limit point of the set of positive periods. Instead it must be a period itself. Now suppose .ω is any other period of r. Then .ω = nω0 + b for some .n ∈ Z and .b ∈ [0, ω0 ). Since b is also a period it must be equal to 0. Finally, for property (6) we first obtain from´du Bois-Reymond’s lemma and the periodicity of r that .R(φ(· + ω)) − R(φ) = C φ for some number C. Now define the distribution P by setting P (φ) = R(φ) +

.

C ω

ˆ xφ(x)dx.

´ ´ Since . xφ(x + ω)dx = (x − ω)φ(x)dx we find that P is periodic. Hence the claim follows if we set .α = −C/ω.

 Remark 2.3 A function .f ∈ L1loc (dx) is called periodic with period .ω, if .f (x + ω) = f (x) for almost all (with respect to Lebesgue measure) .x ∈ R. As mentioned earlier, such a function gives rise to a distribution .f by setting ´ .f(φ) = f φ for any test function .φ. Since ˆ f(φ(· + ω)) =

.

ˆ f φ(· + ω) =

ˆ f (· − ω)φ =

f φ = f(φ)

we see that .f is a periodic distribution with the same periods as f . In the following we will be concerned only with distributions of order 0, i.e., those for which one may choose .k = 0 in inequality (1) regardless of K. They are in close correspondence with functions of locally bounded variation. Specifically, if .R ∈ BVloc (R), then it generates a (Borel) measure dR on compact subsets of .R. It ´ follows that .φ → φdR ´is a distribution of order 0, in fact, it is the derivative of the distribution .φ → Rφdx. Conversely, if r is a distribution of order 0, then Riesz’s representation theorem shows that there is a function .R ∈ BVloc (R) ´ yielding .r(φ) = φdR. For brevity we will frequently identify the distribution r and the local measure dR. We will also identify the antiderivative of r with the ´corresponding ´ function R in .BVloc (R). In particular, we use the designations .r(φ), . rφ, and . φdR interchangeably.

Periodic Distributional Coefficients

221

If .f ∈ L1loc (|dR|) and r is a distribution of order 0 we may define the product of r and f (or f and r) by setting ˆ φ → (rf )(φ) = (f r)(φ) =

.

ˆ f φdR =

rf φ.

rf is again a distribution of order 0. We need the following substitution rule when dealing with integrals. Lemma 2.4 Suppose .(a, b) and .(α, β) are real intervals and .R : (α, β) → C is left-continuous and of bounded variation. If .T : (a, b) → (α, β) is continuous and bijective (and hence strictly monotone), then .R ◦ T : (a, b) → C is also leftcontinuous and of bounded variation. Moreover, if .g ∈ L1 (|dR|), then ˆ .

ˆ gdR = ±

g ◦ T d(R ◦ T )

where one has to choose the positive sign if T is strictly increasing and the negative sign if it is strictly decreasing. This lemma has the following consequence in the context of periodic distributions. Theorem 2.5 Suppose ´ w is´a periodic distribution of order 0 with period .ω and f ∈ L1 (|w|). Then . wf = wf (· + ω).

.

Proof Let .T : R → R : x → x + ω and let W be an anti-derivative of w. Note that by property (6) of Theorem 2.2 we have .W (x) = αx + P (x) for some periodic function (of locally bounded variation) P . Then .W (T (x)) = W (x) + αω and hence .dW = α + dP = d(W ◦ T ). 

3 Floquet Theory In this section we shall develop a Floquet theory for the differential equation u = ru

.

where r is an .n × n-matrix whose entries are periodic distributions of order 0 all of which have a common period .ω (in this case we call r periodic with period .ω). We seek solutions among balanced .Cn -valued functions of locally bounded variation. Theorem 3.1 Suppose u is a balanced function of locally bounded variation such that .u = ru. If v is defined by .v(x) = u(x + ω), then we also have .v  = rv.

222

K. Campbell and R. Weikard

Proof Let .T (x) = x − ω, let .φ be a test function, and set .ψ = φ ◦ T = φ(· − ω). This and Theorem 2.5 give ˆ (rv)(φ) =

ˆ rvφ =

.

ˆ r(vφ) ◦ T =

ruψ = (ru)(ψ).

We also have, by the translation invariance of Lebesgue measure, v  (φ) = −

.

ˆ

v(x)φ  (x)dx = −

ˆ

u(x)ψ  (x)dx = u (ψ).

Since the rightmost expressions are the same so are the leftmost.



Thus the operator which assigns .u(·+ω) to u is a map from the space of solutions of .u = ru to itself. It is  called 2 0   the monodromy operator. 0 The examples .r = 20 −2 k∈Z δk and .r = 0 −2 k∈Z (δ2k+1 − δ2k ), which are periodic distribution with period 1 and 2, respectively, show that the solution space of .u = ru may not be n-dimensional. Indeed, in the former case the solution space is trivial while, in the latter case it is infinite-dimensional. This is due to the fact, that the existence and uniqueness theorem for initial value problems fails for these equations. To proceed we define the matrix .Δr (x) = R + (x) − R − (x) and add the following hypothesis. Hypothesis 3.2 Let .ω > 0. Assume r is an .n × n-matrix of .ω-periodic distributions of order 0 such that the matrices .1 ± 12 Δr (x) are invertible for every .x ∈ R. It was shown in [5] that, under this hypothesis, existence and uniqueness of balanced solutions of initial value problems holds. It follows immediately that the solution space of .u = ru is n-dimensional and hence that we have a fundamental matrix U of solutions. The determinant of .U (x) is different from 0 for any .x ∈ R. Theorem 3.1 shows that .U (· + ω) is also a fundamental matrix of solutions. Hence there is a constant matrix M such that U (x + ω) = U (x)M.

.

The matrix M, called a monodromy matrix, depends on the choice of U . If V is another fundamental matrix of solutions so that .V = U S for a constant invertible ˜ −1 , i.e., M matrix S and .M˜ is the associated monodromy matrix, then .M = S MS ˜ and .M are similar matrices. In particular, they have the same eigenvalues.

Periodic Distributional Coefficients

223

It is known from Linear Algebra that we may choose S so that .M˜ is a matrix in Jordan normal form, i.e., .M˜ is a block diagonal matrix where the diagonal blocks, called Jordan blocks, are square matrices of the form ⎛ ρk ⎜0 ⎜ . ˜k = ⎜ .M ⎜ .. ⎜ ⎝0

1 0 ··· ρk 1 · · · .. . . .. . . . 0 · · · ρk 0 0 ··· 0

⎞ 0 0⎟ ⎟ .. ⎟ . .⎟ ⎟ 1⎠ ρk

Here .ρk is an eigenvalue of .M˜ (and of M) and hence non-zero. If thenumber of Jordan blocks is s and if their size is .μk , .k = 1, ..., s, we have, of course . sk=1 μk = n. Theorem 3.3 Suppose r satisfies Hypothesis 3.2. The differential equation .u = ru has a fundamental system of solutions of the form

.

eαk x

 

qk,j (x)pk,−j (x) for  = 0, ..., μk − 1 and k = 1, ..., s

j =0 ω where .eαk ω = ρk , .qk,0 (x) = 1, .qk,j +1 (x) = qk,j (x) (j x−j +1)ρk ω , and the .pk,j are n balanced, periodic .C -valued function with period .ω.

Proof We are adapting Hochstadt’s proof in [6] which avoids introducing the logarithm of M.2 Let .vm be the m-th column of V , the fundamental matrix whose monodromy matrix is in Jordan normal form. There are unique k−1numbers .k ∈ {1, ..., s} and . ∈ {0, ..., μk − 1} such that .m = 1 +  + h=1 μh . Let k−1 .m0 = 1 + μ . Then we will prove, by induction over ., that there are .ωh h=1 periodic functions .pk,j such that vm0 + (x) = eαk x

 

.

qk,j (x)pk,−j (x)

(2)

j =0

for . = 0, ..., μk − 1. If . = 0 define .pk,0 by .pk,0 (x) = vm0 (x) e−αk x . Then the identity .vm0 (x + ω) = ρk vm0 (x) shows that .pk,0 is .ω-periodic, i.e., .vm0 has the required form. Now assume that (2) has been established for . = 0, ..., r − 1 including the periodicity of .pk,0 , ..., .pk,r−1 for some .0 < r < μk . Let . = r and note that .vm0 + defines the yet undetermined function .pk, . It is only left to prove

2 There

appears to be a flaw in Hochstadt’s reasoning which we tried to circumvent.

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K. Campbell and R. Weikard

that .pk, is .ω-periodic. Since .vm0 + (x + ω) − ρk vm0 + (x) − vm0 +−1 (x) = 0 we get, using the periodicity of .pk,0 , ..., .pk,−1 , pk, (x + ω) − pk, (x) =

   1

.

j =1

ρ

 qk,j −1 (x) + qk,j (x) − qk,j (x + ω) pk,−j (x).

For our choice of the polynomials .qk,j induction shows that each term on the righthand side is 0 proving the periodicity of .pk, . 

The eigenvalues .ρk of a monodromy matrix are called Floquet multipliers while the numbers .αk are called Floquet exponents. The associated (generalized) eigenfunctions are called (generalized) Floquet solutions of .u = ru.

4 Spectral Theory The main goal of this paper is to investigate the spectral theory associated with a periodic first order .2 × 2-system of differential equations. To this end we will recall some basic definitions and results from [5] and [2]. If w is a non-negative .n × n-matrix whose entries are distributions of order 0, 2 .tr w represents a positive scalar measure. By .L (w) we denote the collection of n .C -valued functions f whose components are measurable with respect to .tr w and ´ which satisfy .f 2 = f ∗ wf < ∞. Then .L2 (w) designates the corresponding Hilbert space, i.e., the quotient of .L2 (w) by´ the kernel of . · . The inner product of .L2 (w) is, of course, given by .f, g = f ∗ wg. Now consider the differential equation J u + qu = wf

.

(3)

where J is a constant, invertible and skew-hermitian .n × n-matrix and q is a hermitian .n × n-matrix whose entries are distributions of order 0. Define the linear relations Tmax = {(u, f ) ∈ L2 (w) × L2 (w) : u ∈ BV#loc (R)n , J u + qu = wf }

.

and Tmin = {(u, f ) ∈ Tmax : supp u is compact in R}.

.

Then, in the Hilbert space setting, we represent our differential equation by the relations Tmax = {([u], [f ]) ∈ L2 (w) × L2 (w) : (u, f ) ∈ Tmax }

.

Periodic Distributional Coefficients

225

and Tmin = {([u], [f ]) ∈ L2 (w) × L2 (w) : (u, f ) ∈ Tmin }.

.

∗ =T The cornerstone of spectral theory is the result that .Tmin max , i.e., that .Tmin is a symmetric relation (see [2]). As a consequence we have that

Tmax = Tmin ⊕ Di ⊕ D− i

.

where .Dλ is defined as .{(u, λu) ∈ Tmax }. These are called deficiency spaces if λ ∈ R. The numbers .n± = dim D± i are called deficiency indices. It is important to recall that .dim D(λ) is independent of .λ as long as .λ varies in either the upper or the lower half of the complex plane. The deficiency indices are finite (see [7]) and if they are identical then there exist self-adjoint restrictions of .Tmax (possibly .Tmax itself). Even in the case of constant coefficients two complications arise. Firstly, the space .L0 = {u : J u + qu = 0 and wu = 0} may be non-trivial (note here that .wu = 0 if and only if .u = 0). If this happens the problem is called non-definite. The other issue is that .Tmax may indeed not be a linear operator as our introduction of linear relations already insinuates. If T is a self-adjoint restriction of .Tmax one defines the resolvent set of T by .

(T ) = {λ ∈ C : ker(T − λ) = {0}, ran(T − λ) = L2 (w)}

.

and the spectrum of T by .σ (T ) = (T )c , the complement of . (T ). For .λ ∈ (T ) the linear relation .(T − λ)−1 is, in fact, a linear operator from .L2 (w) to the domain of T . Define the space .H∞ = {f ∈ L2 (w) : (0, f ) ∈ T } and .H0 to be its orthogonal complement. Then the domain of T is a dense subset of .H0 and if .(u, f ) ∈ T , then, of course, .f = f0 + f∞ with .f0 ∈ H0 and .f∞ ∈ H∞ . But since .(0, f∞ ) is in T so is .(u, f0 ). Since .f0 is uniquely determined by u we have that .T0 = T ∩ (H0 × H0 ) is a densely defined self-adjoint linear operator, called the operator part of T . In particular, .H0 = {0} if and only if the spectrum of T is empty.

5 The Case n = 1 Hypothesis 5.1 Throughout this section we assume that ω is a positive real number and J a non-zero purely imaginary number. Moreover, q is a real distribution of order 0 while w is a non-negative but non-zero distribution of order 0. Both q and w are periodic with period ω.

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When λ and x are in R the imaginary part of the number 1 B± (x, λ) = J ± (Δq (x) − λΔw (x)) 2

.

is equal to that of J and hence non-zero. Hypothesis 3.2 is therefore satisfied for n = 1. Let U (·, λ) be the unique solution of the homogeneous equation J u +qu = λwu satisfying the initial condition U + (0, λ) = 1. Then ρ(λ) = U + (ω, λ) is the Floquet multiplier. Thus U (x + nω, λ) = ρ(λ)n U (x, λ) whenever n ∈ Z. Recall from Lemma 3.2 in [5] that U + (x, λ)∗ J U + (x, λ) = U − (x, λ)∗ J U − (x, λ) = J.

.

Hence U + (x, λ)∗ U + (x, λ) = 1 and, in particular, ρ(λ)ρ(λ) = 1. Theorem 5.2 Assume that Hypothesis 5.1 holds. Then the linear relation Tmax is self-adjoint. Its spectrum is purely continuous and fills the entire real line. Proof Fix λ in C and note that ρ(λ) = 0. Using Theorem 2.5 we obtain, for any n ∈ Z, ˆ ˆ . |U (·, λ)|2 w = |ρ(λ)|−2n |U (·, λ)|2 w. [nω,(n+1)ω)

[0,ω)

Hence U (·, λ)2 =



.

n∈Z

|ρ(λ)|−2n

ˆ [0,ω)

|U (·, λ)|2 w

which is finite only when w = 0, a case we have excluded for being trivial. Hence no λ can be an eigenvalue of Tmax , i.e., the deficiency spaces are trivial and Tmax is self-adjoint. Now assume that λ is real. Since ρ(λ)ρ(λ) = 1 this shows that |ρ(λ)| = 1. It follows that λ is an element of the so called stability set S, the set of those λ for which U (·, λ) is bounded. We will prove that S ⊂ σ , the spectrum of Tmax . Then R⊂S⊂σ ⊂R

.

which entails that S = σ = R. Thus assume now that λ ∈ S and, by way of contradiction, that it is also in the resolvent set of Tmax . If we can construct a sequence n → (φn , λφn + fn ) ∈ Tmax such that φn  = 1 and limn→∞ fn  = 0, then φn = Rλ fn . Since Rλ , the

Periodic Distributional Coefficients

227

resolvent operator for Tmax at λ, is bounded,3 we get 1 = φn  ≤ Rλ fn  → 0, a contradiction. Let us construct the required sequence in Tmax . Since λ is fixed we will, in the course of this proof, simply write U in place of U (·, λ). Let W be the left-continuous anti-derivative of w which vanishes at 0. For n ∈ Z let In = (nω, (n + 1)ω] and for n ∈ N define hn = (W + (n + 1)W (ω))χI−n−1 + W (ω)χ(−nω,nω] − (W − (n + 1)W (ω))χIn

.

and φn = an (hn U )# where the numbers an are chosen so that φn  = 1. Note that the functions hn are left-continuous, i.e, hn = h− n. Recall the product rule for functions of locally bounded variation, i.e., (f g) = f + g  + f  g − = f − g  + f  g + . Hence φn = an hn U + + an hn U  . Since (hn U )# = − + hn U + 12 (h+ n − hn )U we get J φn + (q − λw)φn

.

  + 1 − = an hn (J U  + (q − λw)U ) + an J hn + (q − λw)(h+ n − hn ) U . 2 The first term on the right is 0 while the second is a measure supported on the closure of I−n−1 ∪ In . More precisely, we have hn = wχ[−n−1,−n) − wχ[n,n+1) , − + − h+ n (x) − hn (x) = Δw (x) for x ∈ [−n − 1, −n), and hn (x) − hn (x) = −Δw (x) for x ∈ [n, n + 1). Now observe that the discrete measures qΔw and Δq w are identical so that we get J φn + (q − λw)φn = wfn when we define 1 fn = an (J + (Δq − λΔw ))U + (χ[−n−1,−n) − χ[n,n+1) ). 2

.

It follows that (φn , λφn + fn ) is an element of Tmin ⊂ Tmax . Next we show that the norming constants an tend to 0. Indeed, using again Theorem 2.5 and the fact that |ρ(λ)| = 1, ˆ 2 2 2 .φn  ≥ |an | W (ω) |U |2 w = 2n|an |2 C [−nω,nω)

√ ´ where C = W (ω)2 [0,ω) |U |2 w does not depend on n. Hence, |an | ≤ 1/ 2nC. It is now also clear that fn  tends to 0 so that our proof is finished. 

3 Even

in the case of a relation the resolvent is necessarily an operator.

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K. Campbell and R. Weikard

6 The Case n = 2, Real Coefficients The following assumptions are in force throughout this section. Hypothesis 6.1 The .2 × 2-matrix J is real, skew-hermitian, and invertible. The entries of the .2 × 2-matrices q and w are real distributions of order 0 with q hermitian (symmetric) and w non-negative. Both q and w are .ω-periodic where .ω is a positive real number. Moreover, the matrices 1 B± (x, λ) = J ± (Δq (x) − λΔw (x)) 2

.

are invertible for all .λ, x ∈ R. We observe that for all .x ∈ R and all .λ ∈ C we have .det B+ (x, λ) = det B− (x, λ). Let .Λ be the set of all those .λ ∈ C such that, for some .x ∈ R, .det B± (x, λ) = 0. Our assumptions guarantee that .Λ does not intersect .R and that there are only finitely many elements of .Λ in any disk of finite radius. Hence .Λ is a discrete set. It is also symmetric with respect to the real axis. See [7] for more details.   Furthermore, our condition on J implies that .J = r 10 −1 for some number 0 .r ∈ R\{0}. The motivation to restrict our attention to the real case only, is that .v = u satisfies .J v  + (q − λw)v = 0, if u satisfies .J u + (q − λw)u = 0. In particular, if  .U (·, λ), for .λ ∈ Λ, is a fundamental matrix for .J u + (q − λw)u = 0, then .U (·, λ)  is a fundamental matrix for .J u + (q − λw)u = 0. It follows that, for .λ ∈ R, we may choose the fundamental matrix to be real-valued. Moreover, .r det U ± (·, λ) = −U1± (·, λ)∗ J U2± (·, λ) when .U1 and .U2 denote the first and second column of U , respectively. Lemma 3.2 in [5] implies therefore that .det U + (·, λ) = det U − (·, λ) is constant. Thus any monodromy matrix .M(λ) has determinant 1 and this, in turn implies that the Floquet multipliers are reciprocals of each other. Therefore the Floquet multipliers are uniquely determined (up to transposition) by their sum, the Floquet discriminant D(λ) = tr M(λ).

.

Note that .tr M(λ), just like .det M(λ), is invariant under similarity transforms, i.e., it is independent of the specific fundamental matrix used to find .M(λ). Since we may choose .U (·, λ) real for .λ ∈ R, it follows that .D(λ) is then also real. Moreover, by Theorem 2.7 of [5] the entries of .U (x, ·) are analytic in .C \ Λ for any .x ∈ R. Consequently, the Floquet discriminant D is analytic, too. Our first major goal is to show the absence of eigenvalues of .Tmax . First we show that no point in .Λ can be an eigenvalue. Theorem 6.2 If .λ ∈ Λ, then any solution of .J u + qu = λwu is identically equal to 0.

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229

Proof Since .λ ∈ Λ there is a point .x0 for which .B± (x0 , λ) are not invertible. Of course .x0 + ω is then also such a point. First we show that .ran B+ (x0 , λ) and .ran B− (x0 , λ) intersect only trivially. Let us abbreviate .B± (x0 , λ) simply by .B± . Neither .B+ nor .B− is 0. Hence there are two vectors .v+ and .v− spanning, respectively, their kernels. Moreover, .0 = 2J v+ = B− v+ and .0 = 2J v− = B+ v− . Assuming now, by way of contradiction and without loss of generality that .B− v+ = B+ v− we get .J v+ = J v− which contradicts the fact that J is injective and shows that .ran B+ ∩ ran B− = {0}. If u solves the equation .J u + qu = λwu we must have .B+ (x0 , λ)u+ (x0 ) = B− (x0 , λ)u− (x0 ) and hence, by the above, that .B± (x0 , λ)u± (x0 ) = 0. By the same argument we get .B± (x0 + ω, λ)u± (x0 + ω) = 0. It follows now that .v = (uχ(x0 ,x0 +ω) )# is also a solution of .J u + qu = λwu, in fact a solution of finite norm. If the norm were positive we would have a complex eigenvalue of .Tmin which is impossible. Therefore .wv = 0 so that v also solves .J u + qu = λwu for all .λ ∈ C including .λ = 0. However, for .λ = 0 solutions of initial value problems are unique. 

Hence .L0 is trivial and u is identically equal to 0. Lemma 6.3 Fix .λ ∈ C \ Λ. For a solution u of .J u + qu = λwu and .n ∈ Z define ˆ In (u) =

.

[nω,(n+1)ω)

u∗ wu.

If .In (u) = 0 for two consecutive non-zero integers, then .u = 0. Otherwise u has infinite norm. If, in the former case, u is not a Floquet solution and not equal to 0, then all Floquet solutions of .J u + qu = λwu also have norm 0. Proof Suppose first that we have two linearly independent Floquet solutions .ψ1 and ψ2 , the former with multiplier .ρ and the latter with multiplier .1/ρ. Define .Nj2 = ´ ´ ∗ ∗ [0,ω) ψj wψj with .Nj ≥ 0 and .A = [0,ω) ψ1 wψ2 . Note that .|A| ≤ N1 N2 . If .u = αψ1 + βψ2 we obtain .

In (u) = |α|2 |ρ|2n N12 + 2 Re(αβ(ρ/ρ)n A) + |β|2 |ρ|−2n N22

.

≥ (|α|N1 |ρ|n − |β|N2 |ρ|−n )2 . If .u = 0 there is nothing to prove and thus we may assume that .α and .β are not both 0. If one of .α and .β is 0, then u is a Floquet solution and either all of the .In (u) are 0 or all of them are positive. Correspondingly, .u = 0 or .u = ∞. If neither .α nor .β is 0 let us assume that .In (u) = In+1 (u) = 0 for some .n ∈ Z. This implies that either .N1 = N2 = 0 or else .|ρ| = 1. In the former case all solutions of  .J u + qu = λwu have norm 0. In the latter case we have In (u) = 2N 2 (1 + Re(zρ −2n ))

.

where we put .N = |α|N1 = |β|N2 > 0 and .z = αβA/N 2 . Since .|zρ −2n | ≤ 1 and −2n = zρ −2n−2 = −1 which implies that .ρ 2 = 1. .In (u) = In+1 (u) = 0 we get .zρ

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K. Campbell and R. Weikard

But this means that .ψ1 and .ψ2 have the same Floquet multiplier. Hence u itself is a Floquet solution and must have norm 0. We now consider the case when there is only one linearly independent Floquet solution. Then .ρ = ±1 and we treat the case when .ρ = 1; the other one is similar. The general solution of .J u + qu = λwu, according to Theorem 3.3, is u(x) =

.

αx ψ(x) + αp(x) + βψ(x), ω

where .ψ is a Floquet solution and p is some .ω-periodic function. We confine ourselves to the case .α = 1 and define .ϕ = p + βψ, which is .ω-periodic. ´ 2 For .n ≥ 1 we find .In (u) ≥ (Fn − G)2 where .Fn2 = [0,ω) (x+nω) ψ ∗ wψ and ω2 ´ 2 ∗ .G = ϕ wϕ. Now .In (u) = In+1 (u) = 0 implies that .Fn = G = Fn+1 and [0,ω) ´ hence .0 = [0,ω) (2n + 1 + 2x/ω)ψ ∗ wψ. From this we conclude .ψ ∗ wψ = 0 so that .ψ = 0, .Fn = G = 0 and .u = 0. Now the only thing left to prove is that a solution u which does not have norm 0 has infinite norm. This is already known for Floquet solutions so we assume that u is not a Floquet solution. For u to have finite norm it is necessary that .In (u) tends to 0 as n tends to .∞ or .−∞. In the presence of two independent Floquet solutions it is thus necessary that .|α|N1 = |β|N2 and that .|z| = |αβA|/N 2 = 1. In this case we have .In (u) = 2N 2 (1 + cos(t0 + 2nt)) where we set .z = eit0 and .ρ = e−it for some .t0 , t ∈ R. However, the sequence .n → In (u) converges to 0 only if it is identically equal to 0. If only one linearly independent Floquet solution exists and u is not a Floquet solution .In (u) cannot converge to 0 unless .Fn = G = 0. 

An immediate consequence of the previous lemma is the next theorem. Theorem 6.4 The linear relation .Tmax has no eigenvalues. In particular, it is selfadjoint. ´ Lemma 6.5 Fix .λ ∈ C \ Λ. Let B be the set of the vectors . U (·, λ)∗ wf when f varies among the functions in .L2 (w) with support in .[0, 2ω). If .L0 is trivial, then 2 .B = C . Proof Assume, by way of contradiction, that B is less than two-dimensional. Since B = {0} would imply .w = 0 it follows that, in fact, B is one-dimensional. Denote the subspaces of B obtained by restricting f to those functions which are supported only on .I1 = [0, ω) or .I2 = [ω, 2ω) by .B1 and .B2 , respectively. These are also one-dimensional and hence .B1 = B2 = B. Choosing .0 = α ∈ B ⊥ and .f ∈ L2 (w) with support in .I1 we get, using Theorem 2.5 and the periodicity of w,

.

0 = α∗

ˆ

.

I2

U (·, λ)∗ wf (· − ω) = α ∗

ˆ

U (· + ω, λ)∗ wf

I1

= α ∗ M(λ)∗

ˆ I1

U (·, λ)∗ wf.

Periodic Distributional Coefficients

231

This shows .M(λ)α is perpendicular to .B1 and hence a multiple of .α. Therefore u = U (·, λ)α is a Floquet solution of .J u + qu = λwu with norm 0, i.e., .u ∈ L0 which contradicts our hypothesis. 

.

If our problem is non-definite .Tmax is a particularly simple relation as our next theorem shows. Theorem 6.6 For the relation .Tmax to be equal to .{0} × L2 (w) it is necessary and sufficient that .L0 = {0}. In this situation the spectrum of .Tmax is empty. Proof To show necessity assume that .Tmax = {0} × L2 (w) and, by way of contradiction, that .L0 = {0}. For any .f ∈ L2 (w) with support in .[0, 2ω) there is a function u of locally bounded variation such that .J u + qu = wf and .wu = 0. In .(−∞, 0) we have that u must be equal to a linear combination of (generalized) Floquet solutions which we call .u˜ . Since then, in the terminology of Lemma 6.3, .In (u) ˜ = 0 for all negative integers n and since .L0 = {0} we have, in fact, .u˜ = 0 and hence .u = 0 on .(−∞, 0). Similarly we can show that .u = 0 on .[2ω, ∞). For .x > 0 the variation of constants formula (see Lemma 3.3 in [5]) gives u− (x) = U − (x, 0)J −1

ˆ

.

[0,x)

U (·, 0)∗ wf.

´ In particular, . [0,2ω) U (·, 0)∗ wf = 0 regardless of f , contradicting the findings of Lemma 6.5. To show sufficiency we note first that we may assume .w = 0 since otherwise 2 .L (w) = {0}. The conclusion will follow from Theorem 7.3 in [5] once we establish its hypotheses. We have already shown that .n± = 0 and it is clear that .0 < dim L0 < 2, the former inequality following from the hypothesis that .L0 is not trivial and the latter since .w = 0. It remains to show that .ker Δw (x) ⊂ ker Δw (x)J −1 Δq (x) for     all .x ∈ R. Thus fix .x ∈ R and let .Δq (x) = ab db and .Δw (x) = βα βδ . If the rank of .Δw (x) is 0 or 2, then the needed inclusion holds trivially. Hence we assume that 2 .αδ = β with at least one of those numbers non-zero. In this case .det B± (x, λ) is a real polynomial in .λ of degree at most 1. Since .det B± (x, ·) must not have a real zero, it follows that the coefficient of .λ must be 0, i.e., .aδ + αd = 2bβ. Under that condition it turns out that .Δw (x)J −1 Δq (x) is symmetric and hence equal to −1 Δ (x) which proves the desired inclusion and hence the lemma. .−Δq (x)J w For the last claim we simply note that the operator part of .Tmax is .{([0], [0])} which has no spectrum. 

We now define the conditional stability set S to be the set of those .λ ∈ C such that the equation .J u + qu = λwu has a non-trivial bounded solution. Note that Theorem 6.2 shows that S and .Λ do not intersect. It follows that the conditional stability set is the set of those .λ ∈ C \ Λ for which the monodromy matrix .M(λ) has an eigenvalue of absolute value 1. ˜ Lemma 6.7 If .λ is not in .S ∪ Λ, then .M(λ) has distinct eigenvalues .ρ1 = emω − mω ˜ and .ρ2 = e where .m = Re m ˜ > 0. The associated eigenfunctions .ψ1,2 may be

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K. Campbell and R. Weikard

chosen so that .(ψ1± ) J ψ2± = 1. If, for a given .f ∈ L2 (w), we define −

u (x) =

.

ψ2− (x)

ˆ (−∞,x)

ψ1 wf

+ ψ1− (x)

ˆ [x,∞)

ψ2 wf,

then .u = u# satisfies the differential equation .J u + qu = w(λu + f ). Proof The first two statements follow from Theorem 3.3 and Lemma 3.2 in [5]. As before we shall write .B± in place of .B± (·, λ). Since .ψj = (ψj+ + ψj− )/2, + − .B+ ψ j = B− ψj , and .B+ + B− = 2J , our hypothesis on the normalization of .ψ1,2 shows that B+ (ψ2+ ψ1 − ψ1+ ψ2 ) = 1.

(4)

.

Since u+ (x) = ψ2+ (x)

ˆ

.

(−∞,x]

ψ1 wf + ψ1+ (x)

ˆ (x,∞)

ψ2 wf

equation (4) gives .B+ u+ − B− u− = Δw f and hence 1 J u− = B+ u# − Δw f. 2

.

(5)

Now recall that, by the definition of Lebesgue-Stieltjes measures, the derivatives of u− and u are the same. Hence the product rule for functions of locally bounded variation gives

.

J u = J ψ2

ˆ

.

(−∞,·)

ψ1 wf + J ψ1

ˆ [·,∞)

ψ2 wf + J (ψ2+ ψ1 − ψ1+ ψ2 )wf.

−1 Using the identity .J ψk = (λw − q)ψk = (λw − q)B+ J ψk− and Eqs. (4) and (5) we obtain now −1 −1 J u = (λw − q)B+ J u− + J B+ wf

.

1 −1 −1 = (λw − q)u# + (q − λw)B+ Δ w f + J B+ wf. 2 Since .

1 1 −1 −1 −1 (q − λw)B+ Δw f = (Δq − λΔw )B+ wf = (1 − J B+ )wf 2 2

we finally find that .J u + (q − λw)u# = wf .



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Next we show that the function u just constructed is actually an element of L2 (w). In the following we use the notation of Lemma 6.7 and its proof freely. First note that averaging .u+ and .u− gives

.

ˆ

x

u(x) = ψ2 (x)

.

ψ1 wf + ψ1 (x)

ˆ x

ψ2 wf

1 (6) + (ψ2+ (x)ψ1 (x) + ψ1− (x)ψ2 (x) )Δw (x)f (x) 2 ´x ´ ´ ´ where we introduced the notation . and . x as abbreviations for . (−∞,x) and . (x,∞) , respectively. Introduce .W˜ , the Radon-Nikodym derivative of w with respect to the positive scalar measure .tr w. Then .W˜ is a real symmetric matrix and .0 ≤ W˜ ≤ 1 pointwise almost everywhere with respect to .tr w and we may assume it holds indeed in .C2 which we denote by everywhere. Therefore .f ∗ W˜ (x)g is a semi-inner product √ f, f x . .f, gx . The corresponding semi-norm is .|f |x = ˜ p (x) we get .|ψ (x)| = ˜ > 0. Since .ψ1 (x) = emx Recall that .m = Re m 1 1 x mx −mx e |p1 (x)|x . Similarly .|ψ2 (x)|x = e |p2 (x)|x . Since .p1 and .p2 are periodic we may find a constant K such that .|ψj (x)|x ≤ K e±mx where the sign in the exponent is .(−1)j +1 . The norm of a rank one matrix .ab is the product of the norms of a and b. Hence we may estimate the .| · |x -norm of the last term in (6) by 2 .K |Δw (x)f (x)|x . However, the matrices .Δw (x) have to be uniformly bounded by, say, a number .D ≥ 1 since w is periodic and locally finite. Thus our estimate on the last term in (6) becomes .K 2 D|f (x)|x . To deal with the other terms in (6) we define ˆ φ1 (x) =

.

x

ψ1 wf

ˆ and

φ2 (x) = x

ψ2 wf.

Upon using Cauchy’s inequality for the .·, ·x -inner product we obtain the estimates |φj (x)| ≤ KHj (x) where

.

ˆ H1 (x) =

ˆ

x

.

e

my

|f (y)|y tr w(y)

and

H2 (x) =

e−my |f (y)|y tr w(y).

x

Therefore we find |u(x)|x ≤ K 2 e−mx H1 (x) + K 2 emx H2 (x) + K 2 D|f (x)|x .

.

Since .u2 =

´

|u(x)|2x tr w(x) and .Hj2 ≤ Hj2# we obtain ˆ

u ≤ 3K D

.

2

4

2



 e−2mx H12# (x) + e2mx H22# (x) + |f (x)|2x tr w(x).

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´ Next we estimate the integrals .I1 (a, b) = (a,b) e−2mx H12# (x) tr w(x) and ´ 2mx H 2# (x) tr w(x) assuming that a and b are points of continuity .I2 (a, b) = 2 (a,b) e of W , the antiderivative of w. To this end we define ˆ ˆ x .R− (x) = e−2my tr w(y) and R+ (x) = e2my tr w(y). x

We emphasize that .R± are positive functions and that .R+ is non-decreasing while # . Using the periodicity of w and Theorem 2.5 R− is non-increasing. Hence .R± ≤ R± we find .R+ (x) ≤ Cm e2mx and .R− (x) ≤ C−m e−2mx for appropriate constants # 2mx .C±m . The continuity of these upper bounds implies that even .R+ (x) ≤ Cm e # −2mx and .R− (x) ≤ C−m e . Now we see that, using integration by parts, .

ˆ

ˆ

I1 (a, b) = −

.

(a,b)

H12# dR− = −H1 (b)2 R− (b) + H1 (a)2 R− (a) +

(a,b)

# R− dH12 .

The first term on the right is negative and may be omitted. To deal with ´ a the second term we use Cauchy’s inequality to show that .H1 (a)2 ≤ R+ (a) |f |2y tr w(y). Hence .lima→−∞ H1 (a)2 R− (a) = 0. In the last term we use the chain rule .dH12 = 2H1# dH1 (see Vol. pert [8], Section 13.2 or Vol. pert and Hudjaev [9], Chapter 4, §6.3). Thus ˆ ˆ # # # . R− dH12 = 2 R− H1 dH1 (a,b)

(a,b)

ˆ ≤ 2C−m

(a,b)

e−mx H1# (x)|f (x)|x tr w(x).

Cauchy’s inequality and the fact that .Hj#2 ≤ Hj2# give next that I1 (a, b) ≤ H1 (a)2 R− (a) + 2C−m I1 (a, b)1/2 f .

.

Taking now the limit as a tends to .−∞ and b to .∞ shows that .I1 (−∞, ∞) ≤ 2 f 2 . Treating .I in a similar way we finally obtain 4C−m 2 2 2 u2 ≤ 3K 4 D 2 (4C−m + 4Cm + 1)f 2 .

.

This proves that .(u, λu+f ) ∈ Tmax , i.e., .λ ∈ . Equation (6) means that the Green’s function for .Tmax is given by

G(x, y) =

.

⎧  ⎪ ⎪ ⎨ψ2 (x)ψ1 (y)

1  2 (ψ2 (x)ψ1 (x) ⎪ ⎪ ⎩ψ (x)ψ (y) 1 2

if y < x + ψ1 (x)ψ2

(x) )

if y = x if x < y.

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235

We have now shown that .(S ∪ Λ)c ⊂ . Since .Λ is also a subset of . we have established the following lemma. Lemma 6.8 The complement .S c of the conditional stability set S is contained in . , the resolvent set of .Tmax . Next we will address the converse statement. Lemma 6.9 If .L0 is trivial the conditional stability set S is contained in .σ , the spectrum of .Tmax . Proof Suppose .λ ∈ S and, by way of contradiction, that it is also in . . Theorem 6.2 shows that .λ ∈ Λ. Hence there is a balanced, bounded Floquet solution .ψ of the differential equation .J ψ  + (q − λw)ψ = 0 with Floquet multiplier .ρ. Of course, .ψ = U (·, λ)c0 for some vector .c0 which is an eigenvector of .M(λ) associated with ∗ .ρ. Lemma 3.2 in [5] shows that .J u0 is an eigenvector of .M(λ) associated with the eigenvalue .1/ρ. Lemma 6.5 provides the existence of a function .g0 compactly supported in .[0, 2ω) satisfying ˆ .

[0,2ω)

U (·, λ)∗ wg0 = −J c0 .

For .n ∈ N we put .gn = ρ n g0 (· − nω), a function which is supported in .[nω, (n + 2)ω), and use the variation of constants formula to obtain a solution .vn of the equation .J u + qu = w(λu + gn ), i.e., − .vn (x)

−



= U (x, λ) c0 + J

−1

ˆ [0,x)

 U (·, λ)∗ wgn .

The function .vn coincides with .ψ on .(−∞, nω). Since ˆ ˆ U (·, λ)∗ wg0 . U (·, λ)∗ wgn = ρ n M(λ)∗n [0,(n+2)ω)

[0,2ω)

= −ρ n M(λ)∗n J c0 = −J c0 it follows that .vn is identically equal to 0 beyond .(n + 2)ω. With a similar device for the negative half-line we may now construct a sequence .(un , λun + fn ) ∈ Tmax with the following properties: (i) .un = 0 outside .[−(n + 2)ω, (n + 2)ω], (ii) .un = ψ on .(−nω, nω), and (iii) .fn  is independent of n. Since .|ρ| = 1 we have that ˆ 2 .un  ≥ ψ ∗ wψ = 2nC [−nω,nω)

where .C =

´

[0,ω) ψ

∗ wψ

does not depend on n.

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Arguing similarly to the conclusion in the proof of Theorem 5.2 we now have un  ≤ (T − λ)−1 fn  where the left-hand side tends to infinity while the righthand side is constant, the desired contradiction. 

.

Combining Theorem 6.4, Lemma 6.8, and Lemma 6.9 we have now the following result. Theorem 6.10 Assume that .L0 is trivial. Then the spectrum .σ of .Tmax is purely continuous and coincides with the conditional stability set S. We end this section by an investigation of the properties of the Floquet discriminant D. Suppose .x0 is a point of continuity for Q and W , the antiderivatives of q and w, and .U (·, λ) is a fundamental matrix for .J u + qu = λwu such that .U (x0 , λ) = 1. Observe that J U˙  (·, λ) + (q − λw)U˙ (·, λ) = wU (·, λ)

.

where the accent ˙denotes . the derivative with respect to .λ. Since .M = U (x0 + ω, ·) and .U˙ (x0 , ·) = 0 the variation of constants formula shows that ˙ M(λ) = U (x0 + ω, λ)J −1

ˆ

.

(x0 ,x0 +ω)

U (·, λ)∗ wU (·, λ) = M(λ)J −1 T (λ)

where the last equation defines the matrix T . Assume that .λ ∈ R. Then .M(λ) and .D(λ) are real and .T (λ) is real and positive semi-definite. Since .det M = 1 one may now check that .D 2 − 4 = (M11 − M22 )2 + 4M12 M21 . Moreover, setting .a = (M11 − M22 , 2M21 ) and .b = (−2M12 , M11 − M22 ) one obtains 4rM21 D˙ = −T11 (D 2 − 4) + a ∗ T a and 4rM12 D˙ = T22 (D 2 − 4) − b∗ T b.

.

(7)

Now suppose that .D(λ)2 − 4 < 0. It is then clear that .M12 (λ) and .M21 (λ) are ˙ different from 0 and thus the above identities show that .D(λ) = 0 provided that at least one of .T11 (λ) and .T22 (λ) is different from 0. But .T11 (λ) = T22 (λ) = 0 may only happen when .w = 0, a case excluded by our hypothesis. Lemma 6.11 The Floquet discriminant D is constant if and only if .Tmax equals {0} × L2 (w). Moreover, if .dim L0 = 1, D does not take its value in .(−2, 2).

.

Proof Suppose D is constant. If .D 2 > 4, then .R ⊂ S c ⊂ , the resolvent set of .Tmax . Since the spectrum is empty the domain of .Tmax must be .{[0]}. Since .Tmax is self-adjoint its range must be .L2 (w). If .D 2 ≤ 4, then .S = C \ Λ in view of the analyticity of D. If .L0 were trivial, Lemma 6.9 would show that .σ = C. Now Theorem 6.6 shows that .Tmax = {0} × L2 (w). Conversely, if .Tmax = {0} × L2 (w) and, consequently, .L0 = {0}, there exists a non-trivial solution of .J u + qu = 0 with norm 0. Lemma 6.3 shows that then at least one of the Floquet solutions must have norm 0. Such a solution is then a

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solution of .J u + qu = λwu for all .λ. If .ρ is the associated Floquet multiplier, then .D(λ) = ρ + 1/ρ for all .λ ∈ C. Finally, suppose .dim L0 = 1 and, by way of contradiction, that D is in .(−2, 2). Then at least one of the diagonal entries of the matrix T in (7) is positive. Thus ˙ .D(λ) 

= 0 (for .λ ∈ R), a contradiction. Theorem 6.12 Suppose .L0 is trivial. Then, the Floquet discriminant D, restricted to the real line, is strictly monotone in intervals where .D 2 < 4. If .D(λ)2 = 4, we ˙ have .D(λ) = 0 if and only if .M(λ) = ±1. Such a point is not a strict local minimum if .D(λ) = 2 nor a strict local maximum if .D(λ) = −2. Proof We already mentioned above that .λ → D(λ) is real-valued and analytic as .λ ˙ varies in .R. We have also proved that .D(λ) = 0 when .D(λ)2 < 4. 2 ˙ Now, assume .D(λ) = 4 and .D(λ) = 0. Then (7) shows that .a ∗ T a = b∗ T b = 0. Since .L0 is trivial T is positive definite and hence a and b are 0 proving .M(λ) = ˙ ±1. Conversely, if .M(λ) = ±1, then .D(λ) = ± tr(J −1 T (λ)) = 0. Finally, if D has a strict local minimum of 2 at a point .λ, then .λ is an isolated point in the spectrum of .Tmax and hence an eigenvalue which is impossible. Of course, a similar argument prevents D to have a local maximum of .−2 anywhere. 

7 Examples   In all examples presented below we assume that .n = 2, .J = 10 −1 and .ω = 1 0 (except in our last example). Our first three examples have constant coefficients. √  0    • .q = 00 −1 and .w = 10 00 . Here we have .L0 = {0} and .D(λ) = 2 cos( λ) = √ 2 cosh −λ. Hence .S = σ = [0, ∞). This is the system describing the second  orderequation .−y = λy. √ √  a b • .q = b d and .w = 0. We get .D = 2 cosh b2 − ad = 2 cos ad − b2 . Hence D is constant (as anticipated) and may take any value in .[−2, ∞). If .ad − b2 > 0 we have   .S = R but .σ = ∅. • .q = ab b0 and .w = 10 00 . Now .L0 is spanned by the vector .(0, ebx ) . We have .D = 2 cosh b. Note that D cannot be in .(−2, 2). These last two examples show that the hypothesis of a trivial .L0 is necessary in Lemma 6.9.  Our next three examples involve discrete measures as coefficients. Let .μ = k∈Z δk where .δk is the Dirac measure concentrated on .{k}.   αμ 0  0  • .q = aμ . If .α > 0 this problem is definite, i.e., .L0 is trivial. 0 −1 and .w = 0 0 The discriminant is .D(λ) = 2 + a − αλ. Hence .σ = [a/α, (4 + a)/α]. However, if .α = 0, we have .dim L0 = 2. In this case D is constant and the constant may take any real value.

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    • .q = ab b0 μ and .w = 10 00 μ where .b2 = 4 to avoid an intersection of .Λ and .R. Here .L0 is one dimensional. We get .D = 2(4 + b2 )/(4 − b2 ) which may attain any value  outside .[−2, 2). • .q = ab db μ and .w = 1μ where we require .(a − d)2 + 4b2 < 16 to avoid that .Λ intersects .R. Then D(λ) =

.

λ2

16 − 2. − (a + d)λ + ad − b2 + 4

Our condition on a, b and d guarantees that the denominator is always positive. Thus .D(λ) > −2 for all .λ ∈ R but .D(λ) approaches .−2 as .λ tends to .±∞. There is one maximum for D at .λ = (a + d)/2. Its value is at least 2 and is equal to 2 precisely when .a = d and .b = 0. Except for this special situation the spectrum consists of two rays separated by one gap. Finally we mention one example which violates our condition that .Λ ∩ R = ∅. The example is taken from [7] where more details may be found.   2 0 • .q = 02 20 k∈Z (δ2k − δ2k+1 ) and .w = 0 0 k∈Z δk . Note that here .ω = 2. In this case .Λ = C, i.e., unique continuation of solutions fails for any .λ. Since there are compactly supported solutions of .J u + qu = 0, it turns out that 0 is an eigenvalue of infinite multiplicity. The resolvent set is .C \ {0}.

References 1. B.M. Brown, M.S.P. Eastham, K.M. Schmidt, in Periodic Differential Operators, vol. 230 of Operator Theory: Advances and Applications (Birkhäuser/Springer Basel AG, Basel, 2013) 2. K. Campbell, M. Nguyen, R. Weikard, On the spectral theory for first-order systems without the unique continuation property. Linear Multilinear Algebra 69(12), 2315–2323 (2021). Published online: 04 Oct 2019 3. M.S.P. Eastham, in The Spectral Theory of Periodic Differential Equations. Texts in Mathematics (Edinburgh) (Scottish Academic Press, Edinburgh; Hafner Press, New York, 1973) 4. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. de l’Ecole Norm. Superieure 12, 47–88 (1883). 2e série 5. A. Ghatasheh, R. Weikard, Spectral theory for systems of ordinary differential equations with distributional coefficients. J. Differ. Equ. 268(6), 2752–2801 (2020) 6. H. Hochstadt, in Differential Equations (Dover Publications, New York, 1975). A modern approach, Republication, with minor corrections, of the 1964 original 7. S. Redolfi, R. Weikard, Green’s functions for first-order systems of ordinary differential equations without the unique continuation property. Integr. Equ. Oper. Theory 94, 16 (2020) 8. A.I. Vol’pert, Spaces BV and quasilinear equations. Mat. Sb. (N.S.) 73 (115), 255–302 (1967) 9. A.I. Vol’pert, S.I. Hudjaev, in Analysis in Classes of Discontinuous Functions and Equations of Mathematical physics, vol. 8 of Mechanics: Analysis (Martinus Nijhoff Publishers, Dordrecht, 1985)

Asymptotic Analysis of Operator Families and Applications to Resonant Media Kirill D. Cherednichenko, Yulia Yu. Ershova, Alexander V. Kiselev, Vladimir A. Ryzhov, and Luis O. Silva

In memoriam Sergey Naboko

Abstract We give an overview of operator-theoretic tools that have recently proved useful in the analysis of boundary-value and transmission problems for secondorder partial differential equations, with a view to addressing, in particular, the asymptotic behaviour of resolvents of physically motivated parameter-dependent operator families. We demonstrate the links of this rich area, on the one hand, to functional frameworks developed by S. N. Naboko and his students, and on the other hand, to concrete applications of current interest in the physics and engineering communities. Keywords Dissipative operators · Functional models · Generalised resolvents · Quantum graphs · Inverse scattering problem · Zero-range models · Resonant media · Homogenisation · Wave propagation

K. D. Cherednichenko () Department of Mathematical Sciences, University of Bath, Bath, UK e-mail: [email protected] Yu. Yu. Ershova Department of Mathematics, Texas A&M University, College Station, USA e-mail: [email protected] A. V. Kiselev Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, UK e-mail: [email protected] L. O. Silva Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Ciudad de México, México e-mail: [email protected] V. A. Ryzhov Unity Technologies, San Francisco, CA, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_12

239

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1 Introduction It has transpired recently that a number of operator-theoretical techniques which have been under active development for the past 60 years or so are extremely useful in the asymptotic analysis of highly inhomogeneous media. Apart from yielding sharp asymptotics of the corresponding Hamiltonians in the norm-resolvent topology, this research has resulted in a number of new important, yet mostly unexplored, connections between certain areas of the modern operator and spectral theory. These include the theory of dilations and functional models of dissipative and non-selfadjoint operators in Hilbert spaces, the boundary triples theory in the analysis of symmetric operators, zero-range models with internal structure and, finally, the theory of generalised resolvents and their out-of-space “dilations”. The present survey, based on our results published in [42–49, 82, 133, 134, 136], attempts to shed some light on these connections and to thus present the subject area of strongly inhomogeneous media under the spotlight of modern spectral theory. We aim to show that in many ways this novel outlook allows one to gain a better understanding of the mentioned area by providing a universal abstraction layer for all the main objects to be found in the asymptotic analysis. Moreover, in most cases one can then proceed in the analysis on a purely abstract level surprisingly far, essentially postponing the use of the specific features of the problem at hand till the very last stages. For readers’ convenience, we have included a rather detailed exposition of the relevant areas of operator and spectral theory, keeping in mind that some papers laying the foundations of these areas have been poorly accessible to date. We start with Sect. 2, devoted to the now-classical theory of dilations of dissipative operators. The role of dissipative operators as opposed to self-adjoint ones is that whereas the latter represent physical systems with the energy conservation law (“closed”, or conservative, systems), the former allow for the consideration of a more realistic setup, where the loss of the total energy is factored in. The importance of dissipative systems has been a common place since at least the works of I. Prigogine; it is well-known that such systems may possess certain rather unexpected properties. The main difference between the self-adjoint and dissipative theories can be clarified, following M. G. Kre˘ın, as follows: the major instruments of self-adjoint spectral analysis arise from the Hilbert space geometry, whereas this geometry doesn’t work very well in the non-selfadjoint situation, with modern complex analysis taking the role of the main tool in the investigation. Since the seminal contribution of B. Sz.-Nagy and C. Foia¸s, the main object of dissipative spectral analysis has been the so-called dilation, representing an out-ofspace self-adjoint extension, in the sense of (2.1) below, of the original dissipative operator L. Our argument actually goes as far as to suggest that this concept underpins the whole set of ideas and notions presented in the paper. B. S. Pavlov’s explicit construction of dilation relies upon the second major ingredient, which is the characteristic function .S(z), see (2.5), which is an analytic operator-valued contraction in .C+ . The analysis of the dissipative operator L is reduced to the study

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of the function .S(z), and hence from this point onward it belongs to the domain of complex analysis. It turns out that the sole knowledge of .S(z) yields an explicit spectral representation of the dilation. Moreover, S. N. Naboko has shown that, in the same representation, a whole family of operators “close” to L, self-adjoint and non-selfadjoint alike, are modelled in an effective way. This idea in particular led to the description of absolutely continuous subspaces of all the operators considered as the closure of the so-called smooth vectors set. This latter is characterised as the collection of vectors such that the resolvent of the operator in question in the spectral representation maps them as the multiplication operator. An explicit construction of wave operators and scattering matrices then follows almost immediately. In Sect. 2.3, we give a systematic exposition of this approach applied to the family of extensions of a symmetric densely defined operator on a Hilbert space H possessing equal deficiency indices. In so doing, we follow closely the strategy suggested by Sergey Naboko which he had applied in the analysis of additive relatively bounded perturbations of self-adjoint operators. We thus hope to provide a coherent presentation of the major contribution by Naboko to the spectral analysis of non-selfadjoint operators. Our analysis is facilitated by the boundary triples theory, being an abstract framework, from which the extensions theory of symmetric operators, especially differential operators, greatly benefits. That is why we start our exposition by introducing the main concepts of this theory. The formula obtained for the scattering operator in the functional model representation allows us to derive an explicit formula for the scattering matrix, formulated in terms of Weyl-Titchmarsh Mmatrices, i.e., in the natural terms associated with the problem. In Sect. 3, we consider an application of this technique to an inverse scattering problem on a quantum graph, where we are able to give an explicit solution to the problem of reconstructing matching conditions at graph vertices. In Sect. 2.4, we consider the possible generalisation of the approach described above to the case of partial differential operators (PDO), associated with boundary value problems (BVP). Although the theory of boundary triples has been successfully applied to the spectral analysis of BVP for ordinary differential operators and related setups, in its original form this theory is not suited for dealing with BVP for partial differential equations (PDE), see [35, Section 7] for a relevant discussion. Recently, when the works [18, 35, 69, 74, 75, 136] started to appear, it has transpired that, suitably modified, the boundary triples approach nevertheless admits a natural generalisation to the BVP setup, see also the seminal contributions by J. W. Calkin [38], M. S. Birman [27], L. Boutet de Monvel [31], M. S. Birman and M. Z. Solomyak [28], G. Grubb [73], and M. Agranovich [7], which provide the analytic backbone for the related operator-theoretic constructions. In all cases mentioned above, one can see the fundamental rôle of a certain Herglotz operator-valued analytic function, which in problems where a boundary is present (and sometimes even without an explicit boundary [13]) turns out to be a natural generalisation of the classical notion of a Dirichlet-to-Neumann map. Moreover, it is precisely this object that permits to define the characteristic function which in turn facilitates the functional model construction.

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In Sect. 4, we pass over to the discussion of zero-range models with internal structure. The idea of replacing a model of short-range interactions by an explicitly solvable one with a zero-radius potential (possibly with internal structure) [24, 26, 37, 87, 88, 127, 157] has paved the way for an influx of methods of the theory of extensions (both self-adjoint and non-selfadjoint) of symmetric operators to problems of mathematical physics. In particular, we view zero-range potentials with internal structure as a particular case of out-of-space self-adjoint extensions of symmetric operators, the theory of which is intrinsically related to the analysis of generalised resolvents. The latter area is introduced in Sect. 2.5. We argue that the theory of out-of-space self-adjoint extensions corresponding to generalised resolvents naturally supersedes the dilation theory as presented in Sect. 2.2. On yet another level, we claim that zero-range perturbations (and more precisely, zero-range perturbations with internal structure) appear naturally as the normresolvent limits of Hamiltonians in the asymptotic analysis of inhomogeneous media. This relationship is established using the apparatus of generalised resolvents, as explained in Sect. 4.4. Finally, we mention here that the theory of functional models as presented in Sect. 2 is directly applicable to the treatment of models of zero-range potentials with internal structure. Its development yields a complete spectral analysis and an explicit construction of the scattering theory for the latter. Two different models are considered in Sect. 4, one of these being a onedimensional periodic model with critical contrast, unitary equivalent to the double porosity one. The PDE counterpart of the latter is discussed in Sect. 5. The second mentioned model pertains to the problem with a low-index inclusion in a homogeneous material. Our argument shows that the leading order term in the asymptotic expansion of its resolvent admits the same form as expected of a zerorange model; the difference is that here the effective model of the media is no longer “zero-range” per se; rather it pertains to a singular perturbation supported by a manifold. Therefore, this allows us to extend the notion of internal structure to the case of distributional perturbations supported by manifolds. The discussion started in Sect. 4 is then continued in Sect. 5. We note that in every model considered so far, the internal structure of the limiting zero-range model is necessarily the simplest possible, i.e., pertains to the out-of-space extensions defined on .H ⊕ C1 , where H is the original Hilbert space. It turns out that this is due to the fact that we only consider norm-resolvent convergence when the spectral parameter z is restricted to a compact set in .C. Passing over to a generic setup with z not necessarily in a compact, we are able to claim that in some sense the internal structure can be arbitrarily complex, provided that the spectral parameter z is allowed to grow with the large parameter a, which describes the inhomogeneity, increasing to .+∞. This allows us to present an explicit example of a non-trivial internal structure in the leading order term of the normresolvent asymptotics in Sect. 4.4, which is supplemented with the discussion of the so-called scaling regimes which we introduce in Sect. 5.1. The remainder of Sect. 5 is devoted to the analysis of a double porosity model of high-contrast homogenisation, where the leading order term of the asymptotic

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expansion is obtained by an application of the operator-theoretical technique based on the generalised resolvents, as in Sect. 4.

2 Functional Models for Dissipative and Nonselfadjoint Operators Functional model construction for a contractive linear operator T acting on a Hilbert space H is a well developed domain of the operator theory. Since the pioneering works by M. S. Livšic [97, 99, 101], M. S. Brodski˘ı [33], B. Sz.-Nagy, C. Foia¸s [149], followed by the further research of P. D. Lax, R. S. Phillips [95], L. de Branges, and J. Rovnyak [52, 53], this research field has attracted many specialists in operator theory, complex analysis, system control, Gaussian processes and other disciplines. Multiple studies culminated in the development of a comprehensive theory complemented by various applications, see [12, 56, 67, 117– 119, 122] and references therein. The underlying idea of a functional model is the fundamental theorem of B. Sz.Nagy and C. Foia¸s establishing the existence of a unitary dilation for any contractive (linear) Hilbert space operator T , .T  ≤ 1; see also earlier constructions of triangular models for non-selfadjoint operators [33, 100, 101] as well as the recent book [104]. The unitary dilation U of T is a unitary operator on a Hilbert space .H ⊃ H such that .PH U n |H = T n for all .n = 1, 2, . . . . Here .PH : H → H is an orthoprojection from .H to its subspace H . The dilation U is called minimal if the linear set .∨n>0 U n H is dense in .H. The minimal dilation U of a contraction is unique up to unitary equivalence. The spectrum of U is absolutely continuous and covers the unit circle .T = {z ∈ C : |z| = 1}. If one denotes by .μ the spectral measure of U , the spectral theorem yields that operator T is unitarily equivalent to its model .T = PH z|H , where .f → zf is the operator of multiplication on the spectral representation space .L2 (T, μ) of U . Due to its abstract nature, a significant part of the functional model research for contractions took place among specialists in complex analysis and operator theory. The parallel theory for unbounded operators is based on the Cayley transform .T → −i(T + I )(T − I )−1 applied to a contraction T . Assuming that clos ran .(T − I ) = H , the Cayley transform of T is a dissipative densely defined operator L not necessarily bounded in H . It is easily seen that the spectrum of L is situated in the closed upper half plane .C+ of the complex plane. The imaginary part of L (defined in the sense of forms if needed) is a non-negative operator. Alongside the developments in operator theory, the second half of the twentieth century witnessed a huge progress in the spectral analysis of linear operators pertaining to physical disciplines. The principal tool of this was the method of Riesz projections, i.e., the contour integration of the operator’s resolvent in the complex plane of spectral parameter. The spectral analysis of self-adjoint operators of quantum mechanics can be viewed as the prime example of highly successful

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application of contour integration in the study of conservative systems, i.e., closed systems with the energy preserved in the course of evolution. Topical questions concerning the behavior of non-conservative systems, where the total energy is not preserved, and of resonant systems motivated in-depth studies of (unbounded) non-selfadjoint operators. The analysis of non-conservative systems and of non-selfadjoint operators especially relevant to the functional model theory was pioneered in the works of M. S. Brodskiˇı, M. S. Livšic and their colleagues, see [33, 34, 101] and references therein. Starting with a (bounded) self-adjoint operator .A = A∗ as the main operator of a closed conservative system, these authors considered the coupling of this system to the outside world by means of externally attached channels. This construction represents a model of the so-called “open system”, that is, of a physical system connected to its external environment. The energy of such modified system can dissipate through the external channels, while at the same time the energy can be fed into the system from the outside in the course of its evolution. In works of M. S. Brodskiˇı and M. S. Livšic, the external channels are modelled as an additive perturbation of the main self-adjoint operator A by a (bounded) non-selfadjoint perturbation: .A → L = A + iV , ∗ .V = V . The “channel vectors” form the Hilbert space .E = clos ran |V |. If .V ≥ 0, the operator L is dissipative (i.e., .Im(Lu, u) > 0, .u ∈ H ); it describes a nonconservative system losing the total energy. In turn, and quite analogously to the case of contractions, under the assumption .C− ⊂ ρ(L) (recall that dissipative operators satisfying this condition are called maximal), the self-adjoint dilation of L is a selfadjoint operator .L on a wider space .H ⊃ H such that (L − zI )−1 = PH (L − zI )−1 |H ,

.

z ∈ C− ,

(2.1)

where .PH is an orthogonal projection from .H onto H . The operator .L describes a (larger) system with the state space .H, in which the energy is conserved, whereas L describes its subsystem losing its total energy. In the general case, a non-dissipative L corresponds to an open system where both the energy loss and the energy supply coexist. The analysis of a non-selfadjoint operator L relies on the notion of its characteristic function discovered by M. S. Livšic [97] and later, in a different form, by A. V. Strauss [146]. It is a bounded analytic operator-function .(z), .z ∈ ρ(L∗ ) defined on the resolvent set of .L∗ and acting on the “channel vectors” from the space E. For dissipative L the function . coincides with the characteristic function of a contraction .T = (L − iI )(L + iI )−1 (the inverse Cayley transform of L), featured prominently in the works by B. Sz.-Nagy and C. Foia¸s. The characteristic function of a non-selfadjoint operator L (or, alternatively, of its Cayley transform) determines the original operator L uniquely up to a unitary equivalence (see [99, 149]), provided L has no non-trivial self-adjoint “parts”. Therefore, the study of non-selfadjoint operators is reduced to the study of operator-valued analytic functions. In and of itself, this does not mean much as these functions might be as complicated as the operators themselves. A simplification is achieved when the values of these functions are either matrices or belong to Schatten-von Neumann classes of compact operators, which is often the case in physical applications.

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Closely related to the Sz.-Nagy-Foia¸s model for contractions and to the open systems framework are the Lax-Phillips scattering theory [95] and the “canonical model” due to L. de Branges and J. Rovnyak [53]. The latter is developed for completely non-isometric contractions and their adjoints with quantum-mechanical applications in mind. The Lax-Phillips theory was originally developed to facilitate the analysis of scattering problems for hyperbolic wave equations in exterior domains to compact scatterers. It provides useful intuition into the underpinnings of the functional model construction and this connection will be exploited in the next section. It was realized very early [2] that the three theories, i.e., the open systems theory, the Sz.-Nagy-Foia¸s model, and the Lax-Phillips scattering, all deal with essentially the same objects. In particular, the characteristic function of a contraction (or of a dissipative operator) emerges, albeit under disguise, in all three theories. Being a purely theoretical abstract object in the Sz.-Nagy-Foia¸s theory, the characteristic function emerges as a transfer function of a linear system according to M. S. Brodskiˇı and M. S. Livšic, and as the scattering matrix in the Lax-Phillips theory. It has to be stressed that the characteristic function of M. S. Livšic is a useful tool in the analysis of concrete problems in system theory, in the theory of operator colligations, and in scattering theory. As a matter of fact, M. S. Livšic was the first to establish the connection between the Heisenberg scattering matrix and the characteristic function of a relevant dissipative operator [100]. The characteristic function of a contraction is also the central component in the L. de Branges and J. Rovnyak model theory [53]. Deep connections between the Sz-NagyFoia¸s and the de Branges-Rovnyak models are clarified in a series of papers by N. Nikolskii and V. Vasyunin [120–122]. For more information related to the subject of characteristic functions, see [32, 98, 102, 103, 152–154].

2.1 Lax-Phillips Theory The Lax-Phillips scattering theory [95] for the acoustic waves by a smooth compact obstacle in .Rn with .n ≥ 3 odd provides an excellent illustration of the intrinsic links between the operator theory and mathematical physics. A number of concepts found in the theory of functional models of dissipative operators find their direct counterparts here, expressed in the language of realistic physical processes. For instance, the characteristic function of the operator governing the scattering process is realized as the scattering matrix, the self-adjoint dilation corresponds to the operator of “free” dynamics, i.e., the wave propagation process observed in absence of the obstacle, and the scattering channels are a direct analogue of the channels found in the Brodskiˇı-Livšic constructions. In this section we briefly recall the main concepts of Lax-Phillips scattering theory. Let .H be a Hilbert space with two mutually orthogonal subspaces .D± ⊂H , .D− ⊕ D+ = H. Denote by .K the orthogonal complement of .D− ⊕ D+

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in .H. Assume the existence of a single parameter evolution group of unitary operators .{U (t)}t∈R with the following properties

.

U (t)D− ⊆ D− ,

t ≤ 0,

U (t)D+ ⊆ D+ ,

t ≥ 0, (2.2)

∩t∈R U (t)D± = {0}, clos ∪t∈R U (t)D± = H.

In the acoustic scattering, the space .H consists of solutions to the wave equation (i.e., acoustic waves) and is endowed with the energy norm. The group .U (t) describes the evolution of “free” waves in .H, that is, the group .U (t) maps the Cauchy data of solutions at the time .t = 0 to their Cauchy data at the time t. Since .U (t) is unitary for all .t ∈ R, the energy of solutions is preserved under the time evolution .f = U (0)f → U (t)f , for any Cauchy data .f ∈ H. Correspondingly, the infinitesimal generator .B of .U (t) is a self-adjoint operator in .H with purely absolutely continuous spectrum covering the whole real line. The subspaces .D± are called incoming and outgoing subspaces of .U (t). These terms “incoming” and “outgoing” are well justified. Indeed, the subspace .D− consists of solutions that do not interact with the obstacle prior to the moment .t = 0, whereas .D+ consists of scattered waves which do not interact with the obstacle after .t = 0. There exist two representations for the generator .B associated with .D± (the so-called incoming and outgoing translation representations), in which the group .U (t) acts as the right shift operator .U (t) : u(x) → u(x − t) on .L2 (R, E) with some auxiliary Hilbert space E. In these representations the subspaces .D± are mapped to .L2 (R± , E). It is not difficult to see that this construction satisfies the assumptions (2.2). Denote by .P± : H → [D± ]⊥ the orthogonal projections to the complements of .D± in .H. It is worth mentioning that the translation representations appear in [142] and can be deduced from the celebrated von Neumann uniqueness theorem [158]. The elements of .K are the scattering waves that are neither incoming in the past, nor outgoing in the future, i.e., the waves localized in the vicinity of the obstacle. The interaction of incoming waves with the obstacle, i.e., the scattering process, is then described by the compression of the group .U (t) to the neighborhood of the obstacle: Z(t) = P+ U (t)P− = PK U (t)PK ,

.

t ≥ 0.

Here .PK = P− P+ is an orthoprojection on .H. The operator family .{Z(t)}t≥0 forms a strongly continuous semigroup acting on .K. Since .Z(t) is a compression of the unitary group, it is clear that .Z(t) ≤ 1 for all .t ≥ 0. The infinitesimal generator B of the semigroup .{Z(t)}t≥0 turns out to be a linear operator with purely discrete spectrum .{λk } with .Re(λk ) < 0, .k = 1, 2, . . . . The poles .{λk } of the resolvent

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of B and the corresponding eigenvectors are interpreted as the scattering resonances. These resonances correspond to the poles of the scattering matrix defined as an operator-valued function acting in the space .L2 (R− , E), i.e., the space of vector functions taking values in E. The scattering matrix is mapped by the Fourier transform to the analytic in the lower half-plane operator function .S(z), .z ∈ C− with zeroes at .zk = −iλk . The boundary values .S(k − i0) on the real axis exist almost everywhere in the strong operator topology and are unitary for almost all .k ∈ R. The function .S(z) permits an analytic continuation .S(z) = [S ∗ (¯z)]−1 to the upper half-plane, where it is meromorphic. The results of [2] show that .θ (z) := S ∗ (¯z) coincides with the Livšic characteristic function of a dissipative operator being unitary equivalent to the infinitesimal  generator of .Z(t). Consequently, .θ (z + i)/(z − i) is the characteristic function of its Cayley transform as defined by B. Sz.-Nagy and C. Foia¸s [149]. Finally, the resolvents of operators .L = −iB and .L = −iB satisfy the dilation equation (2.1). In other words, the operator corresponding to the free dynamics is the self-adjoint dilation of the dissipative operator that governs the scattering process.

2.1.1

Minimality, Non-selfadjointness, Resolvent

The beautiful geometric interpretation of scattering processes provided by the LaxPhillips theory is not entirely transferable to the modelling of a general dissipative (or contractive) operator. For instance, given an arbitrary dissipative operator L on a Hilbert space K, its selfadjoint dilation .L is not available a priori and must be explicitly constructed first. In addition, such a dilation .L should be minimal, that is, it must contain no reducing self-adjoint parts unrelated to the operator L. Mathematically, the minimality condition is expressed by the equality .

clos



(L − zI )−1 |K = H

z∈R /

where .H is the dilation space .H ⊃ K. The construction of a dilation satisfying this condition is a highly non-trivial task which was successfully addressed for contractions by B. Sz.-Nagy and C. Foia¸s [149], aided by a theorem of M. A. Na˘ımark [116], and then by B. Pavlov [125, 126] in two important cases of dissipative operators arising in mathematical physics. Later, this construction was generalized to a generic setting. We dwell on this further in the following sections. For obvious reasons, the functional model theory of non-selfadjoint operators deals with operators possessing no non-trivial reducing self-adjoint parts. Such operators are called completely non-selfadjoint or, using a somewhat less accurate term, simple. The rationale behind this condition is easy to illustrate within the Lax-Phillips framework. Let the dissipative operator .L = −iB governing the wave dynamics in a vicinity of an obstacle possess a non-trivial self-adjoint part. This part is then a self-adjoint operator acting on the subspace spanned by the eigenvectors

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of L corresponding to its real eigenvalues. The restriction of .Z(t) to this subspace is an isometry for all .t ∈ R, and the energy of these states remains constant (recall that the space K is equipped with the energy norm). Therefore these waves stay in a bounded region adjacent to the obstacle at all times. These bound states do not participate in scattering, as they are invisible to the scattering process describing the asymptotic behaviour as .t → ±∞, so that the matrix S (or the characteristic function of L) contains no information pertaining to them. In operator theory, this is known as the claim that the characteristic function of a dissipative operator is oblivious to its self-adjoint part. It is also well-known, that the characteristic function uniquely determines the completely non-selfadjoint part of a dissipative operator. The interaction dynamics of the Lax-Phillips scattering supposes neither minimality nor complete non-selfadjointness. One can envision incoming waves that do not interact with the obstacle during their evolution and eventually become outgoing as .t → +∞. It is also easy to conceive of trapping obstacles preventing waves from leaving the neighbourhood of the obstacle at .t → +∞. The geometry of such obstacles cannot be fully recovered from the scattering data because, as explained above, these standing waves do not participate in the scattering process. In the applications discussed below, unless explicitly stated otherwise, all nonselfadjoint operators are assumed closed, densely defined with regular points in both lower and upper half planes. The latter condition can be relaxed but is adopted in what follows for the sake of convenience.

2.2 Pavlov’s Functional Model and Its Spectral Form Functional models for prototypical dissipative operators of mathematical physics (as opposed to the model for contractions), alongside explicit constructions of selfadjoint dilations, were investigated by B. Pavlov in his works [124–126]. Two classes of dissipative operators were considered: the Schrödigner operator in .L2 (R3 ) with a complex-valued potential, and the operator generated by the differential expression .−y  +q(x)y on the interval .[0, ∞) with a dissipative boundary condition at .x = 0. In both cases the self-adjoint dilations are constructed explicitly in terms of the problem at hand, and supplemented by the model representations known today as “symmetric” and commonly referred to as the Pavlov’s model. The results of [124– 126] were extensively employed in various applications and provided a foundation for the subsequent constructions of self-adjoint dilations and functional models for general non-selfadjoint operators.

2.2.1

Additive Perturbations [124, 125]

Let .A = A∗ be a selfadjoint unbounded operator on a Hilbert space K and V a bounded non-negative operator .V = V ∗ = α 2 /2 ≥ 0, where .α := (2V )1/2 .

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The paper [125] studies the dissipative Schrödinger operator .L = A + (i/2)α 2 in .R3 defined by the differential expression .− + q(x) + (i/2)α 2 (x) with real continuous functions q and .α such that .0 ≤ α ≤ C < ∞. The operators .A = − + q and .V = α 2 /2 are the real and imaginary parts of L defined on .dom(L) = dom(A). Assuming the operator L has no non-trivial self-adjoint components and the resolvent set of L contains points in both upper and lower half planes, the operator L is a maximal completely non-selfadjoint, densely defined dissipative operator on K. According to the general theory, there exists a minimal dilation of L, which is a self-adjoint operator .L on a Hilbert space .H ⊃ K such that (L − zI )−1 = PK (L − zI )−1 |K ,

.

z ∈ C− ,

(2.3)

where .PK is the orthogonal projection from .H onto its subspace K. The dilation constructed in [125] closely resembles the generator .B of the unitary group .U (t) in the Lax-Phillips theory: Pavlov realised that a natural way to construct a self-adjoint dilation would be to add the missing “incoming” and “outgoing” energy channels to the original non-conservative dynamics, thus mimicking the starting point of Lax and Phillips. The challenge here is to determine the operator that describes the “free” evolution of the dynamical system given only its “internal” part, thus in some sense “reversing” the Lax-Phillips approach. Denote .E := clos ran α and define the dilation space as the direct sum of K and the equivalents of incoming and outgoing channels .D± = L2 (R± , E), H = D− ⊕ K ⊕ D+

.

Elements of .H are represented as three-component vectors .(v− , u, v+ ) with .v± ∈ D± and .u ∈ K. The Lax-Phillips theory suggests that the dilation .L restricted to .D− ⊕ {0} ⊕ D+ should be the self-adjoint generator A of the continuous unitary group of right shifts .exp(iAt) = U (t) : v(x) → v(x − t) in .L2 (R, E). By Stone’s theorem, one has   iAv = lim t −1 [U (t)v − v] = lim t −1 v(x − t) − v(x) = −v  (x),

.

t↓0

t↓0

so that the generator of .U (t) is the operator .A : v → idv/dx. Hence, the action  , 0, iv  ). The selfof .L on the channels .D± is defined by .L : (v− , 0, v+ ) → (iv− + adjointness of .L = L ∗ and the requirement (2.3) yield the form of dilation .L as found in [125], ⎛

⎞ dv− i ⎛ ⎞ ⎜ ⎟ dx v− ⎜ ⎟ α ⎜ ⎟ ⎝ ⎠ .L u = ⎜Au + [v+ (0) + v− (0)]⎟ , ⎜ ⎟ 2 ⎝ ⎠ v+ dv+ i dx

(2.4)

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defined on the domain  1 . dom(L ) = (v− , u, v+ ) ∈ H | v± ∈ W2 (R± , E), u ∈ dom(A), v+ (0) − v− (0) = iαu} Embedding theorems for the Sobolev space .W21 guarantee the existence of boundary values .v± (0). The “boundary condition” .v+ (0) − v− (0) = iαu can be interpreted as a concrete form of coupling between the incoming and outgoing channels .D± realized by the imaginary part of L acting on .E. When .α = 0, the right hand side of (2.4) is the orthogonal sum of two selfadjoint operators, that is, the operator A on K and the operator .id/dx acting in the orthogonal sum of channels .L2 (R, E) = D− ⊕ D+ . The characteristic function of L is the contractive operator-valued function defined by the formula S(z) = IE + iα(L∗ − zI )−1 α : E → E,

.

z ∈ C+

(2.5)

According to the fundamental result of Adamyan and Arov [2], the function S coincides with the scattering operator of the pair .(L , L0 ) where .L0 is defined by (2.4) with .u = 0 and .α = 0.

2.2.2

Extensions of Symmetric Operators [126]

Consider the differential expression y = −y  + q(x)y

.

in .K = L2 (R+ ) with a real function q such that the Weyl limit point case takes place. Denote by .ϕ and .ψ the standard solutions to the equation .y = zy with .z ∈ C+ , satisfying the boundary conditions ϕ(0, z) = 0,

.

ϕ  (0, z) = −1,

ψ(0, z) = 1,

ψ  (0, z) = 0

Then the Weyl solution .χ = ϕ + m∞ (z)ψ ∈ L2 (R+ ), where .m∞ (z) is the Weyl function pertaining to . and corresponding to the boundary condition .y(0) = 0, is defined uniquely. The function .m∞ (z) is analytic with positive imaginary part for .z ∈ C+ . Define the operator L in .K = L2 (R+ ) by the expression . supplied with the non-selfadjoint boundary condition at .x = 0 (y  − hy)|x=0 = 0,

.

where

Im h =

α2 , 2

A short calculation ascertains that L is dissipative indeed.

α>0

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The Pavlov’s dilation of L is the operator .L in the space .H = D− ⊕ K ⊕ D+ , where .D± = L2 (R± ), defined on elements .(v− , u, v+ ) ∈ H which satisfy v± ∈ W21 (R± ), u, u ∈ L2 (R+ ), . ¯ = αv+ (0) u − hu 0 = αv− (0), u − hu 0

(2.6)

The action of the operator .L on this domain is set by the formula ⎛ dv ⎞ − i ⎜ ⎟ dx v− ⎜ ⎟ ⎜ ⎝ ⎠ .L u = ⎜ u ⎟ ⎟. ⎝ dv ⎠ v+ + i dx ⎛

⎞

(2.7)

The characteristic function of L is the scalar analytic in the upper half-plane function .S(z) given by S(z) =

.

m∞ (z) − h , m∞ (z) − h¯

Im(z) > 0.

Note that since .Im h = α 2 /2, the function .S(z) can be rewritten in a form similar to that of the characteristic function (2.5), i. e.,  −1 S(z) = 1 + iα h¯ − m∞ (z) α.

.

2.2.3

(2.8)

Pavlov’s Symmetric Form of the Dilation

According to the general theory [149], once the characteristic function S is known, the analysis of the completely non-selfadjoint part of the operator L is reduced to the analysis of S. Hence, the typical questions of the operator theory (the spectral analysis, description of invariant subspaces) are reformulated as problems pertaining to analytic (operator-valued) functions. Assume that L is completely non-selfadjoint and .ρ(L) ∩ C± = ∅. Let .L be its minimal self-adjoint dilation, S being the characteristic function of L. Owing to the general theory [149], the operator L is unitary equivalent to its model acting in the spectral representation of .L in accordance with (2.3). Recall that the characteristic function .S(z), .z ∈ C+ is analytic in the upper half-plane taking values in the set of contractions of E, S(z) : E → E,

.

S(z) ≤ 1,

z ∈ C+

Due to the operator version of Fatou’s theorem [149], the nontangential boundary values of the function S exist in the strong operator topology almost everywhere

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on the real line. Put .S = S(k) := s-limε↓0 S(k + iε) and .S ∗ = S ∗ (k) := s-limε↓0 [S(k + iε)]∗ , both limits existing for almost all .k ∈ R. The Fatou theorem guarantees that the operators .S(k) and .S ∗ (k) are contractions on E for almost all .k ∈ R. The symmetric form of the dilation is obtained by completion of the dense linear set in .L2 (E) ⊕ L2 (E) with respect to the norm

       ˆ 

g˜ 2 g˜ g˜ I S∗

, . dk,

g := S I g g E⊕E H R

(2.9)

followed by factorisation by the elements of zero norm. In the symmetric representation, the incoming and outgoing subspaces .D± admit their simplest possible form. On the other hand, calculations related to the space K can meet certain difficulties, since the “weight” in (2.9) can be singular. Also note that the elements of .H are not individual functions from .L2 (E) ⊕ L2 (E) but rather equivalence classes [119, 121]. Despite these complications, the Pavlov’s symmetric model has been widely accepted in the analysis of non-selfadjoint operators, and in particular of the operators of mathematical physics. Two alternative and equivalent forms of the norm . · H that are easy to derive, .

  2

g˜

2 2 ∗ 2

˜ 2L2 (E) ,

g = S g˜ + gL2 (E) + ∗ gL2 (E) = g˜ + S g L2 (E) + g H

  √ √ where . := I − S ∗ S and .∗ := I − SS ∗ , show that for each . gg˜ ∈ H expressions .S g˜ + g, .g˜ + S ∗ g, .g, ˜ and .∗ g are in fact usual square summable vector-functions from .L2 (E). Moreover, due to these equalities the form (2.9) is positive-definite indeed and thus represents a norm. The space  H = L2

.

I S∗ S I



with the norm defined by (2.9) is the space of spectral representation for the selfadjoint dilation .L of the operator L. Henceforth we will denote the corresponding unitary mapping of .H onto .H by . . It means that the operator of multiplication by the independent variable acting on .H , i.e., the operator .f (k) → kf (k), is unitary     equivalent to the dilation .L . Hence, for .z ∈ C \ R, the mapping . gg˜ → (k − z)−1 gg˜ is unitary equivalent to the resolvent .(L − z)−1 and therefore (L − zI )−1  PK (k − z)−1

.

K

,

z ∈ C− ,

where the . sign is utilised to denote unitary equivalence.

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The incoming and outgoing subspaces of the dilation space .H admit the form   2 H+ (E) , .D+ := 0

 0 , D− := H−2 (E) 

K := H  [D+ ⊕ D− ]

where .H2± (E) are the Hardy classes of .E-valued vector functions analytic in .C± . As usual [131], the functions from vector-valued Hardy classes .H2± (E) are identified with their boundary values existing almost everywhere on the real line. They form two complementary mutually orthogonal subspaces so that .L2 (E) = H+2 (E) ⊕ H−2 (E). The image .K of K under the spectral mapping . of the dilation space .H to .H is the subspace K =

.

   g˜ ∈ H : g˜ + S ∗ g ∈ H−2 (E), S g˜ + g ∈ H+2 (E) g

The orthogonal projection .PK from .H onto .K is defined by formula (2.10) on a dense set of functions from .L2 (E) ⊕ L2 (E) in .H     g˜ g˜ − P+ (g˜ + S ∗ g) = , .PK g g − P− (S g˜ + g)

g˜ ∈ L2 (E), g ∈ L2 (E).

(2.10)

Here .P± are the orthogonal projections of .L2 onto the Hardy classes .H2± . Further information on model representations can be found in the series of papers [119–122] and the treatise [118].

2.2.4

Naboko’s Functional Model of Non-selfadjoint Operators

The development of the functional model approach for contractions inspired the search for such models of non-dissipative operators. The attempts to follow the blueprints of Sz.-Nagy-Foia¸s and Lax-Phillips meet serious challenges rooted in the absence of a proper self-adjoint dilation for non-dissipative operators: the dilatation in this case is a self-adjoint operator acting on a space with an indefinite metric [50]. Consequently, the characteristic function of a non-dissipative operator is an analytic operator-function, contractive with respect to an indefinite metric [51], which considerably hinders any further progress in this direction. We mention the works [17, 106], the monograph [91] and references therein for more details and examples. An alternative approach was suggested in the late seventies with the publication of papers [109, 110] and especially [111] by S. Naboko who found a way to represent a non-dissipative operator in a model space of a suitably chosen dissipative one. We refer the reader to the relevant section of the paper on Sergey Naboko’s mathematical heritage in the present volume for the details of the mentioned

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approach and the relevant references. In the next section of the present paper we outline the main ingredients of an adaptation of the latter to the setting of extensions of symmetric operators, which was developed by Sergey’s students. We mention that this set of techniques allows one to significantly advance the spectral analysis of non-selfadjoint operators, including the definition of the absolutely continuous and singular subspaces and the study of spectral resolutions of identity. In particular, of major importance is the possibility to construct the wave and scattering operators in a natural representation. It should be noted that the selfadjoint scattering theory (and all the major versions of the latter) turns out to be included as a particular case of a much more general non-selfadjoint one.

2.3 Functional Model for a Family of Extensions of a Symmetric Operator The generic model constructed in [105, 134] lends itself as a powerful and universal tool for the analysis of (completely) non-selfadjoint and (completely) non-unitary operators. Since characteristic functions of such operators are essentially unique and define the operators up to unitary equivalence, all model considerations are immediately available once this function is known. In many applications, however, the results sought need to be formulated in terms of the problem itself (i.e., in the natural terms), rather than in the abstract language of characteristic functions (and their transforms). One prominent example when the general theory is not sufficient is the setting of extensions of symmetric operators and the associated setting of operators pertaining to boundary-value problems. Within this setup, the results are expected to be formulated as statements concerning the symmetric operator itself and the relevant properties of the extension parameters. Some results in this direction were obtained by B. Solomyak in [143], but the related calculations tend to be rather tedious due to the reduction to the case of contractions which is required. The extension theory of symmetric operators, especially differential operators, greatly benefits from the abstract framework known as the boundary triples theory. The basic concepts of this operator-theoretic approach can be found in the textbook [140] by K. Schmüdgen. The recent monograph [22] contains a detailed treatment of this area. It is therefore quite natural to utilise this approach in conjunction with the functional models techniques outlined above in the analysis of non-selfadjoint extensions of symmetric operators. This section briefly outlines the results pertaining to the functional model construction for dissipative and non-dissipative extensions of symmetric operators and the related developments, including an explicit construction of the wave and scattering operators and of the scattering matrices. Since all the considerations in this area are essentially parallel to the ones of Naboko in his development of spectral theory for additive perturbations of selfadjoint operators, one can consider this narrative as a rather detailed exposition of Naboko’s ideas and results in a particular case, important for applications.

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255

Boundary Triples

To the best of our knowledge, the theory of boundary triples originated in the ideas of M. G. Krein and was developed by V. I. Gorbachuk, M. L. Gorbachuk [72] and A. N. Koˇcube˘ı [83, 84]. Further details in relation to differential equations were worked out by V. A. Derkach, M. M. Malamud, J. Behrndt, M. Langer, M. Brown, M. Marletta, S. Naboko, I. Wood and others, see [55] and references therein. A general exposition of this subject can be found in the books [22, 140]. Denote by A a closed and densely defined symmetric operator on the separable Hilbert space H with the domain .dom A, having equal deficiency indices .0 < n+ (A) = n− (A) ≤ ∞. Definition 1 ([83]) A triple .{K, 0 , 1 } consisting of an auxiliary Hilbert space .K and linear mappings .0 , 1 defined everywhere on .dom A∗ is called a boundary triple for .A∗ if the following conditions are satisfied: 1. The abstract Green’s formula is valid (A∗ f, g)H − (f, A∗ g)H = (1 f, 0 g)K − (0 f, 1 g)K ,

.

f, g ∈ dom A∗ (2.11)

2. For any .Y0 , Y1 ∈ K there exist .f ∈ dom A∗ , such that .0 f = Y0 , .1 f = Y1 . In other words, the mapping .f → 0 f ⊕ 1 f , .f ∈ dom A∗ to .K ⊕ K is surjective. It can be shown (see [83]) that a boundary triple for .A∗ exists assuming only .n+ (A) = n− (A). Note also that a boundary triple is not unique. Given any bounded self-adjoint operator . = ∗ on .K, the collection .{K, 0 , 1 + 0 } is a boundary triple for .A∗ as well, provided that .1 + 0 is surjective. Definition 2 Let .T = {K, 0 , 1 } be a boundary triple of .A∗ . The Weyl function of .A∗ corresponding to .T and denoted .M(z), .z ∈ C \ R is an analytic operatorfunction with a positive imaginary part for .z ∈ C+ (i.e., an operator R-function) with values in the algebra of bounded operators on .K such that M(z)0 fz = 1 fz ,

.

fz ∈ ker(A∗ − zI ),

z∈ /R

For .z ∈ C \ R we have .(M(z))∗ = (M(¯z)) and .Im(z) · Im(M(z)) > 0. Definition 3 An extension .A of a closed densely defined symmetric operator A is called almost solvable (a.s.) and denoted .A = AB if there exist a boundary triple .{K, 0 , 1 } for .A∗ and a bounded operator .B : K → K defined everywhere in .K such that f ∈ dom AB ⇐⇒ 1 f = B0 f

.

This definition implies the inclusion .dom AB ⊂ dom A∗ and that .AB is a restriction of .A∗ to the linear set .dom AB := {f ∈ dom A∗ : 1 f = B0 f }. In this context, the operator B plays the role of a parameter for the family of extensions .{AB | B : K → K}.

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It can be shown (see [48] for references) that if the deficiency indices .n± (A) are equal and .AB is an almost solvable extension of A, then the resolvent set of .AB is not empty (i.e. .AB is maximal), both .AB and .(AB )∗ = AB ∗ are restrictions of ∗ .A to their domains, and .AB and B are selfadjoint (dissipative) simultaneously. The spectrum of .AB coincides with the set of points .z0 ∈ C such that .(M(z0 ) − B)−1 does not admit analytic continuation into it.

2.3.2

Characteristic Functions

Assume that the parameter B of an almost solvable extension .AB is completely non-selfadjoint. It can be represented as the sum of its real and imaginary part B = BR + iBI ,

.

BR = BR∗ ,

BI = BI∗

These parts are well defined since B is bounded. The Green’s formula implies that for .BI = 0 the imaginary part of .AB (in the sense of its form) is non-trivial, i. e., .Im(AB u, u) = 0 at least for some .u ∈ dom(AB ). Hence .AB in this case is not a self-adjoint operator. It appears highly plausible that complete non-selfadjointness of .AB can be derived solely from complete non-selfadjointness of B, assuming that A has no reducing self-adjoint parts. However, no direct proof of this assertion seems to be available in the existing literature. According to (2.5), the characteristic function of B has the form B (z) = IE + iJ α(B ∗ − zI )−1 α : E → E,

.

z ∈ ρ(B ∗ ),

√ where .α := 2|BI |, .J := signBI , and .E := clos ran(α). On the other hand, direct calculations according to [146] lead to the following representation for the characteristic function .AB : E → E of the non-selfadjoint part of the extension .AB AB = IE + iJ α(B ∗ − M(z))−1 α,

.

z ∈ ρ(A∗B ).

These two formulae confirm an earlier observation that goes back to B. S. Pavlov’s work [126], see (2.8) above. The function .AB is obtained from .B by the substitution of .M(z) for .zIE , .z ∈ C+ AB (z) = B (M(z)),

.

z ∈ ρ(A∗B ).

Alongside .AB introduce the dissipative almost solvable extension .A+ parametrised by .B+ := BR + i|BI |. Note that the characteristic function S of .A+ is given by (cf. (2.5)) ∗ S(z) = IE + iα(B+ − zI )−1 α : E → E,

.

z ∈ C+ .

(2.12)

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Calculations of [133] (cf. [109]) show that the characteristic functions of .AB and .A+ are related via an operator linear-fractional transform known as PotapovGinzburg transformation, or PG-transform [15]. This fact is essentially geometric. It connects contractions on Kre˘ın spaces (i.e., the spaces with an indefinite metric defined by the involution .J = J ∗ = J −1 ) with contractions on Hilbert spaces endowed with the regular metric. The PG-transform is invertible and the following assertion pointed out in [109] holds. Proposition 1 The characteristic function .AB is J -contractive on its domain and the PG-transform maps it to the contractive characteristic function S of .A+ as follows: AB → S = −(χ + − AB χ − )−1 (χ − − AB χ + ), .

S → AB = (χ − + χ + S)(χ + + χ − S)−1 ,

(2.13)

where .χ ± = (IE ± J )/2 are orthogonal projections onto subspaces of .χ + E and − .χ E, respectively. It appears somewhat unexpected that two operator-valued functions connected by formulae (2.13) can be explicitly written down in terms of their “main operators” .AB and .A+ . This relationship between the characteristic functions of .AB and .A+ goes in fact much deeper, see [14, 15]. In particular, the self-adjoint dilation of .A+ and the J -self-adjoint dilation of .AB are also related via a suitably adjusted version of the PG-transform. Similar statements hold for the corresponding linear systems or “generating operators” of the functions .AB and S, cf. [14, 15]. This fact is crucial for the construction of a model of a general closed and densely defined nonselfadjoint operator, see [134]. 2.3.3

Functional Model for a Family of Extensions

Formulae of the previous section are essentially the same as the formulae of [109] connecting characteristic functions of non-dissipative and dissipative operators. Reasoning by analogy, this suggests an existence of certain identities that would connect the resolvents of .AB and .A+ corresponding to parameters .B = BR + iBI and .B+ = BR + i|BI |. Such identities indeed exist; they are the celebrated Kre˘ın formulae for resolvents of two extensions of a symmetric operator. Their variant is readily derived within the framework of boundary triplets, see [133] for calculations, where all details of the following results can also be found. The functional model of the dissipative extension .A+ begins with the derivation of its minimal selfadjoint dilation .A . It is constructed following the recipe of B. Pavlov [124–126] and takes a form quite similar to (2.6), (2.7) ⎛ ⎞ ⎛ iv  ⎞ − v− ⎜ ∗ ⎟ ⎝ ⎠ .A u = ⎝A+ u⎠ , v+ iv  +

⎛ ⎞ v− ⎝ u ⎠ ∈ dom(A ) v+

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where .dom(A ) consists of vectors .(v− , u, v+ ) ∈ H = D− ⊕ H ⊕ D+ , with .v± ∈ W21 (R± , E), .u ∈ dom(A∗+ ) under two “boundary conditions” imposed on .v± and u: 1 u − B+ 0 u = αv− (0),

.

∗ 1 u − B+ 0 u = αv+ (0)

The functional model construction for .A+ follows the recipe by S. Naboko [111]. The following theorem holds. Theorem 2.1 There exists a mapping . from the dilation space .H onto Pavlov’s model space .H defined by (2.9) with the following properties 1. 2. 3. 4. 5.

is isometric.   g˜ + S ∗ g = F+ h, .S g˜ + g = F− h, where . gg˜ = h, .h ∈ H −1 = (k − z)−1 ◦ , . ◦ (L − zI ) z∈C\R . H = H , . D± = D± , . K = K −1 = (k − z)−1 ◦ F , .F± ◦ (L − zI ) z ∈ C \ R. ± . .

where bounded maps .F± : H → L2 (R, E) are defined by the formulae 1 ∗ F+ : h → − √ α0 (A+ − k + i0)−1 u + S+ (k)vˆ− (k) + vˆ+ (k), 2π 1 F− : h → − √ α0 (A∗+ − k − i0)−1 u + vˆ− (k) + S+ (k)vˆ+ (k), 2π

.

where .h = (v− , u, v+ ) ∈ H and .vˆ± are the Fourier transforms of .v± ∈ L2 (R± , E). Using the Kre˘ın formulae, which play the same role as Hilbert resolvent identities in the case of additive perturbations, one can obtain results similar to those of [111] and obtain an explicit description of .(A+ − z)−1 in the functional model representation. An analogue of this result for more general extensions of A corresponding to a choice of parameter B in the form .BR +αα/2 with bounded . : E → E and .BR = BR∗ is proven along the same lines. This program was realized in [134] for a particular case . = iJ and in [48] for the family of extensions .A parametrised by .B = αα/2. The latter form of extension parameter utilizes the possibility of “absorbing” the part .BR = BR∗ into the map .1 , i. e., passing from the boundary triple .{K, 0 , 1 } to the triple .{K, 0 , 1 + BR 0 }.

2.3.4

Smooth Vectors and the Absolutely Continuous Subspace

Here we characterise the absolutely continuous spectral subspace for an almost solvable extension of a densely defined symmetric operator with equal (possibly infinite) deficiency indices. The procedure we follow is heavily influenced by the ideas of Sergey Naboko, see [111, 113] and is carried out essentially in parallel to the exposition of [111]. The notion of the absolutely continuous subspace of a dissipative operator was introduced in the paper [137], which also includes a

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discussion of the scattering theory for dissipative operators. In contrast to the mentioned works, dealing with additive perturbations of self-adjoint operators, we consider the case of extensions, self-adjoint and non-self-adjoint alike. The narrative below follows the argument presented in our papers [47, 48]. Since we are not limiting the consideration to the case of self-adjoint operators, we first require the notion of the absolutely continuous spectral subspace applicable in the non-self-adjoint setup. In the functional model space .H introduced in Sect. 2.2.3 constructed based on the characteristic function .S(z) introduced in Sect. 2.3.2 consider two subspaces .N± defined as follows: N± :=

.

      g ∈ H : P± χ+ ( g + S ∗ g) + χ− (S g + g) = 0 , g

where χ± :=

.

I ± i . 2

and .P± are orthogonal projections onto their respective Hardy classes, as above. These subspaces have a characterisation in terms of the resolvent of the operator .A . This, again, can be seen as a consequence of a much more general argument (see e.g. [132, 134]). Theorem 2.2 Suppose that .ker α = 0. The following characterisation holds: N± =

.

        g  g 1  g = PK for all z ∈ C± . ∈ H : (A − zI )−1 ∗ PK g k−z g g

Here . denotes the unitary mapping of the dilation space .H onto .H , as above. Consider the counterparts of .N± in the original Hilbert space .H : ± := ∗ PK N± , N

.

which are linear sets albeit not necessarily subspaces. In a way similar to [111], one introduces the set + ∩ N − e := N N

.

e ). of so-called smooth vectors and its closure .Ne := clos(N The next assertion (cf. e.g. [132, 134], for the case of general non-selfadjoint ± . operators), is an alternative non-model characterisation of the linear sets .N ± are described as follows: Theorem 2.3 The sets .N ± = {u ∈ H : α0 (A − zI )−1 u ∈ H±2 (E)}. N

.

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Moreover, one shows that for the functional model image of .N˜ e the following representation holds:     g   ∈H : . Ne = PK g         g  g  g 1 ∈ H satisfies (A − zI )−1 ∗ PK = PK ∀ z ∈ C− ∪ C+ , g g k−z g (2.14) e . (Note that which motivates the term “the set of smooth vectors” used for .N  ˜ the inclusion of the right-hand side of (2.14) into . Ne follows immediately from Theorem 2.2.) The above Theorem together with Theorem 2.4 below motivates generalising the notion of the absolutely continuous subspace .Hac (A ) to the case of non-selfadjoint extensions .A of a symmetric operator .A, by identifying it with the set .Ne . This generalisation follows in the footsteps of the corresponding definition by Naboko [111] in the case of additive perturbations (see also [132, 134] for the general case). Definition 4 For a symmetric operator .A, in the case of a non-selfadjoint extension .A the absolutely continuous subspace .Hac (A ) is defined by the formula  .Hac (A ) := Ne . In the case of a self-adjoint extension .A , we understand .Hac (A ) in the sense of the classical definition of the absolutely continuous subspace of a self-adjoint operator. It turns out that in the case of self-adjoint extensions a rather mild additional condition guarantees that the non-self-adjoint definition above is equivalent to the classical self-adjoint one. Namely, we have the following Theorem 2.4 Assume that . =  ∗ , .ker(α) = {0} and let .α0 (A − zI )−1 be a Hilbert-Schmidt operator for at least one point .z ∈ ρ(A ). If A is completely non-selfadjoint, then the definition .Hac (A ) = Ne is equivalent to the classical definition of the absolutely continuous subspace of a self-adjoint operator, i.e. Ne = Hac (A ) .

.

Remark 1 Alternative conditions, which are even less restrictive in general, that guarantee the validity of the assertion of Theorem 2.4 can be obtained along the lines of [113]. 2.3.5

Wave and Scattering Operators

The results of the preceding section allow us, see [47, 48], to calculate the wave operators for any pair .A1 , A2 , where .A1 and .A2 are two different extensions of

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a symmetric operator A, under the additional assumption that the operator .α has a trivial kernel. For simplicity, in what follows we set .2 = 0 and write . instead of .1 . Note that .A0 is a self-adjoint operator, which is convenient for presentation purposes. In order to compute the wave operators of this pair, one first establishes the model representation for the function .exp(iA t), .t ∈ R, of the operator .A , evaluated on e . Due to (2.14), it is easily shown that on this set the set of smooth vectors .N .exp(iA t) acts as an operator of multiplication by .exp(ikt). We then utilise the following result. g   g ∗  0 Proposition 2 ([111, Section 4]) If . ∗ PK  g ∈ Ne and . PK g ∈ Ne (with the same element1 g), then .

   

g  g ∗

exp(−iA t) ∗ PK  t) P − exp(−iA 0 K

g g

−−−−→ 0.

H t→−∞

g   g ∗  0 It follows from Proposition 2 that whenever . ∗ PK  g ∈ Ne and . PK g ∈ Ne (with the same second component g), formally one has .

iA0 t −iA t

lim e

t→−∞

e

     g  g ∗ = PK

PK g g   −(I + S)−1 (I + S ∗ )g ∗ = PK . g ∗

In view of the classical definition of the wave operator of a pair of self-adjoint operators, see e.g. [81],  W± (A0 , A ) := s-lim eiA0 t e−iA t Pac ,

.

t→±∞

 is the projection onto the absolutely continuous subspace of .A , we where .Pac g   obtain that, at least formally, for . ∗ PK  g ∈ Ne one has

     g −(I + S)−1 (I + S ∗ )g ∗ = PK . .W− (A0 , A ) PK g g ∗

1 Despite

(2.15)

g  the fact that .  g and g can be identified with vectors g ∈ H is nothing but a symbol, still .

in certain .L2 (E) spaces with operators “weights”, see details below in Sect. 2.3.6. Further, we g recall that even then for .  g and g are not, in general, independent of each g ∈ H , the components . other.

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By considering the case .t → +∞, one also obtains W+ (A0 , A ) ∗ PK

.

     g  g = lim eiA0 t e−iA t ∗ PK t→+∞ g g    g ∗ = PK −(I + S ∗ )−1 (I + S) g

g   again for . ∗ PK  g ∈ Ne . Further, the definition of the wave operators .W± (A , A0 ) .

   

−iA t g  g −iA0 t ∗

e  W± (A , A0 ) ∗ PK  − e

P K

g g

yields, for all . ∗ PK

 g

W− (A , A0 ) ∗ PK

.

g

−−−−→ 0

H t→±∞

e0 , ∈N

     g −(I + χ− (S − I ))−1 (I + χ+ (S ∗ − I ))g = ∗ PK g g

and

     g  g ∗ . = PK .W+ (A , A0 ) PK g g −(I + χ+ (S ∗ − I ))−1 (I + χ− (S − I )) (2.16) ∗

In order to rigorously justify the above formal argument, i.e. in order to prove the existence and completeness of the wave operators, one needs to first show that the right-hand sides of the formulae (2.15)–(2.16) make sense on dense subsets of the corresponding absolutely continuous subspaces, which is done in a similar way to [113]. Below, we show how this argument works in relation to the wave operator (2.15) only, skipping the technical details in view of making the exposition more transparent. Let .S(z) − I be of the class .S∞ (C+ ), i.e. a compact analytic operator function in the upper half-plane up to the real line. Then so is .(S(z) − I )/2, which is also uniformly bounded in the upper half-plane along with .S(z). We next use the result of [113, Theorem 3] about the non-tangential boundedness of operators of the form −1 for .T (z) compact up to the real line. We infer that, provided .(I + .(I + T (z)) (S(z0 ) − I )/2)−1 exists for some .z0 ∈ C+ (and hence, see [33], everywhere in .C+ except for a countable set of points accumulating only to the real line), one has nontangential boundedness of .(I + (S(z) − I )/2)−1 , and therefore also of .(I + S(z))−1 , for almost all points of the real line. On the other hand, the latter inverse can be computed in .C+ :  −1  1 I + S(z) = I + iαM(z)−1 α/2 . 2

.

(2.17)

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It follows from (2.17) and the analytic properties of .M(z) that the inverse .(I + S(z))−1 exists everywhere in the upper half-plane. Thus, Theorem 3 of [113] is indeed applicable, which yields that .(I + S(z))−1 is .R-a.e. nontangentially bounded and, by the operator generalisation of the Calderon theorem (see [144]), which was extended to the operator context in [113, Theorem 1], it admits measurable nontangential limits in the strong operator topology almost everywhere on .R. As it is easily seen, these limits must then coincide with .(I + S(k))−1 for almost all .k ∈ R. Then the correctness of the formula (2.15) for the wave operators follows: indeed, consider .1n (k), the indicator of the set .{k ∈ R : (I + S(k))−1  ≤ n}. Clearly, .1n (k) → 1 as .n → ∞ for almost all .k ∈ R. Next, suppose that .PK (g, ˜ g) ∈ N˜ e . Then .PK 1n (g, ˜ g) is shown to be a smooth vector as well as   −(I + S)−1 1n (I + S ∗ )g ∈H. . 1n g It follows, by the Lebesgue dominated convergence theorem, that the set of vectors PK 1n (g, ˜ g) is dense in .Ne . Thus the following theorem holds.

.

Theorem 2.5 Let A be a closed, symmetric, completely nonselfadjoint operator with equal deficiency indices and consider its extension .A under the assumptions that .ker(α) = {0} and that .A has at least one regular point in .C+ and in .C− . If .S − I ∈ S∞ (C+ ), then the wave operators .W± (A0 , A ) and .W± (A , A0 ) exist on dense sets in .Ne and .Hac (A0 ), respectively, and are given by the formulae (2.15)– (2.16). The ranges of .W± (A0 , A ) and .W± (A , A0 ) are dense in .Hac (A0 ) and .Ne , respectively.2 Remark 2 1. The condition .S(z) − I ∈ S∞ (C+ ) can be replaced by the following equivalent condition: .αM(z)−1 α is nontangentially bounded almost everywhere on the real line, and .αM(z)−1 α ∈ S∞ (C+ ) for .z ≥ 0. 2. The latter condition is satisfied [70], as long as the scalar function αM(z)−1 αSp

.

is nontangentially bounded almost everywhere on the real line for some .p < ∞, where .Sp , p ∈ (0, ∞], are the standard Schatten–von Neumann classes of compact operators.

2 In

the case when .A is self-adjoint, or, in general, the named wave operators are bounded, the claims of the theorem are equivalent (by the classical Banach-Steinhaus theorem) to the statement of the existence and completeness of the wave operators for the pair .A0 , A . Sufficient conditions of boundedness of these wave operators are contained in e.g. [111, Section 4], [113] and references therein.

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3. An alternative sufficient condition is the condition .α ∈ S2 (and therefore .B ∈ S1 ), or, more generally, .αM(z)−1 α ∈ S1 , see [112] for details. Finally, the scattering operator . for the pair .A , .A0 is defined by  = W+−1 (A , A0 )W− (A , A0 ).

.

The above formulae for the wave operators lead (cf. [111]) to the following formula for the action of . in the model representation:

 ∗ PK

.

    −(I + χ− (S − I ))−1 (I + χ+ (S ∗ − I ))g g˜ = PK , g (I + S ∗ )−1 (I + S)(I + χ− (S − I ))−1 (I + χ+ (S ∗ − I ))g (2.18)

  e0 . In fact, as explained above, this representation holds on whenever . ∗ PK gg˜ ∈ N 0 e within the conditions of Theorem 2.5, which guarantees that a dense linear set in .N all the objects on the right-hand side of the formula (2.18) are correctly defined.

2.3.6

Spectral Representation for the Absolutely Continuous Part of the Operator A0 and the Scattering Matrix

The identity

  2

  g˜

= (I − S ∗ S)g, . PK ˜ g˜

g H

  e0 allows us to consider the which is derived in [111, Section 7] for all .PK gg˜ ∈ N e0 → L2 (E; I − S ∗ S) defined by the formula isometry .F : N F PK

.

  g˜ = g. ˜ g

Here .L2 (E; I − S ∗ S) is the Hilbert space of E-valued functions on .R square summable with the matrix “weight” .I − S ∗ S. Under the assumptions of Theorem 2.5 one can show that the range of the operator F is dense in the space .L2 (E; I − S ∗ S). Thus, the operator F admits an extension to the unitary mapping between . Ne0 and .L2 (E; I − S ∗ S). e0 acts as It follows that the self-adjoint operator .(A0 − z)−1 considered on .N −1 2 ∗ the multiplication by .(k − z) , .k ∈ R, in .L (E; I − S S). In particular, if one considers the absolutely continuous “part” of the operator .A0 , namely the (e) (e) operator .A0 := A0 |Ne0 , then .F A0 ∗ F ∗ is the operator of multiplication by the independent variable in the space .L2 (E; I − S ∗ S).

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In order to obtain a spectral representation from the above result, it is necessary to diagonalise the “weight” in the definition of the above .L2 √ -space. The corresponding transformation is straightforward when, e.g., .α = 2I. (This choice of .α satisfies the conditions of Theorem 2.5 e.g. when the boundary space .K is finitedimensional). In this particular case one has S = (M − iI )(M + iI )−1 ,

.

and consequently I − S ∗ S = −2i(M ∗ − iI )−1 (M − M ∗ )(M + iI )−1 .

.

Introducing the unitary transformation G : L2 (E; I − S ∗ S) → L2 (E; −2i(M − M ∗ )),

.

by the formula .g → (M + iI )−1 g, one arrives at the fact that .GF A0 ∗ F ∗ G∗ is the operator of multiplication by the independent variable in the space 2 ∗ .L (E; −2i(M − M )). (e)

Remark 3 The weight .M ∗ − M can be assumed to be naturally diagonal in many physically relevant settings, including the setting of quantum graphs considered in Sect. 3. The above result only pertains to the absolutely continuous part of the self-adjoint operator .A0 , unlike e.g. the passage to the classical von Neumann direct integral, under which the whole of the self-adjoint operator gets mapped to the multiplication operator in a weighted .L2 -space (see e.g. [29, Chapter 7]). Nevertheless, it proves useful in scattering theory, since it yields an explicit expression for the scattering  for the pair .A , .A0 , which is the image of the scattering operator . in the matrix . spectral representation of the operator .A0 . Namely, one arrives at: Theorem 2.6 The following formula holds:  = GF (GF )∗ = (M − )−1 (M ∗ − )(M ∗ )−1 M, 

.

(2.19)

where the right-hand side represents the operator of multiplication by the corresponding function in the space .L2 (E; −2i(M − M ∗ )).

2.4 Functional Models for Operators of Boundary Value Problems The surjectivity condition in Definition 1 is a strong limitation that excludes many important problems for extensions of symmetric operators with infinite deficiency

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indices. The standard textbook version of a boundary value problem for the Laplace operator in a bounded domain . ⊂ R3 with smooth boundary .∂ is a typical example. The “natural” boundary maps .0 and .1 are two trace operators .0 : u → u|∂ , .1 : u → −∂u/∂n|∂ , where .∂/∂n denotes the derivative along the exterior normal to the boundary .∂. The ranges of these operators do not coincide with .H = L2 () (the simplest possible Hilbert space of functions defined on the boundary) so the assumption of surjectivity does not hold. A simple argument reveals the source of this problem: it appears due to the limited compatibility of the Green’s formula required to hold on all of .dom(A∗ ) and the required surjectivity of both boundary maps .0 , .1 also defined on the same domain .dom(A∗ ). This limitation of the boundary triples formalism can be relaxed and the framework extended to cover more general cases, albeit at the cost of increased complexity, see [18, 19], the book [22] and the references therein for a detailed account. Formally a more restrictive approach applicable to semibounded symmetric operators A and not based on the description of .dom A∗ was developed by M. Birman, M. Kre˘ın and M. Vishik. Despite its limited scope, this theory proves to be indispensable in applications to various problems of ordinary and partial differential operators. The publication [6] contains a concise exposition of these results. It was realized later that the Birman-Kre˘ın-Vishik method is closely related to the theory of linear systems with boundary control and to the original ideas of M. Livšic from the open systems theory, see e. g. [135, 136] in this connection. Let us give a brief account of relevant results derived from the works cited above and tailored to the purposes of current presentation.

2.4.1

Boundary Value Problem

Let H , E be two separable Hilbert spaces, .A0 an unbounded closed linear operator on H with the dense domain .dom A0 and . : E → H a bounded linear operator defined everywhere in E. Theorem 2.7 Assume the following: • .A0 is self-adjoint and boundedly invertible; • There exists the left inverse .˜ 0 of . so that .˜ 0 ϕ = ϕ for all .ϕ ∈ E; • The intersection of .dom A0 and .ran  is trivial: .dom A0 ∩ ran  = {0}. Since .dom A0 and .ran  have trivial intersection, the direct sum .dom A0  ran  form a dense linear set in H that can be described as .{A−1 0 f +ϕ | f ∈ H, ϕ ∈ E}. Define two linear operators A and .0 with the common domain .dom A0  ran  as “null extensions” of .A0 and .˜ 0 to the complementary component of .dom A0 ran  A : A−1 0 f + ϕ → f,

.

0 : A−1 0 f + ϕ → ϕ,

f ∈ H, ϕ ∈ E

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The spectral “boundary value problem” associated with the pair .{A0 , } satisfying these conditions is the system of two linear equations for the unknown vector .u ∈ dom A := dom A0  ran  :  (A − zI )u = f . (2.20) f ∈ H, ϕ ∈ E, 0 u = ϕ where .z ∈ C is the spectral parameter. Let .z ∈ ρ(A0 ), .f ∈ H , .ϕ ∈ E. Then the system (2.20) admits the unique f,ϕ solution .uz given by the formula f,ϕ

uz

.

−1 = (A0 − zI )−1 f + (I − zA−1 0 ) ϕ

If the expression on the right hand side is null for some .f ∈ H , .ϕ ∈ E, then .f = 0 and .ϕ = 0. Let . be a linear operator on E with the domain .dom  ⊂ E not necessarily dense in E. Define the linear operator .1 on .dom 1 := {A−1 0 f + ϕ | f ∈ H, ϕ ∈ dom } as the mapping ∗ 1 : A−1 0 f + ϕ →  f + ϕ,

.

f ∈ H,

ϕ ∈ dom 

This definition implies . = 1 |dom  . Denote .D := dom 1 = {A−1 0 f + ϕ | f ∈ H, ϕ ∈ dom }. Obviously .D ⊂ dom A. The next theorem is a form of the Green’s formula for the operator A. Theorem 2.8 Assume that . is selfadjoint (and therefore densely defined) in E. Then (Au, v) − (u, Av) = (1 u, 0 v)E − (0 u, 1 v)E ,

.

u, v ∈ D

Notice the difference with the boundary triples version (2.11), where the operator on the left hand side is the adjoint of a symmetric operator. In contrast, Theorem 2.8 has no relation to symmetric operators. The Green’s formula is valid on a set defined by the selfadjoint .A0 and an arbitrarily chosen selfadjoint operator .. Under the assumptions of Theorems 2.7 and 2.8 the operator-valued analytic function −1 M(z) =  + z∗ (I − zA−1 0 ) ,

.

z ∈ ρ(A0 )

defined on .dom M(z) = dom  is the Weyl function (cf. [22]) of the boundary value problem (2.20) in the sense of equality (cf. Definition 2) M(z)0 uz = 1 uz ,

.

f,ϕ

where .uz = uz

z ∈ ρ(A0 )

is the solution to (2.20) with .f = 0 and .ϕ ∈ dom .

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For the boundary value problem pertaining to the Laplace operator in a bounded domain . ⊂ R3 with a smooth boundary .∂ the boundary maps .0 , .1 are defined as .0 : u → u|∂ , .1 : u → −∂u/∂n|∂ . Then .A0 is the Dirichlet Laplacian in .L2 () and . is the operator of harmonic continuation from the boundary space .E = L2 (∂) to .. The conditions .dom(A0 ) ∩ ran() = {0} and .0  = IE are satisfied by virtue of the embedding theorems for Sobolev classes. In this setting .M(·) is known as the Dirichlet-to-Neumann map, which is a pseudodifferential operator defined on .H 1 (∂). A special role of the operator . = M(0) for the study of boundary value problems was pointed out by M. Vishik in his work [157] and sometimes . in the settings of elliptic partial differential operators is referred to as the Vishik operator. 2.4.2

Family of Boundary Value Problems

General boundary value problems for the operator A have the form (cf. [157]) 

(A − zI )u = f

.

(α0 + β1 )u = ϕ

f ∈ H,

ϕ ∈ E.

(2.21)

Here .α, .β are linear operators on E such that .β is bounded (and defined everywhere in E) and .α can be unbounded in which case .dom α ⊃ dom  = D. Under certain verifiable conditions the solutions to (2.21) exist and are described by the following theorem. Theorem 2.9 (See [136]) Assume that the conditions of Theorems 2.7 and 2.8 are satisfied and that the operator sum .α + β is correctly defined on .dom  and closable in E. Then .α + βM(z), .z ∈ ρ(A0 ) is also closable as an additive perturbation of .α + β by the bounded operator .M(z) − . Denote by .B(z) the closure of .α + βM(z), .z ∈ ρ(A0 ) and let .B = B(0). • Consider the Hilbert space .HB formed by the vectors .

  u = A−1 f + ϕ | f ∈ H, ϕ ∈ dom B 0

and endowed with the norm 1/2  uB = f 2 + ϕ2 + Bϕ2

.

The formal sum .α0 + β1 is a bounded map from the Hilbert space .HB to E. Note that the summands in .(α0 + β1 )u, .u ∈ HB need not be defined individually.

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• Assume that for some .z ∈ ρ(A0 ) the operator .B(z) has a bounded inverse −1 . Then the problem (2.21) is uniquely solvable. Under this condition .[B(z)] there exists a closed operator .Aα,β with dense domain .

dom Aα,β = {u ∈ HB | (α0 + β1 )u = 0} = Ker(α0 + β1 )

and the resolvent (Kre˘ın) formula holds: −1 −1 −1 −1 ∗ (Aα,β − zI )−1 = (A0 − zI )−1 − (I − zA−1 0 ) [B(z)] β (I − zA0 )

.

• Denote by .A00 the restriction of .A0 to the set .Ker 1 , that is, .A00 = A|Ker 0 ∩Ker 1 . Then .A00 is a symmetric operator with its domain not necessarily dense in H and A00 ⊂ Aα,β ⊂ A

.

Notice that .A0 is a self-adjoint extension of .A00 contained in A. It is not difficult to recognize the parallel with the von Neumann theory of self-adjoint extensions of symmetric operators. The operator .A00 is the “minimal” operator with the “maximal” equal to .A∗00 (whenever the latter exists) and all self-adjoint extensions .As.a. of .A00 satisfy .A00 ⊂ As.a. ⊂ A∗00 . Within the framework of Theorem 2.9 the equivalent of .A∗00 is the operator A of the boundary value problem (2.20) defined on the domain .dom(A). The semiboundness condition for .A00 is relaxed and replaced by the bounded invertibility of .A0 , i. e., the existence of a regular point of .A0 on the real line.

2.4.3

Functional Model

The results of previous sections hint at the possibility of a functional model construction for the family of operators .Aα,β with a suitably chosen pair .(α, β). Having in mind the model space (2.9), the selection of a “close” dissipative operator is typically guided by the properties of the problem at hand. In the most general case when parameters .(α, β) are unspecified, a reasonable approach seems to be to construct a model suitable for the widest possible range of .(α, β). In accordance with the work by S. Naboko [111], the action of the operator .Aα,β will then be explicitly described in the functional model representation. This program is realized in the recent paper [49]. The “model” dissipative operator .L = A−iI,I corresponds to the boundary condition .(1 − i0 )u = 0 for .u ∈ dom(L) in the notation (2.21). The characteristic function of L then coincides with the Cayley transform of the Weyl function, S(z) = (M(z) − iI )(M(z) + iI )−1 : E → E,

.

z ∈ C+ .

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Under the mapping of the upper half plane to the unit disk, the function .S((z − i)/(z + i)) is the Sz.Nagy-Foia¸s characteristic function of a contraction, namely, the Cayley transform .V0 of .A00 extended to .H  dom(V0 ) by the null operator, hence resulting in a partial isometry. If .A00 has no non-trivial self-adjoint parts, the dissipative operator .L = A−iI,I is also completely non-selfadjoint. The standard assumptions of complete non-selfadjointness and maximality of L are thus met. The minimal self-adjoint dilation .A of L formally coincides with the dilation obtained ∗ = −iI in Sect. 2.3.3 for the case of boundary triples with .A∗+ = A, .B+ = iIE , .B+ E √ and .α = 2IE . The description of operators .Aα,β in the spectral representation of dilation .A , i.e. in the model space (2.9), cannot be easily obtained for arbitrary .(α, β). However, under certain conditions imposed on the parameters .(α, β) the model construction becomes tractable. Namely, one assumes that the operator .β is boundedly invertible, the operator .B = β −1 α is bounded in E and that the operator-valued function .M(z) is invertible and .BM(z)−1 is compact at least for some z in the upper and lower half-plane of .C (and therefore for all .z ∈ C \ R). It follows, that the operator .Aα,β has at most discrete spectrum in .C \ R with possible accumulation to the real line only. Moreover, the resolvent set of .Aα,β coincides with the open set of complex numbers .z ∈ C such that the closed operator .B + M(z) has a bounded inverse, i. e. .ρ(Aα,β ) = {z ∈ C | 0 ∈ ρ(B + M(z))}. Finally, the model representation of the resolvent .(Aα,β − zI )−1 , .z ∈ ρ(Aα,β ) is explicitly computed in the model space (2.9). Once the latter are established, it is natural to expect that the absolutely continuous subspaces can be characterised for the operators of boundary value problems in the case of exterior domains and the scattering theory can then be constructed following the recipe of Naboko, as presented in Sect. 2.2. If this programme is pursued, this would yield a natural representation of the corresponding scattering matrix purely in terms of the .M−operator defined above. A paper devoted to this subject is presently being prepared for publication.

2.5 Generalised Resolvents In the present section, we briefly recall the notion of generalised resolvents (see [8] for details; we also remark that generalised resolvents had essentially appeared already in [100, 101] and a comprehensive list of references in [79]) of symmetric operators, which play a major role in the asymptotic analysis of highly inhomogeneous media as presented in the present paper. It turns out that generalised resolvents and their underlying self-adjoint operators in larger (dilated) spaces feature prominently in our approach; moreover, their setup turns out to be natural in the theory of time-dispersive and frequency-dispersive media. On the mathematical level, this area is closely interrelated with the theory of dilations and functional models of dissipative operators, the latter (at least, in the case of

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dissipative extensions of symmetric operators) being an important particular case of the former. We start with an operator-function .R(z) in the Hilbert space H , analytic in .z ∈ C+ . Assuming that .Im R(z) ≥ 0 for .z ∈ C+ , and under the well-known asymptotic condition lim sup τ R(iτ ) < +∞,

.

τ →+∞

one has due to the operator generalisation of Herglotz theorem by Neumark [116]: ˆ R(z) =

∞

.

−∞

1 dB(t), t −z

where .B(t) is a uniquely defined left-continuous operator-function such that B(−∞) = 0, .B(t2 ) − B(t1 ) ≥ 0 for .t2 > t1 and .B(+∞) bounded. By the argument of [63], it follows from the Neumark theorem [115, 116] (cf., e.g., [112]) that there exists a bounded operator .X : H → H with an auxiliary Hilbert space .H and a self-adjoint operator .A in .H such that

.

R(z) = X(A − z)−1 X∗

.

with XX∗ = s- lim B(t) = s- lim sup τ Im R(iτ ).

.

t→+∞

τ →+∞

A particular case of this result, see [145], holds when (a) for some .z0 ∈ C+ there exists a subspace .L ⊂ H such that (i) for all non-real z and all .f ∈ L one has R(z)f − R(z0 )f = (z − z0 )R(z)R(z0 )f,

.

(ii) for any .z ∈ C+ and any .g ∈ L⊥ one has R(z)g2 ≤

.

1 ImR(z)g, g , Im z

(iii) for all .g ∈ L⊥ the function .R(z)g is regular in .C+ ; (b) .R(z0 )L = H . Under these assumptions, the function .R(z) is ascertained to be a generalised resolvent of a densely defined symmetric operator A in H . Moreover, the deficiency index of A in .C+ is equal to .dim L⊥ . Precisely, this means that .H ⊃ H and .X = P , where P is the orthogonal projection of .H to H , i.e., R(z) = P (A − z)−1 |H ,

.

Im z = 0,

(2.22)

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where .A is a self-adjoint out-of-space extension of the symmetric operator A (or, alternatively, a zero-range model with an internal structure, see Section 4 below).  Moreover, under the minimality condition . Im z=0 (A − z)−1 H = H, it is defined uniquely up to a unitary transform which acts as unity on H , see [115]. The latter representation takes precisely the same form as the dilation condition (2.3) in the case of maximal dissipative extensions of symmetric operators, with the generalised resolvent .R(z) replacing the resolvent of a dissipative operator. It is in fact shown that the property (2.22) generalises (2.3). Namely, it turns out [145, 147] that R(z) = (AB(z) − z)−1 for z ∈ C+ ∪ C− .

.

Here in the particular case of equal deficiency indices, which is of interest to us from the point of view of zero-range models with internal structure, .AB(z) is a .z−dependant extension of A such that there exists a boundary triple .(K , 0 , 1 ) defining this extension as follows: .

dom AB(z) = {u ∈ dom A∗ |1 u = B(z)0 u}

with .B(z) being a .−R function (i.e., an analytic operator-function with a nonpositive imaginary part in .C+ ). Because of .B ∗ (¯z) = B(z), which is the standard extension of an .R−function into .C− implied here, the extension .AB(z) turns out to be dissipative for .z ∈ C− and anti-dissipative for .z ∈ C+ . We henceforth refer to .A as the Neumark-Strauss dilation of the generalised resolvent .R(z) = (AB(z) − z)−1 . In a particular case of constant .B(z) = B such that .Im B ≤ 0, we have (AB − z)−1 = P (A − z)−1 |H for z ∈ C− and (AB ∗ − z)−1

.

= P (A − z)−1 |H for z ∈ C+ , which are precisely (2.3) for both .AB and .AB ∗ at the same time. From what has been said above it follows that generalised resolvents appear when one conceals certain degrees of freedom in a conservative physical system, either for the sake of convenience or because these are not known. In particular, we refer the reader to the papers [63, 64], where systems with time dispersion are analysed, with prescribed “memory” term. It turns out that passing over to the frequency domain one ends up with a generalised resolvent. It then proves possible to explicitly restore a conservative Hamiltonian (the operator .A in our notation) which yields precisely the postulated time dispersion. In a nutshell, the idea here is to work with an explicit and simple enough model of the part of the space pertaining to the “hidden degrees of freedom” instead of the unnecessarily complicated physical equations which govern them. Similar ideas have been utilised in [150, 151]. The same technique has found its applications in numerics, and in particular in the so-called theory of absorbing boundary conditions, see, e.g., [57, 76].

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The problem of constructing a spectral representation for a Neumark-Strauss dilation of a given generalised resolvent thus naturally arises. In a number of special cases, where .H and .A admit an explicit construction (and in particular one has k .H = H ⊕ C for .k ≥ 1), this can be done following essentially the same path as outlined in Sect. 2.3 above. This is due to the fact that in this case the operator .A can be realised as a von Neumann extension of a symmetric operator in .H with equal and finite deficiency indices. The corresponding construction in the case when .k = 1 is presented in [46]. Surprisingly, this rather simple model already has a number of topical applications, see Sects. 4 and 5 of the present paper, and also the papers [61, 89, 90], where a generalised resolvent of precisely the same class appears in the setting of thin networks converging to quantum graphs. The generic case has been studied by Strauss in [148], where three spectral representations of the dilation are constructed, analogous to the ones of L. de Branges and J. Rovnyak, B. S. Pavlov, and B. Sz.-Nagy and C. Foia¸s. These results however present but theoretical interest, as they are formulated in terms which apparently cannot be related to the original problem setup and are therefore not usable in applications. A different approach was suggested by M. D. Faddeev and B. S. Pavlov in [129], where a problem originally studied by P. D. Lax and R. S. Phillips in [96] was considered. In [129], a five-component representation of the dilation was constructed, which further allowed to obtain the scattering matrix in an explicit form. It therefore comes as no surprise that, precisely as in the Lax-Phillips approach, the resonances are revealed to play a fundamental role in this analysis (cf. the analysis of the so-called Regge poles in the physics literature). Later on, and again motivated in particular by applications to scattering, Neumark-Strauss dilations were constructed in some special cases by J. Behrndt et al., see [20, 21]. We remark that all the above results, except [46] and [148], have stopped short of attempting to construct a spectral form of the Neumark-Strauss dilation. Any generic construction leading to the latter and formulated in “natural” terms is presently unknown, to the best of our knowledge.

2.6 Universality of the Model Construction The general form of the functional model of an unbounded closed operator [134] is a generalization of the special cases, as developed in the papers by B. Pavlov and S. Naboko cited above. This section aims to clarify the relationship between the models pertaining to different representations of the characteristic function of a non-selfadjoint operator. As an illustration, we consider two special cases of operators of mathematical physics described above and link them to the general model construction of [134].

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Characteristic Function of a Linear Operator [146]

Let L be a closed linear operator on a (separable) Hilbert space H with the domain .dom(L). Consider the form .L (·, ·) defined on .dom(L) × dom(L): 1 [(Lf, g)H − (f, Lg)H ] , i

L (f, g) =

.

f, g ∈ dom(L)

(2.23)

Definition 5 The boundary space of L is a linear space E with a possibly indefinite scalar product .(·, ·)E such that there exists a closed linear operator . defined on .dom() = dom(L) and the following identity holds L (f, g) = (f, g)E ,

.

f, g ∈ dom(L)

(2.24)

The operator . is called the boundary operator of L. This definition is meaningful for any linear operator on H . For the purposes of model construction, it is sufficient to focus only on the case when L is densely defined and dissipative. The model representation of a non-dissipative operator is given in the model space of an auxiliary dissipative one, as explained above. When L is dissipative, one has .L (f, f ) ≥ 0, .f ∈ dom(L) and therefore the space E can be chosen as the Hilbert space obtained by factorization and completion of .{f | f ∈ dom(L)} with respect to the norm .f E , .f ∈ dom(L). Note that the boundary operator . defined in (2.24) is not uniquely defined. Due to the Hilbert structure of E, for any isometry .π on E the operator .π  also satisfies the condition (2.24). Moreover, if (2.24) holds for some operator .  and space .E  , then there exists an isometry .π : E  → E such that . = π   . Denote by .E∗ and .∗ the boundary space and the boundary operator for the dissipative operator .−L∗ endowed with the Hilbert metric. Assume that L is maximal, i. e., .C− ⊂ ρ(L). Then .L∗ is also maximal and .C+ ⊂ ρ(L∗ ). The following definition is valid for general non-selfadjiont operators. Definition 6 Let E and . be the boundary space and the boundary operator for a closed densely defined operator L. Let .E∗ and .∗ be the boundary space and the boundary operator for the operator .−L∗ . The characteristic function of the operator L is the analytic operator-valued function .S(z) : E → E∗ defined by S(z)f = ∗ (L∗ − zI )−1 (L − zI )f,

.

f ∈ dom(L),

z ∈ ρ(L∗ )

If L is dissipative, then the spaces E and .E∗ are Hilbert spaces, and the operator .S(z) is a contraction on E for each .z ∈ C+ . Note that the actual form of .S(z) depends on the choice of boundary spaces and boundary operators. If .  : dom(L) → E  and .∗ : dom(L∗ ) → E∗ satisfy the condition (2.24) and .S  (z) is the corresponding characteristic function, then there exist two isometries .π∗ : E∗ → E∗ and .π : E → E  such that

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π∗ S(z) = S  (z)π , .z ∈ ρ(L∗ ). Such characteristic functions of the operator L are often called equivalent. All steps involved in the model construction outlined above do not depend on the concrete form of the characteristic function [36]. In particular cases when the characteristic function can be expressed in terms of the original problem, the model admits a “natural” form in relation to the problem setup. Examples of such calculations are provided towards the end of this section. In order to compute a characteristic function of L one has to come up with a suitable definition of boundary spaces and operators. Consider first the general case where no specific assumptions on the operator L are made, and introduce the Cayley transform of L, i.e., .T = (L − iI )(L + iI )−1 . The operator T is clearly contractive. The operators

.

1 Q := √ (I − T ∗ T )1/2 , 2

.

1 Q∗ := √ (I − T T ∗ )1/2 2

are thus non-negative. A straightforward calculation [134] shows that the boundary spaces E, .E∗ and the boundary operators ., .∗ can be defined as follows: E = clos ran(Q), E∗ = clos ran(Q∗ ),

.

 = clos Q(L + iI ), ∗ = clos Q∗ (L∗ − iI ). Here the operators . and .∗ are the closures of the respective mappings initially defined on .dom(L) and .dom(L∗ ). This choice leads to the following expression for the characteristic function of the operator .L :   S(z) = T − (z − i)∗ (L∗ − zI )−1 Q ,

z ∈ C+ .

.

E

(2.25)

An explicit calculation reveals that .S(z) is closely related to the characteristic function of T ,   z−i , z ∈ C+ , .S(z) = −ϑT z+i where ϑT (λ) =

.



 − T + 2λQ∗ (I − λT ∗ )−1 Q , E

|λ| < 1

is the Sz.-Nagy-Foia¸s characteristic function of the contractive operator T . Therefore, the formula (2.25) is the abstract form of the characteristic function of L regardless the “concrete” realization of the operator L.

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Examples

The actual choice of boundary spaces and operators is guided by the specifics of the problem at hand. Let us demonstrate the “natural” selection for these objects for some of the models introduced in Sect. 2.

Additive Perturbations This is the simplest (and canonical) case of the characteristic function calculations included here solely for the completeness of exposition. Let L be a dissipative operator of Sect. 2.2.1, defined as an additive perturbation of a self-adjoint operator. Then for .f, g ∈ dom(A) = dom(L) one has L (f, g) =

.

1 1 [(Lf, g) − (f, Lg)] = i i

 i

   α2 α2 f, g − f, i g = (αf, αg) 2 2

and therefore, the boundary space can be chosen as .E = clos ran(α) with the boundary operator . defined as a mapping .f → αf . In a similar way, .E∗ = E and .∗ = . The characteristic function of L corresponding to this selection of boundary spaces and operators is then computed as (2.5). As explained above, this characteristic function is equivalent to the function (2.25).

Almost Solvable Extensions In the notation of Sect. 2.3, let .L = AB be a dissipative almost solvable extension of a symmetric operator A corresponding to the bounded operator .B =√BR + iBI with .BR = BR∗ and .BI = BI∗ ≥ 0 defined on the space .K. Denote .α := 2(BI )1/2 . From the Green’s formula (2.11) and the condition .1 f = B0 f , .f ∈ dom(L) we obtain for .f, g ∈ dom(L): L (f, g) =

.

 1 1 [(Lf, g) − (f, Lg)] = (B − B ∗ )0 f, 0 g K = (α0 f, α0 g)K . i i

Next we demonstrate two alternative approaches to the derivation of the characteristic function of L.

Approach 1 Define the boundary space .E of the operator .L = AB as the factorization and completion of the linear set .L = {0 f, | f ∈ dom(L)} endowed with the norm .uE = α 2 uK , .u ∈ L. The norm . · E is degenerate if .ker(α) is non-trivial,

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thus the factorization becomes necessary. The corresponding boundary operator . is the map . : f → 0 f on the domain .dom() = dom(L). In a similar way, .E∗ is defined as the factorization and completion of the linear set .L∗ = {0 g, | g ∈ dom(L∗ )} with respect to the norm .uE∗ = α 2 uK , .u ∈ L∗ . The boundary operator .∗ is the mapping .∗ : g → 0 g defined on .dom(∗ ) = dom(L∗ ). Thus, both boundary spaces .E and .E∗ are Hilbert spaces with the norm associated with the “weight” equal to .α 2 . An explicit computation then yields the following expression for the characteristic function .S :   −1  B − M(z) , S (z) = B ∗ − M(z)

.

z ∈ ρ(L∗ ),

where .M(z) is the Weyl-Titchmarsh .M−function of Sect. 2.3.

Approach 2 An alternative form of the characteristic function is obtained based on the boundary operators . and .∗ introduced as the closures of the mappings .f → α0 f and .g → α0 g defined on the linear sets .dom(L) and .dom(L∗ ), respectively. The boundary spaces E and .E∗ in this case are chosen as   E = clos ran α0 |dom(L) ,

.

  E∗ = clos ran α0 |dom(L∗ ) .

In all applications considered in this paper, these spaces coincide: .E = E∗ . Similarly to the situation of additive perturbations, it is often convenient (and common) to extend these spaces to .clos ran(α). The corresponding characteristic function is then represented by the formula (2.12) repeated here for the sake of readers’ convenience:  −1 S(z) = IE + iα B ∗ − M(z) α : E → E,

.

z ∈ ρ(L∗ ).

In contrast to Approach 1, this form captures the specifics of the extension parameter B. In particular, the dimension of the space E equals the dimension of the range of .α. If the operator B is a compact perturbation of a self-adjoint, i. e., .BI = α 2 /2 ∈ S∞ , then the characteristic function .S(z) is an operator-valued function of the form .I + S∞ defined on the (unweighted) Hilbert space .clos ran(α). Equivalence of S and S α : f → αf , .f ∈ E is an isometry from the “weighted” space .E The Mapping . to the space .K. The equality . α S (z) = S(z) α , .z ∈ ρ(L∗ ) expresses equivalence of the characteristic functions .S and S corresponding to different choices of boundary

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spaces and operators. Both .S and S functions are equivalent to the characteristic function of L written in its abstract form (2.25). It is easy to see that the boundary operators of Approach 1 and Approach 2 are also related by means of the isometric mapping . α. In conclusion, we point out the recent paper [36] where the construction of the selfadjoint dilation and the functional model of a dissipative operator is based entirely on the concept of Strauss boundary spaces and operators (2.23), (2.24) with no reference to their “concrete” realizations.

3 An Application: Inverse Scattering Problem for Quantum Graphs In the present section, we present an application of the theory introduced in Sect. 2.3, and in particular of the explicit construction of wave operators and scattering matrices facilitated by the approach based on the functional model due to Sergey Naboko. We first obtain an explicit expression for the scattering matrix of a quantum graph which we take to be the Laplacian on a finite non-compact metric graph, subject to .δ-type coupling at graph vertices. Then we present an explicit constructive solution to the inverse scattering problem for this graph, i.e., explicit formulae for the coupling constants at the graph vertices. The narrative of this section is based upon the papers [47, 48]; for an alternative approach not based on functional model see also [85] and references therein. For simplicity of presentation we will only consider the case of a finite noncompact quantum graph, when the deficiency indices are finite. However, the same approach allows us to consider the general setting of infinite deficiency indices, which in the quantum graph setting leads to an infinite graph. In particular, one could consider the case of an infinite compact part of the graph. In what follows, we denote by .G = G(E, σ ) a finite metric graph, i.e. a collection of a finite non-empty set .E of compact or semi-infinite intervals .ej = [x2j −1 , x2j ] (for semi-infinite intervals we set .x2j = +∞), .j = 1, 2, . . . , n, which we refer to as edges, and of a partition .σ of the set of endpoints .V := {xk : 1 ≤ k ≤ 2n, xk < +∞} into N N equivalence classes .Vm , .m = 1, 2, . . . , N, which we call vertices: .V = m=1 Vm . The degree, or valence, .deg(Vm ) of the vertex .Vm is defined as the number of elements in .Vm , i.e. .card(Vm ). Further, we partition the set .V into the two non-overlapping sets of internal .V(i) and external .V(e) vertices, where a vertex V is classed as internal if it is incident to no non-compact edge and external otherwise. Similarly, we partition the set of edges .E = E(i) ∪ E(e) , into the collection of compact (.E(i) ) and non-compact (.E(e) ) edges. We assume for simplicity that the number of non-compact edges incident to any graph vertex is not greater than one.

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For a finite metric graph .G, we consider the Hilbert spaces n

n

L2 (G) :=

L2 (ej ),

.

W 2,2 (G) :=

j =1

W 2,2 (ej ). j =1

Further, for a function .f ∈ W 2,2 (G), we define the normal derivative at each vertex along each of the adjacent edges, as follows:  ∂n f (xj ) :=

.

f  (xj ),

if xj is the left endpoint of the edge,

−f  (xj ),

if xj is the right endpoint of the edge.

In the case of semi-infinite edges we only apply this definition at the left endpoint of the edge. Definition 7 For .f ∈ W 2,2 (G) and .am ∈ C (below referred to as the “coupling constant”), the condition of continuity of the function f through the vertex .Vm (i.e. .f (xj ) = f (xk ) if .xj , xk ∈ Vm ) together with the condition ! .

∂n f (xj ) = am f (Vm )

xj ∈Vm

is called the .δ-type matching at the vertex .Vm . Remark 4 Note that the .δ-type matching condition in a particular case when .am = 0 reduces to the standard Kirchhoff matching condition at the vertex .Vm , see e.g. [25]. Definition 8 The quantum graph Laplacian .Aa , .a := (a1 , . . . , aN ), on a graph .G with .δ-type matching conditions is the operator of minus second derivative .−d 2 /dx 2 in the Hilbert space .L2 (G) on the domain of functions that belong to the Sobolev space .W 2,2 (G) and satisfy the .δ-type matching conditions at every vertex .Vm , .m = 1, 2, . . . , N. The Schrödinger operator on the same graph is defined likewise on the same domain in the case of summable edge potentials (cf. [58]). If all coupling constants .am , .m = 1, . . . , N, are real, it is shown that the operator Aa is a proper self-adjoint extension of a closed symmetric operator A in .L2 (G) [60]. Note that, without loss of generality, each edge .ej of the graph .G can be considered to be an interval .[0, lj ], where .lj := x2j − x2j −1 , .j = 1, . . . , n is the length of the corresponding edge. Throughout the present Section we will therefore only consider this situation. In [58], the following result is obtained for the case of finite compact metric graphs.

.

Proposition 3 ([58]) Let .G be a finite compact metric graph with .δ-type coupling at all vertices. There exists a closed densely defined symmetric operator A and a boundary triple such that the operator .Aa is an almost solvable extension of A, for which the parametrising matrix . is given by . = diag{a1 , . . . , aN }, whereas the

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Weyl function is an .N × N matrix with elements mj k (z) √   ⎧ & & zlp √ √ ⎪ ⎪ − z cot zl − 2 tan , j = k, p ⎪ ⎪ 2 ⎪ ep ∈Ek ep ∈Lk ⎨ 1 = √z & j=  k; Vj , Vk adjacent, √ , ⎪ ⎪ ⎪ ep ∈Cj k sin zlp ⎪ ⎪ ⎩ 0, j=  k; Vj , Vk non-adjacent. (3.1)

.

√ Here the branch of the square root is chosen so that . z ≥ 0, .lp is the length of the edge .ep , .Ek is the set of non-loop graph edges incident to the vertex .Vk , .Lk is the set of loops at the vertex .Vk , and .Cj k is the set of graph edges connecting vertices .Vj and .Vk . In [47] this is extended to non-compact metric graphs as follows. Denote by .G(i) the compact part of the graph .G, i.e. the graph .G with all the non-compact edges removed. Proposition 3 yields an expression for the Weyl function .M (i) pertaining to the graph .G(i) . Lemma 1 The matrix functions .M, .M (i) described above are related by the formula √ M(z) = M (i) (z) + i zPe ,

.

z ∈ C+ ,

(3.2)

where .Pe is the orthogonal projection in the boundary space .K onto the set of (e) (e) , , i.e. the matrix .Pe such that .(Pe )ij = 1 if .i = j, .Vi ∈ VG external vertices .VG and .(Pe )ij = 0 otherwise. √ The formula (3.2) leads to .M(s) − M ∗ (s) = 2i sPe a.e. .s ∈ R, and the  leads to the classical scattering matrix . e (k) of the pair expression (2.19) for . of operators .A0 (which is the Laplacian on the graph .G with standard Kirchhoff matching at all the vertices) and .A , where . =  = diag{a1 , . . . , aN } : e (s) = Pe (M(s) − )−1 (M(s)∗ − )(M(s)∗ )−1 M(s)Pe , 

.

s ∈ R,

(3.3)

√ which acts as the operator of multiplication in the space .L2 (Pe K; 4 sds). We remark that in the more common approach to the construction of scattering matrices, based on comparing the asymptotic expansions of solutions to spectral e as the scattering matrix. Our approach yields equations, see e.g. [62], one obtains . e into expressions involving the matrices M and . only, an explicit factorisation of . sandwiched between two projections. (Recall that M and . contain the information about the geometry of the graph and the coupling constants, respectively.) From the same formula (3.3), it is obvious that without the factorisation the pieces of

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information pertaining to the geometry of the graph and the coupling constants at the vertices are present in the final answer in an entangled form. We reiterate that the analysis above pertains not only to the cases when the coupling constants are real, leading to self-adjoint operators .Aa , but also to the case of non-selfadjoint extensions, cf. Theorem 2.5. In what follows we often drop the argument .s ∈ R of the Weyl function M and  , . e . the scattering matrices . e into a product of .−dependent and It is easily seen that a factorisation of . .−independent factors (cf. (2.19)) still holds in this case in .Pe K, namely (' ( ' e = Pe (M − )−1 (M ∗ − )Pe Pe (M ∗ )−1 MPe . 

(3.4)

.

We will now exploit the above approach in the analysis of the inverse scattering problem for Laplace operators on finite metric graphs, whereby the scattering matrix e (s), defined by (3.4), is assumed to be known for almost all positive “energies” . .s ∈ R, along with the graph .G itself. The data to be determined is the set of coupling constants .{aj }N j =1 . For simplicity, in what follows we treat the inverse problem for graphs with real coupling constants, which corresponds to self-adjoint operators. e we reconstruct the expression .Pe (M (i) −)−1 Pe for almost First, for given .M, . all .s > 0 :     1 (i) −1 ∗ −1 −1 −1  .Pe (M − ) Pe = √ 2 Pe + e [Pe (M ) MPe ] − I Pe . (3.5) i s In particular, due to the property of analytic continuation, the expression .Pe (M (i) − )−1 Pe is determined uniquely in the whole of .C with the exception of a countable set of poles, which coincides with the set of eigenvalues of the self-adjoint Laplacian (i) (i) .A on the compact part .G of the graph .G with matching conditions at the graph vertices given by the matrix ., cf. Proposition 3. Definition 9 Given a partition .V1 ∪ V2 of the set of graph vertices, for .z ∈ C consider the linear set .U (z) of functions that satisfy the differential equation .−uz = zuz on each edge, subject to the conditions of continuity at all vertices of the graph and the .δ-type matching conditions at the vertices in the set .V2 . For each function .f ∈ U (z), consider the vectors 1V1 uz :=

 !

.

xj ∈Vm

 ∂n f (xj )

Vm ∈V1

,

) * 0V1 uz := f (Vm ) V

m ∈V1

.

The Robin-to-Dirichlet map of the set .V1 maps the vector .(1V1 −  V1 0V1 )uz to V1 V1 . := diag{am : Vm ∈ V1 }. (Note that the function .uz ∈ U (z) is 0 uz , where . determined uniquely by .(1V1 −  V1 0V1 )uz for all .z ∈ C except a countable set of real points accumulating to infinity).

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The above definition is a natural generalisation of the corresponding definitions of Dirichlet-to-Neumann and Neumann-to-Dirichlet maps pertaining to the graph boundary, considered in e.g. [25, 92]. We argue that the matrix .Pe (M (i) − )−1 Pe is the Robin-to-Dirichlet map for the set .V(e) . Indeed, assuming .φ := 1 uz − 0 uz and .φ = Pe φ, where the latter condition ensures the correct .δ-type matching on the set .V(i) , one has .Pe φ = (M (i) −)0 uz and hence .0 uz = (M (i) −)−1 Pe φ. Applying .Pe to the last identity yields the claim, in accordance with Definition 9. We thus have the following theorem. Theorem 3.10 The Robin-to-Dirichlet map for the vertices .V(e) is determined e (s), .s ∈ R, via the formula (3.5). uniquely by the scattering matrix . The following definition, required for the formulation of the next theorem, is a generalisation of the procedure of graph contraction, well studied in the algebraic graph theory, see e.g. [155]. Definition 10 (Contraction Procedure for Graphs and Associated Quantum Graph Laplacians) For a given graph .G vertices V and W connected by an edge e are “glued” together to form a new vertex .(V W ) of the contracted graph G while simultaneously the edge e is removed, whereas the rest of the graph . remains unchanged. We do allow the situation of multiple edges, when V and W are connected in .G by more than one edge, in which case all such edges but the edge e become loops of their respective lengths attached to the vertex .(V W ). The corresponding quantum graph Laplacian .Aa defined on .G is contracted to the a by the application of the following rule pertaining to quantum graph Laplacian .A the coupling constants: a coupling constant at any unaffected vertex remains the same, whereas the coupling constant at the new vertex .(V W ) is set to be the sum of the coupling constants at V and .W. Here it is always assumed that all quantum graph Laplacians are described by Definition 8. Theorem 3.11 Suppose that the edge lengths of the graph .G(i) are rationally independent. The element3 .(1, 1) of the Robin-to-Dirichlet map described above (i) G obtained from the graph yields the element .(1, 1) of the “contracted” graph . (i) .G by removing a non-loop edge e emanating from .V1 . The procedure of passing (i) G is given in Definition 10. from the graph .G(i) to the contracted graph . Proof Due to the assumption that the edge lengths of the graph .G(i) are rationally independent, the element (1,1), which we denote by .f1 , is expressed explicitly as √ a function of . z and all the edge lengths .lj , .j = 1, 2, . . . , n, in particular, of the length of the edge .e, which we assume to be .l1 without loss of generality. This is an immediate consequence of the explicit form of the matrix .M (i) , see (3.1). Again without loss of generality, we also assume that the edge e connects the vertices .V1 and .V2 .

3 By

renumbering if necessary, this does not lead to loss of generality.

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√ Further, consider the expression .liml1 →0 f1 ( z; l1 , . . . , ln ; a). On the one hand, this limit is known from the explicit expression for .f1 mentioned above. On the √ other hand, .f1 is the ratio of the determinant .D(1) ( z; l1 , . . . , ln ; a) of the principal minor of the matrix .M (i) (z) −  obtained by removing its first row and first column and the determinant of .M (i) (z) −  itself: √ √ D(1) ( z; l1 , . . . , ln ; a)   .f1 ( z; l1 , . . . , ln ; a) = det M (i) (z) −  Next, we multiply by .−l1 both the numerator and denominator of this ratio, and pass to the limit in each of them separately:

.

√ lim f1 ( z; l1 , . . . , ln ; a) =

l1 →0

√ lim (−l1 )D(1) ( z; l1 , . . . , ln ; a)   lim (−l1 )det M (i) (z) − 

l1 →0

(3.6)

l1 →0

The numerator of (3.6) is easily computed as the determinant .D(2) (z; l1 , . . . , ln ; a) of the minor of .M (i) (z) −  obtained by removing its first two rows and first two columns. As for the denominator of (3.6), we add to the second row of the matrix .M (i) (z)− √  its first row multiplied by .cos( zl1 ), which leaves the determinant unchanged. This operation, due to the identity .

√ √ − cot( zl1 ) cos( zl1 ) +

√ 1 = sin( zl1 ), √ sin( zl1 )

cancels out the singularity of all matrix elements of the second row at the point l1 = 0. We introduce the factor .−l1 (cf. 3.6) into the first row and pass to the limit as .l1 → 0. Clearly, all rows but the first are regular at .l1 = 0 and hence converge to their limits as .l1 → 0. Finally, we add to the second column of the limit its first column, which again does not affect the determinant, and note that the first row of the resulting matrix has one non-zero element, namely the .(1, 1) entry. This procedure reduces the denominator in (3.6) to the determinant of a matrix of the size reduced by one. As in [59], it is checked that this determinant is nothing but  (i) −   (i) and . .det(M  ), where .M  are the Weyl matrix and the (diagonal) matrix of (i) coupling constants pertaining to the contracted graph . G . This immediately implies that the ratio obtained as a result of the above procedure coincides with the entry  (i) −  (1,1) of the matrix .(M  )−1 , i.e. .

.

(1)

√ (1) √ lim f1 ( z; l1 , . . . , ln ; a) = f1 ( z; l2 , . . . , ln ; a ),

l1 →0

where .f1 is the element (1,1) of the Robin-to-Dirichlet map of the contracted (i) graph . G , and  .a is given by Definition 10.

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The main result of this section is the theorem below, which is obtained as a corollary of Theorems 3.10 and 3.11. We assume without loss of generality that √ (e) .V1 ∈ V and denote by .f1 ( z) the (1,1)-entry of the Robin-to-Dirichlet map for the set .V(e) . We set the following notation. Fix a spanning tree .T (see e.g. [155]) of the graph .G(i) . We let the vertex .V1 to be the root of .T and assume, again without loss of generality, that the number of edges in the path .γm connecting .Vm and the root is a non-decreasing function of .m. Denote by .N (m) the number of vertices in ) * the path .γm , and by . lk(m) , .k = 1, . . . , N (m) −1, the associated sequence of lengths of the edges in .γm , ordered along the path from the root .V1 to .Vm . Note that each (m) of the lengths .lk is clearly one of the edge lengths .lj of the compact part of the original graph .G. Theorem 3.12 Assume that the graph .G is connected and the lengths of its compact e (s), .s ∈ R, the edges are rationally independent. Given the scattering matrix . (e) Robin-to-Dirichlet map for the set .V and the matrix of coupling constants . are determined constructively in a unique way. Namely, the following formulae hold for .l = 1, 2, . . . , N and determine .am , .m = 1, . . . , N : ! .

 am = lim

τ →+∞

m:Vm ∈γl

−τ

 !

 deg(Vm ) − 2(N (l) − 1) −

Vm ∈γl

1 f1(l) (iτ )

 ,

where (l) √ f1 ( z) :=

lim

.

l

(l) →0 N (l) −1

√ . . . lim lim f1 ( z), (l)

(l)

(3.7)

l2 →0 l1 →0

where in the case .l = 1 no limits are taken in (3.7).

4 Zero-Range Potentials with Internal Structure 4.1 Zero-Range Models In many models of mathematical physics, most notably in the analysis of Schrodinger operators, an explicit solution can be obtained in a very limited number of special cases (essentially, those that admit separation of variables and thus yield solutions in terms of special functions). This deficit of explicitly solvable models has led physicists, starting with E. Fermi in 1934, to the idea to replace potentials with some boundary condition at a point of three-dimensional space, i.e., a zero-range potential.

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The rigorous mathematical treatment of this idea was initiated in [24, 107]. It was shown that the corresponding model Hamiltonians are in fact self-adjoint extensions of a Laplacian which has been restricted to the set of .W 2,2 functions vanishing in a vicinity of a fixed point .x0 in .R3 . These ideas were further developed in a vast series of papers and books, culminating in the monographs [4, 5], which also contain comprehensive lists of references. Physical applications of zero-range models have been treated in, e.g., [54]. It has been conjectured that zero-range models provide a good approximation of realistic physical systems in at least a far-away zone, where the concrete shape of the potential might be discarded, making them especially useful in the analysis of scattering problems. Here we also mention the celebrated Kronig-Penney model where a periodic array of zero-range potentials is used to model the atoms in a crystal lattice. Despite the obvious success of the idea explained above, it still carries a number of serious drawbacks. In particular, it can be successfully applied to model spherically symmetric scatterers only. If one attempts to model a scatterer of a more involved structure, i.e., possessing a richer spectrum, by a finite set of zero-range potentials, the complexity of the model grows rapidly, essentially eliminating the main selling point of the model, i.e., its explicit solvability. In the 1980s, based in part on earlier physics papers by Ju. M. Shirokov [139], where the idea was presented in an implicit form, B. S. Pavlov [127] rigorously introduced a model of zero-range potential with an internal structure, see also his later survey [128]. This idea was further developed by Pavlov, his students, and collaborators, see e.g. [3, 130] and references therein. In the mentioned works of Pavlov, one starts by considering the operator .A0 being a Laplacian restricted to the set of .W 2,2 functions vanishing in a vicinity of a fixed point in .R3 , precisely as in [24]. Then, instead of considering von Neumann self-adjoint extensions of the latter, one passes over to the consideration of the so-called out-of-space extensions, i.e., extensions to self-adjoint operators in a larger Hilbert space. The theory of out-of-space extensions generalising that of J. von Neumann was constructed by M. A. Neumark in [114, 115] in the case of densely defined symmetric operators and by M. A. Krasnoselskii [86] and A. V. Strauss [147] in the case opposite, see also [148] for the connections with the theory of functional models. In fact, Pavlov, being quite possibly unaware of these theoretical developments, has reinvented this technique in the following way. Alongside the original Hilbert space .H = L2 (R3 ), consider an auxiliary internal Hilbert space E (which can be in many important cases considered to be finite-dimensional) and a self-adjoint operator A with simple spectrum in it. Let .φ be its generating vector and consider the restriction .Aφ of A (non-densely defined) to the space .

dom Aφ := {(A − i)−1 ψ : ψ ∈ E, φ, ψ = 0}.

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This leads to the symmetric operator .A0 on the Hilbert space .H ⊕ E, defined as A0 ⊕ Aφ on the domain

.

.

dom A0 :=

   f : f ∈ dom A0 , v ∈ dom Aφ , v

where .A0 is the restricted Laplacian on H introduced above. The operator .A0 is then a symmetric non-densely defined operator with equal deficiency indices, and one can consider its self-adjoint extensions .A. Among them, we will single out those which non-trivially couple the spaces H and E by feeding the boundary data at .x0 of a function .f ∈ W 2,2 (R3 ) to the “part” of operator acting in E. The latter then serves as the operator of the “internal structure”, which can be chosen arbitrarily complex. We elect not to dwell on the precise way in which the extensions .A are constructed as an explicit examples of an operator of this class will be presented below in Sect. 4.2.

4.2 Connections with Inhomogeneous Media Leaving the subject of zero-range models with internal structure aside for a moment, let us briefly consider a number of physically motivated models giving rise to zerorange potentials in general. In particular, we will be interested in those models which lead to a distribution “potential” .δ  , where .δ is the Dirac delta function. It is wellknown, see, e.g., [5], that the question of relating an operator of the form .− + αδ  to one of self-adjoint von Neumann extensions of a properly selected symmetric restriction .A0 of .− is far from being trivial, as .δ  , unlike .δ, is form-unbounded. For the same reason, it is non-trivial to construct an explicit “regularisation” of a .δ  −perturbed Laplacian, i.e., a sequence of operators .Aε being either potential perturbations of the Laplacian or, in general, any perturbations of the latter which converge in some sense (say, in the sense of resolvent convergence) to the Laplacian with a .δ  perturbation. In particular, we point out among many others the paper [40] where .Aε are chosen as first-order differential non-self-adjoint perturbations of the Laplacian of a special form and the paper [71] where the perturbation is assumed to admit the form .ε−2 v(x/ε). It turns out that additive .ε−dependant perturbations are not the most straightforward choice for the task described, as zero-range perturbations (and more precisely, zero-range perturbations with internal structure) appear naturally in the asymptotic analysis of inhomogeneous media. In particular, in the paper [42] we studied the norm-resolvent asymptotics of differential operators .Aε with periodic coefficients with high contrast, defined by their resolvents .(Aε − z)−1 : f → u as follows: .

    − a ε x u − zu = f,

f ∈ L2 (R), ε > 0, z ∈ C,

where, for all .ε > 0, the coefficient .a ε is 1-periodic and

(4.1)

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⎧ −2 ⎪ ⎨ aε , y ∈ [0, l1 ), ε .a (y) := 1, y ∈ [l1 , l1 + l2 ), ⎪ ⎩ −2 aε , y ∈ [l1 + l2 , 1), with .a > 0, and .0 < l1 < l1 + l2 < 1. Here in (4.1) the “natural” matching conditions are imposed at the points of discontinuity of the symbol .a ε (x), i.e., the continuity of both the function itself and of its conormal ´derivative, so that the operators .Aε can be thought as being defined by the form . a ε (x)|u (x)|2 dx. We remark that the operators .Aε are unitary equivalent to the operators of the doubleporosity model of homogenisation theory in dimension one, see, e.g., [77]. The main result of the named paper can be reformulated as follows. Theorem 4.13 The norm-resolvent limit of the sequence .Aε is unitarily equivalent to the operator .Ahom in .L2 (R) given by the differential expression .−l2−2 d 2 /dx 2 on ) dom(Ahom ) = U : U ∈ W 2,2 (n, n + 1) ∀n ∈ Z, U  ∈ C(R),

.

* U (n + 0) − U (n − 0) = l2−1 (l1 + l3 )U  (n) ∀n ∈ Z , where .l3 := 1 − (l1 + l2 ). Moreover, for all z in a compact set .Kσ such that the distance of the latter from the positive real line is not less than a fixed .σ > 0, this norm resolvent convergence is uniform, with the (uniform) error bound .O(ε2 ). By inspection, the operator .Ahom defined above corresponds to the formal differential expression .

− l2−2

d2 (l1 + l3 ) !  δ (x − n), + 2 l2 dx n∈Z

i.e., it is the operator of a Kronig-Penney dipole-type model on the real line. It is also quite clear that the periodicity of the model considered has nothing to do with the fact that the effective operator acquires the .δ  -type potential perturbation. Thus it leads to the understanding that strong inhomogeneities in the media in generic (i.e., not necessarily periodic) case naturally give rise to zero-range potentials of .δ  -type. In order to relate this exposition to our subject of zero-range potentials with internal structure, let us describe the main ingredient leading to the result formulated above. As usual in dealing with periodic problems, we apply the Gelfand transform Uˆ (y, τ ) = (2π )−d/2 .

!

  U (y + n) exp − iτ · (y + n) ,

n∈Z

d

(4.2) y ∈ [0, 1], τ ∈ [−π, π ),

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to the operator family .Aε , which yields the operator family .Aε the differential expression  .

−

corresponding to

   d d + iτ a ε (x) + iτ dx dx

on the interval .[0, 1] with periodic boundary conditions at the endpoints. Here .τ ∈ [−π, π ) is the quasimomentum. As above, the matching conditions at the points of discontinuity of the symbol .a ε (x) are assumed to be natural. (τ ) The asymptotic analysis of the operator family .Aε , as shown in [42, 45], yields the following operator as its norm-resolvent asymptotics. Let .Hhom = L2 [0, l2 ] ⊕ ) C1 . For all values .τ ∈ [−π, π ), consider a self-adjoint operator .A(τ hom on the space (τ ) .Hhom , defined as follows. Let the domain .dom A hom be defined as .

  + (τ ) dom Ahom = (u, β)! ∈ Hhom : u ∈ W 2,2 (0, l2 ), u(0) = ξ (τ ) u(l2 ) = β/ l1 + l3 . ) On .dom A(τ hom the action of the operator is set by

⎛   ⎜ (τ ) u =⎜ .A hom ⎝ β

⎞ 2 1 d +τ ⎟ i dx ⎟. ⎠   (τ ) (τ ) ∂ u − ξ (τ ) ∂ u

 1 −√ l1 + l3

0

l2

Here 

  ξ (τ ) := exp i(l1 + l3 )τ ,

∂ (τ ) u :=

.

 d + iτ u. dx

) −1 admits the following estimate in the Theorem 4.14 The resolvent .(A(τ ε − z) uniform operator norm topology: ) −1 (A(τ −  ∗ (Ahom − z)−1  = O(ε2 ), ε − z)

.

(τ )

where . is a partial isometry from .H = L2 (0, 1) to .Hhom . This estimate is uniform in .τ ∈ [−π, π ) and .z ∈ Kσ . (τ )

It is clear now that the operator .Ahom is nothing but the simplest possible example of Pavlov’s zero-range perturbations with internal structure, corresponding to the case where the dimension of the internal space E is equal to one. The (τ ) definition of .Ahom implies that the support of the zero-range potential here is located at the point .x0 = 0, which is identified due to quasi-periodic (of Datta–Das Sarma type) boundary conditions with the point .x = l2 .

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Next it is shown (see [42] for details) that under an explicit unitary transform )   the operator family .A(τ hom is unitary equivalent to the family .Ahom (τ ) at the    quasimomentum point .τ = τ + π(mod 2π ). Here .Ahom (τ ) acts in the space (τ ) 2 .L [0, l2 ] and is defined by the same differential expression as .A hom , with the  parameter .τ replaced by .τ :  .

1 d + τ i dx

2 ,

on the domain described by the conditions   u(0) + e−i(l1 +l3 )τ u(l2 ) = (l1 + l3 )∂ (τ ) u 0 ,

.

   ∂ (τ ) u 0 = −e−i(l1 +l3 )τ ∂ (τ ) u l . 2

An application of the inverse Gelfand transform then yields Theorem 4.13. This (τ ) shows that the operator .Ahom which is an operator of a zero-range model with the internal space E of dimension one is in fact a differential operator with a .δ  potential, up to a unitary transformation. In view of [93, 141] it is plausible that by a similar argument it could be shown, that an operator with a .δ (n) -potential can be realized as a zero-range model with .dim E = n, for any natural n. ) It is interesting to note that an operator admitting the same form as .A(τ hom (with .τ = 0) appears naturally in the setting of [61, 89, 90], who discuss the behaviour of the spectra of operator sequences associated with domains “shrinking” as .ε → 0 to a metric graph embedded into .Rd . Here the rate of shrinking of the “edge” parts is assumed to be related to the rate of shrinking of the “vertex” parts of the domain via .

ε vol(Vvertex ) ε ) → α > 0, ε → 0. vol(Vedge

(4.3)

It is shown in the above works that the spectra of the corresponding Laplacian operators with Neumann boundary conditions converge to the spectrum of a quantum graph associated with a Laplacian on the metric graph obtained as the (τ ) limit of the domain as .ε → 0. The “weight” .l1 + l3 in .Ahom plays the rôle of the constant .α in (4.3). By a similar argument to the one presented above one can show, that in the case of domains shrinking to a graph under the “resonant” condition (4.3) one obtains, under a suitable unitary transform, the matching condition of .δ  -type at the internal graph vertices, with the corresponding coupling constant equal to .α.

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4.3 A PDE Model: BVPs with a Large Coupling 4.3.1

Problem Setup

In [82], we studied a prototype large-coupling transmission problem, posed on a bounded domain . ⊂ Rd , .d = 2, 3, see Fig. 1, containing a “low-index” (equivalently, “high propagation speed”) inclusion .− , located at a positive distance to the boundary .∂. Mathematically, this is modelled by a “weighted” Laplacian .−a± , where .a+ = 1 (the weight on the domain .+ :=  \ − ), and .a− ≡ a (the weight on the domain .− ) is assumed to be large, .a− # 1. This is supplemented by the Neumann boundary condition .∂u/∂n = 0 on the outer boundary .∂, where n is the exterior normal to .∂, and “natural” continuity conditions on the “interface” . := ∂− . For each .a, we consider time-harmonic vibrations of the physical domain represented by ., described by the eigenvalue problem for an appropriate operator in .L2 (). It is easily seen that eigenfunction sequences for these eigenvalue problems converge, as .a → ∞, to either a constant or a function of the form v−

.

1 ||

ˆ v, +

where v satisfies the spectral boundary-value problem (BVP) .

  ˆ 1 − v = z v − v in + , || +

v| = 0,

∂v = 0. ∂n ∂

(4.4)

Here the spectral parameter z represents the ratio of the size of the original physical domain to the wavelength in its part represented by .+ . Fig. 1 Domain with a “stiff” inclusion

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The problem (4.4) is isospectral to the so-called “electrostatic problem” discussed in [161, Lemma 3.4], see also [10] and references therein, namely the eigenvalue problem for the self-adjoint operator Q defined by the quadratic form ˆ ∇v · ∇v, u = v + c,

q(u, u) =

.

+

) * 1 := v ∈ H 1 (+ ), v| = 0 , v ∈ H0,

c∈C

on the Hilbert space .L2 (+ )  C, treated as a subspace of .L2 (). Denote by .A+ 0 the Laplacian .− on .+ , subject to the Dirichlet condition on + + . and the Neumann boundary condition on .∂ and let .λ , .φ , .j = 1, 2, . . . , be j j the eigenvalues and the corresponding orthonormal eigenfunctions, respectively, of + .A . 0 It is then easily shown, that the spectrum of the electrostatic problem is the union of two sets: (a) the set of z solving the equation ˆ  ∞ ! −1 z || + z (λ+ − z) j

.

j =1

+

2  φj+ = 0.

and (b) the set of those eigenvalues .λ+ j for which the corresponding eigenfunction + .φ has zero mean over .+ . j 4.3.2

Norm-Resolvent Convergence to a Zero-Range Model with an Internal Structure

Suppose that . is a bounded .C 1,1 domain, and . ⊂  is a closed .C 1,1 curve, so that . = ∂− is the common boundary of domains .+ and .− , where .− is strictly contained in ., such that .+ ∪ − = , see Fig. 1. For .a > 0, .z ∈ C we consider the “transmission” eigenvalue problem (cf. [138]) ⎧ ⎪ −u+ = zu+ in + , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −au− = zu− in − , ∂u+ ∂u− . ⎪ u+ = u− , +a = 0 on , ⎪ ⎪ ∂n+ ∂n− ⎪ ⎪ ⎪ ⎪ ∂u+ ⎪ ⎩ = 0 on ∂, ∂n+

(4.5)

where .n± denotes the exterior normal (defined a.e.) to the corresponding part of the boundary. The above problem is understood in the strong sense, i.e. .u± ∈ H 2 (± ), the Laplacian differential expression . is the corresponding combination of secondorder weak derivatives, and the boundary values of .u± and their normal derivatives

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are understood in the sense of traces according to the embeddings of .H 2 (± ) into s s .H (), .H (∂), where .s = 3/2 or .s = 1/2. Denote by .Aa the operator of the above boundary value problem. Its precise definition is given on the basis of the boundary triples theory in the form of [136]. Consider the space .Heff = L2 (+ )⊕C and the following linear subset of .L2 () :    u+ η ∂u+ 2 ∈ Heff : u+ ∈ H (+ ), u+ | = √ 1 , . dom Aeff = =0 , η ∂n+ ∂ |− | where .u| is the trace of the function u and .1 is the unity function on .. On dom Aeff we set the action of the operator .Aeff by the formula

.

 Aeff

.

u+ η



⎛ ⎜ =⎝

−u+

⎞

⎟ 1 ´ ∂u+ ⎠ . √ |− |  ∂n+

(4.6)

Theorem 4.15 The operator .Aeff is the norm-resolvent limit of the operator family Aa . This convergence is uniform for .z ∈ Kσ , with the error estimated by .O(a −1 ).

.

This theorem yields in particular the convergence (in the sense of Hausdorff) of the spectra of .Aa to that of .Aeff . This convergence is uniform in .Kσ , and its rate is estimated as .O(a −1 ). Moreover, it is shown that the spectrum of .Aeff coincides with the spectrum of the electrostatic problem (4.4). Note that the form of .Aeff is once again identical to that of a zero-range model with an internal structure in the case when the internal space E is one-dimensional. The obvious difference is that here the effective model of the medium is no longer “zero-range” per se; rather it pertains to a singular perturbation supported by the boundary .. Therefore, the result described above allows one to extend the notion of internal structure to the case of distributional perturbations supported by a curve, see also [94] where this idea was first suggested, although unlike above no asymptotic regularisation procedure was considered. Moreover, well in line with the narrative of preceding sections, the internal structure appears owing exclusively to the strong inhomogeneity of the medium considered. We remark that a “classical” zero-range perturbation with an internal structure can still be obtained by a rather simple modification of the problem considered. Namely, let a be fixed, and let the volume of the inclusion .− now wane to zero as the new parameter .ε → 0. This represents a model that has been studied in detail, see, e.g., [9] and references therein. In this modified setup, a virtually unchanged argument leads to the inclusion being asymptotically modelled by a zero-range potential with an internal structure. Moreover, the dimension of the internal space E is again equal to one, provided that a uniform norm-resolvent convergence is sought for the spectral parameter belonging to the compact .Kσ .

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293

Internal Structure with Higher Dimensions of the Internal Space E

A natural question must therefore be posed: can strongly inhomogeneous media only give rise to simplest possible zero-range models with internal structure, pertaining to the case of .dim E = 1, or is it possible to obtain effective models with more involved internal structures? It turns out that the second mentioned possibility is realised, which we will demonstrate briefly using the material of the preceding section. Recall that in all the results formulated above the uniform convergence was claimed under the additional assumption that the spectral parameter belongs to a fixed compact. If one drops this assumption, within the setup of the previous section one has the following statement. Theorem 4.16 Up to a unitary equivalence, for .k ∈ N there exists a self-adjoint operator .Aeff of a zero-range model with an internal structure on the space .Heff := L2 (+ ) ⊕ Ck such that   (Aa − z)−1  P(Aeff − z)−1 P + O max{a −1 , |z|k+1 a −k }

.

(4.7)

in the uniform operator norm topology. Here .P is the orthogonal projection of .Heff onto .L2 (+ ) ⊕ C (i.e., the space .Heff of the previous section). Note that unlike the results pertaining to the situation of the spectral parameter contained in a compact, here the leading-order term of the asymptotic expansion of the resolvent of the original operator .Aa is not a resolvent of some self-adjoint operator (unless .k = 1), but rather a generalised resolvent. It is also obvious that the concrete choice of k to be used in the last theorem depends on the concrete relationship between z and a and on the error estimate sought. In essence, this brings about the understanding that despite the fact that on the face of it the problem at hand is one-parametric, it must be treated as having two parameters, z and a. The operator .Aeff of the last Theorem admits an explicit description for any .k ∈ N, but this description is rather involved. In view of better readability of the paper, we only present its explicit form in the case .k = 2: ⎛

⎞

⎛

−u+

⎞

u+ ⎜ 1 ´ ∂u ⎟ + ⎜ 2 D −1 η + Bη ) ⎟ + a(B Aeff ⎝ η1 ⎠ := ⎜ ⎟. 1 2 ⎝ κ  ∂n+ ⎠ η2 a(Bη1 + Dη2 )

.

Here .B, D and .κ are real parameters, which are explicitly computed. It should be noted that similar results can be obtained in the homogenisationrelated setup of the previous section, see also Sect. 5. We can therefore conclude that zero-range models with internal structure appear naturally in the asymptotic analysis of highly inhomogeneous media. Moreover, in the generic case they appear as Neumark-Strauss dilations (see [115, 116, 145, 148])

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of main order terms in the asymptotic expansions of the resolvents of problems considered. The complexity of the internal structure can be arbitrarily high (i.e., the dimension of the internal space E can be made as high as required), provided that the spectral parameter z is allowed to grow with the parameter .a → ∞ (or (n) .ε → 0). Further, owing to the remark made above that an operator with a .δ potential could be realized as a zero-range model with .dim E = n for any natural n, we expect models with strong inhomogeneities to admit the role of the tool of choice in the regularisation of singular and super-singular perturbations, beyond the form-bounded case and including the case of singular perturbations supported by a curve or a surface.

4.4 The Rôle of Generalised Resolvents We close this section with a brief exposition of how precisely the asymptotic results formulated above are obtained. The analysis starts with the family of resolvents, say (in the case of Sect. 4.2) .(Aε − z)−1 , describing the inhomogeneous medium at hand. One then passes over to the generalised resolvent .Rε (z) := P (Aε − z)−1 P ∗ , where P denotes the orthogonal projection onto the “part” of the medium which is obtained by removing the inhomogeneities. Note that the generalised resolvent thus defined is a solution operator of a BVP pertaining to homogeneous medium, albeit subject to non-local .z−dependant boundary conditions. The problem considered therefore reduces to the asymptotic analysis of the operator .Bε (z), parameterising these conditions. As such, it becomes a classical problem of perturbation theory. Assuming now, for the sake of argument, that .Rε (z) has a limit, as .ε → 0, in the uniform operator topology for z in a domain .D ⊂ C, and, further, that the resolvent −1 also admits such limit, one clearly has .(Aε − z)  −1 P Aeff − z P ∗ = R0 (z),

.

z ∈ D ⊂ C,

(4.8)

where .R0 and .Aeff are the limits introduced above. The idea of simplifying the required analysis by passing to the resolvent “sandwiched” by orthogonal projections onto a carefully chosen subspace is in fact the same as in [95], where the resulting sandwiched operator is shown to be the resolvent of a dissipative operator. The function .R0 defined by (4.8) is a generalised resolvent, whereas .Aeff is its out-of-space self-adjoint extension (or Neumark-Strauss dilation [145]). By a theorem of Neumark [115] (see Sect. 2.5 of the present paper) this dilation is defined uniquely up to a unitary transformation of a special form, provided that the minimality condition holds. The latter can be reformulated along the following lines: one has minimality, provided that there are no eigenmodes in the effective media modelled by the operator .Aε , and therefore in the medium modelled by the operator .Aeff as well, such that they “never enter” the part of the medium without inhomogeneities. A quick glance at the setup of our models helps one immediately

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convince oneself that this must be true. It then follows that the effective medium is completely determined, up to a unitary transformation, by .R0 (z). Once this is established, it is tempting to construct its Neumark-Strauss dilation and conjecture, that it is precisely this dilation that the original operator family converges to in the norm-resolvent sense (of course, up to a unitary transformation). This conjecture in fact holds true, although it is impossible to prove it on the abstract level: taking into account no specifics of problems at hand, one can claim weak convergence at best. Still, the approach suggested seems to be very transparent in allowing to grasp the substance of the problem and to almost immediately “guess” correctly the operator modelling the effective medium.

5 Applications to Continuum Mechanics and Wave Propagation Parameter-dependent problems for differential equations have traditionally attracted much interest within applied mathematics, by virtue of their potential for replacing complicated formulations with more straightforward, and often explicitly solvable, ones. This drive has led to a plethora of asymptotic techniques, from perturbation theory to multi-scale analysis, covering a variety of applications to physics, engineering, and materials science. While this subject area can be viewed as “classical”, problems that require new ideas continue emerging, often motivated by novel wave phenomena. One of the recent application areas of this kind is provided by composites and structures involving components with highly contrasting material properties (stiffness, density, refractive index). Mathematically, such problems lead to boundary-value formulations for differential operators with parameter-dependent coefficients. For example, problems of this kind have arisen in the study of periodic composite media with “high contrast” (or “large coupling”) between the material properties of the components, see [44, 77, 160]. In what follows, we outline how the contrast parameter emerges as a result of dimensional analysis, using a scalar elliptic equation of second order with periodic coefficients as a prototype example.

5.1 Scaling Regimes for High-Contrast Setups We will consider the physical context of elastic waves propagating through a medium with whose elastic moduli vary periodically in a chosen plane (say .(x1 , x2 )plane) and are constant in the third, orthogonal, direction (say, the .x3 direction). For example, one could think of a periodic arrangement of parallel fibres of a homogeneous elastic material within a “matrix” of another homogeneous elastic material. We will look at the “polarised” anti-plane shear waves, which can be

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described completely by a scalar function representing the displacement of the medium in the .x3 direction. In the case of the fibre geometry mentioned above, the relevant elastic moduli .G then have the form   , G0 , y ∈ Q0 , G0 =: .G(y) = (y). G1 G1 , y ∈ Q1 , where .Q0 , .Q1 are the mutually complementary cross-sections of the fibre and matrix components, respectively, so that .Q0 ∪ Q1 = [0, 1]2 . The mass density of the described composite medium is assumed to be constant. (The constants .G0 , .G1 are the so-called shear moduli of the materials occupying .Q0 , .Q1 .) This physical setup was considered in [108, 123]. Denote by d the period of the original “physical” medium and consider timeharmonic wave motions, i.e. solutions of the wave equation that have the form U (x, t) = eiωt u(x),

.

x ∈ R2 , t ≥ 0,

(5.1)

where .ω is a fixed frequency. In the setting of time-harmonic waves, see (5.1), the function .u = u(x) satisfies the following equation, written in terms of the original physical units:  .

− ∇x ·

G0 G1

, (x/d)∇x u = ρω2 u.

(5.2)

Multiply both sides by .G−1 1 and denote .δ := G0 /G1 . The parameter .δ represents the “inverse contrast”, which will be assumed “small” later, and corresponds to the value −1 in terms of the “large” parameter a of Sects. 4.3.2 and 4.3.3. Equation (5.2) .a takes the form   ρ 2 δ ω u. . − ∇x · (x/d)∇x u = 1 G1 Note that ω=

.

2π c1 2π c0 = , λ1 λ0

(5.3)

where .cj , .λj are the wave speed and wavelength in the relevant media (.j = 0, 1). Introduce a non-dimensional spatial variable .x˜ = 2π x/λ1 : .

−

   ρ 2 4π 2 x˜ δ ∇x˜ u = ∇ · ω u, x˜ 1 2π d/λ1 G1 λ21

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equivalently, with .ε := 2π d/λ1 : − ∇x˜ ·

.

  ρ 2 δ c u, (x/ε)∇ ˜ x˜ u = 1 G1 1

√ where we have used (5.3). Note that .c1 ρ/G1 = 1 and relabel .x˜ by .x :   δ . − ∇x · (x/ε)∇x u = u, 1 Let us “scale to the period one” i.e. consider the change of variable .y = x/ε ˜ = x/d :   δ −2 .−ε ∇y · (y)∇y u = u, 1 or .

− ∇y ·

    2π d 2 δ u. (y)∇y u = 1 λ1

The different scaling regimes, ranging from what we know as “finite frequency, high contrast” to “high frequency, high contrast”, are described by setting ε2 = δ ν z,

.

(5.4)

where  .z, which is obviously dimensionless, is assumed to vary over the compact Kσ , and .0 ≤ ν ≤ 1. Note that  .z can be alternatively expressed as

.

−ν 2  .z = δ ρG−1 1 (dω) .

(5.5)

In particular, the setup analysed in the paper [44] corresponds to the case .ν = 1 : .

− δ −1 ∇y ·

  δ zu. (y)∇y u =  1

In terms of the original spatial variable x the Eq. (5.6) takes the form .

− d 2 δ −1 ∇x ·

  δ zu, (x)∇x u =  1

or   δ . − ∇x · (x)∇x u = k 2 u, 1

(5.6)

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√ z, so that where the wavenumber (i.e. “spatial frequency”) is given by .k := d −1 δ (kd)2 =

.

 z , n2

where .n2 = δ −1 = G0 /G1 is the shear modulus of the material occupying .Q0 relative to the material occupying .Q1 . The setup (5.6) is the “periodic” version of the formulation discussed in Sect. 4.3, see also Sect. 4.2 for the one-dimensional version of a high-contrast homogenisation problem that gives rise to the same formulation. Similarly, choosing the values .ν = 1/k, .k = 2, 3, . . . , in (5.4) gives rise to “high-frequency large-coupling” formulations, which in turn lead, in the limit as .δ → 0, to effective operators with an “internal space” of dimension .k, see Sect. 4.3.3. The parameter  .z is related to the spectral parameter z in (4.7) via z = zδ ν−1 =  zδ

.

1−k k

= za

k−1 k

,

k = 2, 3, . . . .

5.2 Homogenisation of Composite Media with Resonant Components 5.2.1

Physical Motivation

The mathematical theory of homogenisation (see e.g. [16, 23, 80]) aims at characterising limiting, or “effective”, properties of small-period composites. Following a non-dimensionalisation procedure as described above, a typical problem here is to study the asymptotic behaviour of solutions to equations of the type .

  ω2 uε = f, −div Aε (·/ε)∇uε −

f ∈ L2 (Rd ),

d ≥ 2,

 ω2 ∈ / R+ ,

(5.7)

where for all .ε > 0 the matrix .Aε is Q-periodic, .Q := [0, 1)d , non-negative, bounded, and symmetric. The parameter . ω here represents a “non-dimensional frequency”: . ω2 =  z, where  .z is the spectral parameter introduced in (5.4), so for √ example for .ν = 1 one can set . ω = d ρ/G0 ω, see (5.5). One proves (see [30, 159] and references therein) that when A is uniformly elliptic, there exists a constant matrix .Ahom such that solutions .uε to (5.7) converge to .uhom satisfying .

  ω2 uhom = f. − div Ahom ∇uhom − 

(5.8)

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In what follows we write .ω, z in place of . ω,  .z, implying that either the dimensional or non-dimensional version of the equation is chosen. In recent years, the subject of modelling and engineering a class of composite media with “unusual” wave properties (such as negative refraction) has been brought to the forefront of materials science. Such media are generically referred to as metamaterials, see e.g. [39]. In the context of homogenisation, the result sought (i.e., the “metamaterial” behaviour in the limit of vanishing .ε) belongs to the domain of the so-called time-dispersive media (see, e.g., [63, 64, 150, 151]). For such media, in the frequency domain one faces equations of the form .

  − div A∇u + B(ω2 )u = f,

f ∈ L2 (Rd ),

(5.9)

where A is a constant matrix and .B(ω2 ) is a frequency-dependent operator in 2 d 2 .L (R ) taking the place of .−ω in (5.7), if, for the sake of argument, in the time domain we started with an equation of second order in time. If, in addition, the matrix function .B is scalar, i.e., .B(ω2 ) = β(ω2 )I with a scalar function .β, the problem of the type .

  − div Ahom (ω2 )∇u = ω2 u

(5.10)

appears in place of the spectral problem after a formal division by .−β(ω2 )/ω2 , with frequency-dependent (but independent of the spatial variable) matrix .Ahom (ω2 ). In the Eq. (5.10), in contrast to (5.8), the matrix elements of .Ahom , interpreted as material parameters of the medium, acquire a non-trivial dependence on the frequency, which may lead to their taking negative values in some frequency intervals. The possibility of electromagnetic media exhibiting negative refraction was envisaged in an early work [156], which showed theoretically that the material properties of such media must be frequency-dependent, and the last two decades have seen a steady advance towards realising such media experimentally. One may hope that upon relaxing the condition of uniform ellipticity on .Aε one may be able to achieve a metamaterial-type response to wave propagation for sufficiently small values of .ε. It is therefore important to understand how inhomogeneity in the spatial variable in (5.7) can lead, in the limit .ε → 0, to frequency dispersion as in (5.9).

5.2.2

Operator-Theoretic Motivation

Already in the setting of finite-dimensional matrix algebra, equations of the form (see (5.9)) Au + B(z)u = f,

.

u ∈ Cd ,

(5.11)

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where .A = A∗ ∈ Cd×d , .f ∈ Cd , .B is a Herglotz function with values in .Cd×d , emerge when one seeks solutions to the standard resolvent equation for a block matrix: .- . - . - . - . A B u u f u . (5.12) −z = , ∈ Cd+k , ∗ B C v v 0 v where .B ∈ Ck×d , .C = C ∗ ∈ Ck×k . Indeed, it is the result of a straightforward calculation that (5.12) implies Au − (B(C − z)−1 B ∗ + zI )u = f,

.

whenever .−z is not an eigenvalues of .C, so (5.12) implies (5.11) with .B(z) = −B(C − z)−1 B ∗ − zI. Another consequence of the above calculation is that for any vector .f ∈ Ck one has u=P

.

A B B∗ C

. + zI

−1

P ∗ f,

where P is the orthogonal projection of .Cd+k onto .Ck . The above argument, in the more general setting of block operator matrices in a Hilbert space, likely appeared for the first time in [63]. “Generalised resolvents”, i.e. objects of the form P (A − z)−1 P ∗ ,

.

(5.13)

where .A is an operator in a Hilbert space .H and P is an orthogonal projection of .H onto its subspace H , have already been discussed in the present survey, see Sects. 2.5 and 4.4. As discussed in Sect. 2.5, abstract results of Neumark and Strauss [115, 145] establish that solution operators for formulations (5.11), where A is a selfadjoint operator in a Hilbert space H , can be written in the form (5.13) for a suitable “out-of-space” extension .A. Therefore, a natural question is whether formulations (5.9) can be viewed as generalised resolvents obtained by an asymptotic analysis of some parameter-dependent operator family describing a heterogeneous medium. One piece of evidence pointing at the validity of such a conjecture is the result of Sect. 4.3.2, where the role of the operator .A in (5.13) is played by .Aeff , see (4.6). In [42, 43, 45] a model of a high-contrast graph periodic along a single direction was considered. A unified treatment of critical-contrast homogenisation was proposed and carried out in three distinct cases: (i) where neither the soft nor the stiff component of the medium is connected; (ii) where the stiff component of the medium is connected; (iii) where the soft component of the medium is connected. The analytical toolbox presented in these works was then amplified to the PDE setting in [44]. In the wider context of operator theory and its applications,

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this provides a route towards constructing explicit spectral representations and functional models for both homogenisation limits of critical-contrast composites and the related time-dispersive models, as well as towards solving the related direct and inverse scattering problems.

5.2.3

Prototype Problem Setups in the PDE Context

Consider the problem (5.7) under the following assumptions: Aε (y) =

.

⎧ ⎨aI,

y ∈ Qstiff ,

⎩ε2 I,

y ∈ Qsoft ,

where .Qsoft (.Qstiff ) is the soft (respectively, stiff) component of the unit cube .Q = [0, 1)d ⊂ Rd , so that .Q = Qsoft ∪ Qstiff , and .a > 0. Two distinct setups were studied in [44]. For one of them (“Model I”), which is unitary equivalent to the model of [66, 77], the component .Qsoft ⊂ Q is simply connected and and its distance to .∂Q is positive, cf. [41, 160]. For the other one (“Model II”) the component .Qstiff has the described properties (Fig. 2). It is assumed that the Dirichlet-to-Neumann maps for .Qsoft and .Qstiff , which map the boundary traces of harmonic functions in .Qsoft and .Qstiff to their boundary normal derivatives, are well-defined as pseudo-differential operators of order one in the .L2 space on the boundary [1, 11, 65, 78]. For both above setups, [44] deals with the resolvent .(Aε − z)−1 of a self-adjoint operator in .L2 (Rd ) corresponding to the problem (5.7), so that its solutions are

Fig. 2 Model setups. Model I: soft component .Qsoft in blue, stiff component .Qstiff in green. Model II: soft component .Qsoft in green, stiff component .Qstiff in blue

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expressed as .uε = (Aε − z)−1 f with .z = ω2 . For each .ε > 0, the operator .Aε is defined by the forms ˆ . Aε (·/ε)∇u · ∇u, u, v ∈ H 1 (Rd ). Rd It is assumed that .z ∈ C is separated from the spectrum of the original operator family, in particular .z ∈ Kσ , where .Kσ is defined in Theorem 4.13. In order to deal with operators having compact resolvents, it is customary to apply Gelfand transform [68], which we review next. 5.2.4

Gelfand Transform and Direct Integral

The version of the Gelfand transform convenient for the analysis of the operators Aε is defined on functions .u ∈ L2 (Rd ) by the formula4 (cf. (4.2))  2 d/2 !     ε u ε(y + n) exp − iτ · (y + n) , .Gε u(y, t) := 2π d n∈Z

.

y ∈ Q, τ ∈ Q := [−π, π )d , This yields a unitary operator .Gε : L2 (Rd ) −→ L2 (Q × Q ), and the inverse with the inverse mapping given by     ˆ x x −d/2 dτ, x ∈ Rd , .u(x) = (2π ) Gε u , τ exp iτ · ε ε Q where .Gε u is extended to .Rd × Q by Q-periodicity in the spatial variable. As in [30], an application of the Gelfand transform G to the operator family (τ ) .Aε corresponding to the problem (5.7) yields the two-parametric family .Aε of 2 operators in .L (Q) given by the differential expression .

− (∇ + iτ )Aε (x/ε)(∇ + iτ ),

ε > 0,

τ ∈ Q ,

subject to periodic boundary conditions on .∂(εQ) and defined by the corresponding closed coercive sesquilinear form. For each .ε > 0, the operator .Aε is then unitary ) equivalent to the von Neumann integral (see e.g. [29, Chapter 7]) of .A(τ ε : Aε =

.

4 The

G∗ε





ˆ ⊕

Q

) A(τ ε dτ

Gε .

formula (4.2) is first applied to continuous functions U with compact support, and then extended to the whole of .L2 (Rd ) by continuity.

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Similar to [66] and facilitated by the abstract framework of [136], the operator ) A(τ ε can be associated to transmission problems [138], akin to those considered in Sect. 4.3.2. To this end, consider Q as a torus with the opposite parts of .∂Q identified, and view .Qsoft and .Qstiff as subsets of this torus. Furthermore, in line with the notation of Sect. 4.3.2, denote by . the interface between .Qsoft and .Qstiff . For each .ε, .τ, .f ∈ L2 (Q), the transmission problem is formulated as finding a function .u ∈ L2 (Q) such that .u|Qsoft ∈ H 1 (Qsoft ), .u|Qstiff ∈ H 1 (Qstiff ), that solves, in the weak sense, the boundary-value problem (cf. (4.5))

.

⎧ ⎪ −ε−2 (∇ + iτ )2 u+ − zu+ = f in Qstiff , ⎪ ⎪ ⎪ ⎨ −(∇ + iτ )2 u− − zu− = f in Qsoft , .     ⎪ ⎪ ⎪ ∂ ∂ ⎪ −2 ⎩ u+ = u− , + iτ · n+ u+ + ε + iτ · n− u− = 0, ∂n+ ∂n−

on .

where .n+ and .n− = −n+ are the outward normals to . with respect to .Qsoft and Qstiff . By a classical argument the weak solution of the above problem is shown to ) −1 coincide with .(A(τ ε − z) f.

.

5.2.5

Homogenised Operators and Convergence Estimates

Throughout this section, .Hhom := L2 (Qsoft ) ⊕ C1 , .H := L2 (), and .∂nτ u := −(∂u/∂n + iτ · nu)| is the co-normal boundary derivative for .Qsoft . MODEL I. Set  (τ ) . dom A = (u, β)! ∈ Hhom : u ∈ H 2 (Qsoft ), hom      u| = u| , ψ0 H ψ0 and β = κ u| , ψ0 H , where .κ := |Qstiff |1/2 /||1/2 , .ψ0 (x) = ||−1/2 , .x ∈ , and define (τ ) .A hom

.   −(∇ + iτ )2 u u = ,   β −κ −1 ∂nτ u| , ψ0 H − κ −2 ε−2 (μ∗ τ · τ )β

  u (τ ) ∈ dom Ahom , β

where .μ∗ τ · τ is the leading-order term (for small .τ ) of the first Steklov eigenvalue for .−(∇ + iτ )2 on .Qstiff . Model II. Set  (τ ) . dom A = (u, β)! ∈ Hhom : u ∈ H 2 (Qsoft ), hom      u| = u| , ψτ H ψτ and β = κ u| , ψτ H ,

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where .κ is as above and .ψτ (x) = ||−1/2 exp(−iτ · x)| , .x ∈ . The action of the operator is set by (τ ) .A hom

  −(∇ + iτ )2 u u =   β −κ −1 ∂nτ u| , ψτ

. , H

  u (τ ) ∈ dom Ahom . β

CONVERGENCE ESTIMATE. Set .γ = 2/3 for the case of Model I and .γ = 2 for (τ ) the case of Model II. The resolvent .(Aε − z)−1 admits the following estimate in the uniform operator-norm topology:  (τ )  (τ ) −1 −1 Aε − z − ∗ Ahom − z  = O(εγ ),

.

(5.14)

where . is a partial isometry from .L2 (Q) onto .Hhom : on the subspace .L2 (Qsoft ) it coincides with the identity, and each function from .L2 (Qstiff ) represented as an orthogonal sum cτ stiff ψτ −1 stiff ψτ ⊕ ξτ ,

.

cτ ∈ C1 ,

is mapped to .cτ . Here .stiff maps .ϕ on . to the solution .uϕ of the problem −(∇ + iτ )2 uϕ = 0 in Qstiff ,

.

uϕ | = ϕ. The estimate (5.14) is uniform in .τ ∈ Q and .z ∈ Kσ . Acknowledgments KDC, AVK have been supported by EPSRC (Grants EP/L018802/2, EP/V013025/1.) The work of all authors has been supported by CONACyT CF-2019 No. 304005.

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133. V. Ryzhov, Functional model of a class of non-selfadjoint extensions of symmetric operators. Oper. Theory Adv. Appl. 174, 117–158 (2007) 134. V. Ryzhov, Functional model of a closed non-selfadjoint operator. Integr. Equ. Oper. Theory 60(4), 539–571 (2008) 135. V. Ryzhov, Weyl-Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control. Compl. Anal. Oper. Theory 3, 289– 322 (2009) 136. V. Ryzhov, Linear operators and operator functions associated with spectral boundary value problems. Oper. Theory: Adv. Appl. 276, 576–626 (2020). 137. L.A. Sahnoviˇc, Dissipative operators with absolutely continuous spectrum (Russian). Trudy Moskov. Mat. Obsc. 19, 211–270 (1968) 138. M. Schechter, A generalization of the problem of transmission. Ann. Scuola Norm. Sup. Pisa 14(3), 207–236 (1960). 139. J.M. Shirokov, Strongly singular potentials in three-dimensional quantum mechanics (Russian). Teoret. Mat. Fiz. 42(1):45–49 (1980) 140. K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol. 265 (Springer, Dordrecht, 2012) 141. A.A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions. J. Soviet. Math. 33(6), 1311–1342 (1983) 142. J.G. Sina˘ı, Dynamical systems with countable Lebesgue spectrum I (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 25, 899–924 (1961) 143. B.M. Solomyak, On the functional model for dissipative operators. The coordinate-free approach. J. Soviet Math. 61(2), 1981–2002 (1992) 144. I.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser. vol. 30 (Princeton University Press, Princeton, 1970) 145. A.V. Štraus, Generalised resolvents of symmetric operators (Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 18, 51–86 (1954) 146. A.V. Štrauss, Characteristic functions of linear operators. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 24(1), 43–74 (1960) 147. A.V. Strauss, Extensions and generalized resolvents of a non-densely defined symmetric operator. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 34, 175–202 (1970) 148. A.V. Štraus, Functional models and generalized spectral functions of symmetric operators. St. Petersbg. Math. J. 10(5), 733–784 (1999) 149. B. Sz.-Nagy, C. Foia¸s, H. Bercovici, L. Kérchy, Harmonic Analysis of Operators on Hilbert Space (Springer, New York, 2010) 150. A. Tip, Linear absorptive dielectrics. Phys. Rev. A 57, 4818–4841 (1998) 151. A. Tip, Some mathematical properties of Maxwell’s equations for macroscopic dielectrics. J. Math. Phys. 47, 012902 (2006) 152. E. Tsekanovskii, Characteristic function and sectorial boundary value problems (Russian), in Proceedings of the Institute of Mathematics, Siberian branch of RAS, Research on Geometry and Math. Analysis, Nauka vol. 7 (1987), pp. 180–194 153. E. Tsekanovskii, Triangular models of unbounded accretive operators and regular factorization of their characteristic operator-functions (Russian). Soviet Math. Dokl. 36(3), 512–515 (1988) 154. E. Tsekanovskii, Y. Shmuljan, The theory of bi-extensions of operators in rigged Hilbert spaces. Unbounded operator colligations and characteristic functions. Russ. Math. Surv. 32, 73–131 (1977) 155. W.T. Tutte, Graph Theory. Encyclopedia of Mathematics and its Applications, vol. 21 (Addison-Wesley Publishing Company, Advanced Book Program, Reading, 1984) 156. V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ. Sov. Phys., Usp. 10(4), 509–514 (1968) 157. M. I. Višik, On general boundary problems for elliptic differential equations. Trudy Moskov. Mat. Obšˇc. 1, 187–246 (1952)

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On the Number and Sums of Eigenvalues of Schrödinger-type Operators with Degenerate Kinetic Energy Jean-Claude Cuenin and Konstantin Merz

Dedicated to the memory of Sergey N. Naboko

Abstract We estimate sums of functions of negative eigenvalues of Schrödingertype operators whose kinetic energy vanishes on a codimension one submanifold. Our main technical tool is the Stein–Tomas theorem and some of its generalizations. Keywords Degenerate kinetic energy · Eigenvalue estimates · Eigenvalue sums

1 Introduction For .d ≥ 1 we consider Schrödinger-type operators of the form H = T (−i∇) − V

.

in L2 (Xd ),

X ∈ {R, Z}

(1.1)

where the kinetic energy .T (ξ ) vanishes on a codimension-one submanifold. A prime example is .T = | + 1|, which naturally appears, e.g., in the BCS theory of superconductivity and superfluidity, see, e.g., Frank, Hainzl, Naboko, and Seiringer [20], Hainzl, Hamza, Seiringer, and Solovej [28], and Hainzl and Seiringer

J. -C. Cuenin Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, UK e-mail: [email protected] K. Merz () Institut für Analysis und Algebra, Technische Universität Braunschweig, Universitätsplatz, Braunschweig, Germany Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_13

313

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[29, 30], as well as the Hartree–Fock theory of the electron gas (jellium), see, e.g., Gontier, Hainzl, and Lewin [26]. The potential V is assumed to be real-valued and sufficiently regular, so that H can be realized as self-adjoint operator. In this note we are interested in estimates for sums of functions of negative eigenvalues of H when .V ∈ Lq for some .q < ∞. We now state our assumptions on T . Assumption 1.1 Assume that .T (ξ ) ≥ 0 attains its minimum on a smooth compact codimension one submanifold .S = {ξ ∈ Rd : T (ξ ) = 0}. Assume that there exists an open, precompact neighborhood . ⊆ Rd of S such that the following holds. (1) There exists .P ∈ C ∞ () such that .T (ξ ) = |P (ξ )|. Let .τ := maxξ ∈ T (ξ ). (2) There exist .cP > 0 such that .|∇P (ξ )| ≥ cP for all .ξ ∈ . (3) There exist constants .C1 , C2 > 0 and .s ∈ (0, d) such that .T (ξ ) ≥ C1 |ξ |s + C2 for .ξ ∈ Rd \ . For .t > 0, consider the level set .St := {ξ ∈ Rd : |P (ξ )| = t} which is a smooth compact codimension one submanifold embedded in .Rd with corresponding surface measure .dSt . We set .dσSt (ξ ) := dSt (ξ )/|∇P (ξ )| and assume that (4) there is .r > 0 such that .supt∈(0,τ ) |(dσSt )∨ (x)| τ (1 + |x|)−r , where ´ 2π ix·ξ ∨ .(dσSt ) (x) = dσSt (ξ ) denotes the Fourier transform of .dσSt . St e Assumptions (1)–(3) also appear in the work of Hainzl and Seiringer [31], where it is assumed that .V ∈ L1 ∩ Ld/s . These assumptions imply that the quadratic form ∞ d . u, (T (−i∇)−V )u is bounded from below, whenever .u ∈ Cc (R ). The Friedrichs extension then provides us with a self-adjoint operator .H = T (−i∇) − V . Note that the constants .τ, cP , C1 , C2 in Assumption 1.1 are fixed .O(1)-quantities. Assumption (4) is related to the curvature of .St and is crucial since it allows d/s us to consider .V ∈ Lq ∩ Lloc with .q > 1. Littman [43] showed that if S has .2r ∈ {0, 1, . . . , d − 1} non-vanishing principles curvatures, then one has the decay ∨ −r ). In particular, Assumption (4) holds for .T = | + 1| with .|(dσS ) (x)| = O(|x| .r = (d − 1)/2. Note also that this assumption is always guaranteed in the nonzero curvature case, whenever one has the decay .(dσS )∨ (x) = O(|x|−(d−1)/2 ) for .t = 0. (See, e.g., [9, Proposition 4.1].) For .V ∈ Lq with .q ∈ [d/s, ∞) the essential spectrum .σess (H ) = [0, ∞) coincides with that of .T (−i∇). The discrete spectrum of the operator .Hλ := T (−i∇) − λV for .0 < λ  1 has recently received considerable interest. For 1 d/s (Rd ) it has been shown, e.g., by Frank, Hainzl, Naboko, and Seiringer .V ∈ L ∩L j [20] and Hainzl and Seiringer [29, 31] that for any eigenvalue .aS > 0 of the operator L2 (S, dσS ) → L2 (S, dσS ), ˆ . Vˆ (ξ − η)u(η)dσS (η), u → S

u ∈ L2 (S, dσS ),

(1.2)

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315

there is a corresponding eigenvalue .−ej (λ) < 0 of .T − λV which satisfies  ej (λ) = exp −

.



1 j

2λaS

(1 + o(1))

as λ → 0.

(1.3)

´ Here, .Vˆ (ξ ) = Rd e−2π ix·ξ V (x) dx denotes the Fourier transform of V in .Rd . Recently, the authors [9] extended this result to a substantially larger class of potentials, such as .V ∈ Lq (Rd ) with .q ∈ [d/s, r + 1], whenever .T (−i∇) satisfies also the curvature assumption (4) with .r + 1 ≥ d/s in Assumption 1.1. This is clearly the case for .T = | + 1| with .r = (d − 1)/2. On the other hand, Laptev, Safronov, and Weidl [37] studied the asymptotic behavior of the eigenvalues .−ej < 0 of .T − V as .j → ∞, when V is of the form −1− , where .v ∈ L∞ (Rd ) satisfies .v(x) = w(x/|x|)(1+o(1)) .V (x) = v(x)(1+|x|) as .|x| → ∞ with .w ∈ C ∞ (Sd−1 ). Similarly as in (1.3), the eigenvalue asymptotics j is determined by that of the eigenvalues .aS > 0 of the operator in (1.2). Their main result [37, Theorem 4.4] essentially relied on an abstract theorem (Theorem 3.4 there) which connected the spectral asymptotics of H and (1.2) with each other. In j turn, the limit .limj →∞ aS is well understood thanks to the works [6] of Birman and Solomjak on singular values of (asymptotically) homogeneous pseudodifferential operators with symbol .h1 (x)a(x, ξ )h2 (x). Here .h1 , h2 ∈ Cc∞ , and .a(x, tξ ) = t −β a(x, ξ ) for all .|ξ | ≥ 1 and .t > 1. By a change of coordinates, the operator in (1.2) can be transformed into this operator modulo “error operators” which do not change the leading order of the spectral asymptotics of (1.2). We refer to Birman and j Yafaev [5] for a detailed exposition and for the explicit expression for .limj →∞ aS . d q For V merely in .L (R ), the results of Birman and Solomjak are not applicable. It would be interesting to study the asymptotics .limj →∞ ej in this case. The purpose of this note is to prove estimates for sums of functions .f (x) on q .R+ of the absolute values of the negative eigenvalues of .T − V when .V ∈ L . For γ .f (x) = x this will lead to modifications of the celebrated Lieb–Thirring inequality [38, 41, 42] ˆ γ + d2 γ dx (1.4) .tr[(− − V )− ] ≤ cd,γ V (x) + Rd with .γ ≥ 1/2 if .d = 1, .γ > 0 if .d = 2, and .γ ≥ 0 if .d ≥ 3, and a constant .cd,γ > 0 which is independent of V . Here we denote the positive and negative parts of a real number or a self-adjoint operator by .X+ := max{X, 0} and .X− := max{−X, 0}, respectively. We refer to Frank [19] for a recent review of its history, applications, and generalizations. Observe that the right side of (1.4) is homogeneous in V . Since the assumptions on .T (ξ ), i.e., the constants .τ, cP , C1 , C2 appearing in Assumption 1.1 are fixed .O(1)-quantities, we do not expect scaleinvariant inequalities.

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Nevertheless, non-scale-invariant inequalities relating sums of eigenvalues with Lq -norms of V , such as Daubechies’ inequality [12]

.

√ tr[( − + 1 − 1 − V )− ] ≤ cd

.

ˆ Rd

  1+ d V (x)+ 2 + V (x)1+d dx +

(1.5)

for the pseudorelativistic Chandrasekhar operator, are important in the analysis of many-particle quantum systems. In fact, using the techniques of [38], Daubechies extended (1.5) to a larger class of operators .T (−i∇). However, these results are not applicable in the present situation, since they require .T (ξ ) to be a spherically symmetric and strictly increasing function with .T (0) = 0. This condition is not satisfied by the operators T we consider here, such as .T = |+1|. Further examples of eigenvalue estimates involving a sum of two terms were proved, e.g., by Lieb, Solovej, and Yngvason [40] in the context of the Pauli operator and by Exner and Weidl [15] in the context of Schrödinger operators in wave guides .ω × R with d−1 . For two-term estimates for eigenvalue sums of Schrödinger operators .ω ⊂ R on metric trees, we refer to Frank and Kovaˇrík [21, Theorem 6.1], see also Ekholm, Frank, and Kovaˇrík [13], Molchanov and Vainberg [45], and the references therein for further results. Finally, we refer to Frank, Lewin, Lieb, and Seiringer [22] for two-term estimates for eigenvalue sums of Schrödinger operators in presence of a constant positive background density. Besides sums of powers of eigenvalues, we also prove estimates for sums of powers of logarithms (i.e., .f (x) = (log(2 + 1/x))−γ ) of eigenvalues of .T − V . This is natural, as (1.3) indicates that the eigenvalues of .T − V cluster with an exponential rate at zero. In particular, the proofs of these results yield estimates on the eigenvalues .ej and show how fast they cluster at zero as .j → ∞, see (3.16). However, we do not investigate the asymptotics for .limj →∞ ej here. The idea of deriving estimates for logarithms of eigenvalues is not new and has already been considered by Kovaˇrík, Vugalter, and Weidl [36] in the context of two-dimensional Schrödinger operators .− − V , whose eigenvalues also cluster exponentially fast at the bottom of the essential spectrum, see Simon [52]. If T degenerates sublinearly, we are able to prove Cwikel–Lieb–Rosenbljumtype estimates [11, 38, 47] for the number of negative eigenvalues. We illustrate this using .T = | + 1|1/s with .s > 1. Finally, we generalize our results to lattice Schrödinger-type operators on .2 (Zd ). Under the same curvature assumption we obtain better estimates than in .L2 (Rd ) due to the absence of high energies. Organization and Notation In Sect. 2 we collect facts about Schatten spaces and Fourier restriction theory that are used in the subsequent sections. In Sect. 3 we prove estimates for the number of eigenvalues of .T − V in .L2 (Rd ) below a fixed threshold .−e < 0 (Theorem 3.1). Then we prove inequalities for sums of powers of eigenvalues (Theorem 3.2), and for sums of powers of logarithms of eigenvalues of .T − V (Theorem 3.3). We conclude with a Cwikel–Lieb–Rosenbljum bound for .| + 1|1/σ − V with .σ > 1 (Theorem 3.4). In Sect. 4 we consider the corresponding problems for Schrödinger

Number and Sums of Eigenvalues

317

operators on .2 (Zd ). We first recall two versions of a discrete Laplace operator and a modification of the “BCS operator” .| + 1| − V to .2 (Zd ). In Sect. 4.2 we prove estimates on the number of negative eigenvalues of .T − V in .2 (Zd ) below a threshold .−e < 0 (Theorem 4.5), ordinary and logarithmic Lieb–Thirring-type inequalities (Theorems 4.6 and 4.7), and a Cwikel–Lieb–Rosenbljum bound for powers of the modified BCS operator in .2 (Zd ) (Theorem 4.9). We write .A  B for two non-negative quantities .A, B ≥ 0 to indicate that there is a constant .C > 0 such that .A ≤ CB. If .C = Cτ depends on a parameter .τ , we write .A τ B. The dependence on fixed parameters like d and s is sometimes omitted. Constants are allowed to change from line to line. The notation .A ∼ B means .A  B  A. All constants are denoted by c or C and are allowed to change from line to line. We abbreviate .A ∧ B := min{A, B} and .A ∨ B := max{A, B}. The Heaviside function is denoted by .θ (x). We use the convention .θ (0) = 1. The indicator function and the Lebesgue measure of a set . ⊆ Rd are denoted by .1 and .||, respectively. For .x ∈ Rd we write . x := (2 + x 2 )1/2 .

2 Preliminaries 2.1 Trace Ideals We collect some facts on trace ideals that are used in this note, see also, e.g., Birman–Solomjak [4, Chapter 11] or Simon [53]. Let .(B, ·) denote the Banach space of all linear, bounded operators on a Hilbert space .H. The p-th Schatten space of all compact operators .T ∈ S∞ (H) whose order, appearing according to their singular values .{sn (T )}n∈N (in non-increasing  p multiplicities) satisfy .T  p := n≥1 sn (T )p < ∞ for .p > 0 is denoted by S (H ) p .S (H). We denote the p-th weak Schatten space over .H by Sp,∞ (H) := {T ∈ S∞ (H) : T  p,∞ := sup λp n(λ, T ) < ∞} ⊇ Sp (H), S (H ) λ>0 (2.1) p

.

where n(λ, T ) := #{n : sn (T ) > λ},

.

λ > 0.

(2.2)

Note that 1

T Sp,∞ (H) = sup sm (T )m p ,

(2.3)

.

m

which together with (2.1) implies in particular sm (T ) ≤ T Sp,∞ (H) m

.

− p1

and

n(λ, T ) ≤ T  p,∞ λ−p . S (H) p

(2.4)

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If .T : H → H is a linear operator between two Hilbert spaces .H and .H we denote its p-th Schatten norm by .T Sp (H,H ) . If .H = H , we either write p p .T  p S (H) , .T Sp , or .T p , and abbreviate .S (H) = S . Analogous notation is p,∞ . used for .S

2.2 Fourier Restriction and Extension Let .X ∈ {R, Z}, .Xˆ = R when .X = R, and .Xˆ = T when .X = Z, where .Td := (R/Z)d denotes the d-dimensional torus with Brillouin zone .[−1/2, 1/2)d . If .X = Z, then the .Lq (Xd )-spaces are equipped with counting measure so that .Lq (Zd ) ≡ q (Zd ) for any .q > 0. Let S be a smooth, compact codimension one submanifold embedded in .Xˆ d with induced Lebesgue surface measure .dS . If S is the level set of a smooth real-valued function .P ∈ C ∞ (Xˆ d ), i.e., .S = {ξ ∈ Xˆ d : P (ξ ) = 0}, then the Leray measure [25] is .dσS (ξ ) = |∇P (ξ )|−1 dS (ξ ). We introduce the Fourier restriction operator FS : S(Xd ) → L2 (S, dσS ),

.

 (ξ ) = φ → (FS φ)(ξ ) = φ S

ˆ Xd

 e−2π ix·ξ φ(x) dx S (2.5)

and its adjoint, the Fourier extension operator ∗ .FS



: L (S, dσS ) → S (X ), 2

d

u →

(FS∗ u)(x)

ˆ =

u(ξ )e2π ix·ξ dσS (ξ ).

(2.6)

S

Under the additional assumption that the Gaussian curvature of S is non-zero everywhere, the Stein–Tomas theorem [7, 54, 56] asserts that .FS : Lp (Xd ) → L2 (S) is bounded for all .p ∈ [1, 2(d + 1)/(d + 3)]. Its proof relies on the bound − d−1 ∨ 2 . By duality, the Stein–Tomas theorem is equivalent to the .|(dσS ) (x)|  x

operator norm bound .W1 FS∗ FS W2 L2 (Xd )→L2 (Xd )  W1 L2q (Xd ) W2 L2q (Xd ) for all .W1 , W2 ∈ L2q , whenever .1/q = 1/p − 1/p and .p ∈ [1, 2(d + 1)/(d + 3)], i.e., .q ∈ [1, (d + 1)/2]. Frank and Sabin [24, Theorem 2] upgraded this to a Schatten norm estimate. For smooth compact hypersurfaces .S ⊆ Xˆ d with everywhere nonvanishing Gaussian curvature and σ (q) :=

.

(d − 1)q , d −q

q ∈ [1, d),

(2.7)

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Frank and Sabin proved W1 FS∗ FS W2  σ (q) 2 d d,S,q W1 L2q (Xd ) W2 L2q (Xd ) , S (L (X ))

.

d +1 . q ∈ 1, 2 (2.8)

Note that .σ (1) = 1, .σ ((d + 1)/2) = d + 1, and .σ (q) ≥ q. As discussed in the introduction, if S has .2r ∈ {0, 1, . . . , d − 1} non-vanishing principle curvatures, then one has the weaker decay .|(dσS )∨ (x)|  x −r , which, as Greenleaf [27] showed, implies that .FS : Lp (Xd ) → L2 (S) is bounded for all ∨ .p ∈ [1, (2 + 2r)/(2 + r)]. For a given decay rate of .|(dσS ) (x)| the first author proved the following generalization of (2.8). Proposition 1 ([8, Proposition A.5]) Let .S ⊆ Xˆ d be a smooth compact hypersurface with normalized defining function1 .P : Xˆ d → R and Lebesgue surface measure .dS and Leray measure .dσS (ξ ) = |∇P (ξ )|−1 dS (ξ ). Assume that .

sup (1 + |x|)r |(dσS )∨ (x)| < ∞

(2.9)

x∈Xd

for some .r > 0. Let .1 ≤ q ≤ 1 + r and define  2(d−1−r)q σ (q, r) :=

.

d−q 2rq+ 2rq−d(q−1)

if

d d−r

≤ q ≤ 1 + r,

if 1 ≤ q
0 arbitrarily small but fixed. Then for all W1 , W2 ∈ L2q (Xd ), we have

.

W1 FS∗ FS W2  σ (q,r)  W1 L2q (Xd ) W2 L2q (Xd ) , S

.

(2.11)

where the implicit constant is independent of .W1 , W2 . Remark 1 (1) We have .σ (q, r) ≥ q when .r ≤ (d − 1)/2 and .σ (q, (d − 1)/2) = σ (q) with .σ (q) as in (2.7). (2) The estimates (2.8) and (2.11) were proved for .Rd , but their (Fourier-analytic) proofs readily generalize to .Zd . (3) The estimate in [8, Proposition A.5] involved the resolvent of .P (−i∇). As usual, this implies (2.11) since the imaginary part of the limiting resolvent equals the spectral measure. (4) Littman’s bound .|(dσS )∨ (x)|  x −r is rarely optimal except when the surface is completely flat in the vanishing curvature direction. (For a more detailed

1 This

means that .S = {P = 0} and .|∇P | = 1 on S

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discussion and references to generic results, see, e.g., [10] by Schippa and the first author.) (5) As is discussed, e.g., in Ikromov, Kempe, and Müller [33–35], sharp decay estimates do not always imply .L2 → Lp Fourier restriction bounds with optimal p.

3 Bounds on Number and Sums of Functions of Eigenvalues in L2 (Rd ) Suppose that the kinetic energy .T (−i∇) satisfies (1)–(3) in Assumption 1.1. Let −e1 ≤ −e2 ≤ · · · < 0 denote the negative eigenvalues of .H = T − V in nondecreasing order (counting multiplicities) and

.

Ne (V ) :=



.

1

(3.1)

ej >e

denote the number of negative eigenvalues of H below .−e ≤ 0. Let 1

1

BS(e) := |V | 2 (T + e)−1 V 2

.

on L2 (Rd )

(3.2)

where .V 1/2 (x) := |V (x)|1/2 sgn(V (x)) with .sgn(V (x)) = 1 if .V (x) = 0. By the Birman–Schwinger principle (cf. [53, Proposition 7.2], [11, p. 99], [39, Proposition 6]), one has Ne (V ) = n(1, BS(e)) ≤ BS(e)m m,∞ ≤ BS(e)m m S S

.

for all m > 0.

(3.3)

As a consequence of the variational principle, i.e., Ne (V ) ≤ Ne (V+ ) = Ne/2 (V+ − e/2) ≤ Ne/2 ((V+ − e/2)+ ),

.

(3.4)

one can estimate for any .γ > 0, γ .tr(T (−i∇) − V )−

ˆ ≤γ 0

∞

eγ −1 Ne/2 ((V+ − e/2)+ ) de.

(3.5)

3.1 Number of Eigenvalues below a Threshold We first prove estimates for Schatten norms of .BS(e). Theorem 3.1 Let .e > 0 and suppose .T (ξ ) satisfies (1)–(3) in Assumption 1.1.

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321

1. Let .m > d/s. If .V ∈ Lm (Rd ), then there exists a constant .cS > 0 (which also depends on .d, s, m, τ ) such that .

1−m BS(e)m θ (1 − e) + ed/s−m θ (e − 1))V m m ≤ cS (e m.

(3.6)

2. Suppose T also satisfies (4) in Assumption 1.1 with .r > 0. Let .q ∈ [1, r + 1] and .m = σ (q, r) be as in (2.10). Suppose additionally .m > d/s and let .V ∈ Lm ∩Lq (Rd ). Then there is a constant .cS (which also depends on .d, s, m, τ, q, r) such that

 m m m BS(e)m m ≤ cS V m + log(2 + 1/e) V q θ (1 − e) .

 −m + cS ed/s−m V m min{V m , V q }m θ (e − 1). m+e (3.7) 3. In addition to the assumptions in (2), suppose .q > d/s. Then m . BS(e)m

≤

cS V m q

md 1 m −m sq log(2 + ) θ (1 − e) + e θ (e − 1) . e

(3.8)

m If .q = d/s, then (3.8) holds with . · m m on the left side replaced by . · m,∞ .

Proof We begin with the proof of (3.6). Hölder’s inequality yields W1 FS∗ FS W2  1 2 d ≤ W1 FS∗  2 2 S (L (R )) S (L (S,dσS ),L2 (Rd )) × W2 FS∗  2 2 S (L (S,dσS ),L2 (Rd ))

.

(3.9)

= W1 L2 W2 L2 σS (S). For .τ > 0 as in Assumption 1.1 we separate high and low energies using a bump function .χ ∈ Cc∞ (R+ : [0, 1]) with .suppχ ⊆ [0, 1]. By the Kato–Seiler–Simon inequality [53, Theorem 4.1] (with .m > d/s), we obtain BS(e)m m 1

.

1

1

m (|V | 2 (T + e)−1 χ(T /τ )|V | 2 m + |V | 2 (T + e)−1 (1 − χ(T /τ ))|V |1/2 m )m 1

1

m d/s−m  |V | 2 (T + e)−1 χ(T /τ )|V | 2 m }. m + V m min{1, e

(3.10)

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To treat the low energy part we use the Lieb–Thirring trace inequality [42, Theorem 9] .B

1/2

AB 1/2 m m ≤ B m/2 Am B m/2  1 , S S

m≥1

(3.11)

for linear operators .A, B ≥ 0 in a separable Hilbert space, the spectral theorem, and (3.9). We obtain 1

1

m

m

−m 2 |V | 2 (T + e)−1 χ(T /τ )|V | 2 m χ(T /τ )m |V | 2 1 m ≤ |V | (T + e) ˆ τ ˆ τ |V |m/2 FS∗t FSt |V |m/2 1 m ≤ dt ≤ V  dt (t + e)−m σSt (St ) . m (t + e)m 0 0 ˆ τ dt 1−m −m ≤ cS,τ V m ≤ cS,τ,m V m , e }, m m min{e m 0 (t + e)

(3.12)

where we used Assumption 1.1 to estimate .σSt (St )

≤ sup sup t∈[0,τ ] ξ ∈St

St (St ) ≤ cS,τ . |∇P (ξ )|

(3.13)

Combining (3.10) and (3.12) proves (3.6). To prove (3.7), we proceed as in the proof of (3.6) but estimate the low energies using the Stein–Tomas estimate for trace ideals (2.11) instead. For .q ∈ [1, r + 1], we obtain 1

1

|V | 2 (T + e)−1 χ(T /τ )|V | 2 σ (q,r) ≤ .

ˆ 0

τ

dt |V |1/2 FS∗t FSt |V |1/2 σ (q,r) t +e

≤ cS min{log(1 + τ/e), τ/e}V q .

(3.14) Setting .m = σ (q, r) on the left side of (3.14) and combining it with (3.6) yields (3.7). The final estimate (3.8) follows from the proof of (3.7) by replacing the estimate for the high energies in the second and third line of (3.10) by the following estimate, 1 .|V | 2 (T

1

−1 1/2 m 2 + e)−1 (1 − χ(T /τ ))|V |1/2 m q m ≤ |V | (T + e) (1 − χ(T /τ ))|V | md

sq  V m q min{1, e

−m

},

where .m ≥ q > d/s . (Here we used the Kato–Seiler–Simon inequality again.) This concludes the proof of (3.8). For .q = d/s we use Cwikel’s bound (see [53, Theorem 4.2] or (3.37) below), which is applicable since .q > 1 in this case.  

Number and Sums of Eigenvalues

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Remark 2 md

−m

sq (1) The terms proportional to .V m in m in (3.7) and the term that scales like .e (3.8) are due to high energies. (2) If .r = (d − 1)/2, .d > s ≥ d/q and .0 < e < 1, then (3.8) implies for .q ∈ (1, (d + 1)/2] and .m = σ (q),

σ (q)

Ne (V ) ≤ cS log(1/e)σ (q) V q

.

.

(3.15)

Thus, the n-th negative eigenvalue .−1 < −en < 0 satisfies  en ≤ exp −

.



n1/σ (q) 1/σ (q)

cS

V q

.

(3.16)

We close this subsection by proving a slight refinement of the bound for .Ne (V ) that follows immediately from (3.3) and (3.6). To that end we apply Fan’s inequality [16] (see also [53, Theorem 1.7]), which asserts sj ++1 (A + B) ≤ sj +1 (A) + s+1 (B)

.

(3.17)

for all .j,  ∈ N0 and all .A, B ∈ S∞ . Corollary 1 Let .e, τ > 0, .m1 , m2 ≥ 1. Let .L1loc (Rd )  T (ξ ) ≥ 0 and .V ∈ 1

L1loc (Rd ) so that .BS< (e) := |V | 2 (T + e)−1 1{T (e) := |V | 2 (T +

1

e)−1 1{T >τ } V 2 are compact operators. Then

Ne (V ) ≤ 2 · [2m1 BS< (e)m1m1 ,∞ + 2m2 BS> (e)m2m2 ,∞ ]. S S

.

(3.18)

In particular, for .T (ξ ) satisfying (1)–(3) in Assumption 1.1, .m1 > 1 and .m2 > d/s, there is .cS > 0 (which also depends on .d, s, m1 , m2 , τ ) such that m2 d/s−m2 1 2 Ne (V ) ≤ cS [(e1−m1 V m V m m1 + V m2 )θ (1 − e) + e m2 θ (e − 1)]. (3.19)

.

Proof We first prove (3.18). By (3.17), we have .sn+1 (BS(e)) ≤ sn/2+1 (BS< (e)) + sn/2+1 (BS> (e)) for even n and .sn+1 (BS(e)) ≤ s(n+1)/2+1 (BS< (e)) + s(n−1)/2+1 (BS> (e)) for odd n. Thus, {n ∈ 2N0 : sn+1 (BS(e)) > 1} .

⊆ {n ∈ 2N0 : sn/2+1 (BS< (e)) > 1/2} ∪ {n ∈ 2N0 : sn/2+1 (BS> (e)) > 1/2} (3.20)

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and a similar statement holds for odd n. Combining this with {n ∈ N0 : sn+1 (BS(e)) > 1}

.

= {n ∈ 2N0 : sn+1 (BS(e)) > 1} ∪ {n ∈ (2N0 + 1) : sn+1 (BS(e)) > 1} and (3.3) yields (3.18), because Ne (V ) = #{n ∈ N : sn (BS(e)) > 1} ≤ 2 (#{n ∈ N0 : sn+1 (BS< (e)) > 1/2} + #{n ∈ N0 : sn+1 (BS> (e)) > 1/2})   ≤ 2 2m1 BS< (e)m1m1 ,∞ + 2m2 BS< (e)m2m2 ,∞ . S S

.

(3.21)

To prove (3.19), we first write .Ne (V )

= n(1, BS(e))θ (1 − e) + n(1, BS(e))θ (e − 1).

(3.22)

The second summand is estimated using the Kato–Seiler–Simon inequality by .n(1, BS(e))θ (e

m2 d/s−m2 2 − 1) ≤ BS(e)m θ (e − 1). m2 θ (e − 1)  V m2 · e

(3.23)

The first summand in (3.22) is estimated using (3.18) by m2 1 n(1, BS(e))θ (1 − e)  (BS< (e)m m1 + BS> (e)m2 )θ (1 − e) .

m2 1  (e1−m1 V m m1 + V m2 )θ (1 − e),

where we used the steps in the proof of (3.6).

(3.24)  

3.2 Sums of Powers of Eigenvalues We now use (3.3)–(3.5) and Theorem 3.1 to obtain estimates for sums of powers of eigenvalues of .T − V . Theorem 3.2 Suppose .T (ξ ) satisfies (1)–(3) in Assumption 1.1. 1. If .γ > 0 and .V ∈ Lγ +1 ∩ Lγ +d/s (Rd ) then there exists a constant .cS > 0 (which also depends on .d, s, m, γ ) such that ˆ γ .tr 2 (V+ (x)γ +1 + V+ (x)γ +d/s ) dx. (3.25) d (T (−i∇) − V )− ≤ cS L (R ) Rd 2. Suppose T also satisfies (4) in Assumption 1.1 with .r > 0. Let .q ∈ [1, r + 1] and q .m = σ (q, r). Suppose additionally .m > d/s. If .γ > m − d/s and .V ∈ L ∩

Number and Sums of Eigenvalues

325

Lγ +d/s (Rd ), then there is a constant .cS (which also depends on .d, s, m, q, γ , r) such that γ +d/s

γ

trL2 (Rd ) (T (−i∇) − V )− ≤ cS (V+ m q + V+ γ +d/s ).

.

(3.26)

Proof By the variational principle we can assume .V = V+ ≥ 0. To prove (3.25) we apply (3.19) in Corollary 1 for any .m1 > 1 and .m2 > d/s and obtain Ne/2 ((V (x) − e/2)+ )

ˆ ˆ m2 d/s−m2 1 ≤ cS e1−m1 d (V (x) − e/2)m + e (V (x) − e/2) + + . R Rd

.

Plugging this into (3.5) with .γ > max{m1 − 1, m2 − d/s} yields γ

tr(T (−i∇) − V )− ˆ ∞ ˆ ≤ cS de eγ −m1

.

Rd

0

+e ˆ ≤ cS

Rd

1 dx (V (x) − e/2)m +

γ −1+d/s−m2



ˆ Rd

dx

2 (V (x) − e/2)m +

dx (V (x)γ +1 + V (x)γ +d/s ),

where .cS also depends on .d, s, m, γ . This proves (3.25). To prove (3.26) we instead use (3.7) in Theorem 3.1 and plug the right side of .Ne/2 ((V (x)

− e/2)+ )

ˆ

≤ cS (ed/s−m θ (e − 1) + θ (1 − e))

Rd ˆ

+ cS θ (1 − e) log(2 + 1/e)m ˆ ≤ cS ed/s−m

(V (x) − e/2)m + m q

q

R

(V (x) − e/2)+ d

ˆ

m (V (x) − e/2)m + + cS θ (1 − e) log(2 + 1/e) Rd

R

m q q (V (x) − e/2) + d

into (3.5). For .γ + d/s > m > d/s the first summand gives again rise to ˆ

∞

de e

.

0

γ −1+d/s−m

ˆ

ˆ (V (x) − e/2)m + dx d

R

≤ cS

Rd

V (x)γ +d/s dx,

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J. -C. Cuenin and K. Merz

whereas the second summand contributes with ˆ

1

.

de eγ −1 log(2 + 1/e)m

ˆ

m q

(V (x) − e/2)+ Rd

0

q

≤ cS V m q

to the left hand side of (3.26) for all .γ > 0. (As before, .cS also depends on ) This concludes the proof.  

.d, s, m, q, γ , r

Remark 3 γ +d/s

(1) The term .V+ γ +d/s on the right sides of (3.25)–(3.26) comes from high energies as can be seen from the proofs of (3.6)–(3.7). In Theorem 4.6 we will see that this term is absent for operators in .2 (Zd ) since only low energies are present. γ +d/s (2) The term .V+ γ +d/s is necessary, which can be seen by repeating the arguments in the proof of Theorem 3.1 with .| + μ| and letting .μ → 0. (3) Estimate (3.25) also holds in case the level sets of T are not curved and can be seen as a Lieb–Thirring inequality since the right hand side is “local” in the sense that it involves only integrals over V . In contrast, (3.26) requires nonvanishing Gaussian curvature of the level sets. Moreover, (3.26) is non-local in the sense that it involves powers of integrals of V .

3.3 Sums of Logarithms of Eigenvalues Suppose .r = (d − 1)/2, let .s ∈ [2d/(d + 1), d), .V ∈ L(d+1)/2 , and assume that T satisfies (1)–(4) in Assumption 1.1. By Cuenin and Merz [9, Theorem 4.2], for any eigenvalue .ajS > 0 of the operator .VS = FS V FS∗ in .L2 (S), there exists a negative eigenvalue .−ej (λ) < 0 of .Hλ = T − λV with weak coupling limit  ej (λ) = exp −

.

1 2λajS

 (1 + o(1)) ,

λ → 0.

(3.27)

(Eigenvalues .−ej < 0 corresponding to zero-eigenvalues of .VS obey .ej (λ) = −2 e−cj λ for some .cj > 0, cf. [9, Theorem 4.4]). On the other hand, as we have seen in (3.16) in Remark 2, if the j -th eigenvalue .−ej (λ) is greater than .−1, then it satisfies  

j 1/σ (q) d d +1 , . (3.28) .ej (λ) ≤ exp − , q ∈ 1/σ (q) s 2 c λV q S

Number and Sums of Eigenvalues

327

Formulae (3.27)–(3.28) illustrate that the eigenvalues of .Hλ approach inf σess (Hλ ) = 0 exponentially fast. This suggests to compute logarithmic moments of eigenvalues,

.



1 log( 1/ej )

.

j

γ ,

γ > 0.

(Note that .1/ log(1/x) ≥ x for .0 < x < 1/2, say.) For those eigenvalues .ej (λ) corresponding to the .ajS in the asymptotics (3.27), estimate (2.8) implies, for .λ in a sufficiently small open neighborhood of 0,  .

j

1 log( 1/ej (λ) )

σ (q) ∼

j

 1+

1

−σ (q)

j 2λaS

σ (q)

∼ λσ (q) tr(VS )+

σ (q)

 λσ (q) V Lq , (3.29) where .σ (q) is as in (2.7) and .q ∈ [1, (d + 1)/2]. We now prove analogous estimates for .λ = 1, in which case we cannot use the results in the weak coupling regime. Theorem 3.3 Let .H = T − V with T satisfying (1)–(4) in Assumption 1.1 with r > 0. Let .q ∈ [1, r + 1] and .m = σ (q, r). Suppose additionally .m > d/s and let d m q .V ∈ L ∩ L (R ). Then for any .γ > m there is a constant .cS (which also depends on .d, s, m, q, γ , r) such that .

.

m [log( 1/ej )]−γ ≤ cS V+ m m + V+ q .

(3.30)

j

Moreover, if .q ≥ d/s, then .

[log( 1/ej )]−γ ≤ cS V+ m q.

(3.31)

j

Proof By the variational principle we can again assume .V = V+ . To estimate the left side of (3.30) we use

.



1 log( e−1 )

ˆ γ = γ

0

e



log( r −1 )

−γ −1  1 −2 dr · r r3

(3.32)

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J. -C. Cuenin and K. Merz

for .γ > 0. Thus, j .

ˆ

1

 γ = γ log( ej−1 )

∞

log( r

−1

0

−γ −1  1 −2 dr

) · θ (ej − r) 3 r r j

ˆ ∞ −γ −1  1 −2 dr −1 log( r ) =γ · Nr (V ) 3 . r r 0 (3.33)

By (3.3) and (3.7) in Theorem 3.1 for .m > d/s, we estimate    1 m V m Nr (V )  V m + log 2 + m q. r

.

(3.34)

Thus, the left side of (3.30) can be estimated by  j

1 log( 1/ej )

γ

ˆ 

V m m

∞

0

ˆ + V m q

.

0

−γ −1 dr −1 −2  −1 log( r r

) r3 ∞

  1 · log 2 + r

−γ −1 dr −1 −2  −1 log( r r

) r3 m

(3.35)

m  V m m + V q .

This concludes the proof of (3.30). The proof of (3.31) is completely analogous, but uses (3.8) instead of (3.7). Thus, estimate (3.34) is replaced by   

1 m q . Nr (V )  V m 1 + log 2 + r

.

Proceeding as in the proof of (3.30) concludes the proof of (3.31).

(3.36)  

Remark 4 (1) In contrast to the right side of (3.25), the powers of V appearing on the right side of (3.30) are all the same. (2) For .r = (d − 1)/2 and .m = d + 1 the power .d + 1 on the right side of (3.30) is consistent with that on the right side of (3.29). However, (3.30) is slightly weaker than (3.29) due to the assumption .γ > d + 1 and, if .q < d/s, the additional .V d+1 d+1 term on the right of (3.30). (3) We do not know whether the restriction .γ > m (especially .γ > d + 1 for .r = (d − 1)/2 and .m = d + 1) is necessary.

Number and Sums of Eigenvalues

329

3.4 CLR Bounds in L2 (Rd ) Recall that .N0 (V+ ) equals the number of eigenvalues of .V+ T −1 V+ above one, which can be estimated by (3.3). Formula (2.4), Cwikel’s bound [11], i.e., 1/2

f (−i∇)g(x)Sp,∞ (L2 (Rd )) p f Lp,∞ (Rd ) gLp (Rd ) ,

.

1/2

p ∈ (2, ∞), (3.37)

and (3.3) yield the classical Cwikel–Lieb–Rosenbljum (CLR) bound [11, 38, 47] V+ (−)−1 V+  d/2,∞ 2 d d |ξ |−1 Ld,∞ V+ Ld/2 S (L (R ))

.

1/2

1/2

for the number of negative eigenvalues .N0 (V ) of .T − V when .T = − in .d ≥ 3. Such bounds can never hold in .d = 1, 2, or for T satisfying Assumption 1.1 in .d ≥ 2 due to the existence of weakly coupled bound states [9, 20, 29, 31, 32, 37, 52]. Interestingly, using (3.37), one does obtain a CLR bound for powers .T = | + 1|1/σ of the BCS operator .| + 1| in .L2 (R2 ) when .σ > 1. This follows from the ´1 uniform bound . 0 (t + e)−1/σ dt s 1 for all .e ≥ 0 and .σ > 1. The following theorem generalizes this observation to powers .T 1/σ with T satisfying (1)–(3) in Assumption 1.1 to all .d ∈ N. The proof is inspired by Frank [18], which, in turn, uses ideas of Rumin [48, 49]. Theorem 3.4 Let .d ∈ N, .σ > 1, and suppose .T (ξ ) satisfies (1)–(3) in Assumpσd tion 1.1 with the weaker assumption .s ≤ d. Then, for .V ∈ Lσ ∩ L s (Rd ), one has 1

1

n(1, |V | 2 T −1/σ V 2 ) S,σ,s,d,τ V+ σ σ

.

σ d/s

L (R ) d

+ V+ 

Lσ d/s (R ) d

.

(3.38)

Proof By the variational principle we can assume .V = V+ . We first show how to prove (3.38) for .s = d ∈ N using Cwikel’s estimate (3.37). For .β > 0 a straightforward computation shows T (ξ )−1/(2σ ) 

2p L2p,∞ (R ) d

.

    = sup β −2p {ξ ∈ Rd : T (ξ )−1/(2σ ) > 1/β} β>0

   sup β −2p β 2σ 1{β≤1} + β 2σ ·d/s 1{β≥1} .

(3.39)

β>0

For the right side to be finite we need .p = σ and .s = d. Thus, by (3.3) and Cwikel’s estimate (3.37), we obtain for .σ > 1, 1

1

1

1

1

n(1, |V | 2 T −1/σ V 2 ) ≤ V 2 T − σ V 2 σ σ,∞ 2 d S (L (R ))

.

1

1

≤ T − 2σ V 2 2σ2σ,∞ 2 d  V σ σ d , L (R ) S (L (R ))

330

J. -C. Cuenin and K. Merz

which concludes the proof for .s = d. We will now show (3.38) for .s < d by proceeding as in [18]. Let . be an arbitrary operator in .L2 (Rd ) satisfying .0 ≤  ≤ T −1/σ and .ρ (x) := (x, x). Let .PE := 1(E,∞) (T 1/σ ) and .PE⊥ = 1 − PE . We shall now estimate ˆ ˆ ∞ 1/2 1/σ 1/2 .tr( T  )= dx dE (PE PE )(x, x) (3.40) 0 Rd from below. By a density argument it suffices to consider the case where . has finite-rank and smooth eigenfunctions. For any subset . ⊆ Rd of finite measure we have ˆ

1/2 ρ (x) dx

.



=  1/2 1 2 ≤  1/2 PE 1 2 +  1/2 PE⊥ 1 2 ≤ 

1/2

PE 1 2 + ||

1/2



(3.41)

F (E)

where we used . ≤ T −1/σ and defined ˆ T −1/(2σ ) PE⊥ 1 22 dξ = F (E) := 1{T (ξ )1/σ 0 and suppose .T (ξ ) satisfies (1) and (2) in Assumption 1.1. 1. Let .m ≥ 1. If .V ∈ m (Zd ), then there exists a constant .cS > 0 (which also depends on .d, τ, m) such that BS(e)m m 2 d ≤ cS min{e1−m , e−m }V mm d . S ( (Z ))  (Z )

.

(4.5)

2. Suppose T also satisfies (4) in Assumption 1.1 with .r ∈ (0, (d − 1)/2]. Let q d m q d .q ∈ [1, r + 1] and .m = σ (q, r). If .V ∈  (Z ) =  ∩  (Z ), then there is a constant .cS > 0 (which also depends on .d, τ, m, q, r) such that .

m BS(e)m m 2 d ≤ cS V m q min{log(2 + 1/e), 1/e} . S ( (Z ))

(4.6)

In particular, BS(e)m m 2 d S ( (Z )) .

 −m m ≤ cS (log(2 + 1/e))m V m θ (1 − e) + e V  θ (e − 1) . q m

(4.7)

Proof The proofs of (4.5) and (4.6) are exactly the same as those of (3.6) and (3.7) in the continuum case with two exceptions. Due to the absence of high energies in the estimate involving the Kato–Seiler–Simon inequality, any .m ≥ 1 becomes admissible and the .ed/s -factors for .e > 1 are absent. Secondly, by the nestedness

334

J. -C. Cuenin and K. Merz

of the .p spaces, we may estimate .V m  V q since .σ (q, r) ≥ q (cf. (1) in Remark 1) to dispose of .V m -norms. Estimate (4.7) follows from (4.5)–(4.6).  

4.3 Sums of Powers of Eigenvalues The previous estimates allow us to prove an analog of Theorem 3.2 for the lattice Schrödinger operators considered here. Theorem 4.6 Suppose .T (ξ ) satisfies (1) and (2) in Assumption 1.1. 1. If .γ > 0 and .V ∈ γ +1 (Zd ), then there exists a constant .cS > 0 (which also depends on .d, τ, γ ) such that γ

tr2 (Zd ) (T (−i∇) − V )− ≤ cS

.

x∈Z

V+ (x)γ +1 .

(4.8)

d

2. Suppose T also satisfies (4) in Assumption 1.1 with .r ∈ (0, (d − 1)/2]. Let m+γ −δ ∩ .q ∈ [1, r + 1] and .m = σ (q, r). Suppose .δ ∈ [0, m], .γ > δ, and .V ∈  d q  (Z ). Then .q < m + γ − δ and there is a constant .cS (which also depends on .d, τ, m, q, δ, γ , r) such that m+γ −δ

γ

tr2 (Zd ) (T (−i∇) − V )− ≤ cS (V+ m q + V+ m+γ −δ ).

.

(4.9)

γ +1

For .δ = m − 1, the bound in (4.9) restores .V+ γ +1 in (4.8). Proof By the variational principle we can assume .V = V+ ≥ 0. The proof of (4.8) is the same as that of (3.25) and we omit it. To prove (4.9) we use (4.7) in Theorem 4.5 and .m ≥ q to bound ⎛ 1 m ) θ (1 − e) ⎝ .Ne (V ) S log(2 + e + e−m θ (e − 1)

x∈Z

⎞m

x∈Z

e ⎠ (V (x) − )m + 2 d

q

e (V (x) − )m + 2 d

−δ  log(2 + 1/e)m θ (1 − e)V m q +e



e (V (x) − )m 2 + d x∈Z

for any .0 ≤ δ ≤ m. For .γ > δ the second term on the right contributes with ˆ .

0

∞

de eγ −1−δ

x

m+γ −δ

(V (x) − e/2)m + m,δ,γ V m+γ −δ ,

Number and Sums of Eigenvalues

335

whereas the first term contributes with ˆ .

0

1

m de eγ −1 log(2 + 1/e)m V m q m,q,γ V q

to the left side of (4.9). This concludes the proof.

 

Remark 5 Bach, Lakaev, and Pedra [2] proved CLR bounds in .d ≥ 3 when the symbol .T ∈ C 2 (Td ) is a Morse function, i.e., it satisfies .T (ξ ) ∼ |ξ − ξ0 |2 near a minimum´.ξ0 ∈ Td . This is needed [2, p. 21] to apply [18, Theorem 3.2] when computing . T −1 ((0,E]) T (ξ )−1 dξ .

4.4 Sums of Logarithms of Eigenvalues Theorem 4.7 Let H = T − V in 2 (Zd ) with T satisfying (1), (2), and (4) in Assumption 1.1 with r ∈ (0, (d − 1)/2]. Let q ∈ [1, r + 1], m = σ (q, r), and V ∈ m ∩ q (Zd ) = q (Zd ). Then for any γ > m there is a constant cS (which also depends on d, s, m, q, γ , r) such that  .

j

1 | log( 1/ej )|

γ m m ≤ cS (V+ m m + V+ q )  cS V+ q .

(4.10)

Proof Without loss of generality let V = V+ . Using (3.4), the representation (3.32), and (4.7) in Theorem 4.5 for γ > 0 and m ≥ 1, i.e.,    1 m m Nr (V )  V m + V  · log 2 + , m q r

.

lets us proceed as in the proof of Theorem 3.3.

(4.11)  

Remark 6 We make an observation similar to that after Theorem 3.3. The right side of (4.10) is bounded by a constant times V+ m q which is consistent with the right side of (3.29) when r = (d − 1)/2, m = d + 1, and q = (d + 1)/2. However, we need to restrict ourselves again to γ > d + 1 which makes (4.10) weaker compared to (3.29). A similar question arises whether (4.10) can hold for γ = m.

4.5 A CLR Bound for Powers of the BCS Operator in 2 (Zd ) Let .d ≥ 1. We generalize Theorem 3.4 to .|+μ|1/s with .s > 1 and .μ ∈ [−1, 1]\Z on .2 (Zd ). To that end we use that Cwikel’s estimate continues to hold in .2 (Zd ). This is a consequence of an abstract theorem by Birman, Karadzhov, and Solomyak [3, Theorem 4.8], which also includes an extension of the Kato–Seiler–Simon

336

J. -C. Cuenin and K. Merz

inequality. Recall that the discrete unitary Fourier transform .F : 2 (Zd ) → L2 (Td ) obeys .F1 (Zd )→L∞ (Td ) ≤ 1. Adapted to our setting, their result reads as follows. Theorem 4.8 ([3, Theorem 4.8]) Let .q > 2, .f ∈ q (Zd ), and .g ∈ Lq,∞ (Td ). Then f F ∗ gSq,∞ (L2 (Td )→2 (Zd )) q f q (Zd ) gLq,∞ (Td ) .

.

(4.12)

In combination with (3.3) (as in the proof of Theorem 3.4), (4.12) yields Theorem 4.9 Let .d ≥ 1, .μ ∈ [−1, 1] \ Z, .σ > 1, .p ∈ (1, σ ], and .Tμ (ξ ) be defined as in (4.4) with the ordinary Laplace operator. Then .Tμ (ξ )−1/(2σ ) ∈ L2p,∞ (Td ) (not necessarily uniformly in .μ, σ, p, d). Moreover, the number of negative eigenvalues of .(Tμ )1/σ − V is bounded by a constant (possibly depending on .μ, σ, p, d) times p .V+  d .  p (Z ) Proof The proof is analogous to that of Theorem 3.4. The bound for the number of negative eigenvalues follows from the variational principle, (3.3), and (4.12) (with 1/2 and .g(ξ ) = (T (ξ ))−1/(2σ ) for .x ∈ Zd and .ξ ∈ Td ). Thus, we .f (x) = |V (x)| μ are left with showing .Tμ (ξ )−1/(2σ ) ∈ L2p,∞ (Td ) with .p ≤ σ . Since .|Td | = 1, it suffices to check |{ξ ∈ Td : Tμ (ξ ) ≤ β 2σ }| μ,d,σ,p β 2p

.

for β ≤ 1.

(4.13)

Since .μ is a given, fixed parameter, we may even suppose .β 2σ < 1 − μ in the following. Then .Tμ (ξ ) ≤ β 2σ is equivalent to the bounds

.

− β 2σ ≤ d −1

d

cos(2π ξj ) − μ ≤ β 2σ ,

j =1

1 1 ξj ∈ (− , ). 2 2

(4.14)

Since .1 − x 2 /2 ≤ cos x ≤ 1 − x 2 /(2π ) for all .x ∈ (−π, π ), (4.14) implies 1 − μ − β 2σ ≤

.

2π 2 2 |ξ | ≤ 1 − μ + β 2σ . d

Thus, the left side of (4.13) is bounded from above by .β 2σ ≤ β 2p since .p ≤ σ and .β < 1. This concludes the proof.  

5 Alternative Proof of Theorem 3.2 We now give an alternative proof of Theorem 3.2 (1) for .γ > 0 using an observation made by Frank [17, p. 794], together with Theorem 3.4.

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Theorem 5.10 Suppose .T (ξ ) satisfies (1)–(3) in Assumption 1.1. If .γ > 0 and V ∈ Lγ +1 ∩ Lγ +d/s (Rd ) then there exists a constant .cS > 0 (which also depends on .d, s, γ ) such that

.

ˆ

γ

trL2 (Rd ) (T (−i∇) − V )− ≤ cS

.

Rd

(V+ (x)γ +1 + V+ (x)γ +d/s ) dx.

(5.1)

Proof Without loss of generality we assume .V ≥ 0. For .E > 0 and .σ > 1 we record T (−i∇) + E ≥ cσ · T (−i∇)1/σ · E 1/σ



(5.2)

.

for .σ  = (1 − 1/σ )−1 and some .cσ > 0. This observation and Theorem 3.4 imply that the number of eigenvalues .N (2E, T − V ) of .T − V below .−2E < 0 is bounded by 

N (2E, T − V ) = N (0, T + E − (V − E)) ≤ N (0, cσ E 1/σ T 1/σ − (V − E)) 

1

1

= n(1, |V − E| 2 (cσ E 1/σ T 1/σ )−1 (V − E) 2 ) .



= N (0, cσ T 1/σ − E −1/σ (V − E)) σ

σ E − σ  (V − E)+ σ σ

L (R ) d

+ E−

σ d/s σ

σ d/s

(V − E)+ 

Lσ d/s (R ) d

.

(5.3) Thus, we obtain for any .γ > dσ/(sσ  ), tr(T

γ − V )−

ˆ = ˆ

∞

0 ∞

 0 .

ˆ ∼

Rd

dE E γ −1 · N (E, T − V ) ˆ σ E dx (V (x) − )σ+ dE E γ −1− σ  d 2 R

ˆ dσ E dσs ) +E γ −1− sσ  dx (V (x) − 2 + Rd

(5.4)

(V (x)γ +1 + V (x)γ +d/s ) dx.

This concludes the proof.

 

Remark 7 1. We do not know whether the CLR bounds in Theorem 3.4 and an argument similar to that in the proof of Theorem 5.10 can be used to prove estimates for sums of logarithms of eigenvalues as in Theorem 3.3.

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2. Theorem 5.10 for .γ = 1 can be proved using Rumin’s method, see also [23, Proposition 4] or [19, Section 6]. The case .γ > 1 then follows from this together with the argument of Aizenman and Lieb [1] and the observation ˆ .

R

d

γ (T (ξ ) − V (x))−

ˆ dξ =

V (x)

ˆ dt (V (x) − t)

γ

0

ˆ ∼

St V (x)

dSt (ξ ) |∇P (ξ )|

dt (V (x) − t)γ · (1 + t)d/s−1

0

∼ V (x)γ +1 + V (x)γ +d/s .

Acknowledgments We are grateful to Volker Bach for valuable discussions and to Kouichi Taira for providing helpful comments. Special thanks go to Rupert Frank for providing critical remarks on Theorems 3.2 and 3.4, and for pointing out that the methods of Rumin and [17, 18] provide an alternative proof of Theorem 3.2 (1). Support by the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF) (K.M.) is acknowledged.

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Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators David Damanik and Jake Fillman

Dedicated to the memory of Sergey Naboko

Abstract In this survey, we give an introduction to and proof of the gap labelling theorem for discrete one-dimensional ergodic Schrödinger operators via the Schwartzman homomorphism. To keep the paper relatively self-contained, we include background on the integrated density of states, the oscillation theorem for 1D operators, and the construction of the Schwartzman homomorphism. We illustrate the result with some examples. In particular, we show how to use the Schwartzman formalism to recover the classical gap-labelling theorem for almostperiodic potentials. We also consider operators generated by subshifts and operators generated by affine homeomorphisms of finite-dimensional tori. In the latter case, one can use the gap-labelling theorem to show that the spectrum associated with potentials generated by suitable transformations (such as Arnold’s cat map) is an interval. Keywords Schrödinger operators · Gap labelling · Density of states · Topological dynamics · Oscillation theory

D. Damanik () Department of Mathematics, Rice University, Houston, TX, USA e-mail: [email protected] J. Fillman Department of Mathematics, Texas State University, San Marcos, TX, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_14

341

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1 Introduction 1.1 Setting We will discuss Schrödinger operators of the form .H = HV =  + V acting in 2 (Z), where

.

[ψ](n) = ψ(n − 1) + ψ(n + 1),

.

[V ψ](n) = V (n)ψ(n).

The main case of interest is that in which the potential, V , is dynamically defined over a topological dynamical system. Given a compact metric space ., a homeomorphism .T :  → , and a continuous function .f :  → R, we consider Hf,ω =  + Vf,ω ,

.

(1.1)

where the potential .Vf,ω is defined by Vf,ω (n) = f (T n ω).

.

(1.2)

The Schrödinger operators that can be defined in this manner have been heavily studied over the years; see for example the books [21, 23, 24, 27, 62, 77], the surveys [25, 26, 44, 53, 59, 74], and references therein. One reason for the popularity of this perspective is that this setup contains as special cases numerous models of particular interest including, inter alia, periodic operators (. = Zp , .T ω = ω + 1 mod p), quasi-periodic operators (. = Td , T is translation by a vector with rationally independent coordinates), almost-periodic operators (. is a compact monothetic group and T is a minimal translation), random operators (. is a suitable product space and T is a shift thereupon), and subshift operators (. is a subshift of low complexity and T is the shift on .). We will discuss each of these examples in more detail later in the paper. It is often fruitful to equip the topological dynamical system .(, T ) with additional structure. A Borel probability measure .μ is said to be T -invariant if −1 B) for every Borel .B ⊆  and T -ergodic if it is T -invariant and .μ(B) = μ(T −1 B = B. We will also simply say that .(, T , μ) is .μ(B) ∈ {0, 1} whenever .T ergodic. It is well known and not hard to show that for a given .(, T ), the space of invariant Borel probability measures is nonempty, convex, and compact (in the weak.∗ topology), that the ergodic measures are the extreme points of the set of invariant probability measures, and that there is always at least one ergodic measure [79]. Ergodic systems enjoy many helpful statistical properties; for the purpose of this text, the chief consequences of ergodicity that we employ are the almost-sure constancy of invariant functions and the Birkhoff pointwise ergodic theorem. That

Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

343

is, for any Borel function g, if .g ◦ T = g, then there is a constant c for which g(ω) = c for .μ-a.e. .ω ∈ , and, if .h ∈ L1 (, μ), then

.

.

 N −1 1  h(T n ω) → h dμ N 

for μ-a.e. ω ∈ .

(1.3)

n=0

Given .(, T ) and .f ∈ C(, R), and defining .{Hf,ω } as above, if .μ is a T ergodic Borel probability measure on ., then one sees that the spectrum is .μ-almost everywhere constant, that is, there exists a compact set .μ,f ⊆ R such that σ (Hf,ω ) = μ,f for μ-a.e. ω ∈ .

.

(1.4)

In a similar way, the absolutely continuous, singular continuous, and pure point spectra of .Hf,ω are .μ-almost everywhere constant [24, 54]. If .(, T ) is uniquely ergodic, then we simply write .f = μ,f where .μ is the unique invariant measure. Similarly, if .(, T ) is minimal, then there exists .f such that .σ (Hf,ω ) = f for all .ω ∈  and again there is no need to note the dependence on the measure. The density of states measure .κ = κμ,f associated with the family .{Hω }ω∈ is given by 

 δ0 , g(Hω )δ0  dμ(ω).

g dκ =

.



It is well known that μ,f = supp(κμ,f ).

.

(1.5)

The accumulation function of .κ,  k(E) = kμ,f (E) =

.

χ(−∞,E] dκ,

is called the integrated density of states (IDS). We will discuss proofs of some fundamental facts about the IDS in Sect. 3. Our presentation is based on Benderskii– Pastur [17], Pastur [63], Avron–Simon [5], and Delyon–Souillard [34]. We direct the reader to the surveys [51, 52, 73] for more background about the density of states and related objects. The present survey aims to discuss the gap labelling theorem for onedimensional operators: Given an ergodic topological dynamical system .(, T , μ), there is a countable group .A = A(, T , μ) ⊆ R such that for any continuous .f ∈ C(, R), kμ,f (E) ∈ A ∩ [0, 1]

.

(1.6)

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for all .E ∈ R \ μ,f , and moreover, this group can be explicitly calculated in many examples of interest (e.g. for almost-periodic and random operator families). The precise formulation follows. The relevant definitions take some work to set up. For definitions of the terms used in the statement of the theorem (suspension, homotopy classes, and the Schwartzman homomorphism), we refer the reader to Definitions 4.6, 4.9, 4.16. Theorem 1.1 (Gap-Labelling Theorem) Given an ergodic topological dynamical system .(, T , μ) such that .supp μ = , let .(X, T , μ) denote its suspension, let .C (X, T) denote the set of homotopy classes of maps .X → T, let .Aμ : C (X, T) → R denote the Schwartzman homomorphism, and denote by A = A(, T , μ) := Aμ (C (X, T))

.

(1.7)

the range of the Schwartzman homomorphism. Then, for any continuous .f ∈ C(, R), kμ,f (E) ∈ A ∩ [0, 1]

.

(1.8)

for all .E ∈ R \ μ,f . One can show (cf. Remark 5.2) that the group .A(, T , μ) always contains .Z; furthermore, the reader can check from the definitions that the IDS obeys .0 ≤ k ≤ 1. Consequently, the gap labelling theorem has the following important consequence. If one can show that .A = Z, then the only labels corresponding to open gaps are 0 and 1, which correspond to the trivial gaps .(−∞, min μ,f ) and .(max μ,f , ∞). Thus, if .A(, T , μ) = Z, then .μ,f is an interval for every continuous .f :  → R. Corollary 1.2 Suppose .(, T , μ) is an ergodic topological dynamical system. If A(, T , μ) = Z, then .μ,f is an interval for every .f ∈ C(, R). More generally, if .A(, T , μ) = n−1 Z for some .n ∈ N, then .μ,f is a finite union of no more than n intervals for every .f ∈ C(, R).

.

Examples with connected almost-sure spectra include certain affine torus automorphisms (see Sect. 8) and the doubling map [28]. Remark 1.3 Let us make some additional comments about Theorem 1.1. 1. The Schwartzman homomorphism derives its name from the study of the asymptotic cycle by Schwartzman [70]. 2. The theorem as formulated here and the approach to the proof we give are due to Johnson [46]. There is a more general gap-labelling framework obtained by Bellissard and coauthors that uses K-theory of C.∗ -algebras and was in fact obtained earlier than Johnson’s work [8–11, 13]. See Sect. 1.4 for further discussion. 3. Each connected component of .R \ μ,f is called a gap of the spectrum. In view of (1.5), the IDS is constant on every gap of the spectrum. As such, Theorem 1.1 implies that there is a function that associates an element of .A ∩ [0, 1] to each gap of .μ,f . Accordingly, elements of .A(, T , μ) are called gap labels.

Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

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4. In the event in which .(, T ) is uniquely ergodic, we suppress .μ and simply write .A(, T ) instead of .A(, T , μ). 5. One can show that .C (X, T) is countable (see Proposition 4.10). Consequently, the set .A(, T , μ) is countable. In particular, the set of gap labels is always countable, and therefore could possibly be in one-to-one correspondence with the set of gaps. 6. The theorem is optimal in the sense that there exist models for which every possible gap label corresponds to an open gap. That is, there exist topological dynamical systems .(, T , μ) and .f ∈ C(, R) such that for every .a ∈ A(, T , μ) ∩ [0, 1], there is a nonempty open interval .I ⊆ R such that .k|I ≡ a. In this situation, one says all gaps are open. 7. In some situations, one can try to show that all gaps are open for generic choices of .f ∈ C(, R) [4, 36, 65, 71]. See Sect. 1.3 for further discussion. 8. One may also try to prove that all gaps are open for specific dynamical systems and explicit sampling functions. The most famous version of this problem concerns the almost-Mathieu operator, where . = T, .T ω = ω + α with .α irrational, and .f (ω) = 2λ cos(2π ω). In this case, it is known that the set of possible labels is .A = {n + mα : n, m ∈ Z} (see Theorem 6.2 for details). It is widely expected that for any .λ = 0 and any irrational .α, all gaps are open. This is known as the Dry Ten Martini Problem, and has been discussed in [1, 20, 22, 56, 64, 68] and in the forthcoming work [2]. A similar result (all possible gaps open) has been presented for the Extended Harper’s Model in the non-self-dual region for Diophantine .α as well [40]. 9. This problem (showing that all possible gap labels correspond to open gaps) has also been heavily studied for substitution Hamiltonians [6, 58], such as those generated by the Fibonacci [30, 31, 67], period-doubling [12, 55], and Thue– Morse [9] Hamiltonians. Beyond Schrödinger operators, gap-labelling is also relevant in other contexts, such as the study of the asymptotic distribution of the Lee–Yang zeros in the Ising model; compare [7, 32, 39].

1.2 Examples Let us see some examples. We will give proofs in later sections. The first class of examples we discuss are those generated by minimal translations of compact groups. Recall that a toplogical dynamical system .(, T ) is called minimal if the T -orbit of every point is dense in .. Minimal translations of groups are important since they are precisely the dynamical systems by which one recovers almost-periodic sequences. Example 1.4 (Group Translations) If G is an abelian group and .g ∈ G, we write Rg : G → G for the map .Rg : x → g + x. If G is a compact abelian group with a dense cyclic subgroup, we say that G is monothetic. In the event that G is

.

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monothetic and g generates a dense subgroup, it is known that .(G, Rg ) is uniquely ergodic, with normalized Haar measure supplying the unique invariant measure. These examples will be addressed in detail with proofs in Sect. 6. 1. For .p ∈ N, let .Zp := Z/pZ denote the integers modulo p; for .k ∈ Z, we write .[k] ∈ Zp for the residue class of k. We have A(Zp , R[1] ) = p−1 Z.

(1.9)

.

This gives the first precise instance of a general heuristic: that .A(, T ) represents the “frequencies” of .(, T ) in a suitable sense. 2. Given .d ∈ N, let .Td = Rd /Zd denote the d-dimensional torus. We say .α ∈ Rd has rationally independent coordinates if the set .{1, α1 , α2 , . . . , αd } is linearly independent over .Q. Equivalently, for .k ∈ Zd , d  .

kj αj ∈ Z ⇒ k1 = k2 = · · · = kd = 0.

j =1

In this case, .(Td , Rα ) is strictly ergodic1 and (defining .α0 = 1 for convenience) A(Td , Rα ) = Z + αZd =

.

⎧ d ⎨ ⎩

kj αj : kj ∈ Z ∀0 ≤ j ≤ d

j =0

⎫ ⎬ ⎭

.

(1.10)

3. In general, suppose . is a compact monothetic group and .α ∈  generates a dense cyclic subgroup .α = {nα : n ∈ Z}. Assuming . is not cyclic, .α ∼ = Z.

denote the dual group (see Sect. 6 for definitions), we have an injective Letting .

→ T given by .

homomorphism .

ϕα :  ϕα (χ ) = χ (α). Denoting .π : R → T the canonical projection, one has

)). A(, Rα ) = π −1 (

ϕα (

.

(1.11)

This allows one to recover the classical gap-labelling theorem for almost-periodic potentials; compare [33, 45]. Example 1.5 (Subshifts) The next class of examples come from subshifts, which give a natural setting in which one may study ergodic potentials taking finitely many values. Given a finite set .A, called the alphabet, the full shift is the space .AZ equipped with the product topology. The shift acts on .AZ by .[T ω](n) = ω(n + 1). A subshift is a nonempty closed (hence compact) and T -invariant subset . ⊆ AZ . These examples will be addressed in detail with proofs in Sect. 7.

employ a minor but standard abuse of notation, writing .α both for the element of .Rd and its projection to .Td . This will be convenient later on when we talk about flows.

1 We

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347

1. Suppose .(, T ) is a subshift with ergodic measure .μ. Then  A(, T , μ) =

.

f dμ : f ∈ C(, Z) .

(1.12)

Thus, for a given subshift, the problem of determining the set of gap labels amounts to identifying integrals of elements of .C(, Z) against .μ which in turn can be reduced to computing the .μ-measure of cylinder sets (compare Theorem 7.1). 2. Let .A = {0, 1}, define .S : A → A∗ by .S : 0 → 01 and .1 → 0, let .un = S n (0), and define 

 = ω ∈ AZ : for every interval I ⊆ Z, ω|I is a subword of some un .

.

This is a strictly ergodic subshift, called the Fibonacci subshift. For additional background about subshifts generated by substitions, see Sect. 7.2 and [66]. One has A(, T ) = Z + αZ = {m + nα : m, n ∈ Z},

.

(1.13)

√ where .α = 12 ( 5 − 1) is the inverse of the golden mean. 3. Consider . = {1, 2, . . . , m}Z , .T :  →  the shift .[T ω](n) = ω(n + 1), .μ0 a measure on .{1, 2, . . . , m}, and .μ = μZ0 the product measure. Then .A(, T , μ) is the .Z-algebra generated by .{μ({j }) : 1 ≤ j ≤ m}, that is, A(, T , μ) = Z[w1 , . . . , wn ] ⎧ ⎫ ⎨ ⎬ = ck w k : ck ∈ Z and ck = 0 for all but finitely many k , ⎩ n ⎭

.

k∈Z+

(1.14) where .wj = μ0 ({j }) and .w k = w1k1 w2k2 · · · wnkn for .k ∈ Zn+ . Example 1.6 (Affine Toral Homeomorphisms) Given .d ∈ N, consider . = Td = Rd /Zd , the d-dimensional torus. We now consider maps of the form .T = TA,b given by .T ω = Aω + b where .A ∈ SL(d, Z) and .b ∈ Td . Note that Lebesgue measure is T -invariant for all .A ∈ SL(d, Z) and .b ∈ Td . These examples will be addressed in detail with proofs in Sect. 8. 1. If .μ is a .TA,b -ergodic measure with .supp μ = Td , then   A(Td , TA,b , μ) = n + k, b : n ∈ Z, k ∈ Zd ∩ Ker(I − A∗ ) .

.

(1.15)

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2. If A does not have any root of unity as an eigenvalue, .b = 0, and .μ is Lebesgue measure, then .(Td , TA,0 , μ) is ergodic [79, Corollary 1.10.1] and A(Td , TA,0 , μ) = Z.

.

(1.16)

As mentioned in Corollary 1.2, the conclusion .A(, T , μ) = Z implies that μ,f is an interval for every .f ∈ C(, R). The cat map, defined by .Tcat = TA,0 , where   21 .A = (1.17) , 11

.

supplies a concrete example of a transformation with the property that the underlying linear transformation has no root of unity as an eigenvalue. 3. The skew-shift is defined by taking .d = 2,  A=

.

 10 , 11

b=

  α 0

(1.18)

with .α irrational. In this case, it is known that .(T2 , TA,b ) is strictly ergodic [37]. One has A(T2 , TA,b ) = Z + αZ.

.

(1.19)

1.3 Deriving or Preventing Cantor Spectrum A very important application of the gap labelling theorem involves the ability to show the presence or absence of Cantor-type structures in the spectrum in concrete settings. This is motivated by the discovery in the 1980s that the spectra of Schrödinger operators with almost-periodic potentials can be Cantor sets, that is, they can be nowhere dense while not having any isolated points. In fact, an ergodicity argument rules out the existence of isolated points [63], so that the challenge in establishing a Cantor-type structure revolves around finding a proof that the gaps of the spectrum are dense. In other words, any real number can be approximated by sequence of elements of the complement of the spectrum in .R. Gap labelling enters this discussion as follows. As the set of growth points of the integrated density of states coincides with the almost sure spectrum, gaps can be dense only if the possible gap labels are dense. Thus, the first application of the gap labelling theorem is the general criterion that the non-denseness of gap labels implies the absence of a Cantor spectrum. Thus, it is of interest to identify settings

Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

349

where the range of the Schwartzman homomorphism is not dense in .[0, 1]. We have seen some examples above and will provide more detailed explanations below. If on the other hand the range of the Schwartzman homomorphism is dense in .[0, 1], this does not by itself imply that the spectrum is a Cantor set, as one does not know which elements of .A ∩ [0, 1] actually appear as gap labels corresponding to open gaps. In other words, the denseness of the range of the Schwartzman homomorphism in .[0, 1] suggests that Cantor spectrum is in principle possible, but does not establish it. The gap labels that do appear depend on f . Thus, a natural goal is to show that for a dense subset of the dense set of all possible labels, there are f ’s for which the labels in this subset all occur on gaps of .μ,f . In typical applications, one actually works with the full set of labels, which is natural given the reasoning employed. Namely, one usually fixes an potential gap label and considers the set of f ’s for which .μ,f has a gap with that label. If one is able to show that this set of f ’s is open and dense in a suitable function space, then by Baire’s theorem, Cantor spectrum is generic in that function space. As this argument only uses that the set of labels one intersects over is countable, one may as well intersect over all possible labels, that is, the option to intersect over a countable dense subset does not give any stronger conclusion. We refer the reader to [36], [65], and [4] for results obtained along the lines just discussed. We also mention that the somewhat involved approach of [4] can be simplified significantly if one is only interested in the conclusion, generic Cantor spectrum, rather than the sufficient condition for it in terms of gap labelling; see [3].

1.4 Gap Labelling via K-Theory We want to note that there is a beautiful approach to gap labeling, which is in fact substantially more general in that it is not restricted to operators in one space dimension and which also can handle more general covariant families of operators in one dimension. This alternative approach is based on K-theory of C.∗ -algebras and it was developed by Bellissard and coworkers. Let us describe this approach briefly. For proofs, more details, and further discussion, we refer the reader to [8–11, 13, 50]. A C.∗ -algebra .A is a Banach algebra together with an involution .x → x ∗ satisfying, among other things, .x ∗ x = x2 . An element .p ∈ A is an orthogonal projection (henceforth simply a projection) if .p2 = p∗ = p. Denote by .P(A) the set of all projections. Let us assume for simplicity that .A is a unital (i.e., it has a multiplicative identity) C.∗ -algebra. For each .n ∈ N, the set .Mn (A) of .n × n matrices with entries from .A is a C.∗ -algebra as well with the natural .∗ operation, and we set Pn (A) := P(Mn (A)),

.

P∞ (A) :=

 n∈N

Pn (A).

(1.20)

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On .P∞ (A), the relation .∼0 is defined as follows: if .p ∈ Pn (A) and .q ∈ Pm (A), then .p ∼0 q if and only if there is .v ∈ Mm,n (A) (the .m × n matrices with entries from .A) with .p = v ∗ v and .q = vv ∗ . It turns out that .∼0 is an equivalence relation and we set K0+ (A) := P∞ (A)/ ∼0 .

.

(1.21)

We also define .⊕ : P∞ (A) × P∞ (A) → P∞ (A) by  p ⊕ q := diag(p, q) =

.

 p0 . 0q

(1.22)

The operation .⊕ on .P∞ (A) descends to an operation on .K0+ (A) and makes it an abelian semigroup. A standard (Grothendieck) construction then associates an abelian group .K0 (A) whose elements are equivalence classes of formal differences of elements of the semigroup, along with a natural extension of the group operation. The next step is to associate a C.∗ -algebra with an ergodic family .{Hω }ω∈ of Schrödinger operators. Let us consider the case where . ⊆ I Z is closed and shiftinvariant for some compact interval I in .R and T is the shift on this sequence space. We can arrive at this setting by pushing forward under .ω → Vω and using the evaluation at zero as the sampling function. The associated C.∗ algebra is then given by the crossed product A := C ∗ (, T ) := C() T Z

.

(1.23)

of .C() by the .Z action defined by T , that is, the closure of .Cc ( × Z) with respect to the C.∗ norm g := sup πω (g),

(1.24)

.

ω∈

where .πω is the family of .∗ -representations on .2 (Z) defined by [πω (g)ψ](n) :=



.

g(T −n ω, n)ψ(m + n),

g ∈ Cc ( × Z).

(1.25)

m∈Z

An ergodic measure .μ on . (again, we would need to push forward under ω → Vω if we start from the abstract setting) defines a trace .τμ on .C ∗ (, T ) in the following way:

.

 τμ (g) =

g(ω, 0) dμ(ω),

.

g ∈ Cc ( × Z).



It can then be shown that there is an induced group homomorphism

(1.26)

Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

τμ∗ : K0 (A) → R

.

351

(1.27)

such that for every projection .p ∈ A, we have .τμ (p) = τμ∗ ([p]), where .[p] is the class of p in .K0 (A). Theorem 1.7 (Gap labeling via K-theory) Consider an ergodic family .{Hω }ω∈ of Schrödinger operators and let .A be the associated C.∗ -algebra. Then for each gap of ., the value the integrated density of states takes inside it belongs to the countable set of real numbers given by .τμ∗ (K0 (A)) ∩ [0, 1]. In view of this discussion, it is natural to ask whether the two approaches yield identical results. That is: Question 1.8 Given an ergodic topological dynamical system .(, T , μ), is it true that τμ∗ (K0 (A)) = A(, T , μ)?

.

(1.28)

This is addressed in higher dimensions for minimal .Zd actions on totally disconnected spaces and repetitive Delone sets in [14–16, 47, 78].2 See also [38] for relevant work. When specialized to dimension one as in the present work, one essentially ends up with minimal subshifts as in Sect. 7. Given that there is a more powerful and more general approach to gap-labelling (which applies both to operators in higher dimensions and to more general operators in dimension one), we should reflect on why we felt compelled to elucidate the current perspective. Indeed, it should be understood as a pedagogical and aesthetic choice: presenting in this fashion allows us to give a self-contained presentation modulo tools and techniques from linear algebra, measure theory, and a few elementary aspects of ergodic theory.

1.5 Organization The remainder of the paper is divided into two parts: general theory and examples. The first part of the paper builds up the necessary machinery and proves the onedimensional version of the gap-labelling theorem formulated in Theorem 1.1. We give a brief review of the oscillation theorem for one-dimensional Schrödinger operators in Sect. 2, define and recall basic properties of the integrated density of states in Sect. 3, and then recall basic facts about flows and the definition of the Schwartzman homomorphism in Sect. 4. This is all put together in Sect. 5 to prove Theorem 1.1. The second part of the paper then deals with computing .A(, T , μ) 2 To be more precise, while .A does not have an immediate analogue in higher dimensions in general, in the specific setting considered in these papers, .A can be identified as a set, compare (1.12), that does have a natural higher-dimensional counterpart.

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for examples of interest. We deal with almost-periodic potentials in Sect. 6, subshift and random potentials in Sect. 7, and operators generated by general affine toral homeomorphisms in Sect. 8.

2 Oscillation Theory Let us briefly leave the ergodic setting and consider a general potential .V : Z → R, and the associated difference equation u(n − 1) + u(n + 1) + V (n)u(n) = Eu(n),

.

n ∈ Z.

(2.1)

We will seek to connect oscillation of solutions of (2.1) with eigenvalue counting functions of Dirichlet truncations of H ; for .N ∈ N, define .HN ∈ RN ×N by ⎤

⎡

V (1) 1 ⎢ 1 V (2) ⎢ ⎢ .. .HN = ⎢ . ⎢ ⎣

⎥ 1 ⎥ ⎥ .. .. ⎥. . . ⎥ 1 V (N − 1) 1 ⎦ 1 V (N )

(2.2)

Let .uE : Z → R denote the sequence defined by uE solves (2.1) with uE (0) = 0 and uE (1) = 1.

.

(2.3)

The main theorem of the present section states that the number of sign flips of .uE on the interval .[1, N + 1] coincides with the number of eigenvalues of .HN that exceed E. Oscillation and comparison theorems for linear second-order differential operators date back to Sturm [75, 76]. To the best of our knowledge, the earliest appearance of discrete versions of these results is due to Bôcher [19]. The approach to oscillation theory here is inspired by Simon [72]. See also Teschl [77, Chapter 4] and Zettl [80]. In the present setting, one may readily derive the main results by noting that .uE (n) (as a function of energy) may be realized as the characteristic polynomial of a suitable Dirichlet truncation, and as such a sign flip of .uE between n and .n + 1 can be formulated in terms of eigenvalue counts. More precisely: Proposition 2.1 Let .V : Z → R be given. For any .E ∈ C and any .n ∈ N, uE (n + 1) = det(E − Hn ).

.

(2.4)

Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

353

Proof Both sides are readily seen to be monic polynomials of degree n in E that vanish whenever .E ∈ σ (Hn ). The reader can check that every eigenvalue of .Hn is simple, so the result follows.   The discussion of sign flips becomes slightly more delicate when E is a Dirichlet eigenvalue, i.e., an eigenvalue of some .Hn . In particular, for such E, we have .uE (n+ 1) = 0 and uE (n + 2) = −uE (n),

.

so we should only count this as a single sign flip. To accomplish this, linearly interpolate .uE to obtain a continuous real function .uE (x), .x ∈ R. For each .N ∈ N, let .FN (E) denote the number of zeros of .uE (x) in the interval .(1, N + 1). Equivalently, FN (E) = #{1 ≤ j ≤ N : sgn(uE (j )) = sgn(uE (j + 1))}

.

(2.5)

whenever .E ∈ σ (HN ) and FN (E) = #{1 ≤ j ≤ N − 1 : sgn(uE (j )) = sgn(uE (j + 1))}

.

(2.6)

for .E ∈ σ (HN ), where we adopt the convention .sgn(0) = 1 to avoid over-counting sign flips at the Dirichlet eigenvalues. Theorem 2.2 (Oscillation Theorem) Fix a potential .V : Z → R. For all .N ∈ N and .E ∈ R, .FN (E) is equal to the number of eigenvalues of .HN that exceed E. That is, # (σ (HN ) ∩ (E, ∞)) = FN (E) for all E ∈ R.

.

(2.7)

The following lemma will be useful. Lemma 2.3 (Interlacing Lemma) For every .N ≥ 2, the eigenvalues of .HN −1 N −1 1 strictly interlace those of .HN . More precisely, if .EN −1 < · · · < EN −1 and N 1 .E N < · · · < EN denote the eigenvalues of .HN −1 and .HN , respectively, then j

j +1

j

EN < EN −1 < EN

.

(2.8)

for all .1 ≤ j ≤ N − 1. Proof The reader can check that all eigenvalues of .HN and .HN −1 are simple, so it suffices to show that for any .1 ≤ j ≤ N − 1, there exists an eigenvalue .β of j j +1 .HN −1 with .E N < β < EN . To that end, let .v1 , . . . , vN be the (orthonormal) eigenvectors of .HN , chosen so that j

HN vj = EN vj ,

.

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and define a meromorphic function h by det(z − HN −1 ) . det(z − HN )

h(z) =

.

(2.9)

  By Cramer’s rule, .h(z) = δN , (z − HN )−1 δN for all .z ∈ / σ (HN ). If we expand .δN in the basis .{v1 , . . . , vN }, we see that   h(z) = δN , (z − HN )−1 δN

.

N N   = vj , δN vj , j =1

=

j =1



1 j

z − EN

vj , δN vj

N  |vj , δN |2 j =1

j

z − EN

for all .z ∈ / σ (HN ). One can readily check that .vj , δN  = 0. Thus, for each .1 ≤ j ≤ N − 1, we have .

lim h(x) = +∞ and j

x↓EN

lim h(x) = −∞, j +1

x↑EN

j

j +1

so h vanishes somewhere in the open interval . EN , EN j

! . It follows that .HN −1

j +1

has an eigenvalue strictly between .EN and .EN , as desired.

 

With the interlacing lemma in hand, we are ready to prove the desired oscillation theorem. Proof (Proof of Theorem 2.2) Let .BN (E) denote the number of eigenvalues of .HN that exceed E, i.e., the left-hand side of (2.7). In the case .N = 1, one has E < E11 ⇐⇒ E < V (1)

.

⇐⇒ sgn(E − V (1)) = −1. Since .uE (1) = 1 and .uE (2) = E − V (1), .B1 (E) = F1 (E) for every .E = V (1). If E = E11 = V (1), then one has .B1 (E) = F1 (E) = 0. Inductively, assume that .BN ≡ FN for some .N ∈ N, and consider .E ∈ R. 0 = −∞ and .E N +1 = +∞, choose .0 ≤ k ≤ N so that Adopting the conventions .EN N

.

k+1 k EN ≤ E < EN .

.

By definition of .BN , one has .BN (E) = N − k, and then .FN (E) = N − k by the inductive hypothesis. From here, we analyze a few cases separately.

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Case 1 .E = E kN . It follows that .uE (N + 1) = 0. Consequently, exactly one of .uE (N), .uE (N + 2, E) is positive and the other must be negative. In particular, we have FN +1 (E) = FN (E) + 1 = N − k + 1.

.

k On the other hand, the interlacing lemma (Lemma 2.3) implies that .EN +1 < k+1 E < EN +1 , which yields

BN +1 (E) = N − k + 1 = FN +1 (E),

.

concluding this case. First, notice that this gives .BN +1 (E) = BN (E) + Case 2 .E kN < E < E k+1 N+1 . 1, as before. Then, since E is neither an eigenvalue of .HN nor of .HN +1 , Proposition 2.1 yields sgn(uE (N + 2)) = sgn(det(E − HN +1 ))

.

= (−1)BN+1 (E)

= (−1)BN (E) = sgn(det(E − HN )) = sgn(uE (N + 1), and hence .FN +1 (E) = FN (E) + 1 = BN +1 (E). k+1 k+1 By the interlacing lemma, the remaining cases are .E = EN +1 or .EN +1 < E < k+1 , which are similar and left to the reader.   EN

3 The Integrated Density of States In this section we discuss two closely related fundamental objects associated with an ergodic family of Schrödinger operators .{Hω }ω∈ : the density of states measure and the integrated density of states. Definition 3.1 The density of states measure (DOSM) is the measure .κ defined by  .

 δ0 , g(Hω )δ0  dμ(ω)

g dκ = 

(3.1)

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for bounded measurable g. The function k defined by  k(E) =

.

χ(−∞,E] dκ

(3.2)

is called the integrated density of states (IDS). Note that by definition, .κ is the .μ-average of the spectral measure corresponding to the pair .(Hω , δ0 ). We first establish one of the central properties of the IDS/DOSM, namely its relation with the almost sure spectrum. Theorem 3.2 The almost sure spectrum is given by the points of increase of k. Equivalently, . = supp κ. Proof We begin with the inclusion .supp κ ⊆ . If .E0 ∈ , there is an open interval I containing .E0 with .I ∩  = ∅. For .μ-almost every .ω, we have . = σ (Hω ) and hence .χI (Hω ) = 0. It follows that   . χI dκ = δ0 , χI (Hω )δ0  dμ(ω) = 0. 

Thus, .E0 ∈ supp κ. Let us now prove .supp κ ⊇ : If .E0 ∈ supp κ, there is an open interval I containing .E0 such that .I ∩ supp κ = ∅. But this implies  0=

.

 χI dκ =

δ0 , χI (Hω )δ0  dμ(ω) 



δ0 , χI (HT n ω )δ0  dμ(ω)

= 



δn , χI (Hω )δn  dμ(ω),

= 

where the second step follows by T -invariance of .μ, and the third follows from the covariance relation .HT ω = U Hω U ∗ , where U is the unitary shift .δn → δn−1 . Thus, for .μ-almost every .ω, we have .

Consequently, .E0 ∈ .

dim Ran χI (Hω ) = Tr χI (Hω ) = 0.  

We now connect the IDS and density of states measure to properties of finite cutoffs of .Hω . To that end, we will show that the density of states measure can also be viewed as the .μ-a.e. weak limit of normalized traces of truncations of .Hω .

Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

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Definition 3.3 For .ω ∈  and .N ≥ 1, define the measure .κω,N by  g dκω,N =

.

1 Tr (g(Hω )χ[1,N ] ) N

(3.3)

for bounded measurable g. Lemma 3.4 For every bounded, measurable function g, there exists a set .g ⊆  of full .μ-measure such that 

 .

g dκω,N =

lim

N →∞

g dκ

(3.4)

for every .ω ∈ g . Proof Fix a bounded measurable function g and define g(ω) = δ0 , g(Hω )δ0 ,

.

ω ∈ .

(3.5)

Since g is bounded and measurable, .g ∈ L∞ (, μ) ⊆ L1 (, μ). Therefore,  .

g dκω,N = =

1 Tr (g(Hω )χ[1,N ] ) N N 1  δn , g(Hω )δn  N n=1

=

N 1  δ0 , g(HT n ω )δ0  N n=1

N 1  g(T n ω) N n=1  → g dμ

=

 =



g dκ.

In the penultimate step, we applied the Birkhoff ergodic theorem, which works for μ-almost every .ω ∈ .  

.

Corollary 3.5 For .μ-almost every .ω, .κω,N converges weakly to .κ as .N → ∞. That is, there exists a set .∗ of full .μ-measure such that (3.4) holds for all .ω ∈ ∗ and all .g ∈ C(R). Proof Since .Hω  ≤ 2 + f ∞ =: M for a.e. .ω, it follows that the interval .J = [−M, M] supports .dkω,N for a.e. .ω and N. Consequently, it suffices to show that

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there is a full-measure set .∗ ⊆  such that (3.4) holds for all .g ∈ C(J, R). Since J is compact, .C(J ) enjoys a countable dense subset, .D. The statement of the corollary then follows with " .∗ = g , (3.6) g∈D

 

by an .ε/3 argument (where .g is as defined in Lemma 3.4).

Theorem 3.6 The density of states measure and the integrated density of states are both continuous. Proof Fix .E0 ∈ R and suppose .gn are continuous compactly supported functions with .0 ≤ gn ≤ 1, .gn (E0 ) = 1, and .gn (E) ↓ 0 for .E = E0 . Then, by dominated convergence,  .

gn dκ → κ({E0 }).

(3.7)

Moreover, .δ0 , gn (Hω )δ0  → δ0 , χ{E0 } (Hω )δ0  for all .ω, and hence, again by dominated convergence, 

 .



gn dκ =

δ0 , gn (Hω )δ0  dμ(ω) → 



δ0 , χ{E0 } (Hω )δ0  dμ(ω).

(3.8)

By a standard argument using ergodicity, .Tr (χ{E0 } (Hω )) is almost-surely constant, and the almost-sure value lies in .{0, ∞}. Since .Tr (χ{E0 } (Hω )) ≤ 1 for every3 .ω, we have .χ{E0 } (Hω ) = 0 for .μ-a.e. .ω. Combining this observation with (3.7) and (3.8), we observe .κ({E0 }) = 0, which implies that .κ is a continuous measure. Since k is the accumulation function of .κ, continuity of k follows.   A different approach to the integrated density of states is obtained if we restrict the operator to a finite box, rather than its spectral projection. Definition 3.7 Denote the restriction of .Hω to .[1, N] with Dirichlet boundary conditions by .Hω,N , that is, ⎡

Hω,N

.

Vω (1) 1 ⎢ 1 Vω (2) ⎢ ⎢ .. =⎢ . ⎢ ⎣

⎤ ⎥ 1 ⎥ ⎥ .. .. ⎥. . . ⎥ 1 Vω (N − 1) 1 ⎦ 1 Vω (N )

(3.9)

trace is at most two since the solution space of .Hω ψ = E0 ψ is two-dimensional; this bound already suffices for this argument. One can reduce the two to one by using constancy of the Wronskian.

3 The

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359

For .ω ∈  and .N ≥ 1, define measures # .κω,N by placing uniformly distributed point (1) (N ) < · · · < Eω,N of .Hω,N (recall that every eigenvalue masses at the eigenvalues .Eω,N of .Hω,N is simple, as observed in Proposition 2.1). That is, we define  .

g d# κω,N =

N ! 1  (n) g Eω,N N

(3.10)

n=1

for bounded measurable g. j

Note that .{g(Eω,N ) : 1 ≤ j ≤ N} is precisely the spectrum of .g(Hω,N ), so we can rewrite the definition of # .κω,N as follows:  .

g d# κω,N =

1 Tr (g(Hω χ[1,N ] )). N

(3.11)

This also makes the connection to .κω,N more transparent (compare (3.3)). To be more explicit, both measures involve cutting off and taking a normalized trace. The measure .κω,N applies g to .Hω and cuts the result off, while # .κω,N cuts .Hω off first and then applies g. The following theorem shows that the two perspectives on the density of states measure indeed are equivalent. Theorem 3.8 For .μ-almost every .ω ∈ , the measures # .κω,N converge weakly to .κ as .N → ∞. Proof Since all spectra are contained in the interval .[−2 − f ∞ , 2 + f ∞ ], it .κω,N converge to suffices to prove, for .μ-almost every .ω ∈ , that the moments of # the moments of .κ as .N → ∞. Given .p ∈ Z+ , notice  .

κω,N (E) = E p d#

N 1  (n) !p Eω,N N n=1

=

$ % 1 Tr (Hω,N )p N

N  1  δn , (Hω,N )p δn = N n=1

1 = N 1 = N

&

&

N  

'



δn , (Hω ) δn + O(1) p

n=1 N   n=1



'

δ0 , (HT n ω ) δ0 + O(1) . p

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By the Birkhoff ergodic theorem, it follows that

.

 N  1  δ0 , (HT n ω )p δ0 = δ0 , (Hω )p δ0  dμ(ω) N →∞ N  lim

n=1

for .μ-almost every .ω ∈ . Thus, for these .ω’s, 

 .

lim

N →∞



κω,N (E) = E d#

δ0 , (Hω ) δ0  dμ(ω) =

p

p

E p dκ(E),



 

as desired.

We conclude the present section by showing how to view the integrated density of states as a rotation number by counting sign flips of Dirichlet eigenfunctions using the discrete version of Sturm oscillation theory described in Sect. 2. Theorem 3.9 If k denotes the integrated density of states of the ergodic family {Hω }ω∈ , one has

.

1 #{1 ≤ j ≤ N : sgn(uE,ω (j )) = sgn(uE,ω (j + 1))} N →∞ N

1 − k(E) = lim

.

for .μ-almost every .ω ∈  and every .E ∈ R, where .uE,ω is the solution to .Hω u = Eu with .u(0) = 0 and .u(1) = 1. Proof Since # .κω,N converges weakly to .κ for almost every .ω ∈  by Theorem 3.8 and .κ is continuous (by Theorem 3.6), we have  .

lim

N →∞

 χ(−∞,E] d# κω,N =

χ(−∞,E] dκ

(3.12)

for every .E ∈ R and almost every .ω. Concretely, let .fn and .gn be continuous functions with .fn ≡ 1 on .(−∞, E − 1/n], .gn ≡ 1 on .(−∞, E], .fn ≡ 0 on .[E, ∞), .gn ≡ 0 on .[E + 1/n, ∞) (and linearly interpolated in between). Then, for any .ω for κω,N → dκ weakly, one obtains which .d#  .

 fn dκ = lim

κω,N fn d#

N →∞



≤ lim inf N →∞

κω,N χ(−∞,E] d# 

≤ lim sup

κω,N χ(−∞,E] d#

N →∞



≤ lim

N →∞

κω,N gn d#



=

gn dκ.

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( By dominated convergence and continuity of .κ, . (gn − fn ) dk → 0 as .n → ∞, concluding the proof of (3.12). Consequently, for a.e. .ω, one has  1 − k(E) =

.

 χ(E,∞) dk = lim

N →∞

= lim

N →∞

kω,N χ(E,∞) d#

1 #(σ (Hω,N ) ∩ (E, ∞)), N

so the desired conclusion follows from Theorem 2.2.

 

Remark 3.10 Let us conclude our discussion of the integrated density of states by pointing out that much of the material in the present section generalizes to the study of Schrödinger operators in higher dimensions. However, Theorem 3.9 is a purely one-dimensional result. Due to the central role this result plays in the subsequent discussion, we restrict attention to the one-dimensional case in the present manuscript.

4 Flows, Suspensions, and the Schwartzman Homomorphism We will briefly recall some facts about continuous flows on compact metric spaces. The overall goal is to build up enough machinery to describe the Schwartzman homomorphism, which provides a version of rotation number for continuous maps to the circle originating from a space on which a flow is defined. Later in the paper, we will relate the Schwartzman homomorphism to the rotation number for a cocycle originating from a Schrödinger operator, which is in turn related to sign flips of Dirichlet solutions via Theorem 3.9 and hence to the IDS for ergodic operators.

4.1 Basics Let X denote a compact metric space. A flow on X will mean an action of .R on X by homeomorphisms. That is, for each .t ∈ R, there is a homeomorphism .T t : X → X so that .T t ◦ T s = T t+s for all .s, t ∈ R and .(x, t) → T t x is a continuous map .X × R → X. If X is in addition equipped with a Borel probability measure .μ, $ % we say that the flow is measure-preserving if one has .μ T t B = μ(B) for every .B ∈ B and every .t ∈ R. We want to emphasize at the outset that we assume no regularity of T beyond mere continuity, and indeed no differentiable structure on X; in particular, X does not need to be a manifold. In our main applications, X will be the suspension of a suitable discrete-time dynamical system and hence will sometimes come equipped with a local differentiable structure along the flow, but we again emphasize that our typical application will be something like the suspension

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of a translation on a compact group or the suspension of the shift on a subshift over a finite alphabet. We say that a Borel probability measure on X is T -ergodic if it is measurepreserving and every flow-invariant function is almost surely constant. That is to say, if g is a measurable function on .X and .g(T t x) = g(x) for every .t ∈ R, .x ∈ X, then there is a constant .g∗ and a set .X∗ ⊆ X of full .μ-measure so that .g(x) = g∗ for every .x ∈ X∗ . Equivalently, we say that the measure preserving flow described by .T t is ergodic if every measurable set .E ⊆ X with .T t E = E for every t satisfies .μ(E) = 0 or .μ(X \ E) = 0. If T denotes a flow on X, then the map .T 1 is often called the time-one map. Thus, associated to any flow, there is a natural discrete-time dynamical system, namely 1 .(X, T ). Theorem 4.1 Suppose .(X, μ) is a probability space and T is a flow on X that is ergodic with respect to .μ. Then 1 . lim t→∞ t



t

 g(T x) ds = s

0

g dμ

(4.1)

X

for every .g ∈ L1 (μ) and .μ-almost every .x ∈ X. Theorem 4.1 is well known and can be deduced from the discrete-time version of the ergodic theorem; compare [18]. We will eventually want to apply the Schwartzman homomorphism to an invariant section derived from the (discrete) cocycle dynamics associated with .{Hf,ω }. In order to do that, we need an appropriate analog of a continuous cocycle over a flow. Definition 4.2 A continuous .SL(2, R) cocycle over the flow T is defined to be a continuous map . : X × R → SL(2, R) with the properties (x, 0) = I .

.

(x, s + t) = (T s x, t) · (x, s)

(4.2) (4.3)

for all .x ∈ X and .s, t ∈ R. We will denote .(x, t) = t (x). A cocycle induces flows on .X × R2 and .X × RP1 via .T t (x, v) = (T t x, t (x)v). One has the following notions of uniform hyperbolicity for continuous cocycles. Definition 4.3 Let . be a continuous cocycle. We say that . exhibits uniform exponential growth if there are constants, .C > 0, .λ > 1 with the property that t (x) ≥ Cλ|t|

.

for all .t ∈ R and .x ∈ X.

(4.4)

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We say that . admits an invariant exponential splitting if there exist constants c > 0, L > 1 and continuous maps .s , u : X → RP1 such that the following statements hold.

.

(a) (Invariance) For all .x ∈ X, and .t ∈ R, one has t (x)s (x) = s (T t x),

.

t (x)u (x) = u (T t x).

(4.5)

(b) (Contraction) For all .t > 0, .x ∈ X, .vs ∈ s (x), and .vu ∈ u (x), one has t (x)vs  ≤ cL−t vs ,

.

−t (x)vu  ≤ cL−t vu .

(4.6)

We say that . enjoys a bounded orbit if there exist .x ∈ X and .v ∈ S1 so that .

sup t (x)v < ∞.

(4.7)

t∈R

Theorem 4.4 Suppose . : X × R → SL(2, R) is a continuous cocycle over a continuous flow T on a compact metric space X. Then the following are equivalent. (a) . exhibits uniform exponential growth. (b) . admits an invariant exponential splitting. (c) . does not enjoy a bounded orbit. Definition 4.5 Whenever any one (and hence every) condition in Theorem 4.4 holds, we say that . is uniformly hyperbolic. Proof (Proof of Theorem 4.4) Consider the discrete cocycle .(T 1 , A) associated with the time-one map, that is, the skew product .(T 1 , A) : X × R2 → X × R2 given by .(x, v) → (T 1 x, 1 (x)v). There are notions of uniform hyperbolicity for such discrete cocycles that are almost identical to those discussed above for continuous cocycles; compare [81]. Then, use continuity of . to interpolate the corresponding discrete results characterizing uniform hyperbolicity of .(T 1 , A) (e.g. [29, 81]). Alternatively, one can prove the equivalence of (a), (b), and (c) directly by following the proofs of corresponding discrete-time results and making minor cosmetic changes. The details are left to the reader.  

4.2 The Suspension of a Dynamical System It is easy to produce a discrete-time dynamical system from a flow by passing to the time-one map. To go in the other direction, one may consider the suspension.

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Definition 4.6 Suppose .(, T ) is a topological dynamical system, that is, . is a compact metric space and .T :  →  is a homeomorphism. Define an action of .Z on . × R by n  (ω, t) = (T n ω, t − n),

(4.8)

.

which induces an equivalence relation on . × R via (ω, t) ∼ (ω , t  ) ⇐⇒ n  (ω, t) = (ω , t  ) for some n ∈ Z.

.

⇐⇒ (t − t  ) ∈ Z, and T

t−t 

ω = ω .

(4.9) (4.10)

The suspension of the dynamical system .(, T ) is the quotient of . × R by this equivalence relation, i.e. X(, T ) := ( × R)/ ∼ .

(4.11)

.

Generally, the homeomorphism and compact space are clear from the context, so we will sometimes simply write X. Naturally, there are homeomorphic copies of . inside X, namely the fibers .s = {[ω, s] : ω ∈ } for each fixed .s ∈ R, where .[ω, t] denotes the equivalence class of .(ω, t) in X. There is a natural .R-flow on X defined by translation in the second factor, which we denote by .T . That is, t .T [ω, s] = [ω, s + t]. For each .ω ∈ , one has 1

T [ω, s] = [ω, s + 1] = [T ω, s].

(4.12)

.

1

1

In other words, .T maps the fiber .s to itself and the action of .T on the fiber .s is the same as the action of T on .. Remark 4.7 One could equivalently realize X as the quotient . × [0, 1]/ ∼ where .(ω, 1) ∼ (T ω, 0). The reader can readily check that the map sending t! ω, {t}] is a homeomorphism from the first version of X to this .[ω, t] → [T version. Henceforth, we freely work with whichever realization of the suspension is more convenient. We will also need the notion of the suspension of a measure .μ on ., which is the measure .μ defined on X by  g dμ =

.

X

 1 g[ω, s] dμ(ω) ds. 0

(4.13)



Lemma 4.8 Fix a topological dynamical system .(, T ), a Borel probability measure .μ on ., and .X = X(, T ). (a) If .μ is T -invariant, then .μ is .T -invariant. (b) If .μ is T -ergodic, then .μ is .T -ergodic.

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365

Proof This follows from direct calculations. We leave the details to the reader.

 

4.3 The Schwartzman Homomorphism Let us return to the setting of general flows to define the Schwartzman homomorphism, which will play a key role in the gap labelling theorem. For the remainder of the section, fix a compact metric space X, a continuous flow T on X, and a T -ergodic probability measure .μ. Consider .C(X, T), the set of continuous maps .X → T endowed with the group operation of pointwise addition, i.e. (φ1 + φ2 )(x) = φ1 (x) + φ2 (x),

.

x ∈ X.

(4.14)

Denote the unit interval by .I = [0, 1]. Let us recall some general definitions. Definition 4.9 Given a topological space .X, continuous functions .φ, φ  : X → X are said to be homotopic if there is a continuous function .F : X × I → X (called a homotopy) so that .

F (x, 0) = φ(x).

(4.15)

F (x, 1) = φ  (x)

(4.16)

for all .x ∈ X. One can confirm that the relation φ ∼ φ  ⇐⇒ φ is homotopic to φ 

.

(4.17)

defines an equivalence relation on .C(X, T); we denote by .[φ] the equivalence class of .φ modulo this relation and refer to it as the homotopy class of .φ. Let .C (X, T) denote the set of equivalence classes in .C(X, T) modulo homotopy. We collect a few significant facts in the following proposition. Proposition 4.10 Let X be a compact metric space. (a) If .φ1 , φ2 , φ1 , φ2 ∈ C(X, T) are such that .φj ∼ φj for .j = 1, 2, then .(φ1 +φ2 ) ∼ (φ1 + φ2 ). (b) The operation [φ1 ] + [φ2 ] = [φ1 + φ2 ],

.

φ1 , φ2 ∈ C(X, T)

is well-defined and gives .C (X, T) the structure of an abelian group. (c) Let d denote the standard metric on .T, given by d(x, y) = dist(x − y, Z).

.

(4.18)

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If .φ1 , φ2 ∈ C(X, T) are such that d(φ1 (x), φ2 (x)) < 1/2

.

for all x, then .φ1 ∼ φ2 . (d) .C (X, T) is countable. Proof If .Fj is a homotopy from .φj to .φj for .j = 1, 2, then one can check that   .F1 + F2 is a homotopy between .φ1 + φ2 and .φ + φ , proving the assertion from 1 2 (a). The assertion in (b) follows. We leave the assertion from (c) to the reader. Since X is compact, .C(X, T) is separable. Denoting a countable dense set by .{φn }, we see from (c) that every homotopy class in .C (X, T) contains at least one .φn , which suffices to prove the assertion from (d).   The following fact about homotopic maps into .T will be useful. Proposition 4.11 Let X be a compact metric space. Then two maps .φ, φ  : X → T # : X → R so that are homotopic if and only if there exists a continuous map .ψ $ % #(x) φ(x) = φ  (x) + π ψ

.

(4.19)

for all .x ∈ X, where .π : R → T denotes the canonical projection. Proof If (4.19) holds, then $ % #(x) F (x, t) = φ(x) − π t ψ

.

defines a homotopy from .φ to .φ  . For the converse, notice that it suffices to deal with the case when .φ  ≡ 0. To that end, assume that F is a homotopy from .φ to the constant function 0. Let . denote the space of continuous functions .γ : I → T with the property that .γ (1) = 0, topologized with the uniform metric dunif (γ , γ  ) = sup d(γ (t), γ  (t)),

.

(4.20)

0≤t≤1

and observe that . is contractible, path-connected, and locally path-connected. The homotopy F induces a continuous map .f : X →  by mapping .x → F (x, ·). If . :  → T is given by .(γ ) = γ (0), then we have the following commutative diagram:

.

(4.21)

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Since . is contractible, path connected, and locally path connected, we may use # :  → R so that the following covering space theory (e.g. [41]) to lift . to a map . diagram commutes:

(4.22)

.

#= # ◦ f , we have the desired result. With .ψ

 

Remark 4.12 The reader should notice that the Proposition 4.11 is not a trivial corollary of basic lifting theorems from geometric topology since we have not assumed that X is path connected, nor have we assumed that it is locally path connected; indeed some of the applications we have in mind will involve suspensions of totally disconnected spaces. Given a continuous map .φ : X → T and .x ∈ X, we obtain a continuous function φx : R → T by following the image of .φ along the orbit of x, i.e.

.

φx (t) = φ(T t x).

.

(4.23)

We then want to talk about the “rotation number” of .φ by tracking the average rate at which .φx (t) winds around .T. To make this precise, recall from the general covering #x : R → R that satisfy space theory that, for each x, there are continuous functions .φ # .π ◦ φx = φx , where .π : R → T is the canonical projection. Indeed, there is a unique such lift satisfying #x (0) = a φ

.

(4.24)

# and .φ # are any two distinct lifts of .φx , for each .a ∈ π −1 {φx (0)}. Moreover, if .ψ # # then there is an integer n such that .φ ≡ ψ + n. We will say that .φ is differentiable #x is a differentiable function for every .x ∈ X. One can check that along the flow if .φ this notion does not depend on the choice of lift. We denote the derivative of .φ with respect to the flow by .∂φ, i.e. (∂φ)(x) =

.

#x dφ (0). dt

(4.25)

Note for later use that one has for .s ∈ R and .x ∈ X the relationship (∂φ)(T s x) =

.

#x dφ (s). dt

(4.26)

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Finally, let us say that .φ is .C 1 along the flow if .∂φ exists everywhere and is a continuous function .X → R. In order to measure the average rate of winding of .φ along the orbit of some point .x ∈ X, the rotation number of .φ along the orbit of x is defined by #x (t) φ , t→∞ t

rot(φ; x) = lim

.

(4.27)

whenever this limit exists. The next crucial step is to define a homomorphism .A : C(X, T) → R by sending a map .φ ∈ C(X, T) to its rotation number. There is a slight complication in that the limit defining the rotation number may not exist uniformly and it may depend on the choice of x. We will presently prove that the limit on the right hand side of (4.27) is defined and constant for a set of full .μmeasure (recall that we are assuming that .μ is a T -ergodic measure). We then define .A = Aμ to be this almost-sure value:  #x (t) φ , μ-a.e. x. .Aμ (φ) = rot(φ; x)) dμ(x) = lim (4.28) t→∞ t X To establish the existence of the limit, we will apply a smoothing argument to show that an arbitrary continuous function on X can be approximated by integrals along the flow. More precisely, we will show that the set of functions that are .C 1 along the flow is uniformly dense in the space of continuous functions. If .φ is .C 1 along the flow, we then show that  .rot(φ; x) = ∂φ dμ a.e. x ∈ X (4.29) X

by Birkhoff’s theorem. Lemma 4.13 Suppose T is a continuous flow on the compact metric space X. Then, the set of functions that are .C 1 along the flow is dense in .C(X, T) with respect to the uniform topology. The lemma is taken from [70]; Schwartzman attributes the statement to Kakutani. Proof (Proof of Lemma 4.13) It is somewhat more straightforward to work with functions X → S1 := {z ∈ C : |z| = 1}.

.

(4.30)

Naturally, one can define the notion of flow-differentiability for elements of C(X, S1 ), and it suffices to prove that .C 1 functions from X to .S1 are dense in 1 1 .C(X, S ). To that end, assume given .φ ∈ C(X, S ) and .ε > 0, and define ε .ψ = ψ ∈ C(X, C) by  1 ε .ψ(x) = φ(T s x) ds, x ∈ X. (4.31) ε 0 .

Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

369

One can verify that .ψ ε converges uniformly to .φ as .ε ↓ 0. In particular, if .ε is small enough, .ψ(x) = 0 for all .x ∈ X. For such small .ε, define .ηε (x) = ψ ε (x)/|ψ ε (x)|. The reader may check that .ηε is .C 1 and .ηε → ψ uniformly as .ε ↓ 0.   Corollary 4.14 Any .φ ∈ C(X, T) is homotopic to a function that is .C 1 along the flow. Proof If .φ and .φ  are sufficiently close in the uniform metric, then they are homotopic by Proposition 4.10. Thus, the result follows from Lemma 4.13.   We are now in a position to define the Schwartzman asymptotic cycle and prove that it has the desired properties. The following result is due to Schwartzman [70]. Theorem 4.15 Let X denote a compact metric space, T a continuous flow on X, and .μ a T -ergodic Borel probability measure. (a) For each .φ ∈ C(X, T), the limit rot(φ; x) = lim

.

t→∞

#x (t) φ t

(4.32)

exists for .μ-almost every .x ∈ X and does not depend on the choice of lift. (b) For each .φ ∈ C(X, T), there exists .Aμ (φ) ∈ R with .rot(φ; x) = Aμ (φ) for .μ-a.e. x. (c) If .φ and .φ  are homotopic, then .Aμ (φ) = Aμ (φ  ). Thus, (4.28) descends to a well-defined homomorphism from .C (X, T) to .R. Definition 4.16 The induced map .Aμ : C (X, T) → R is called the Schwartzman homomorphism. Proof (Proof of Theorem 4.15) Suppose given .φ ∈ C(X, T). If .φ is .C 1 along the flow, then Birkhoff’s theorem (Theorem 4.1) yields #x (t) φ t→∞ t ) t * 1 #x (0) = lim (∂φ)(T s x) ds + φ t→∞ t 0  t 1 (∂φ)(T s x) ds = lim t→∞ t 0  = ∂φ dμ,

rot(φ; x) = lim

.

for .μ-almost every .x ∈ X. Any two lifts of .φx differ by a fixed additive constant, which will wash out of the right hand side of (4.32) in the limit .t → ∞. Thus, independence of the right hand side of (4.32) on the choice of lift follows. Consequently, we have obtained (a) and (b) for functions that are .C 1 along the flow.

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Given a general .φ ∈ C(X, T), there exists a function .ψ that is .C 1 along the flow and to which .φ is homotopic by Corollary 4.14. Using Proposition 4.11, we can produce a continuous (hence bounded!) map .f# : X → R such that .φ = ψ + π ◦ f#. Boundedness of .f# implies that .rot(φ; x) = rot(ψ; x) for all x for which the latter is well-defined. In particular, we obtain (a) and (b) for general .φ. Moreover, the preceding argument also implies that .Aμ is well-defined modulo homotopy, as claimed in (c). Now that the various well-definedness issues above are resolved, one can verify that .Aμ defines a homomorphism, concluding the argument.   Remark 4.17 Since X is assumed to be a compact metric space, .C(X, T) is separable, and so we can reverse the quantifiers: there is a uniform set .X∗ of full .μ-measure so that .rot(φ; x) exists and equals .Aμ (φ) for all .x ∈ X∗ and every continuous .φ : X → T. To construct the desired gap labelling scheme, we will need one final piece: we will relate the Schwartzman homomorphism to a natural notion of rotation number of a uniformly hyperbolic continuous cocycle. To that end, suppose . is a uniformly hyperbolic continuous .SL(2, R)-cocycle. We wish to define the rotation number of t . to be the average winding number of . (x) · v around the origin for some vector 2 .v ∈ R \ {0}. Since the Schwartzman homomorphism was defined in terms of maps into .T and the winding of .t (x) · v is most readily comprehended as a net change in argument in .RP1 , the following definition is helpful. Definition 4.18 For .θ ∈ R, write .eθ = [cos θ, sin θ ]" , and recall that .e¯θ ∈ RP1 denotes the corresponding equivalence class. Write .p : R → RP1 for the universal covering map p : θ → e¯θ ∈ RP1

.

(4.33)

and define the identification .h : RP1 → T by h : e¯π θ → θ ∈ T.

.

(4.34)

Consider a continuous map . : [a, b] → RP1 . As before, one may lift . to the # : [a, b] → R for which universal cover .p : R → RP1 to obtain a continuous . # = . p◦

(4.35)

# # a,b arg  = (b) − (a),

(4.36)

.

With this setup, one defines .

which we refer to as the total change of argument of . as t increases from a to b.

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We are now prepared to define the application of the Schwartzman homomorphism to a uniformly hyperbolic continuous cocycle .. Given such a cocycle ., let 1 u . : X → RP denote the corresponding unstable section. Using the map h from Definition 4.18, we can view .u as a continuous map .u0 = h ◦ u : X → T and define Aμ () = Aμ (u0 ).

(4.37)

.

Using the definitions and (4.34), we have Aμ (u0 ) = lim

.

t→∞

1 0,t $ u s %   (T x) , π t arg

μ-a.e. x.

(4.38)

The content of the final theorem of this section is that (4.37) is indeed an appropriate notion of the winding number of a hyperbolic cocycle, in that it measures the average rate of change of the argument of .t (x) · v for .μ-almost every x and every .v ∈ RP1 . Theorem 4.19 Suppose . is uniformly hyperbolic. For .μ-almost every .x ∈ X and every .v ∈ RP1 , we have % 1 0,t $ s arg  (x) · v . t→∞ π t

Aμ () = lim

.

(4.39)

Proof By invariance, we have .s (x) · u (x) = u (T s x). In particular, (4.38) implies that % 1 0,t $ s arg  (x) · u (x) t→∞ π t

Aμ () = Aμ (u0 ) = lim

.

(4.40)

for .μ-almost every .x ∈ X. Since .s (x) is nonsingular for all s and x, it follows that .

+ $ s % %++ + 0,t $ s u  (x) ·  (x) + 0, .x ∈ X, and every .v ∈ RP1 . Combining (4.41) with (4.40), this implies that .

lim

t→∞

% % 1 0,t $ s 1 0,t $ s    (x) · v = lim  (x) · u (x) , t→∞ π t arg π t arg

(4.42)

for any .v ∈ RP1 and .μ-almost every .x ∈ X, since (4.40) implies that the right-hand   limit exists. Combining (4.42) with(4.38) yields the desired result. Remark 4.20 Let us point out that the use of the unstable section in the previous proof is not essential. One could just as well work with the stable section—the crucial property that is used in the proof is invariance, not exponential decay.

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Remark 4.21 We want to emphasize that we work with “merely continuous” flows, that is, actions of .R on compact metric spaces via homeomorphisms, rather than, say, actions of .R on differentiable manifolds via diffeomorphisms. This is not simply generality for its own sake. Rather, the main application of the Schwartzman homomorphism will involve the suspension of a general topological dynamical system, and some of the main examples we have in mind (like a minimal translation of a totally disconnected compact group or a shift on a suitable sequence space) are given by suspensions of minimal transformations of totally disconnected spaces (which then are not differentiable manifolds).

5 The Gap Labelling Theorem Having discussed the necessary background, we return to the setting of ergodic Schrödinger operators. Recall that .(, T ) is a topological dynamical system, .μ is a T -ergodic Borel probability measure on ., and .f ∈ C(, R). By restricting T if necessary, we will also assume .

supp μ = ,

(5.1)

that is, we assume .μ(U ) > 0 for every nonempty open set .U ⊆ . Recall also that .μ,f denotes the .μ-almost sure spectrum of the family .{Hf,ω }ω∈ defined in (1.1) and (1.2). Since .μ,f is closed and bounded, there exists a countable pairwise disjoint family of open intervals .Gn such that -  , μ,f = min μ,f , max μ,f \ Gn .

.

(5.2)

n

Each such interval is naturally called a gap of .. In Theorem 3.2, we have seen that the almost-sure spectrum . is precisely the set of points of increase of the integrated density of states, k. In particular, k is constant on each gap. As mentioned before, the constant value k assumes on .Gn is referred to as the label of the gap. In this section, we will discuss the proof of Theorem 1.1, and emphasize that it applies to any topological family of Schrödinger operators satisfying the stated restrictions: any gap label must be in the image of the Schwartzman homomorphism of the suspension of the underlying dynamical system .(, T ) (see Definitions 4.6 and 4.16). In particular, the possible gap labels that might appear depend only on the underlying dynamics, and do not depend on the choice of sampling function f . Moreover, we now reap a reward from our work in Sect. 4. In particular, we established the general theory surrounding the Schwartzman homomorphism without additional assumptions on the topology of . (such as connectedness or a smooth structure), so we can now discuss gap labelling for a general topological dynamical system.

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Recall the Schrödinger cocycle (T , AE ) :  × R2 →  × R2

(5.3)

.

defined by .(T , AE )(ω, v) = (T ω, AE (ω)v), where  AE (ω) =

.

 E − f (T ω) −1 . 1 0

(5.4)

As usual, we define .AnE for .n ∈ Z by

n .AE (ω)

=

⎧ n−1 ω)A (T n−2 ω) · · · A (ω) ⎪ ⎪ E E ⎨AE (T I ⎪ ⎪ ⎩[A−n (T n ω)]−1 E

n≥1 n=0

(5.5)

n ≤ −1,

so that iterates of the cocycle obey .(T , AE )n = (T n , AnE ). The definitions of .AE and n .A ensure that one may characterize solutions of the difference equation associated E to .Hω ; that is, .u : Z → C solves .Hω u = Eu if and only if     u(n + 1) u(1) n . = AE (ω) u(n) u(0)

for every n ∈ Z.

(5.6)

By (5.1) and Johnson’s theorem, . is precisely the complement of the energies at which .(T , AE ) is uniformly hyperbolic [46]; see also the surveys [29, 81]. Now, let .X = X(, T ) denote the suspension of .(, T ) (see Definition 4.6). In order to relate the Schwartzman homomorphism on X to the integrated density of states, we want to define an interpolated .AtE :  → SL(2, R) for .t ∈ R in such a way that .AtE induces a well-defined cocycle on X and the winding number of orbits of .AtE corresponds to the rotation number as measured by counting sign flips of Dirichlet eigenfunctions. In particular, one may do this in such a way that .AtE (ω) is a smooth function of t for each fixed .E ∈ R and .ω ∈ . To that end let .θ and .λ be smooth nondecreasing functions so that θ (t) = 0 for 0 ≤ t ≤ 1/6,

.

θ (t) = π/2 for 1/3 ≤ t ≤ 1/2,

(5.7)

and λ(t) = 0 for 1/2 ≤ t ≤ 2/3,

.

λ(t) = 1 for 5/6 ≤ t ≤ 1,

(5.8)

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and then define ' ⎧& ⎪ cos(θ (t)) − sin(θ (t)) ⎪ ⎪ ⎪ ⎪ ⎨ sin(θ (t)) cos(θ (t)) .YE (ω, t) = ' & ⎪ ⎪ λ(t)(E − f (T ω)) −1 ⎪ ⎪ ⎪ ⎩ 1 0

0 ≤ t ≤ 1/2 (5.9) 1/2 ≤ t ≤ 1

With this, we define .AtE (ω) by using .YE to interpolate between .AnE and .An+1 E . More precisely, put $ % AtE (ω) = YE T n ω, t − n AnE (ω),

.

ω ∈ , n ≤ t < n + 1,

(5.10)

where .n ∈ Z. One can check that .AtE (ω) is a smooth function of t for all fixed n .ω ∈  and .E ∈ R that agrees with .A when restricted to the integers. This induces E a continuous cocycle .E on X via s −1 tE ([ω, s]) = At+s E (ω)AE (ω) .

.

(5.11)

The reader may readily check the following: Proposition 5.1 The map .E is a well-defined continuous cocycle on X. Moreover, E is uniformly hyperbolic if and only if the corresponding discrete cocycle .(T , AE ) is uniformly hyperbolic. .

By Johnson’s theorem and the relationship between . and A, Proposition 5.1 demonstrates that .E is uniformly hyperbolic if and only if .E ∈ / . We are heading towards a proof of the identity k(E) = 1 − Aμ (E ),

.

(5.12)

where .Aμ denotes the Schwartzman homomorphism (defined in Definition 4.16). Let us recall some notation from Sect. 4.3: .C (X, T) denotes the set of homotopy classes of maps .X → T, and .Aμ : C (X, T) → R denotes the Schwartzman homomorphism, which is defined by (4.28). Finally, .Aμ () for a hyperbolic cocycle . is defined to be the Schwartzman homomorphism evaluated at the (homotopy class of the) unstable section of said cocycle. Remark 5.2 Notice that the image of the Schwartzman homomorphism in the present case always contains .Z (proof: use the map .X # [ω, t] → nt ∈ T). In particular, (5.12) suffices to establish that .k(E) lies in the image of .Aμ whenever E lies in a gap.

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From Theorem 3.9, we know that k can be related to sign flips of solutions, namely  1 # 1 ≤ j ≤ N : sgn(uE,ω (j )) = sgn(uE,ω (j + 1)) N →∞ N (5.13)

1 − k(E) = lim

.

for .μ-almost every .ω ∈ , where .uE,ω is the Dirichlet solution of .Hω u = Eu (with the convention .sgn(0) = 1). Our main goal is to relate this sign-flip counting to a rotation number in a more explicit fashion in order to invoke the Schwartzman homomorphism. If . : [a, b] → RP1 is continuous, recall that .a,b arg (t) denotes the total change in the argument of .(t) as t increases from a to b as in (4.36). Theorem 5.3 For each .E ∈ R and .μ-almost every .ω, 1 − k(E) = lim

.

N →∞

1 0,N t  (A (ω) · e1 ), π N arg E

(5.14)

where .e1 = [1, 0]" denotes the standard basis vector. Proof Fix .E ∈ R and .ω in the full-measure set for which (5.13) holds and suppress ω from the notation. For each t, define   u(t + 1) .u(t) = (5.15) = AtE · e1 . u(t)

.

In view of (5.6) and (5.10), .{u(n)}n∈Z is the solution of .Hω u = Eu satisfying u(0) = 0 and .u(1) = 1, so .{u(t)}t∈R may be thought of as an “interpolated” solution. Note that invertibility of .AE implies .u(t) = 0 for all .t ∈ R. By our choice of .ω, and our definition of .u(t), (5.13) implies that

.

1 − k(E) = lim

.

N →∞

1 #{j ∈ N : 1 ≤ j ≤ N and sgn(u(j )) = sgn(u(j + 1))} N (5.16)

for all E. The key observation here is that each sign flip of .u(j ) corresponds to a half-rotation of .u(t) about the origin and that the cocycle .AtE is such that one actually sees these rotations reflected in .arg (u(t)); the rest is bookkeeping. Concretely, we claim that #{j ∈ N : 1 ≤ j ≤ N and sgn(u(j )) = sgn(u(j + 1))} =

.

1 0,N +1  (u(t)) + O(1). π arg

Let us make this more precise. Given signs .s, s  ∈ {+, −}, define the quadrant s,s  = {(x, y) ∈ R2 : sx > 0, s  y > 0}. Similarly, denote the four semi-axes by .Q Q0,± = {(0, y) : ±y > 0},

.

Q±,0 = {(x, 0) : ±x > 0}.

(5.17)

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One can show the following: #{j ∈ N ∩ [1, N] : sgn(u(j )) = sgn(u(j + 1))} ⎧/ 0 0,N +1 ⎪ ⎨ π1 arg (u(t)) − 1 uN ∈ Q0,+ = / 0 ⎪ +1 ⎩ 1 0,N (u(t)) otherwise π arg

.

(5.18)

for all .N ∈ N, which suffices to prove (5.14). To prove (5.18), one proceeds inductively as in the proof of Theorem 2.2. Denote the left-hand side of (5.18) by .FN and the right-hand side by .RN . First, consider the case .N = 1. If .u(2) > 0, then .sgn(u(1)) = sgn(u(2)) and .u(2) = 0, so .u(t) > 0 for all .0 < t ≤ 2. In particular, .u(t) is in the open upper half-plane for all .0 < t ≤ 2, and hence 0 ≤ 0,2 arg (u(t)) < π.

.

If .u(2) = 0 (notice that this implies .u(1) ∈ Q0,+ ), then one has .0,2 arg u(t) = π by a direct calculation. On the other hand, if .sgn(u(1)) = sgn(u(2)), one has π ≤ 0,2 arg (u(t)) < 2π.

.

Thus, the claim holds for .N = 1 (keep in mind the convention .sgn 0 = 1). Now, suppose that .FN −1 = RN −1 for some .N ≥ 2. One must consider several cases separately. Case 1 .u(N ) ∈ Q−,0 . First, notice that this implies that .sgn(u(N + 1)) = sgn(u(N)), which means that the sign flip counter increments, i.e., FN = FN −1 + 1.

.

Additionally, in this case, one has .u(N − 1) ∈ Q0,+ and .u(N + 1) is in the open lower half-plane. In particular, 1 RN =

.

2 1 2 1 0,N +1 1 0,N   (u(t)) = RN −1 + 1, (u(t)) = π arg π arg

which proves .RN = FN in this case. Notice that we have used .u(N − 1) ∈ Q0,+ to get the final equality. Case 2 .u(N ) ∈ Q−,+ Since this implies .sgn(u(N )) = sgn(u(N + 1)), one has −,− ∪ Q0,− ∪ Q+,− . Using the .FN = FN −1 + 1. Notice that .u(N + 1) ∈ Q explicit form of the homotopy used to construct .AtE , we see that 1 RN =

.

2 1 2 1 0,N +1 1 0,N   (u(t)) + 1 = RN −1 + 1, (u(t)) = π arg π arg

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377

which shows .FN = RN in this case. The remaining cases are similar and left to the reader. Thus, .FN = RN for every N ∈ N, so (5.18) holds.  

.

Theorem 5.4 With setup as above, let .G ⊆ R \ μ,f be given. We have 1 − k(E) = Aμ (E )

.

(5.19)

for every .E ∈ G. Proof By Theorem 5.3, 1 0,N t  (A (ω)e1 ), π N arg E

1 − k(E) = lim

.

N →∞

μ-a.e. ω ∈ .

(5.20)

1 0,N t  (E (x)e1 ), π N arg

μ-a.e. x ∈ X.

(5.21)

1 0,N t  (A (ω)e1 ), π N arg E

μ-a.e. ω ∈ .

(5.22)

On the other hand, Theorem 4.19 implies Aμ (E ) = lim

.

N →∞

Consequently, Aμ (E ) = lim

.

N →∞

Combining (5.20) and (5.22), we get (5.19).

 

This allows us to complete the proof of Theorem 1.1. Proof (Proof of Theorem 1.1) As discussed above, we know that .A = A(, T , μ) contains .Z. Since .A is a subgroup of .R, if E belongs to a gap, we get k(E) = 1 − (1 − k(E)) ∈ A

.

from Theorem 5.4. Since .0 ≤ k(E) ≤ 1 for all E by definition, the theorem is   proved. Remark 5.5 Recall that we showed in Proposition 4.10 that .C (, T) is countable whenever . is compact. Since (by Theorem 4.15), the Schwartzman homomorphism is constant on homotopy classes, it follows that the range of the Schwartzman homomorphism is a countable set. Thus, there is a fixed countable set of gap labels that only depends on the base dynamics .(, T ).

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6 Almost-Periodic Potentials With the general theorem proved, it is naturally of interest to compute the group A(, T , μ) for specific examples. In this section, we begin this endeavor by computing .A(, T , μ) for almost-periodic potentials. For reference, this section will address the claims from Example 1.4. We will start off by studying two special cases of almost-periodic potentials: periodic and quasi-periodic potentials. In each of those two cases, one can identify the groups .A(, T ) in a few lines, whereas the characterization of .A(, T ) for general almost-periodic hulls takes more work.

.

6.1 Examples As a warm-up, let us see how the gap labels arise from the Schwartzman homomorphism in the setting of periodic Schrödinger operators. Of course, one can produce the gap labels directly from Floquet theory; the point is to demonstrate the general theory in the simplest example first. Suppose .V : Z → R satisfies .V (n + p) = V (n) for some .p ∈ N and every .n ∈ Z. This can be realized as an ergodic operator with  = Zp ,

T [k] = [k + 1],

.

f ([k]) = V (k),

(6.1)

where .[k] denotes the class of .k ∈ Z in .Zp . Of course, .(Zp , T ) is minimal and uniquely ergodic with unique invariant measure given by normalized counting measure. Proposition 6.1 Let .p ∈ N, . = Zp , and .T :  →  be given by .T [k] = [k + 1]. We have  n 1 Z= :n∈Z . .A(, T ) = p p Proof The reader can verify that .X = X(, T ) is homeomorphic to the circle .T via the identification given by , X # [k], t → p−1 (k + t) ∈ T.

.

(6.2)

Since every continuous map .T → T is homotopic to .x → nx for some .n ∈ Z, (6.2) implies that every map .φ : X → T is homotopic to φ (n) : [0, t] → nt/p ∈ T

.

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¯ t] = [0, t + k]). for some .n ∈ Z (notice that this defines .φ (n) for all .x ∈ X, since .[k, (n) (n) For any choice of base point .x ∈ X, one considers .φx : R → T given by .φx (t) = #x(n) is of the form .nt/p + C φ (n) (T t x) as in Sect. 4.3. One readily sees that the lift .φ for some .C = C(x) and thus .

#x(n) (t) φ nt/p + C n = lim = , t→∞ t→∞ t t p lim

which shows that the range of .A is contained in .p−1 Z, as claimed. Moreover, since every .φ (n) is continuous, the previous calculation also shows that the range contains −1 Z. .p   Before considering the case of general almost-periodic potentials, let us consider the class of quasi-periodic potentials, which as we will see in Theorem 6.14 is a strict superset of the set of periodic potentials and a strict subset of the set of all almostperiodic potentials. Here, the base dynamics are given by .Rα : ω → ω + α with d 4 .α ∈ R having rationally independent coordinates. In this case, it is known that d .(T , Rα ) is strictly ergodic with Lebesgue measure supplying the unique invariant measure; see [48, Propositions 1.4.1, 4.2.2, and 4.2.3]. Theorem 6.2 Let .α ∈ Rd have rationally independent coordinates. One has A(Td , Rα ) = Z + αZd =

.

⎧ ⎨ ⎩

k0 +

d 

kj αj : kj ∈ Z ∀0 ≤ j ≤ d

j =1

⎫ ⎬ ⎭

.

Proof The suspension .X = X(Td , Rα ) is homeomorphic to .Td+1 via the map X # [ω, t] → [ω + tα, t].

.

(6.3)

Since every continuous map .Td+1 → T is homotopic to5 .x → k, x for some d+1 , the result follows from a calculation similar to the conclusion of the .k ∈ Z proof of Proposition 6.1.  

6.2 Generalities about Almost-Periodic Sequences We now move to a discussion of general almost-periodic sequences. The first goal is to place a general almost-periodic potential in the setting of dynamically defined

that we write .α both for the vector in .Rd and its projection to .Td . is well known and not hard to show using that any map .T → T is homotopic to .x → nx for some .n ∈ Z. Alternatively, it also follows from the more general discussion below in Sect. 6.3.

4 Recall 5 This

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potentials. One can accomplish this by showing that the shift on the hull of an almost-periodic sequence can be viewed as a translation of a suitable compact group. First, we recall some background without proofs. Let S denote the shift ∞ ∞ .[SV ](n) = V (n + 1) for .V ∈  (Z) and .n ∈ Z. Given .V ∈  (Z), the orbit of V is the set orb(V ) = {V (· − k) : k ∈ Z}

.

of all translates of V . The hull of V is the closure of the orbit in .∞ : ∞ (Z)

hull(V ) = orb(V )

.

.

We say that V is almost-periodic if .hull(V ) is compact in .∞ (Z). The reader can readily check that a given .V ∈ ∞ (Z) is almost-periodic if and only if for all .ε > 0, the .ε almost-periods of V are relatively dense in .Z (recall .p ∈ Z is an .ε almostperiod of V if .V − S p V ∞ < ε). Given .V ∈ ∞ (Z) almost-periodic, it is well known (and not hard to show) that the pairing . : orb(V ) × orb(V ) → orb(V ) given by S k V S  V := S k+ V

(6.4)

.

is uniformly continuous, and thus extends uniquely to a uniformly continuous binary operation on .hull(V ). One has: Proposition 6.3 If V is almost-periodic, then .(hull(V ), ) is a compact abelian topological group with a dense cyclic subgroup (namely .orb(V )). The allows one to characterize almost-periodic sequences as precisely those that are generated by continuously sampling translations of compact monothetic groups: Proposition 6.4 Let .V ∈ ∞ (Z) be given. The following are equivalent: (a) V is almost-periodic. (b) There exist a compact monothetic metrizable group . (with dense cyclic subgroup generated by .α), a continuous function .f :  → R, and an element .ω0 ∈  such that V (n) = f (ω0 + nα),

.

n ∈ Z.

(6.5)

Proof . (a) ⇒ (b). If V is almost-periodic, we can write it in the form (6.5) with . = hull(V ), .ω0 = V , .f (ω) = ω(0) (which is continuous), and .α = SV . The reader should recall that we write the group operation as . in this case. . (b) ⇒ (a). Assume that V is of the form (6.5). For each .ω ∈ , define ∞ .Vω ∈  (Z) by Vω (n) = f (ω + nα),

.

n ∈ Z.

(6.6)

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The reader may readily check that .Vω ∈ hull(V ) for every .ω ∈  and that the mapping .ω → Vω is continuous and maps . onto .hull(V ). Thus, the hull of V is the continuous image of a compact set and hence is compact.   In this section we single out several prominent subclasses of almost-periodic sequences, prove that they are indeed almost-periodic, and characterize them in terms of their hulls. Let us begin with the genuinely periodic sequences. The following is not hard to check and is left to the reader. Proposition 6.5 (a) Every periodic sequence is almost-periodic. (b) An almost-periodic sequence is periodic if and only if its hull is isomorphic to .Zp for some .p ∈ N. (c) An almost-periodic sequence is aperiodic if and only if no point of its hull is isolated. Definition 6.6 We say that .V ∈ ∞ (Z) is limit-periodic if it belongs to the .∞ closure of the set of periodic points of S. More precisely, V is limit-periodic if there ∞ is a sequence .(Vj )∞ j =1 such that .Vj ∈  (Z) for each j , .Vj is periodic for every j , and .

lim V − Vj ∞ = 0.

j →∞

(6.7)

Proposition 6.7 (a) Every limit-periodic sequence is almost-periodic. (b) An almost-periodic sequence is limit-periodic if and only if its hull is totally disconnected. Proof (a) Suppose V is limit-periodic. For each .j ∈ N, choose .Vj ∈ ∞ (Z) and .pj ∈ N such that .S pj Vj = Vj and .limj →∞ V − Vj ∞ = 0. It suffices to prove that .orb(V ) is totally bounded in order to show that V is almost-periodic. Given .ε > 0, choose j such that V − Vj ∞ < ε.

.

We claim that .orb(V ) is contained in the following finite union of .ε-balls: pj −1



.

k=0

B(S k Vj , ε).

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Indeed, given . ∈ Z, choose .0 ≤ k < pj with . ≡ k mod pj . Then, by pj -periodicity of .Vj and isometry of S, we have

.

S  V − S k Vj ∞ = S  V − S  Vj ∞ = V − Vj ∞ < ε.

.

Thus, .hull(V ) is compact and V is almost-periodic. (b) First, supposing that V is limit-periodic, let us show that .hull(V ) is totally disconnected. Since the hull of V is a topological group, it suffices to show that there are arbitrarily small neighborhoods of the identity that are both open and closed. Therefore, given .ε > 0 we will show that .B(V , ε)∩hull(V ) contains a non-empty set that is both closed and open. Since V is limit-periodic, choose a periodic W with period p so that .W − V ∞ ≤ 2ε . Then, hullp (V ) = {S jp V : j ∈ Z}

.

(6.8)

is a compact subgroup of .hull(V ) of index at most p. By construction, .hullp (V ) is closed, but it is also open since it is the complement of the union of no more than .p−1 other closed cosets. Moreover, since .S p W = W and .W −V ∞ ≤ 2ε , every element .V  ∈ hullp (V ) satisfies .V  −W ∞ ≤ ε/2 and hence .hullp (V ) is contained in the .ε-ball around V . Consequently, .hull(V ) is totally disconnected. Conversely, let V be almost-periodic and suppose its hull is totally disconnected. We have to show that V is limit-periodic. Given .ε > 0, we have to find .W ∈ ∞ (Z) and .p ∈ N with .S p W = W and .W − V ∞ < ε. By total disconnectedness, we may choose a compact open neighborhood N of V in .hull(V ), small enough so that (W1 W2 ) − W1 ∞ < ε/2

.

for every W1 ∈ hull(V ) and every W2 ∈ N.

(6.9)

This is possible since . is uniformly continuous, V is the identity of .hull(V ) with respect to ., and .hull(V ) is totally disconnected. Since the sets N and .hull(V ) \ N are compact and disjoint, there exists .δ > 0 so that .X − Y  ≥ δ for all .X ∈ N and all .Y ∈ hull(V ) \ N . By almost-periodicity of V , we can choose .p ≥ 1 so that S p V − V ∞ < δ,

.

hence .S p V ∈ N by our choice of .δ. But then we find inductively that .{S kp V : k ∈ Z} ⊆ N, by isometry of S and the choice of .δ. Now consider the p-periodic W that coincides with V on .[0, p − 1]. Given .n ∈ Z, we write .n = r + p with . ∈ Z and .0 ≤ r ≤ p − 1. Then, it follows from (6.9) that + + + + |V (n) − W (n)| = |V (r + p) − V (r)| = +(S r V S p V )(0) − (S r V )(0)+ < ε/2,

.

since .S p V ∈ N. This shows that the p-periodic W obeys .W − V ∞ ≤ ε/2 < ε,   concluding the proof.

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Definition 6.8 We say that .V ∈ ∞ (Z) is quasi-periodic if and only if it can be obtained by continuously sampling along orbits of a translation on a finitedimensional torus. More precisely, V is quasi-periodic if and only if it can be written as V (n) = f (nα + ω0 ),

.

n ∈ Z,

(6.10)

for some .ω0 , α ∈ Td , .d ∈ N, and some continuous .f : Td → R. Naturally, if .d = 1 and .α = 1/p ∈ T, then .f (ω0 + nα) defines a p-periodic sequence. Moreover, it is clear that any given p-periodic sequence can be obtained in this manner. Proposition 6.9 (a) Any quasi-periodic sequence is almost-periodic. (b) An almost-periodic sequence is quasi-periodic if and only if its hull is isomorphic to .Tr ⊕ A for some .r ≥ 0 and some finite abelian group A. Proof (a) This follows immediately from the definition and Proposition 6.4. (b) To prove one direction, let us assume that V is quasi-periodic and show that the hull of V is indeed a direct sum of a finite-dimensional torus and a finite abelian group. Since V is quasi-periodic, we may write .V (n) = f (nα + ω0 ) with .ω0 , α ∈ Td and .f ∈ C(Td ). Replacing f by a suitable translate thereof, we may assume .ω0 = 0. Define .K(α) to be the closed subgroup of .Td generated by .α, and put Vω (n) = f (nα + ω),

.

n ∈ Z, ω ∈ K(α),

(6.11)

as in the proof of Proposition 6.4. Arguing as before, we see that .(ω) = Vω defines a continuous surjective map . : K(α) → hull(V ). Moreover . is a homomorphism from .K(α) to .hull(V ), and hence is a (topological) quotient map by the open mapping theorem (see, e.g., [61, Chapter 1]). In particular, .hull(V ) is isomorphic to .K(α)/Ker(), which must be of the stated form since it is a quotient of a closed subgroup of .Td by a closed subgroup. Conversely, assume that V is almost-periodic and that .hull(V ) ∼ = Tr ⊕ A. By using the classification of finite abelian groups (for example), it is not hard to see that d #: K → .hull(V ) is isomorphic to a closed subgroup .K ⊆ T for some .d ≥ r. Let .f # hull(V ) be a continuous group isomorphism, and choose .α so that .f (α) = SV . Extending .f# to a continuous map .f#,: Td → - hull(V ), we see that V can be realized   via (6.10) with .ω0 = 0 and .f (ω) = f#(ω) (0).

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6.3 The Frequency Module As discussed in Proposition 6.4, every almost-periodic potential is a dynamically defined potential with base dynamics given by a minimal translation of a compact abelian group; moreover, the structure of the group dictates the structure of the potential. Definition 6.10 Given a (locally) compact abelian topological group G, a character

of all characters of G is a continuous homomorphism .χ : G → T. The collection .G of G is a group under pointwise addition that is itself a topological group in the compact-open topology, called the dual group of G. We freely use well known facts about topological groups, their duals, and harmonic analysis on such groups. For reference and additional background, see any of the textbook treatments in [35, 42, 49, 57, 60, 61, 69]. Definition 6.11 Let V be an almost-periodic sequence, .0 = orb(V ), . = hull(V ), and . : Z →  the natural homomorphism  : k → S k V .

.

(6.12)

Since

.Z ∼ Z with .χ (1) ∈ T), this in turn induces a = T (by identifying .χ ∈

:

→ T by duality. Thus homomorphism .

(χ ) = χ (SV ), 

.

. χ ∈

(6.13)

is an injective homomorphism from .

onto Since .0 is a dense subgroup of ., .

its image in .T, so we may identify . with a countable subgroup of .T. The pullback of this subgroup of .T to .R is called the frequency module of V , and is denoted by .M = M(V ). In particular, since .Z is the kernel of the canonical projection from .R to .T, one has .Z ⊆ M. Using the explicit form of . in (6.13), we see that .t ∈ M(V ) if and only if there

with .χ (SV ) = t (mod Z), which holds if and only if exists .χ ∈  χt (S k V ) = kt (mod Z)

.

(6.14)

extends to a continuous homomorphism .χt :  → T. Theorem 6.12 Suppose . is a compact topological group and .α ∈  generates a dense cyclic subgroup. With .

ϕα , .Rα , and .π defined as in Example 1.4, one has

)). A(, Rα ) = π −1 (

ϕα (

.

(6.15)

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In particular, if V is almost-periodic, . = hull(V ), and .T :  →  denotes the shift .[T ω](n) = ω(n + 1), then A(, T ) = M(V ).

.

(6.16)

Before proving the main theorem, it is instructive to try to understand these definitions when the underlying potential is periodic, quasi-periodic, or limitperiodic. We will show how one can characterize these various subclasses in terms of their frequency modules. To do this adequately, we need one more definition. Let us say that G has the divisor property if, for every pair .g, g  ∈ G, there exist    .n, n ∈ Z and an element .d ∈ G such that .nd = g and .n d = g . Lemma 6.13 Let V be almost-periodic, . = hull(V ), and .M = M(V ).

is finitely generated. (a) .M is finitely generated if and only if .

is a torsion group. (b) .M enjoys the divisor property if and only if . Proof The proof of (a) is left to the reader. For the second, first suppose that every

has finite order, and note that this implies .M ⊆ Q. Thus, given element of .    .r = p/q and .r = p /q in .M, use Bezout’s theorem to choose .n, m ∈ Z with     .d = npq + mp q, where d denotes the greatest common divisor of .p q and .pq . Naturally, nr + mr  =

.

d ∈M qq 

(6.17)

is a divisor of both r and .r  . Conversely, suppose .M has the divisor property. Since .1 ∈ M, one has .M ⊆ Q,

has finite order, concluding the proof which in turn implies that every element of . of part (b).   Theorem 6.14 Let V be almost-periodic and .M = M(V ). Then (a) V is periodic if and only if .M is discrete. (b) V is quasi-periodic if and only if .M is finitely generated. (c) V is limit-periodic if and only if .M enjoys the divisor property. Proof Put . = hull(V ). (a) If V is periodic, one can calculate directly that .M is the subgroup of .R generated by .1/p, where p is the minimal period of V . On the other hand, if .M is discrete,

must be finite. By Pontryagin duality, . is finite, so V is periodic. then .

is (b) By Lemma 6.13, it suffices to prove that V is quasi-periodic if and only if . finitely generated. By the classification of finitely generated abelian groups and

is finitely generated if and only if . is of the form .Tr ⊕ A Pontryagin duality, . for some finite abelian group A and some .r ≥ 0. By Proposition 6.9, .hull(V ) is of this form if and only if V is quasi-periodic.

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(c) By Proposition 6.7, V is limit-periodic if and only if . is totally disconnected.

is a torsion The reader can check that . is totally disconnected if and only if . group, so this statement follows from the second claim in Lemma 6.13.   Corollary 6.15 A sequence is periodic if and only if it is both quasi-periodic and limit-periodic. Proof A periodic sequence is trivially both quasi-periodic and limit-periodic. Conversely, if V is quasi-periodic and limit-periodic, then .M is finitely generated and has the divisor property. As this implies that .M is generated by a single element, it follows that V is periodic by Theorem 6.14.   We conclude by recovering the classical gap labelling theorem for almostperiodic potentials. The main observations are that the suspension .X = X(, Rα ) is itself a topological group and that continuous functions on X are homotopic to characters of X. Proof (Proof of Theorem 6.12) Let . be given and suppose .α ∈  generates a dense cyclic subgroup. We have already discussed . = Zp , so let us assume that . is infinite. Denote by .μ the normalized Haar measure on ., and consider the suspension .X = X(, T ). In X, notice that [ω, s] = [ω , s  ] ⇐⇒ (ω − ω , s − s  ) = (mα, −m) for some m ∈ Z.

.

(6.18)

Thus, X is the quotient of the topological group . × R by the closed subgroup {(mα, −m) : m ∈ Z}, so X is a topological group, and moreover the subgroup .S = {[0, t] : t ∈ R} is dense in X. The topology of X is metrizable, with a metric given by .



dX (x1 , x2 ) = inf d (ω1 , ω2 ) + |t1 − t2 | : ωj ∈ , tj ∈ R with [ωj , tj ] = xj .

.

We will also denote by d the usual metric on .T: dT (t, t  ) = dist(t − t  , Z).

.

(6.19)

Now, let .φ ∈ C(X, T) be given, and denote .ρ = Aμ (φ). Using uniform continuity of .φ, one can check that .rot(φ; x) = ρ for every .x ∈ X, in particular for6 .x = [0, 0]. Recall that .π : R → T denotes the canonical projection, choose a continuous # : R → R so that .φ([0, t]) = π(φ #(t)), and denote function .φ # =φ #(t) − ρt, G(t)

.

6 We

# G = π ◦ G.

are writing the abelian group . additively, so one should understand the zero in the first coordinate as the additive identity element of ..

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Notice that these definitions yield .

# #(t) − ρt φ G(t) = lim = ρ − ρ = 0. t→∞ t t→∞ t lim

(6.20)

). We will need the following claim. To conclude, we must show .π(ρ) ∈

ϕα ( Claim The map .S → T given by .[0, t] → π(ρt) is uniformly continuous.

 

Proof of Claim. Since .φ is continuous, it suffices to prove that .[0, t] → G(t) is uniformly continuous7 on S. To that end, let .0 < ε < 1/2 be given. Since .φ is continuous on the compact space X, it is uniformly continuous, and we may find .δ > 0 so that dT (φ(x), φ(x  )) < ε whenever dX (x, x  ) < δ.

.

Consequently, a typical lifting argument shows that + + #(t) − φ #(s)+ < ε whenever |t − s| < δ. . +φ

(6.21)

# Now, suppose that # is uniformly continuous, and so too is .G. Thus, .φ dX ([0, 0], [0, q]) < δ for some .q ∈ N and .0 < δ < 1 (note that this forces .d (qα, 0) < δ). There then exists an integer n such that + + #(t + q) − φ #(t) − n+ < ε . +φ .

for all .t ∈ R. Equivalently, # + q) − G(t) # − (n − qρ)| < ε |G(t

.

(6.22)

for all t. Next, let us show that .|n − qρ| > ε, produces a contradiction to (6.20). Concretely, if .n − qρ > ε, then # + q) − G(t) # ≥  := n − qρ − ε > 0 G(t

.

# for all t. However, this necessarily implies .lim inf G(t)/t ≥ /q > 0, contradicting (6.20); the case .n − qρ < −ε is similar. Consequently, one has .|n − qρ| ≤ ε and so # + q) − G(t)| # |G(t < 2ε

.

for all .t ∈ R. Putting everything together, dT (G(t), G(s)) < 3ε whenever dX ([0, t], [0, s]) < δ,

.

# since .d([α, t], [α, s]) can that this is stronger than just proving uniform continuity of .G, be small even if .s − t is not small.

7 Notice

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and hence .S # [0, t] → G(t) is uniformly continuous, which in turn implies that [0, t] → π(ρt) is uniformly continuous on S. .♦

.

By the claim, the map .ψ : S → T given by .ψ : [0, t] → π(ρt) is uniformly continuous, and hence extends to a continuous function .ψ : X → T. Restricting .ψ to the fiber {[ω, 0] : ω ∈ } ∼ = ,

.

via .χ (ω) = ψ([ω, 0]). One has we obtain an element of .

ϕα (χ ) = χ (α) = ψ([α, 0]) = ψ([0, 1]) = π(ρ),

.

) and completes the proof of .A(, Rα ) ⊆ π −1 (

)). which shows .π(ρ) ∈

ϕα ( ϕα (

)) be given. Then, there exists To see the reverse inclusion, let .ρ ∈ π −1 (

ϕα (

such that .χ (α) = π(ρ). Notice that .χ can be extended to an element .χ ∈ X .χ ∈  via χ ([ω, t]) = π(ρt) + χ (ω),

ω ∈ , t ∈ R.

.

)) ⊆ A(, Rα ). One can check that .Aμ (χ ) = ρ, and hence .π −1 (

ϕα (

(6.23)  

7 Subshift Potentials Our next class of examples will be those topological dynamical systems given by subshifts. The present section addresses the claims from Example 1.5.

7.1 Schwartzman Group Associated with a General Subshift Suppose .A is a finite set (called the alphabet) and give .AZ the product topology induced by equipping each factor with the discrete topology. A set . ⊆ AZ is called a subshift over .A if it is closed (thus compact) and invariant under the action of the shift .T : AZ → AZ given by .[T ω](n) = ω(n + 1). Abusing notation somewhat, we also write T for the restriction of T to . since this should not cause confusion. The topology on .AZ (hence on .) is metrizable, e.g., by 

d(ω, ω ) = 2− min{|n|:ω(n) =ω (n)} ,

.

ω = ω .

One may naturally ask how to construct continuous functions from .X = X(, T ) to .T whenever .(, T ) is a subshift, and moreover, how to construct a family that is large enough that it represents every homotopy class of maps .X → T. The idea is

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to start with a continuous integer-valued function, g, on the subshift and use those to construct functions on the suspension by winding around the circle .g(ω) times in passing from .[ω, 0] to .[ω, 1] = [T ω, 0]. More precisely, given .g ∈ C(, Z), define .g : X → T by g[ω, t] = tg(ω),

ω ∈ , t ∈ [0, 1].

.

(7.1)

Given a subshift . ⊆ AZ , recall that a cylinder set is a set of the form

 = ω ∈  : ωn+j = uj for all 1 ≤ j ≤ k ,

.

where .n ∈ Z and .u = u1 · · · uk ∈ A∗ , where .A∗ denotes the set of finite words over .A. Theorem 7.1 Let .(, T ) denote a subshift and let .X = X(, T ) denote its suspension. (a) For any .g ∈ C(, Z), .g defined in (7.1) belongs to .C(X, T). (b) .C (X, T) = {[g] : g ∈ C(, Z)}. That is, every .h ∈ C(X, T) is homotopic to .g for some .g ∈ C(, Z). (c) If .μ is a T -ergodic measure on ., then  A(, T , μ) =

.

g dμ : g ∈ C(, Z)

(7.2)

(d) If .μ is a T -ergodic measure on ., then .A is the .Z-module generated by S = {μ( ) :

.

⊆  is a cylinder set} .

Proof (a) Let .g ∈ C(, Z) be given. Let us first note that g[ω, 1] = 0 = g[T ω, 0],

.

so .g is well-defined. Since g is continuous and integer-valued, choose .δ > 0 such that .d(ω, ω ) < δ implies .g(ω) = g(ω ). Suppose one is given .ω, ω such that .d(ω, ω ) < δ, and denote .k = g(ω) = g(ω ). One then can note dT (g[ω, t], g[ω , t  ]) ≤ k|t − t  |,

.

t, t  ∈ [0, 1]

to see that .g is continuous. (b) Let .h : X → T be a given continuous function. Step 1 Using total disconnectedness of ., one can check that there exists .h1 ∈ C(X, T) homotopic to h such that .h1 [ω, 0] = 0 for all .ω ∈ .

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Step 2 The winding of .t → h1 [ω, t] is locally constant. More precisely, for #ω : R → R and define each .ω ∈ , lift .φω : t → h1 [ω, t] to .φ #ω (1) − φ #ω (0). g(ω) = φ

.

One has .g ∈ C(, Z). Indeed, if .ω and .ω are sufficiently close, then .d(φω (t), φω (t)) < 1/2 for all t, so one can choose lifts that preserve this property and hence .g(ω) = g(ω ). Thus, .g ∈ C(, Z), as claimed. Step 3 One can check that .h1 is homotopic to .g, ¯ which concludes the proof of (b). (c) Given .g ∈ C(, Z) and .ω ∈ , consider .φω (t) = g[ω, t] and the corresponding #ω . From the definition of .g, one has, for .μ-a.e. .ω, lift .φ  n−1 #ω (n) φ 1 j = lim . lim g(T ω) = g dμ. n→∞ n→∞ n n j =0

Thus, the conclusion of (c) immediately by noting that the previous calculation implies  Aμ (g) =

.

g dμ.

(d) Note that any integer combination of characteristic functions of cylinder sets lies in .C(, Z), so the .Z-module generated by S is contained in .A(, T , μ). For the other inclusion, consider .g :  → Z continuous. By continuity and compactness, f assumes finitely many values .s1 , . . . , sk and the sets .j = g −1 ({sj }), .1 ≤ j ≤ k are closed ( and open. Writing each .j as a disjoint union of cylinder sets, one sees that . g dμ is in the .Z-module generated by S.   Corollary 7.2 Let .(, T ) denote a strictly ergodic subshift. We have  A(, T ) =

.

g dμ : g ∈ C(, Z) ,

where .μ denotes the unique T -invariant measure on ..

7.2 Subshifts Generated by Substitutions The following result contains the statement from part (2) of Example 1.5.

(7.3)

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√ Theorem 7.3 Let .(, T ) denote the Fibonacci subshift. If .α = 12 ( 5 − 1) denotes the inverse of the golden mean, then A(, T ) = Z[α].

.

(7.4)

We will derive Theorem 7.3 from a more general result for primitive substitution subshifts, since the latter is obtained via a procedure that works in the same fashion for any subshift of that kind. This procedure is due to Bellissard, Bovier, and Ghez [13]. Definition 7.4 A substitution on the finite alphabet .A is given by a map .S : A → A∗ , where .A∗ denotes the set of finite words over .A. The substitution S can be extended by concatenation to .A∗ and .AN . A fixed point u of .S : AN → AN is called a substitution sequence; the associated subshift is given by  = u = {ω ∈ AZ : each finite subword of ω is a subword of u}.

.

In the Fibonacci case considered in Example 1.5 we have .A = {0, 1}, .S(0) = 01, S(1) = 0, and .u = 0100101001001 . . ..

.

Definition 7.5 The substitution matrix .M = MS associated with a substitution .S : A → A∗ , where .A = {a1 , . . . , a }, is given by the .× matrix .M = (mi,j )1≤i,j ≤ , where .mi,j is given by the number of times .ai occurs in .S(aj ). The substitution S is called primitive if its substitution matrix M is primitive, that is, for some .k ∈ N, all entries of .M k are strictly positive. The Fibonacci substitution is primitive, which follows from observing that M2 =

.

  21 . 11

(7.5)

If S is primitive and u is a substitution sequence, then the associated subshift . is known to be strictly ergodic [66]. In fact, even if .S : AN → AN does not have a fixed point, one can check that some power of this map does, and that power will be primitive as well and hence the associated subshift will be strictly ergodic. In addition, the resulting subshift is independent of the power and the fixed point one chooses and therefore only depends on S. We are thus in a setting covered by Corollary 7.2 and wish to determine .A(, T ) via (7.3). In order to do so, we need the following concepts. Applying the PerronFrobenius theorem to M we infer that M has a leading simple eigenvalue .θ > 0 (in the sense that it has multiplicity one and every other eigenvalue .λ of M obeys .|λ| < θ ), along with an eigenvector .vθ with strictly positive entries. We will normalize .vθ so that the sum of its entries equals one. The entries of this normalized vectors then correspond to the frequencies with which the symbols .aj appear in u (or any .ω ∈ ) [66].

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In order to study frequencies of subwords of u of length .m > 1, one can consider the set .Wm of these words and the derived substitution .Sm : Wm → W∗m , which is defined as follows. If .w = w1 . . . wm ∈ Wm and .S(w) = s1 s2 . . . s|S(w)| with .wj , sj ∈ A, then set Sm (w) = (s1 . . . sm )(s2 . . . sm+1 ) . . . (s|S(w1 )| . . . s|S(w1 )|+m−1 ).

.

(7.6)

It can be checked that .Sm is primitive as well, with the same leading eigenvalue .θ (m) and an m-dependent normalized corresponding eigenvector .vθ and, as before, the (m) frequency of a subword .w ∈ Wm is given by the entry of .vθ that corresponds to w [66]. We can now state the general result for subshifts arising from primitive substitutions, which is due to Bellissard, Bovier, and Ghez [13]. Theorem 7.6 If S is a primitive substitution, then with the notation from above, the Schwartzman group .A(, T ) is contained in the .Z[θ −1 ]–module generated by the entries of .vθ and .vθ(2) . Let us first show how Theorem 7.3 follows quickly from Theorem 7.6. Proof (Proof of Theorem 7.3) As pointed out above, in the case of the Fibonacci substitution, the substitution matrix is given by (7.5). A quick calculation then shows that the leading eigenvalue is given by the golden mean θ=

.

√ 1+ 5 = α −1 2

and the corresponding normalized eigenvector is given by    θ −1 θ −1 = .vθ = . θ −2 2−θ 

Another calculation shows that the substitution matrix associated with .S2 is given by ⎡

⎤ 001 .M2 = ⎣1 1 0⎦ , 110 which has the same leading eigenvalue .θ and associated normalized eigenvector ⎡

(2)

vθ

.

⎤ ⎡ ⎤ θ −3 2θ − 3 = ⎣θ −2 ⎦ = ⎣ 2 − θ ⎦ . 2−θ θ −2

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393

Using one more time that .θ −1 = θ − 1 = α, we see that the .Z[θ −1 ]–module generated by the entries of .vθ and .vθ(2) is equal to .Z[α]. Thus, Theorem 7.3 indeed follows from Theorem 7.6.   Proof (Proof of Theorem ( 7.6) Recall that due to (7.3) we need to consider the collection of numbers . f dμ, where .μ denotes the unique T -invariant measure on . and f runs through .C(, Z). Due to the nature of the topology of . and the fact that each f in question takes values in the integers, we can restrict our attention to the consideration of the .μmeasure of cylinder sets; compare Theorem 7.1.(d). This in turn shows that we need to understand the frequencies with which finite subwords of u occur in u (or any .ω ∈  due to unique ergodicity). As pointed out above, the frequency of a subword .w ∈ Wm is given by the entry of .vθ(m) that corresponds to w. Thus, it remains to relate the entries of .vθ(m) for .m ≥ 3 to the entries of .vθ and .vθ(2) . A useful relation of the desired nature is obtained by observing from (7.6) that if p p is large enough, then .Sm (w) only depends on the first two symbols of w, that is, on .w1 , w2 if .w = w1 . . . wm with .wj ∈ A. Indeed, writing S p (w) = S p (w1 ) . . . S p (wm ) = s1 . . . s|S p (w)|

.

and using that .(Sm )p = (S p )m , we find p

Sm (w) = (s1 . . . sm )(s2 . . . sm+1 ) . . . (s|S p (w1 )| . . . s|S p (w1 )|+m−1 ).

.

Thus, if |S p (w1 )| + |S p (w2 )| ≥ |S p (w1 )| + m − 1,

.

then .S p (w) depends only on .w1 and .w2 . By primitivity of S (and hence M), this inequality holds for large enough p, uniformly in .w2 . p Fix such a value of p. As a consequence of this choice, in the matrix .Mm (where .Mm denotes the substitution matrix associated with the substitution .Sm ), all columns with labels w having the same prefix .w1 w2 are equal. Thus, we may as well collapse them into one column. This results in the matrix .Mp,m,2 , where the columns are now labeled by the elements .w1 w2 of .W2 . Denoting the process of passing to the prefix of length 2 by .π2,m : Wm → W2 , .w = w1 . . . wm → w1 w2 and the associated substitution matrix by .P2,m , we therefore have p

Mm = Mp,m,2 P2,m .

.

(7.7)

On the other hand, a moment’s thought yields the additional relations p

M2 = P2,m Mp,m,2

.

(7.8)

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and Mm Mp,m,2 = Mp+1,m,2 = Mp,m,2 M2 .

(7.9)

.

(m)

We are now able to address our actual goal: expressing the entries of .vθ (2) terms of those of .vθ and .vθ . Consider the vector (2)

v˜ (m) := Mp,m,2 vθ .

.

in

(7.10)

By (7.9) we have (2)

(2)

(2)

Mm v˜ (m) = Mm Mp,m,2 vθ = Mp,m,2 M2 vθ = θ Mp,m,2 vθ = θ v˜ (m) .

.

As .θ is the leading eigenvalue of .Mm and it is simple, it follows that .v˜ (m) must be a multiple of .vθ(m) . To determine the multiplier, let us denote by .1k the vector of all ones in .R#Wk , so that  (m) . v˜w = 1m , v˜ (m)  w∈Wm

= 12 P2,m , Mp,m,2 vθ(2)  (2)

= 12 , P2,m Mp,m,2 vθ  p (2)

= 12 , M2 vθ  (2)

= 12 , θ p vθ  = θ p, where we used (7.8) in the fourth step and the normalization of .vθ(2) in the sixth step. As .vθ(m) is normalized as well, we see that the multiplier is .θ p , that is, (m)

v˜ (m) = θ p vθ .

.

(7.11)

It now follows from (7.10) and (7.11) that for every .m ≥ 3, each of the entries of vθ(m) is given by an integral linear combination of the entries of .vθ(2) times a negative power of .θ , concluding the proof.  

.

Two other prominent examples of substitutions on a two-symbol alphabet are given by the Thue–Morse and period-doubling substitutions. The Thue–Morse substitution acts on .A = {0, 1} via .0 → 01 and .1 → 10.

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395

Theorem 7.7 Let .(, T ) denote the Thue–Morse subshift. One has  A(, T ) =

.

k : k ∈ Z, n ∈ Z + . 3 · 2n

(7.12)

Proof The substitution matrix, leading eigenvalue, and normalized eigenvector for the Thue–Morse substitution are given by M=

.

  11 , 11

vθ =

θ = 2,

  1/2 . 1/2

(7.13)

Another quick calculation shows that the substitution matrix associated with .S2 and its leading eigenvector are given by ⎡

⎤ 0010 ⎢1 1 0 1⎥ ⎥ .M2 = ⎢ ⎣1 0 1 1⎦ ,

⎡

(2)

vθ

⎤ 1/6 ⎢1/3⎥ ⎥ =⎢ ⎣1/3⎦ .

0100

(7.14)

1/6

In view of Theorem 7.6, .A(, T ) is the .Z[1/2]-module generated by {1/2, 1/3, 1/6}, which the reader can readily check is equivalent to the set in (7.12).  

.

The period-doubling substitution acts on .A = {0, 1} via .0 → 01 and .1 → 00. Theorem 7.8 Let .(, T ) denote the period-doubling subshift. One has  A(, T ) =

.

k : k ∈ Z, n ∈ Z+ . 3 · 2n

(7.15)

Proof The substitution matrix, leading eigenvalue, and normalized eigenvector of the period-doubling substitution are given by   12 .M = , 10

θ = 2,

  2/3 vθ = . 1/3

(7.16)

Another quick calculation shows that the substitution matrix and leading eigenvector associated with .S2 are given by ⎡

⎤ 002 .M2 = ⎣1 1 0⎦ , 110 The result follows again from Theorem 7.6.

⎡

vθ(2)

⎤ 1/3 = ⎣1/3⎦ . 1/3

(7.17)

 

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7.3 Full Shift over a Finite Alphabet Let us consider the random case in which . = AZ for some finite set .A, .μ = μZ0 for a probability measure .μ0 on .A, and T again is the shift. Without loss, take .A = {1, 2, . . . , m}, and abbreviate .wj = μ({j }) for .1 ≤ j ≤ m. Theorem 7.9 For the random setup here, A(, T , μ) = Z[w1 , w2 , . . . , wm ].

.

In particular, in the case of equal weighting .w1 = · · · = wm = 1/m, one has A(, T , μ) = Z[1/m].

.

Proof Consider .u = a1 . . . an with .aj ∈ A for each j . The function χu (ω) =

3 1

.

0

ω|[1,n] = u otherwise

is a locally constant (hence continuous) integer-valued function. Thus,  .

χu dμ = wa1 wa2 · · · wan ∈ A(, T , μ).

Since .A is a subgroup of .R, the inclusion “.⊇” is proved. Conversely, assume given .f ∈ C(, Z). By compactness, we can write f as a finite sum  .f = kχk , k = f −1 ({k}). By writing each .k as a disjoint union of cylinder sets, we observe the inclusion   “.⊆”.

8 Potentials Generated by Affine Torus Homeomorphisms In this section, we will consider homeomorphisms of the torus induced by suitable affine maps on .Rd and address the claims from Example 1.6. Namely, given .d ∈ N, denote . = Td . Given .A ∈ SL(d, Z), and .b ∈ Td , we recall .T = TA,b :  →  is the homeomorphism .Td → Td given by .TA,b ω = Aω + b. This class includes

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397

several important examples as special cases, among which we single out two (both in dimension .d = 2). The cat map is the homeomorphism .Tcat of .T2 given by A=

.

  21 , 11

b = 0.

Given .α ∈ T \ Q, the associated skew-shift is defined by T (ω1 , ω2 ) = (ω1 + α, ω1 + ω2 )

.

(8.1)

which corresponds to the choices   10 .A = , 11

  α b= . 0

(8.2)

For the affine transformations discussed in this section, the image of the Schwartzman homomorphism may be computed as follows: Theorem 8.1 Let .d ∈ N, .A ∈ SL(d, Z), and .b ∈ Td be given, and let .μ be a .TA,b -ergodic measure. In this case,   A(Td , TA,b , μ) = k, b + n : n ∈ Z and k ∈ Zd ∩ Ker(I − A∗ ) .

.

(8.3)

Here, let us point out that Theorem 8.1 computes .A(Td , TA,b , μ) for any ergodic measure, .μ. However, in order for Theorem 1.1 to guarantee that .A(Td , TA,b , μ) gives the set of labels, one needs the additional assumption .supp μ = Td . Let us explain how the labels in Theorem 8.1 arise. It is instructive to consider first the basic example .A = I so that T is simply the identity map. In this case, one has .X(, T ) = Td+1 . More generally, if A fixes a subspace .V ⊆ Rd of dimension k, this projects to a set .V¯ ⊆ Td that is homeomorphic to .Tk and hence produces a copy of .Tk+1 inside X. This suggests a link between subspaces fixed by A and generators of homotopy classes. The following lemma gives the precise result. Lemma 8.2 Let .d ∈ N and .A ∈ SL(d, Z) be given. For each .b ∈ Td , define d d ∗ .Xb = X(, TA,b ). For each .k ∈ K := Z ∩ Ker(I − A ), .n ∈ Z, and .b ∈ T , define .gk,n,b : Xb → T by gk,n,b [ω, t] = k, ω + nt + tk, b.

.

(8.4)

(a) For each .k ∈ K, .n ∈ Z, and .b ∈ Td , .gk,n,b is a well-defined continuous map .Xb → T. (b) One has .C (X0 , T) = {[gk,n,0 ] : k ∈ K, n ∈ Z}; that is, every .g ∈ C(X0 , T) is homotopic to some .gk,n,0 . (c) For every .b ∈ Td , .Xb is homeomorphic to .X0 .

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(d) For every .b ∈ Td , one has .C (Xb , T) = {[gk,n,b ] : k ∈ K, n ∈ Z}. Proof (a) Since .k ∈ Ker(I − A∗ ), one has .A∗ k = k. Using this, one checks directly that gk,n,b [TA,b ω, t] = k, Aω + b + nt + tk, b

.

= A∗ k, ω + nt + (t + 1)k, b = k, ω + n(t + 1) + (t + 1)k, b = gk,n,b [ω, t + 1], which implies that .gk,n is well-defined and continuous. (b) Let .g ∈ C(X, T) be given. For .m ∈ Zd , let .χm (ω) = m, ω. Step 1 Write .t for the fiber .{[ω, t] : ω ∈ } in X. Since every continuous map .Td → T is homotopic to .χk : ω → k, ω for some .k ∈ Zd , there exists d .k ∈ Z such that .g|0 is homotopic to .χk . By continuity, notice furthermore that .g|t is also homotopic to .χk for every t.   Step 2 Notice that .m, Aω = A∗ m, ω for .m ∈ Zd and .ω ∈ Td . In particular, since .g|0 is homotopic to .g|1 , we deduce .A∗ k = k with k from the previous step, and consequently, .k ∈ K. Step 3 Considering the circle8 .{[0, t] : t ∈ R}, there exists .n ∈ Z such that .t → g[0, t] is homotopic to .t → nt. Step 4 With k and n as in Steps 1 and 3, there is a map .h ∈ C(X, T) homotopic to g with .h[x, 0] = k, x and .h[0, t] = nt. As mentioned before, note that .h|t is homotopic to .k, · for every t. Step 5 Consider .h0 = h − gk,n with the k and n from Steps 1 and 3. By previous steps, .h0 vanishes on .{[ω, 0] : ω ∈ } ∪ {[0, t] : t ∈ R}. The reader can check that .h0 is then nullhomotopic. Since .h0 is nullhomotopic, .[g] = [gk,n ] and the proof is done. (c) View .Xb as .Td × [0, 1]/ ∼ where .(ω, 1) ∼ (TA,b ω, 0). The map φb [ω, t] = [ω + tA−1 b, t]

.

establishes the desired homeomorphism .φb : Xb → X0 . Indeed, since φb [ω, 1] = [ω + A−1 b, 1] = [A(ω + A−1 b), 0]

.

= [Aω + b, 0] = φb [Aω + b, 0] = φb [TA,b ω, 0],

8 This

uses .A0 = 0.

(8.5)

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we see that .φb is well-defined and continuous. The reader can check that the inverse is given by sending .[ω, t] ∈ X0 to .[ω − tA−1 b, t] ∈ Xb and is continuous. (d) This follows from (b) by composing .gk,n,0 with the homeomorphism from (c).   Remark 8.3 With some additional tools from algebraic topology, one can view Lemma 8.2 as an explicit realization of the following diagram chase: First, since .T is a .K(Z, 1)-space, .C (X, T) can be identified with .H 1 (X, Z), the first cohomology group (cf. [41]). Since X is an orientable .d + 1-dimensional manifold, this can further be identified with .Hd (X, Z), the d-th homology group via Poincaré duality. By using a standard exact sequence for mapping tori (cf. [41, Example 2.48]), one arrives at the exact sequence 0 −→ Hd (Td ) −→ Hd (X) −→ Ker(I − A) −→ 0,

.

(8.6)

from which one can realize .Hd (X) is free abelian on .1 + dim(Ker(I − A)) generators. The functions .gk,n,b provide explicit representatives of homotopy classes in .C (X, T) and hence (by chasing the diagram backwards) also give representatives of the cohomology classes in .H 1 (X). Lemma 8.4 Let .d ∈ N, .A ∈ SL(d, Z), and .b ∈ Td be given. For all .n ∈ Z, d d ∗ .k ∈ Z ∩ Ker(I − A ), and .x ∈ X(T , TA,b ), one has .rot(gk,n,b ; x) = n + k, b. In particular, if .μ is any .TA,b -ergodic measure on .Td , then Aμ ([gk,n,b ]) = n + k, b.

.

(8.7)

for every .n ∈ Z and .k ∈ Zd ∩ Ker(I − A∗ ). Proof For the first claim, it suffices to establish .rot(gk,n,b ; [ω, 0]) = n + k, b for every .ω ∈ Td . Given .ω ∈ Td , the map .φω : R → T sending .t ∈ R to .gk,n,b [ω, t] lifts to the map #ω : t → k, ω + nt + tk, b ∈ R. φ

.

One then sees immediately that .

lim

t→∞

#ω (t) φ = n + k, b. t

Since this holds for every .ω ∈ Td , the first statement of the result follows. The   second statement is an immediate consequence. We can now prove the main result. Proof (Proof of Theorem 8.1) This is a consequence of Lemmas 8.2 and 8.4.

 

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Theorem 8.5 Let . = T2 , .α ∈ T \ Q be given, let .T : T2 → T2 denote the associated skew-shift given by T (ω1 , ω2 ) = (ω1 + α, ω1 + ω2 ),

.

(8.8)

and let .μ be Lebesgue measure. One has A(, T ) = Z + αZ = {n + mα : n, m ∈ Z}.

.

(8.9)

Proof Since the skew-shift is strictly ergodic with Lebesgue measure supplying the unique invariant measure [37], the result follows from Theorem 8.1.   Let us point out that the skew-shift is known to be uniquely ergodic (with Lebesgue measure the unique preserve measure), justifying the suppression of invariant measure from the notation. Theorem 8.6 Let . = Td , suppose .A ∈ SL(d, Z) is such that no eigenvalue of A is a root of unity, and let .μ denote Lebesgue measure. One has A(, TA,0 , μ) = Z.

.

(8.10)

Consequently, for every .f ∈ C(, R), .μ,f is an interval. Proof The assumption on the eigenvalues of A implies that .(, TA,0 , μ) is ergodic [79, Corollary 1.10.1]. Thus, the first statement follows from Theorem 8.1. The second statement follows from Corollary 1.2.   In particular, the reader can readily check that the eigenvalues of   21 .A = 11 √ are . 12 (3 ± 5), neither of which is a root of unity, and hence Theorem 8.6 applies to ergodic operators generated by the cat map. Remark 8.7 There is a subtle point here: the ergodic measure must have full support in order for the gap labelling theorem to be applicable. Concretely, the reader may note that any hyperbolic toral automorphism has many invariant measures. Notably, any periodic point of such an automorphism may be used to generate an ergodic measure with finite support, which then leads to periodic potentials, which can in turn be chosen in such a way as to produce a spectral gap.9 On one hand, these measures generated by periodic orbits do not have full support, so the theorem asserting gaplessness does not apply to them. On the other

9 Indeed,

the spectrum of any Schrödinger operator in .2 (Z) with a non-constant periodic potential has at least one gap [43].

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hand, the mechanism at work here is the following: for the gap labelling theorem, one only works on the set of energies at which the associated cocycle is uniformly hyperbolic, which in particular means that the monodromy matrices associated with every periodic orbit must simultaneously be hyperbolic. On a single periodic orbit, we can have hyperbolicity of the associated monodromy without forcing uniform hyperbolicity of the whole cocycle. Acknowledgments We are grateful to Michael Baake, Franz Gähler, Svetlana Jitomirskaya, Johannes Kellendonk, Andy Putman, Lorenzo Sadun, and Hiro Lee Tanaka for helpful conversations and comments. We also want to thank the American Institute of Mathematics for hospitality and support during a January 2022 visit, during which part of this work was completed. We also gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University at which some of this work was performed.

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Degenerate Elliptic Operators and Kato’s Inequality Tan Duc Do and A. F. M. ter Elst

Dedicated to the memory of Sergey Nikolaevich Naboko (1950–2020)

2,∞ (Rd , R), .c ∈ W 1,∞ (Rd , R) and Abstract For all .k, l ∈ {1, . . . , d} let .c kl ∈ W k d d d .c0 ∈ L∞ (R , R). We assume that .Re k,l=1 ckl (x) ξk ξl ≥ 0 for all .x ∈ R and d .ξ ∈ C . We consider the divergence form operator

Bp u = −

d 

.

∂k (ckl ∂l u) +

k,l=1

d 

ck ∂k u + c0 u

k=1

with the maximal domain in .Lp (Rd ) and prove that .−Bp is the generator of a .C0 semigroup for all .p ∈ [1, ∞). We show that the space .Cc∞ (Rd ) of test functions is a core for .Bp for all .p ∈ (1, ∞). Moreover we prove a Kato inequality in the degenerate setting on a domain . for functions in .L1,loc (). Keywords Degenerate elliptic operator · Core property · Kato’s inequality

1 Introduction Kato [10] proved that the inequality Re(sgn u u) ≤ |u|

.

T. D. Do Vietnamese-German University, Binh Duong Campus, Binh Duong, Vietnam e-mail: [email protected] A. F. M. ter Elst () Department of Mathematics, University of Auckland, Auckland, New Zealand e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_15

405

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is valid in the distributional sense for all .u ∈ L1,loc (Rd ) such that .u ∈ L1,loc (Rd ). Kato first used the inequality in proving the self-adjointness of the Schrödinger operator .− + V in the same paper [10], and then provided an .Lp -analysis of .− + V in a subsequent paper [11]. Kato’s inequality can be generalized to a slightly different inequality to characterize generators of positive .C0 -semigroups (cf. [17], [13] and [1]), as well as the sub-Markovian property of a .C0 -semigroup [2]. A parabolic version of Kato’s inequality has been proved in [4]. In this paper we are interested in extending Kato’s inequality to a degenerate setting. Specifically we aim to show that Kato’s inequality remains valid for secondorder differential operators whose matrix of principal coefficients may not be symmetric and its symmetric part is merely positive semidefinite. Lower-order terms are also allowed in the expression of the degenerate operators. The study of degenerate elliptic operators originated from stochastic theory as early as in the 1950’s with the work of Fichera [9]. Other early contributions include [14], [16], [5], [19] and the references therein. Beside establishing a Kato inequality, we also present a core property and .C0 -semigroup generation of the degenerate elliptic operators. We prove that a degenerate elliptic operator generates a .C0 -semigroup on d .L1 (R ). This extends previous work in [8], [6] and [7]. The .L1 -result is the main tool to prove the Kato inequality for degenerate elliptic operators. Our results extend and are in the spirit of [5] and [19]. In order to formulate the main theorems of this paper we have to introduce some notation. For all .k, l ∈ {1, . . . , d} let .ckl ∈ W 2,∞ (Rd , R), .ck ∈ W 1,∞ (Rd , R) and d .c0 ∈ L∞ (R , R), where .d ∈ N. We assume that d 

Re

.

ckl (x) ξk ξl ≥ 0

k,l=1

for all .x ∈ Rd and .ξ ∈ Cd . So if .C(x) is the matrix with coefficients .ckl (x) for all .x ∈ Rd , then we assume that .C(x) is accretive on .Cd for all .x ∈ Rd . Define  d d .A : L1,loc (R ) → D (R ) by Au = −

d 

.

∂k ckl ∂l u +

k,l=1

d 

ck ∂k u + c0 u.

k=1

For all .p ∈ [1, ∞] define the maximal operator .Bp in .Lp (Rd ) by D(Bp ) = {u ∈ Lp (Rd ) : Au ∈ Lp (Rd )}

.

and .Bp u = Au for all .u ∈ D(Bp ). Clearly .W 2,p (Rd ) ⊂ D(Bp ) and Bp u = −

d 

.

k,l=1

∂k ckl ∂l u +

d  k=1

ck ∂u + c0 u

(1)

Degenerate Elliptic Operators and Kato’s Inequality

407

for all .u ∈ W 2,p (Rd ). So .Bp is densely defined for all .p ∈ [1, ∞) and .D(B∞ ) is weakly.∗ dense in .L∞ (Rd ). Also .Bp is closed if .p ∈ [1, ∞) and .B∞ is closed in d ∗ d .(L∞ (R ), w ). For all .p ∈ [1, ∞] define the minimal operator .Hp in .Lp (R ) by ∞ d .D(Hp ) = Cc (R ) and Hp u = −

d 

∂k ckl ∂l u +

.

k,l=1

d 

ck ∂k u + c0 u.

(2)

k=1

Clearly .Hp ⊂ Bp and hence .Hp is closable if .p ∈ [1, ∞) and .H∞ is closable with respect to the weak.∗ -topology. We denote by .Hp the closure of .Hp , where we take the norm-topology on .Lp (Rd ) if .p ∈ [1, ∞) and the weak.∗ -topology if .p = ∞. Throughout this paper we denote the constant ω = c0 ∞ +

d 

.

∂k ck ∞ .

(3)

k=1

Our first main theorem is as follows. Theorem 1.1 For all .k, l ∈ {1, . . . , d} let .ckl ∈ W 2,∞ (Rd , R), .ck ∈ W 1,∞ (Rd , R) and .c0 ∈ L∞ (Rd , R), where .d ∈ N. We assume that Re

d 

.

ckl (x) ξk ξl ≥ 0

k,l=1

for all .x ∈ Rd and .ξ ∈ Cd . Define the maximal operator .Bp and minimal operator .Hp as above for all .p ∈ [1, ∞]. Then one has the following. (a) For all .p ∈ [1, ∞) the operator .Bp + ω I is m-accretive. (b) For all .p ∈ (1, ∞) the space .Cc∞ (Rd ) is a core for .Bp , that is .Hp = Bp . For all .p ∈ [1, ∞) let .S (p) be the .C0 -semigroup generated by .−Bp . (c) The semigroups .S (p) are consistent for all .p ∈ [1, ∞) and extend consistently to a weak.∗ -continuous semigroup .S (∞) on .L∞ (Rd ). (d) The semigroup .S (p) is positivity preserving for all .p ∈ [1, ∞]. (e) The generator of .S (∞) is .−H∞ . In [15] Theorem 5.2 Ouhabaz proved that .B2 +ω I is m-accretive in .L2 (Rd ) with core .Cc∞ (Rd ). Moreover, in Theorem 5.6 he proved that the semigroup generated by .−B2 extends consistently to a quasi-contraction semigroup on .Lp (Rd ) for all .p ∈ [1, ∞], which is a .C0 -semigroup if .p ∈ [1, ∞). In Theorem 1.1 we obtain a new proof for [15] Theorem 5.6, we characterise the generator and show that it is equal to .−Bp if .p ∈ [1, ∞). Moreover, we obtain a weakly.∗ -continuous semigroup on .L∞ (Rd ) and show that for all .p ∈ (1, ∞] the generator of the semigroup on d .Lp (R ) is equal to .−Hp .

408

T. D. Do and A. F. M. ter Elst

Wong-Dzung [19] proved a version of Theorem 1.1 for regular .p ∈ (1, ∞) if ckl ∈ C 2 (Rd , R) with bounded second-order derivatives, .ck ∈ C 1 (Rd , R) with bounded first-order derivatives and .c0 ∈ L∞ (Rd , R). The second main result of this paper is the following Kato inequality for degenerate operators on an open subset . ⊂ Rd .

.

2,∞ Theorem 1.2 Let . ⊂ Rd be open. For all .k, l ∈ {1, . . . , d} let .ckl ∈ Wloc (, R), 1,∞ .ck ∈ W (, R) and .c0 ∈ L∞,loc (, R), where .d ∈ N. We assume that loc d 

Re

.

ckl (x) ξk ξl ≥ 0

k,l=1

for all .x ∈  and .ξ ∈ Cd . Define .A : L1,loc () → D () by Au = −

d 

.

∂k ckl ∂l u +

k,l=1

d 

ck ∂k u + c0 u.

k=1

Let .u ∈ L1,loc () and suppose that .Au ∈ L1,loc (). Then Re(sgn u Au) ≥ A|u|

(4)

.

in .D (). Note that (4) means that ϕ Re(sgn u Au) ≥

.









|u| 

−

d  k,l=1

∂k clk ∂l ϕ −

d 

 ∂k (ck ϕ) + c0 ϕ

k=1

for all .ϕ ∈ Cc∞ () with .ϕ ≥ 0. In Devinatz [5] Theorem 1.2 was proved with .L1,loc () replaced by .Lp,loc () with .p ∈ (1, ∞), under the assumption that .ckl ∈ C 2 (), .ck ∈ C 1 () and .c0 ∈ C(). As an application we can add a positive potential. Theorem 1.3 Adopt the notation and assumptions of Theorem 1.1. Let .p ∈ (1, ∞) and .V ∈ Lp,loc (Rd ) with .V ≥ 0. Then .−(Hp + V ) is closable and the closure generates a .C0 -semigroup on .Lp (Rd ). The structure of this paper is as follows. We prove Theorem 1.1 in a series of statements in Sect. 4. The proof of Theorem 1.1 requires a lot of preparation involving accretivity, energy estimates and a range condition, which we prove in Sect. 2. We also need some density results, which we prove in Sect. 3. In Sect. 5 we prove Theorem 1.2 using Theorem 1.1 and techniques of [13] and [5]. Then Theorem 1.3 is proved using arguments of [11].

Degenerate Elliptic Operators and Kato’s Inequality

409

2 Maximal and Minimal Operators Throughout this section we adopt the notation and assumptions of Theorem 1.1. Define the maximal operator .Bp and minimal operator .Hp as in (1) and (2). Further let .ω ≥ 0 be as in (3). Formally the adjoint of .B2 is of the same type of .B2 with .ck replaced by .−ck for all .k ∈ {1, . . . , d} and .c0 replaced by .c0 − dk=1 ∂k ck . Hence we define the operator #  d d .A : L1,loc (R ) → D (R ) by A# u = −

d 

.

∂k clk ∂l u −

k,l=1

d 

d    ck ∂k u + c0 − ∂k ck u.

k=1

k=1

Also for all .p ∈ [1, ∞] define the maximal operator .Bp# in .Lp (Rd ) by D(Bp# ) = {u ∈ Lp (Rd ) : A# u ∈ Lp (Rd )}

.

and .Bp# u = A# u for all .u ∈ D(Bp ). We define similarly .Hp# .  p We write .|∇u|2 = dj =1 |∂j u|2 and . ∇u p = Rd (|∇u|2 )p/2 for all .p ∈ [1, ∞) and .u ∈ W 1,p (Rd ). We also write .C(x) = (ckl (x))k,l∈{1,...,d} for all .x ∈ Rd and C is the corresponding matrix valued function on .Rd . In the proof of next lemma we do not need the condition .ckl ∈ W 2,∞ (Rd ), but merely .ckl ∈ W 1,∞ (Rd ). Lemma 2.1 Let .p ∈ (1, ∞). Then p

Re Bp u, 1[u =0] |u|p−2 uLp ×Lq ≥ −ω u p

.

for all .u ∈ W 2,p (Rd ), where q is the dual exponent of p. Proof First suppose that u has compact support. Let .δ > 0 if .p ∈ (1, 2) and .δ = 0 p−2 if .p ∈ [2, ∞). Define .v = (δ + |u|2 ) 2 u. We use that .∂k u = 0 almost everywhere on .[u = 0] by Sard’s theorem. Then v is weakly differentiable with weak derivative ∂k v = (p − 2) 1[u =0] (δ + |u|2 )

.

p−4 2

u Re(u ∂k u) + (δ + |u|2 )

for all .k ∈ {1, . . . , d}. Therefore .v ∈ W 1,q (Rd ) and −Re

d  

.

d k,l=1 R

(∂k ckl ∂l u) v

= Re

d   d k,l=1 R

ckl (∂l u) ∂k v

p−2 2

∂k u ∈ Lq (Rd )

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T. D. Do and A. F. M. ter Elst

= (p − 2)

d  

ckl (δ + |u|2 )

k,l=1 [u =0]

+ Re

d   d k,l=1 R

p−4 2

ckl (δ + |u|2 )

Re(u ∂l u) Re(u ∂k u)

p−2 2

(∂l u) ∂k u.

Next let .R > 0 be such that .supp u ⊂ BR (0). The divergence theorem gives Re

d  

.

d k=1 R

ck (∂k u) v

=

d   Rd

k=1

ck (δ + |u|2 )

p−2 2

Re(u ∂k u)

d  1 = ck ∂k ((δ + |u|2 )p/2 ) p BR (0) k=1

=−

d  d  1 1 (∂k ck ) (δ + |u|2 )p/2 + νk ck δ p/2 p p BR (0) ∂BR (0) k=1

≥ −

d 

k=1

 ∂k ck ∞

k=1

+

(δ + |u|2 )p/2 BR (0)

d  1 νk ck δ p/2 , p ∂BR (0) k=1

where .(ν1 , . . . , νd ) is the outer normal derivative on the ball. Finally,  Re



.

Rd

c0 u v ≥ − c0 ∞

|u|2 (δ + |u|2 )

p−2 2

.

BR (0)

Combining the estimates and taking the limit .δ ↓ 0 if .p ∈ (1, 2) gives Re Bp u, 1[u =0] |u|p−2 uLp ×Lq

.

≥ (p − 2)

d   k,l=1 [u =0]

+ Re

ckl |u|p−2 Re(sgn u ∂l u) Re(sgn u ∂k u)

d   k,l=1 [u =0]

p

ckl |u|p−2 (∂l u) ∂k u − ω u p

Degenerate Elliptic Operators and Kato’s Inequality

= (p − 1)

d   k,l=1 [u =0]

+

411

ckl |u|p−2 Re(sgn u ∂l u) Re(sgn u ∂k u)

d  

p

k,l=1 [u =0]

ckl |u|p−2 Im(sgn u ∂l u) Im(sgn u ∂k u) − ω u p

p

≥ −ω u p . This proves the lemma if u has compact support. Now we consider the general case. Let .χ ∈ Cc∞ (Rd ) be such that .0 ≤ χ ≤ 1 and .χ |B1 (0) = 1. For all .n ∈ N define .χn ∈ Cc∞ (Rd ) by .χn (x) = χ (n−1 x). Let 2,p (Rd ). Let .n ∈ N. Then .χ u ∈ W 2,p (Rd ) with compact support. Then the .u ∈ W n above gives p−1

Re Bp (χn u), χn

.

p

|u|p−1 sgn uLp ×Lq ≥ −ω χn u p .

Finally take the limit .n → ∞.

 

Next we shall prove an Ole˘ınik inequality. To this end, we need the following lemma. Note that since .ckl ∈ W 2,∞ (Rd ), one can rewrite Au = −

d 

.

k,l=1

  1 1 (∂l clk −∂l ckl ) ∂k u+c0 u ck − ∂k (ckl +clk ) ∂l u+ 2 2 d

d

k=1

l=1

(5)

 for all .u ∈ L1,loc (Rd ). Moreover, .ck − dl=1 12 (∂l clk − ∂l ckl ) ∈ W 1,∞ (Rd ) for all .k ∈ {1, . . . , d}. Hence the symmetry condition .ckl = clk for all .k, l ∈ {1, . . . , d} does not give a restriction in the next lemmas. Lemma 2.2 Suppose .ckl = clk for all .k, l ∈ {1, . . . , d}. Then    | (∂j C)ζ, τ |2 ≤ Cζ, ζ  + Cτ, τ  ζ 2 + τ 2 ∂j2 C L∞ (Rd ,L(Cd ))

.

for all .j ∈ {1, . . . , d} and .ζ, τ ∈ Cd . Proof Let .f ∈ W 2,∞ (Rd ) and suppose that .f ≥ 0. Let .(ρn )n∈N be a mollifier. Then .ρn ∗ f ∈ C ∞ (Rd ) for all .n ∈ N. Let .x ∈ Rd , .h ∈ R and .n ∈ N. Then 0 ≤ (ρn ∗ f )(x + h ej )

.



= (ρn ∗ f )(x) + h (∂j (ρn ∗ f ))(x) + 0

≤ (ρn ∗ f )(x) + h (∂j (ρn ∗ f ))(x) +

h

(∂j2 (ρn ∗ f ))(x + t ej ) (h − t) dt

h2 2

∂ (ρn ∗ f ) ∞ . 2 j

412

T. D. Do and A. F. M. ter Elst

Now take the limit .n → ∞. Then 0 ≤ f (x) + h (∂j f )(x) +

.

h2 2

∂ f ∞ . 2 j

Minimising over h gives |(∂j f )(x)|2 ≤ 2f (x) ∂j2 f ∞ .

.

Next let .ζ, τ ∈ Cd and .α ∈ R. Then Reeiα Cζ, τ  ≤ | Cζ, τ | ≤ Cζ, ζ 1/2 Cτ, τ 1/2 ≤

.

 1 Cζ, ζ  + Cτ, τ  . 2

Define .f = Cζ, ζ + Cτ, τ −2Reeiα Cζ, τ . Then .f ≥ 0 and the above estimate gives | (∂j C)ζ, ζ  + (∂j C)τ, τ  − 2Reeiα (∂j C)ζ, τ |2   ≤ 2 Cζ, ζ  + Cτ, τ  − 2Reeiα Cζ, τ  ·

.

· sup | (∂j2 C)ζ, ζ  + (∂j2 C)τ, τ  − 2Reeiα (∂j2 C)ζ, τ |   ≤ 4M Cζ, ζ  + Cτ, τ  − 2Reeiα Cζ, τ  ( ζ 2 + τ 2 ), (6) where .M = ∂j2 C L∞ (Rd ,L(Cd )) . Replacing .α by .α + π gives | (∂j C)ζ, ζ  + (∂j C)τ, τ  + 2Reeiα (∂j C)ζ, τ |2    ≤ 4M Cζ, ζ  + Cτ, τ  + 2Reeiα Cζ, τ  ζ 2 + τ 2 . (7)

.

Hence by adding (6) and (7), and using the parallelogram identity one deduces that    2 |2Reeiα (∂j C)ζ, τ |2 ≤ 8M Cζ, ζ  + Cτ, τ  ζ 2 + τ 2 .

.

This is a pointwise estimate on .Rd , valid for all .α ∈ R. Therefore    | (∂j C)ζ, τ |2 ≤ M Cζ, ζ  + Cτ, τ  ζ 2 + τ 2

.

as required. Now we are able to derive an Ole˘ınik inequality.

 

Degenerate Elliptic Operators and Kato’s Inequality

413

Lemma 2.3 Suppose .ckl = clk for all .k, l ∈ {1, . . . , d}. Let U be a complex .d × d matrix with .U t = U . Then |Tr (∂j C) U |2 ≤ 4Tr (U C U ) ∂j2 C L∞ (Rd ,L(Cd ))

.

for all .j ∈ {1, . . . , d}. Proof Use the polar √ decomposition of U to write .U = V |U | with V a unitary matrix and .|U | = U ∗ U . The positive operator .|U | is diagonisable, so there exists a unitary matrix W and a positive diagonal matrix D such that .|U | = W D W −1 . Then Lemma 2.2 gives |Tr (∂j C) U |2

.

= |Tr ((∂j C) V W D W −1 )|2 = |Tr (W −1 (∂j C) V W D)|2 =

d 

2 |(W −1 (∂j C) V W )kk |2 Dkk

k=1

≤

d    (W −1 V −1 C V W )kk + (W −1 C W )kk · k=1

≤2

  2 · V W ek 2 + W ek 2 ∂j2 C L∞ (Rd ,L(Cd )) Dkk

d    Dkk (W −1 V −1 C V W )kk Dkk + Dkk (W −1 C W )kk Dkk · k=1

· ∂j2 C L∞ (Rd ,L(Cd ))

  = 2 Tr (D W −1 V −1 C V W D) + Tr (D W −1 C W D) · ∂j2 C L∞ (Rd ,L(Cd ))

  = 2 Tr (U ∗ C U ) + Tr (U C U ∗ ) ∂j2 C L∞ (Rd ,L(Cd )) . Up to now we have not used that .U t = U . The symmetry of U and C gives Tr (U ∗ C U ) = Tr (U C U ) and .Tr (U C U ∗ ) = Tr ((U C U ∗ )t ) = Tr (U C U ). The proof is complete.  

.

Next we prove an energy estimate. Lemma 2.4 Suppose that .c0 = 0. Let .p ∈ (1, ∞). Then Re

.

d  p ∂j Bp u, 1[∇u =0] |∇u|p−2 ∂j uLp ×Lq ≥ −M ∇u p j =1

(8)

414

T. D. Do and A. F. M. ter Elst

for all .u ∈ W 3,p (Rd ), where M = (3d 2 +

.

  d ) max ∂j ∂k C L∞ (Rd ,L(Cd )) + ∂j ck ∞ p − 1 j,k

and q is the dual exponent of p. Proof First assume that .ckl = clk for all .k, l ∈ {1, . . . , d} and that u has compact support. Let .δ > 0 if .p ∈ (1, 2) and .δ = 0 if .p ∈ [2, ∞). Clearly Re

d 

.

∂j Bp u, (δ + |∇u|2 )

p−2 2

∂j uLp ×Lq

j =1

= −Re

 d  d j,k,l=1 R

   p−2 ∂k ckl ∂j ∂l u + (∂j ckl ) ∂l u (δ + |∇u|2 ) 2 ∂j u

+ Re

= −Re

 d 



d j,k,l=1 R

− Re

d   d j,k=1 R

(∂j ck ∂k u) (δ + |∇u|2 )

p−2 2

∂j u

 p−2 ∂k ckl ∂j ∂l u (δ + |∇u|2 ) 2 ∂j u

 d 

  p−2 ∂k (∂j ckl ) ∂l u (δ + |∇u|2 ) 2 ∂j u

d j,k,l=1 R

d  

+ Re

j,k=1

Rd

(∂j ck ∂k u) (δ + |∇u|2 )

p−2 2

∂j u

= I1 + I2 + I3 . We estimate the three terms separately. p−2 First note that .ckl ∂j ∂l u ∈ W 1,p (Rd ) and .(δ + |∇u|2 ) 2 ∂j u ∈ W 1,q (Rd ) for all .j, k, l ∈ {1, . . . , d}. So integration by parts gives I1 = −Re

 d 



.

= Re

d j,k,l=1 R

 d  d j,k,l=1 R

 p−2 ∂k ckl ∂j ∂l u (δ + |∇u|2 ) 2 ∂j u

ckl (∂j ∂l u) ∂k ((δ + |∇u|2 )

p−2 2

∂j u)

Degenerate Elliptic Operators and Kato’s Inequality

 d 

= Re

+ Re

ckl (∂j ∂l u) (δ + |∇u|2 )

Rd

j,k,l=1

415

 d  d j,k,l=1 R

p−2 2

∂k ∂j u

ckl (∂j ∂l u) ∂j u ∂k ((δ + |∇u|2 )

p−2 2

)

= I1a + I1b . Write .U = (∂k ∂l u)k,l∈{1,...,d} . Then  I1a =

.

Rd

Tr (U C U ) (δ + |∇u|2 )

p−2 2

.

Moreover, I1b = Re

 d 

.

=

d j,k,l=1 R

p−2 Re 2

p−2 Re 2  = (p − 2) =

ckl (∂j ∂l u) ∂j u ∂k ((δ + |∇u|2 )

d 

p−2 2

)

 ckl (∂j ∂l u) ∂j u (δ + |∇u|2 )

2 j,k,l,m=1 [δ+|∇u| >0]





[δ+|∇u|2 >0]

[δ+|∇u|2 >0]

p−4 2

·

  · (∂k ∂m u) ∂m u + ∂k ∂m u ∂m u

 p−4 C U ∇u, U ∇u + C U ∇u, U ∇u (δ + |∇u|2 ) 2

C Re(U ∇u), Re(U ∇u) (δ + |∇u|2 )

p−4 2

.

For the second term we decompose I2 = −Re

 d 

.

= −Re

d j,k,l=1 R

 d  d j,k,l=1 R

− Re

  p−2 ∂k (∂j ckl ) ∂l u (δ + |∇u|2 ) 2 ∂j u

(∂k ∂j ckl ) (∂l u) (δ + |∇u|2 )

 d  d j,k,l=1 R

= I2a + I2b .

p−2 2

∂j u

(∂j ckl ) (∂k ∂l u) (δ + |∇u|2 )

p−2 2

∂j u

416

T. D. Do and A. F. M. ter Elst

Then  d



.|I2a | = Re

d j,k,l=1 R

≤

(∂k ∂j ckl ) (∂l u) (δ + |∇u|2 )

p−2 2



∂j u

 d   p−2 1  |∂k ∂j ckl | (δ + |∇u|2 ) 2 |∂l u|2 + |∂j u|2 2 Rd j,k,l=1

p

≤ d 2 max ∂j ∂k C L∞ (Rd ,L(Cd )) ∇u p . j,k

Moreover, using the Ole˘ınik inequality of Lemma 2.3 we estimate  d



.|I2b | = Re

d j,k,l=1 R

d 



= Re

Tr ((∂j C) U ) (δ + |∇u|2 )

p−2 2

|Tr ((∂j C) U )|2 (δ + |∇u|2 )

p−2 2

d j =1 R

≤ε

d   d j =1 R

+

(∂j ckl ) (∂k ∂l u) (δ + |∇u|2 )

p−2 2



∂j u



∂j u

d  p−2 1  |∂j u|2 (δ + |∇u|2 ) 2 4ε Rd j =1

≤ 4ε

d   d j =1 R

Tr (U C U ) (δ + |∇u|2 )

p−2 2

∂j2 C L∞ (Rd ,L(Cd )) +

for all .ε > 0. Finally, for the third term we write I3 = Re

d  

.

= Re

d j,k=1 R

d   j,k=1

Rd

(∂j ck ∂k u) (δ + |∇u|2 )

(∂j ck ) (∂k u) (δ + |∇u|2 )

d  

+ Re

j,k=1

= I3a + I3b .

p−2 2

Rd

∂j u

p−2 2

∂j u

ck (∂j ∂k u) (δ + |∇u|2 )

p−2 2

∂j u

1 p

∇u p 4ε

Degenerate Elliptic Operators and Kato’s Inequality

417

Then d    p−2 1  |∂j ck | (δ + |∇u|2 ) 2 |∂k u|2 + |∂j u|2 .|I3a | ≤ 2 Rd j,k=1

p

≤ d max ∂j ck ∞ ∇u p . j,k

For the last term the chain rule gives d  

I3b = Re

.

=

d j,k=1 R

ck (∂j ∂k u) (δ + |∇u|2 )

p−2 2

∂j u

d  p 1 ck ∂k ((δ + |∇u|2 ) 2 ) p BR (0) k=1

=−

d  d  p 1 1 (∂k ck ) (δ + |∇u|2 ) 2 + νk ck δ p/2 , p p BR (0) ∂BR (0) k=1

k=1

where .R > 0 is such that .supp u ⊂ BR (0) and .ν is again the normal derivative. Hence |I3b | ≤

d 

.

 ∂k ck ∞

k=1

p

(δ + |∇u|2 ) 2 + δ p/2 BR (0)

d  

|νk ck |.

k=1 ∂BR (0)

Combining the estimates and taking the limit .δ ↓ 0 if .p ∈ (1, 2), one obtains Re

.

d  ∂j Bp u, 1[∇u =0] |∇u|p−2 ∂j uLp ×Lq j =1 d    ≥ 1 − 4ε

∂j2 C L∞ (Rd ,L(Cd )) j =1

+ (p − 2)

[∇u =0]

Tr (U C U ) |∇u|p−2

 [∇u =0]

C Re(U ∇u), Re(U ∇u) |∇u|p−4

  1 p + 2d max ∂j ck ∞ ∇u p − d 2 max ∂j ∂k C L∞ (Rd ,L(Cd )) + j,k j,k 4ε for all .ε > 0. Now 0 ≤ C Re(U ∇u), Re(U ∇u) ≤ C(U ∇u), U ∇u

.

= U C U ∇u, ∇u ≤ Tr (U C U ) |∇u|2 .

418

T. D. Do and A. F. M. ter Elst

Hence if .ε is small enough, then .Re

d  ∂j Bp u, 1[∇u =0] |∇u|p−2 ∂j uLp ×Lq j =1

d    ≥ p − 1 − 4ε

∂j2 C L∞ (Rd ,L(Cd )) j =1

[∇u =0]



− d 2 max ∂j ∂k C L∞ (Rd ,L(Cd )) + j,k

Choosing .ε such that .p − 1 − 4ε(ε + limit .ε ↓ 0 gives the estimate .Re

C Re(U ∇u), Re(U ∇u) |∇u|p−4

 1 p + 2d max ∂j ck ∞ ∇u p . j,k 4ε

d

2 j =1 ∂j C L∞ (Rd ,L(Cd )) )

= 0 and taking the

d  ∂j Bp u, 1[∇u =0] |∇u|p−2 ∂j uLp ×Lq j =1

 ≥ − (d 2 +

 d p ) max ∂j ∂k C L∞ (Rd ,L(Cd )) + 2d max ∂j ck ∞ ∇u p . j,k p − 1 j,k

Next one can drop the symmetry assumption .ckl = clk and use (5) to obtain the estimate (8) for all .u ∈ W 3,p (Rd ) with compact support. Finally we drop the assumption that u has compact support. For all .j ∈ {1, . . . , d} define .Rj : W 1,p (Rd ) → L∞ (Rd ) by  .Rj v

=

|∇v|−1 ∂j v if |∇v| = 0, 0

if |∇v| = 0.

Fix again .χ ∈ Cc∞ (Rd ) such that .0 ≤ χ ≤ 1 and .χ|B1 (0) = 1. For all .n ∈ N define .χn ∈ Cc∞ (Rd ) by .χn (x) = χ(n−1 x). Then .limn→∞ Rj (χn v) = Rj v almost everywhere for all .v ∈ W 1,p (Rd ). Let .u ∈ W 3,p (Rd ). Then we proved that .Re

d  ∂j Bp (χn u), |∇(χn u)|p−1 Rj (χn u)Lp ×Lq j =1

= Re

d  ∂j Bp (χn u), 1[∇(χn u) =0] |∇(χn u)|p−2 ∂j (χn u)Lp ×Lq j =1 p

≥ −M ∇(χn u) p .

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419

Now take the limit .n → ∞. Then .Re

d  p ∂j Bp u, |∇u|p−1 Rj (u)Lp ×Lq ≥ −M ∇u p . j =1

 

This is equivalent to (8).

The next proposition provides sufficient information for the range condition to obtain a semigroup generator in Sects. 4 and 5. It also ensures that the semigroup will be positivity preserving. Proposition 2.5 Suppose .c0 = 0. Let .p0 ∈ (1, 2]. Then there exists a .λ0 > ω such that for all .λ ≥ λ0 and .f ∈ Cc∞ (Rd ) there exists a u∈



.



W 1,p (Rd ) ∩ D(Bp )



p∈[p0 ,∞]

such that .f = (λ I + Bp )u for all .p ∈ [p0 , ∞]. Moreover, if .f ≥ 0, then .u ≥ 0. Proof Define M = (3d 2 +

.

  d ) max ∂j ∂k C L∞ (Rd ,L(Cd )) + ∂j ck ∞ . p0 − 1 j,k

Choose .λ0 = 2 + (ω ∨ M). Let .λ ≥ λ0 . For all .n ∈ N define the sesquilinear form an : W 1,2 (Rd ) × W 1,2 (Rd ) → C by

.

an (u, v) =

d  

.

d k,l=1 R

(ckl + n1 ) (∂l u) ∂k v +

d   d k=1 R

 ck (∂k u) v +

Rd

c0 u v.

Further for all .p ∈ [1, ∞] define .Bp,n to be the operator similarly as .Bp , but with principal coefficients .ckl + n1 instead of .ckl . Let .n ∈ N. Since the sesquilinear form .(u, v) → an (u, v) + λ (u, v)L2 (Rd ) is coercive, there exists a .un ∈ W 1,2 (Rd ) such that an (un , v) + λ (un , v)L2 (Rd ) = (f, v)L2 (Rd )

.

for all .v ∈ W 1,2 (Rd ). Moreover, if .f ≥ 0, then .un ≥ 0 by the Beurling–Deny criteria. Because .c0 = 0 one deduces by elliptic regularity that .un ∈ W 3,p (Rd ) for all .p ∈ (1, ∞). Hence .un ∈ D(Bp,n ) and .(λ I + Bp,n )un = f for all .p ∈ (1, ∞). Let .p ∈ [p0 , ∞) and let q be the dual exponent of p. Then Lemma 2.1 gives p

p

Re f, 1[un =0] |un |p−2 un Lp ×Lq ≥ (λ − ω) un p ≥ 2 un p .

.

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Moreover, Lemma 2.4 implies that Re

.

d  ∂j f, 1[∇u =0] |∇un |p−2 ∂j un Lp ×Lq j =1

= λRe

d 

∂j un , 1[∇u =0] |∇un |p−2 ∂j un Lp ×Lq

j =1

+ Re

d 

∂j Bp un , 1[∇u =0] |∇un |p−2 ∂j un Lp ×Lq

j =1 p

≥ (λ − M) ∇un p p

≥ 2 ∇un p . We use the convention that   p p 1/p

u W 1,p (Rd ) = u p + ∇u p

.

for all .u ∈ W 1,p (Rd ). Then p

p

2 un W

.

d 1,p (R )

p

= 2 un p + 2 ∇un p ≤ Re f, 1[un =0] |un |p−2 un Lp ×Lq + Re

d 

∂j f, 1[∇un =0] |∇un |p−2 ∂j un Lp ×Lq

j =1

≤ f p |un |

p−1

p/q

= f p un p

q + ∇f p |∇un |p−1 q p/q

+ ∇f p ∇un p

  p/q p/q ≤ f W 1,p (Rd ) un p + ∇un p .

Therefore pq

2q un W

.

1,p

(Rd )

  q p/q p/q q ≤ f W 1,p (Rd ) un p + ∇un p   q p p ≤ 2q f W 1,p (Rd ) un p + ∇un p q

p

= 2q f W 1,p (Rd ) un W 1,p (Rd )

Degenerate Elliptic Operators and Kato’s Inequality

421

and

un W1,p (Rd ) ≤ f W 1,p (Rd ) .

.

Moreover, since .Bp,n un = f − λ un one obtains the bound . Bp,n un W 1,p (Rd ) ≤ (1 + λ) f W 1,p (Rd ) . Hence the sequences .(un )n∈N and .(Bp,n un )n∈N are bounded in .W 1,p (Rd ) for all .p ∈ [p0 , ∞). Using a diagonal argument together with interpolation, and passing

to a subsequence if necessary, there exist .u, v ∈ p∈[p0 ,∞) W 1,p (Rd ) such that 1,p (Rd ) for all .p ∈ [p , ∞). .limn→∞ un = u and .limn→∞ Bp,n un = v weakly in .W 0 Clearly .u ≥ 0 if .f ≥ 0, because then .un ≥ 0 for all .n ∈ N. Since .Bp,n un = f −λ un for all .n ∈ N one deduces that .v = f − λ u. If .p ∈ [p0 , ∞), then . u W 1,p (Rd ) ≤ lim infn→∞ un W 1,p (Rd ) ≤ f W 1,p (Rd ) . Now 1/p     p 1/p . f p = |f |p ≤

f ∞ supp f

= |supp f |1/p f ∞ ≤ (1 + |supp f |) f ∞ and similarly . ∇f p ≤ (1 + |supp f |) ∇f ∞ . So   p p 1/p

u W 1,p (Rd ) ≤ (1 + |supp f |)p f ∞ + (1 + |supp f |)p ∇f ∞

.

≤ 2(1 + |supp f |) f W 1,∞ (Rd ) . This is for all .p ∈ [p0 , ∞). Therefore .u ∈ L∞ (Rd ) and .∂k u ∈ L∞ (Rd ) for all 1,∞ (Rd ). Then also .v = f − λ u ∈ W 1,∞ (Rd ). .k ∈ {1, . . . , d}. So .u ∈ W ∞ d # ϕ = H # ϕ strongly Let .ϕ ∈ Cc (R ). Choose .p ∈ [p0 , ∞). Then .limn→∞ Hq,n q # is defined in the natural way. in .Lq (Rd ), where q is the dual exponent of p and .Hq,n Hence # v, ϕLp ×Lq = lim Bp,n un , ϕLp ×Lq = lim un , Hq,n ϕLp ×Lq

.

n→∞

n→∞

= u, Hq# ϕLp ×Lq = Au, ϕLp ×Lq . So .Au = v. Finally, since .u, v ∈ Lp (Rd ) for all .p ∈ [p0 , ∞] one deduces that .u ∈ D(Bp ) and .Bp u = v = f − λ u. In particular, .(λ I + Bp )u = f .  

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T. D. Do and A. F. M. ter Elst

3 Density Throughout this section we adopt the notation and assumptions of Theorem 1.1 and the operators introduced in Sects. 1 and 2. The main result of this section is that elements which are in the intersection of the space .W 1,p (Rd ) and the domain of the maximal operator .Bp can be approximated in the graph norm by test functions. Proposition 3.1 Let .p ∈ [1, ∞]. Then one has the following. (a) .D(Bp ) ∩ W 1,p (Rd ) ⊂ D(Hp ). (b) .{u ∈ D(Bp ) ∩ W 1,p (Rd ) : u ≥ 0} ⊂ Cc∞ (Rd )+

D(Bp )

.

The proof requires two steps. In the first we truncate the domain and in the second we regularise. Lemma 3.2 Let .χ ∈ Cc∞ (Rd ) be such that .0 ≤ χ ≤ 1 and .χ |B1 (0) = 1. For all .n ∈ N define .χn ∈ Cc∞ (Rd ) by .χn (x) = χ (n−1 x). Let .p ∈ [1, ∞] and .u ∈ D(Bp ) ∩ W 1,p (Rd ). Then .χn u ∈ D(Bp ) for all .n ∈ N. Moreover, .limn→∞ Bp (χn u) = Bp u in .Lp (Rd ) if .p ∈ [1, ∞) and .limn→∞ B∞ (χn u) = B∞ u weakly.∗ in .L∞ (Rd ). Proof Let .n ∈ N. Define w n = χn B p u − u

d 

.

∂k ckl ∂l χn

k,l=1

−

d 

(ckl + clk ) (∂l u) ∂k χn + u

k,l=1

d 

ck ∂k χn ∈ Lp (Rd ).

k=1

Then χn u, Hq# vLp ×Lq = wn , vLp ×Lq

.

for all .v ∈ Cc∞ (Rd ), where q is the dual exponent of p. Hence .χn u ∈ D(Bp ) and .Bp (χn u) = wn . Clearly .

lim Bp (χn u) − χn Bp u p = lim wn − χn Bp u p = 0.

n→∞

n→∞

 

Hence the lemma follows. Proposition 3.3 Let .p ∈ [1, ∞]. Then one has the following. (a) .D(Bp ) ∩ W 1,p (Rd ) ∩ C ∞ (Rd ) ⊂ D(Hp ). (b) .{u ∈ D(Bp ) ∩ W 1,p (Rd ) ∩ C ∞ (Rd ) : u ≥ 0} ⊂ Cc∞ (Rd )+

D(Bp )

.

Degenerate Elliptic Operators and Kato’s Inequality

423

 

Proof This follows immediately from Lemma 3.2. Next we regularise.

Lemma 3.4 Let .τ ∈ Cc∞ (Rd ) be a function such that .τ ≥ 0, .supp τ ⊂ B1 (0) and ∞ d d . τ = 1. For all .n ∈ N define .τn ∈ Cc (R ) by .τn (x) = n τ (n x). Let .p ∈ [1, ∞] 1,p d and .u ∈ D(Bp ) ∩ W (R ). Then .τn ∗ u ∈ D(Bp ) for all .n ∈ N. Moreover, d .limn→∞ Bp (τn ∗ u) = Bp u in .Lp (R ) if .p ∈ [1, ∞) and .limn→∞ B∞ (τn ∗ u) = ∗ d B∞ u weakly. in .L∞ (R ). Proof For all .p ∈ [1, ∞] and .y ∈ Rd let .Ly denote the left translation over y in  d .Lp (R ), that is .(Ly u) = u(x−y). Fix .p ∈ [1, ∞]. Define .c ˜k = ck − dm=1 ∂m cmk ∈ W 1,∞ (Rd ). Then d 

Bp u = −

.

ckl ∂k ∂l u +

k,l=1

d 

c˜k ∂k u + c0 u

(9)

k=1

for all .u ∈ W 2,p (Rd ). Let .n ∈ N. Let .u ∈ W 2,p (Rd ). Then as a Perron integral in .Lp (Rd ) one obtains Bp (τn ∗ u)

.

=−

d 

ckl ∂k ∂l (τn ∗ u) +

k,l=1

=−

d 

c˜k ∂k (τn ∗ u) + c0 (τn ∗ u)

k=1

d   d k,l=1 R

ckl τn (y) Ly ∂k ∂l u dy +



+ =−

Rd

d k,l=1 R

τn (y) Ly ckl ∂k ∂l u dy +



+

Rd

d k=1 R

c˜k τn (y) Ly ∂k u dy

c0 τn (y) Ly u dy

d  

+

d  

τn (y) Ly c0 u dy +

d   d k=1 R

d   d k=1 R

d   d k,l=1 R

τn (y) Ly c˜k ∂k u dy

  τn (y) Ly ckl − ckl Ly ∂k ∂l u dy

   τn (y) c˜k − Ly c˜k Ly ∂k u dy +

Rd

  τn (y) c0 − Ly c0 Ly u dy

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T. D. Do and A. F. M. ter Elst

= τn ∗ (Bp u) +

+

d   d k=1 R

   ∂ τn (y) Ly ckl − ckl Ly ∂l u dy ∂yk

d   d k,l=1 R







τn (y) c˜k −Ly c˜k Ly ∂k u dy+

Rd

  τn (y) c0 −Ly c0 Ly u dy,

where we used that .Ly ∂k v = − ∂y∂ k Ly v for all .v ∈ W 1,p (Rd ). Therefore Bp (τn ∗ u) = τn ∗ (Bp u) +

d  

.

−

+

d k,l=1 R

d   d k,l=1 R

d   d k=1 R

 +

Rd

  (∂k τn )(y) Ly ckl − ckl Ly ∂l u dy

τn (y) (Ly ∂k ckl ) Ly ∂l u dy

  τn (y) c˜k − Ly c˜k Ly ∂k u dy

  τn (y) c0 − Ly c0 Ly u dy. (1)

(4)

Define the bounded operators .Tn , . . . , Tn : W 1,p (Rd ) → Lp (Rd ) by Tn(1) u =

d  

.

d k,l=1 R

Tn(2) u = −

Tn(3) u =

d   d k,l=1 R

d   d k=1 R

 Tn(4) u =

  (∂k τn )(y) Ly ckl − ckl Ly ∂l u dy,

Rd

τn (y) (Ly ∂k ckl ) Ly ∂l u dy,

  τn (y) c˜k − Ly c˜k Ly ∂k u dy,

  τn (y) c0 − Ly c0 Ly u dy.

Then we just proved that τn ∗ u, Hq# vLp ×Lq = u, Hq# (τˇn ∗ v)Lp ×Lq + Tn(1) u, vLp ×Lq + . . .

.

+ Tn(4) u, vLp ×Lq

(10)

Degenerate Elliptic Operators and Kato’s Inequality

425

for all .u ∈ W 2,p (Rd ) and .v ∈ Cc∞ (Rd ), where q is the dual exponent and .τˇn ∈ Cc∞ (Rd ) is given by .τˇn (y) = τn (−y). Since .W 2,p (Rd ) is dense in .W 1,p (Rd ) if 2,p (Rd ) is weakly.∗ dense in .W 1,p (Rd ) if .p = ∞, it follows that .p ∈ [1, ∞) and .W (10) is valid for all .u ∈ W 1,p (Rd ) and .v ∈ Cc∞ (Rd ). Hence .τn ∗ u ∈ D(Bp ) and Bp (τn ∗ u) = τn ∗ (Bp u) + Tn(1) u + . . . + Tn(4) u

(11)

.

for all .u ∈ D(Bp ) ∩ W 1,p (Rd ). Next since .ckl ∈ W 1,∞ (Rd ) one estimates . (I − Ly )ckl ∞ ≤ |y| ckl W 1,∞ (Rd ) for all .y ∈ Rd and .k, l ∈ {1, . . . , d}. Therefore

Tn(1) u p ≤

d  

.

=

d k,l=1 R

d 

|(∂k τn )(y)| |y| ckl W 1,∞ (Rd ) u W 1,p (Rd ) dy 

ckl W 1,∞ (Rd ) u W 1,p (Rd )

|(∂k τ )(y)| |y| dy

k,l=1 (1)

for all .n ∈ N and .u ∈ W 1,p (Rd ). So the set of operators .{Tn : n ∈ N} is (2) (4) equibounded. Similarly (and easier) the sets .{Tn : n ∈ N}, . . . , .{Tn : n ∈ N} are equibounded. Suppose .p ∈ [1, ∞). If .u ∈ W 2,p (Rd ), then .limn→∞ τn ∗ u = u in .W 2,p (Rd ), hence by (9) one deduces that .limn→∞ Bp (τn ∗ u) = Bp u in .Lp (Rd ). So (11) gives (1) (4) d .limn→∞ Tn u + . . . + Tn u = 0 in .Lp (R ). Then by equiboundedness and density 2,p d 1,p d of .W (R ) in .W (R ) one establishes that .limn→∞ Tn(1) u + . . . + Tn(4) u = 0 in .Lp (Rd ) for all .u ∈ W 1,p (Rd ). Hence .lim Bp (τn ∗ u) = Bp u in .Lp (Rd ) for all 1,p (Rd ). .u ∈ D(Bp ) ∩ W For .p = ∞ the argument is similar, using the weak.∗ -topology on .L∞ (Rd ).   Now we are able to show that .D(Bp ) ∩ W 1,p (Rd ) ⊂ D(Hp ). Proof of Proposition 3.1 If .u ∈ D(Bp ) ∩ W 1,p (Rd ), then .τn ∗ u ∈ D(Bp ) ∩ W 1,p (Rd ) ∩ C ∞ (Rd ) for all .n ∈ N by Lemma 3.4. Moreover, .τn ∗ u ≥ 0 if .u ≥ 0.   Then the results follow from Lemma 3.4 and Proposition 3.3.

4 Semigroup Generation Throughout this section we adopt the notation and assumptions of Theorem 1.1 and the operators introduced in Sects. 1 and 2. Theorem 4.1 Let .p ∈ (1, ∞). Then the operator .(Bp + ω I ) is m-accretive. Moreover, .Cc∞ (Rd ) is a core for .Bp , that is .Bp = Hp .

426

T. D. Do and A. F. M. ter Elst

Proof First suppose that .c0 = 0. It follows from Lemma 2.1 that .Hp + ω I is accretive. Clearly .Hp + ωI is densely defined. Hence .Hp + ω I is accretive by [12] Lemma 3.4. One deduces from Proposition 3.1(a) that Hp + ω I = (Bp + ω I )|D(Bp )∩W 1,p (Rd ) .

.

By Proposition 2.5 there exists a .λ > ω such that .(Bp + λ I )|D(Bp )∩W 1,p (Rd ) has dense range. Hence the operator .Hp + ω I is m-accretive. Next we show that .Bp = Hp . Clearly .Hp ⊂ Bp . Let q be the dual exponent of p. It follows similarly that there exists a .λ˜ > ω such that .ran(Hq# + λ˜ I ) = Lq (Rd ). Let .u ∈ D(Bp ). There exists a .v ∈ D(Hp ) such that .(Hp + λ˜ I )v = (Bp + λ˜ I )u. Then .v ∈ D(Bp ) and .(Bp + λ˜ I )(u − v) = 0. Hence 0 = (Bp + λ˜ I )(u − v), ϕLp ×Lq = u − v, (Hq# + λ˜ I )ϕLp ×Lq

.

for all .ϕ ∈ Cc∞ (Rd ). Therefore by density 0 = u − v, (Hq# + λ˜ I )ϕLp ×Lq

.

for all .ϕ ∈ D(Hq# ). So . u − v, f Lp ×Lq = 0 for all .f ∈ Lq (Rd ) and .u = v ∈ D(Hp ). (0) Finally we remove the condition .c0 = 0. Let .Bp be the operator with .c0 (0) replaced by 0. Then .Bp = B + c0 I . Hence .Bp is a bounded perturbation of .B (0) . (0) Consequently .Bp is also the generator of a .C0 -semigroup and .D(Bp ) = D(Bp ), (0) ∞ d with equivalent norms. Since .Cc (R ) is dense in .D(Bp ), it is also dense in ∞ d .D(Bp ), that is .Cc (R ) is a core for .Bp . The operator .Hp + ω I is accretive by (0) the same arguments as for .Bp . Hence .Bp + ω I is m-accretive.   For all .p ∈ (1, ∞) let .S (p) be the .C0 -semigroup generated by .−Bp . (p )

(p )

Lemma 4.2 Let .p1 , p2 ∈ (1, ∞). Then .St 1 u = St 2 u for all .t > 0 and .u ∈ Lp1 (Rd ) ∩ Lp2 (Rd ). Moreover, the semigroup .S (p1 ) is positivity preserving. Proof First suppose that .c0 = 0. Let .p0 = p1 ∧ p2 ∧ 2. Let .λ0 > ω be as in Proposition 2.5. Then (Bp1 + λ I )−1 f = (Bp2 + λ I )−1 f

.

(12)

for all .λ ≥ λ0 and .f ∈ Cc∞ (Rd ) by Proposition 2.5. Hence (12) is valid for all d d .f ∈ Lp1 (R ) ∩ Lp2 (R ) and .λ ≥ λ0 . Then the first statement follows from [18] Lemma 2.3.

Degenerate Elliptic Operators and Kato’s Inequality

427

If .f ∈ Cc∞ (Rd ) with .f ≥ 0, then .(Bp1 +λ I )−1 f ≥ 0 by Proposition 2.5. Hence the operator .(Bp1 + λ I )−1 is positivity preserving for all .λ ≥ λ0 . The Euler formula then implies that the semigroup .S (p1 ) is positivity preserving. If .c0 = 0, then use the Trotter product formula.   We also obtain a consistent semigroup on .L1 (Rd ). Theorem 4.3 The operator .−(B1 + ω I ) is the generator of a positivity preserving (2) contractive .C0 -semigroup which is consistent with .(e−ωt St )t>0 . Proof Let .λ > 3ω and .g ∈ Cc∞ (Rd ). Let .p ∈ (2, ∞). Then

(B2# + λ I )−1 g p

.

= (Bp# + λ I )−1 g p ≤

1 1

g p ≤ (1 + |supp g|) g ∞ . λ − 3ω λ − 3ω

So .(B2# + λ I )−1 g ∈ L∞ (Rd ) and

(B2# + λ I )−1 g ∞

.

1 1

g p =

g ∞ . p→∞ λ − 3ω λ − 3ω

= lim (B2# + λ I )−1 g p ≤ lim p→∞

Let .f ∈ L1 (Rd ) ∩ L2 (Rd ). Then | (B2 + λ I )−1 f, gL1 ×L∞ = | f, (B2# + λ I )−1 gL1 ×L∞

.

≤ f 1 (B2# + λ I )−1 g ∞ ≤ So .(B2 + λ I )−1 f ∈ L1 (Rd ) and . (B2 + λ I )−1 f 1 ≤ Let

1

f 1 g ∞ . λ − 3ω

1 λ−3ω

f 1 .

D = D(B1 ) ∩ D(B2 ) = {u ∈ L1 (Rd ) ∩ L2 (Rd ) : Au ∈ L1 (Rd ) ∩ L2 (Rd )}.

.

Let .u ∈ D and .λ > 3ω. Define .f = (B1 + λ I )u = (B2 + λ I )u. Then by the 1 above . (B2 + λ I )−1 f 1 ≤ λ−3ω

f 1 , hence .(λ − 3ω) u 1 ≤ (B1 + λ I )u 1 . So the operator .(B1 + 3ω I )|D is accretive. Obviously .Cc∞ (Rd ) ⊂ D, so D is dense in .L1 (Rd ). Next let .f ∈ L1 (Rd ) ∩ L2 (Rd ). Define .u = (B2 + 4ω I )−1 f . Then d d d d .u ∈ L1 (R ) ∩ L2 (R ) and .B2 u = f − 4ω u ∈ L1 (R ) ∩ L2 (R ). So .u ∈ D and .(B1 + 4ω I )u = f . Therefore the operator .(B1 + 4ω I )|D has dense range. Hence .B1 |D + 3ω I is m-accretive.  Next we show that .B1 = B1 |D . First suppose that .c0 = dk=1 ∂k ck . Let .λ0 be as in Proposition 2.5 with the choice .p0 = 2 and the operators .Bp# with .p ∈ [p0 , ∞]. Let .u ∈ D(B1 ). Choose .λ = λ0 ∨ 4ω. There exists a .v ∈ D(B1 |D ) such that

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T. D. Do and A. F. M. ter Elst

(B1 |D + λ I )v = (B1 + λ I )u. Then .v ∈ D(B1 ). If .ϕ ∈ Cc∞ (Rd ), then

.

# 0 = (B1 + λ I )(u − v), ϕL1 ×L∞ = u − v, (H∞ + λ I )ϕL1 ×L∞ .

.

So # 0 = u − v, (B∞ + λ I )ϕL1 ×L∞

.

# ) ∩ W 1,∞ (Rd ) by Proposifirst for all .ϕ ∈ Cc∞ (Rd ) and then for all .ϕ ∈ D(B∞ tion 3.1(a). Then .0 = u − v, f L1 ×L∞ for all .f ∈ Cc∞ (Rd ) by Proposition 2.5. Therefore .u − v = 0 and.u = v ∈ D(B1 |D ). Hence .B1 = B1 |D . The condition .c0 = dk=1 ∂k ck can be removed by perturbation theory and the Trotter product formula in order to obtain that .−B1 is the generator of a positive quasi-contraction semigroup .S (1) on .L1 (Rd ). (1) It remains to show that . St 1→1 ≤ eωt for all .t > 0. It follows from (p) Theorem 4.1 that . St p→p ≤ eωt for all .t > 0 and .p ∈ (1, ∞). Hence by duality (p)∗ . St

q→q ≤ eωt for all .t > 0 and .p ∈ (1, ∞), where q is the dual exponent of p. (p)∗ (2)∗ is consistent with .St for all .t > 0 and .p ∈ (1, ∞). Let .v ∈ Cc∞ (Rd ) Also .St and .t > 0. Then it follows as at the beginning of this proof that .St(2)∗ v ∈ L∞ (Rd ) and . St(2)∗ v ∞ ≤ eωt v ∞ Now let .t > 0, .u ∈ L1 (Rd ) ∩ L2 (Rd ) and .v ∈ Cc∞ (Rd ). Then (1)

(2)∗

| St u, vL1 ×L∞ | = | u, St

.

(2)∗

vL1 ×L∞ | ≤ u 1 St

v ∞ ≤ eωt u 1 v ∞ .

So . St(1) u 1 ≤ eωt u 1 and by density . St(1) 1→1 ≤ eωt .

 

On .L∞ (Rd ) one obtains a weakly.∗ -continuous semigroup for which .Cc∞ (Rd ) is a core for the generator. We refer to Yosida [20] Chapter IX for background material on continuous semigroups in sequentially complete topological vector spaces. Theorem 4.4 The operator .−H∞ is the generator of a weakly.∗ -continuous semigroup on .L∞ (Rd ) which is consistent with .S (2) . In particular, .Cc∞ (Rd ) is a core for its generator. Proof It follows from Theorem 4.3 that .−B1# is the generator of a .C0 -semigroup, say .S (1)# , on .L1 (Rd ) which is consistent with the semigroup generated by .−B2# , that is .S (2)∗ . Then .S (1)# is a continuous semigroup with respect to the weak topology on .L1 (Rd ) and its generator is equal to the operator .−B1# . Denote by # ∗ d .(B ) the dual operator with respect to the weak topology on .L1 (R ) and the 1 # ∗ d ∗ ∗ weak. topology on .L∞ (R ). Then .−(B1 ) is the generator of a weak. -continuous semigroup on .L∞ (Rd ) which is consistent with .S (2) . It remains to show that # ∗ .H∞ = (B ) . 1 Obviously .H∞ ⊂ (B1# )∗ and hence .H∞ ⊂ (B1# )∗ . Note that one has to consider the graphs of the operators. Suppose that .H∞ = (B1# )∗ . Then there exists a

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v0 ∈ D((B1# )∗ ) such that

.

# ∗

.(v0 , (B1 )

v0 ) ∈ clo{(v, H∞ v) : v ∈ Cc∞ (Rd )}



 in (L∞ (Rd ), w∗ )×(L∞ (Rd ), w∗ ) .

Hence by the Hahn–Banach theorem there exists an .f

∈ ((L∞ (Rd ), w ∗ ) × (L∞ (Rd ), w ∗ ))∗

such that .f (v0 , (B1# )∗ v0 ) = 1 and .f (v, H∞ v) = 0 for all .v ∈ Cc∞ (Rd ). Then there are .u0 , u1 ∈ L1 (Rd ) such that .f (v, w)

= u0 , vL1 ×L∞ + u1 , wL1 ×L∞

for all .v, w ∈ L∞ (Rd ). Hence .0 = f (v, H∞ v) = u0 , vL1 ×L∞ + u1 , H∞ vL1 ×L∞ for all .v ∈ Cc∞ (Rd ), which implies that .A# u1 = −u0 . Since both .u0 , u1 ∈ L1 (Rd ) one deduces that .u1 ∈ D(B1# ) and .B1# u1 = −u0 . But then .1

= f (v0 , (B1# )∗ v0 ) = u0 , v0 L1 ×L∞ + u1 , (B1# )∗ v0 L1 ×L∞ = u0 , v0 L1 ×L∞ + B1# u1 , v0 L1 ×L∞ = u0 , v0 L1 ×L∞ + −u0 , v0 L1 ×L∞ = 0,

which is a contradiction. So .H∞ = (B1# )∗ .

 

5 The Kato Inequality The semigroup generator on .L1 (Rd ) allows us to prove a kind of a Kato inequality. Proposition 5.1 Adopt the notation and assumptions of Theorem 1.1. Then Re(sgn u Au) ≥ A|u|

.

in .D (Rd ) for all .u ∈ D(B1 ). Proof Let .S (1) be the semigroup generated by .−B1 . Then .S (1) is positivity preserving and consistent with .S (2) . Let .u ∈ D(B1 ) and .ϕ ∈ Cc∞ (Rd ) with .ϕ ≥ 0. (1) We argue as in [13]. Let .t > 0. Since .St is positivity preserving one obtains that (1) (1) .|St u| ≤ St |u|. Hence   (1) (1) Re sgn u (u − St u) = |u| − Re(sgn u St u)

.

(1)

≥ |u| − |St u| (1)

≥ |u| − St |u|

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and   (1) Re sgn u t −1 (u − St u) , ϕL1 (Rd )×L∞ (Rd )   ≥ t −1 |u| − St(1) |u| , ϕL1 (Rd )×L∞ (Rd ) .

.

Next we take the limit .t ↓ 0. Clearly .

  (1) lim Re sgn u t −1 (u − St u) , ϕL1 (Rd )×L∞ (Rd ) t↓0

  = Re sgn u B1 u , ϕL1 (Rd )×L∞ (Rd ) .

On the right hand side one obtains . lim t

t↓0

−1



 |u| − St(1) |u| , ϕL1 (Rd )×L∞ (Rd ) = lim |u|, t −1 (I − St(2)∗ )ϕL1 (Rd )×L∞ (Rd ) t↓0

= |u|, H2# ϕL1 (Rd )×L∞ (Rd ) # = |u|, H∞ ϕL1 (Rd )×L∞ (Rd ) .

So   # sgn u Au , ϕL1 (Rd )×L∞ (Rd ) ≥ |u|, H∞ ϕL1 (Rd )×L∞ (Rd )

. Re

as required.

 

Now we are able to prove a Kato inequality on a domain .. Proof of Theorem 1.2 Let .u ∈ L1,loc () and .ϕ ∈ Cc∞ (). Suppose that .Au ∈ L1,loc () and .ϕ ≥ 0. Let .χ, ˜ χ ∈ Cc∞ (, R) be such that .χ˜ |supp ϕ = 1 and .χ |supp χ˜ = 1. We identify functions in .L1 () with functions in .L1 (Rd ) by extending them with zero. Similarly, we identify an element (with compact support) of .Wc2,∞ () with an element of .W 2,∞ (Rd ) by extending it by zero, and analogously with .Wc1,∞ () and .L∞ (). Then .χ u ∈ L1 () ⊂ L1 (Rd ) and  be the operator on .L1,loc (Rd ) .χ ˜ A(χ u) = χ˜ Au ∈ L1 () ⊂ L1 (Rd ). Let .A  with coefficients .χ˜ ckl ∈ Wc2,∞ () ⊂ W 2,∞ (Rd ), .χ˜ ck − dm=1 (∂m χ˜ ) cmk ∈ Wc1,∞ () ⊂ W 1,∞ (Rd ) and .χ˜ c0 ∈ L∞ () ⊂ L∞ (Rd ) instead of .ckl , .ck and .c0 . Then these coefficients satisfy the assumptions of Theorem 1.2. Moreover, 1 ), where .B 1 is the  u) = χ˜ A(χ u) ∈ L1 () ⊂ L1 (Rd ). So .χ u ∈ D(B .A(χ  on .L1 (Rd ). Therefore Proposition 5.1 gives operator associated with .A   # ∞  u) , ϕL (Rd )×L (Rd ) ≥ |χ u|, H Re sgn(χ u) A(χ ϕL1 (Rd )×L∞ (Rd ) , ∞ 1

.

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# . Hence ∞ with the obvious definition for .H

    Re sgn u Au , ϕL1 ()×L∞ () = Re sgn(χ u) χ˜ A(χ u) , ϕL1 ()×L∞ ()    u) , ϕL (Rd )×L (Rd ) = Re sgn(χ u) A(χ ∞ 1

.

# ∞ ≥ |χ u|, H ϕL1 (Rd )×L∞ (Rd ) # = |u|, H∞ ϕL1 ()×L∞ () ,

which proves the Kato inequality.

 

Finally we add a potential and prove that the closure of .−(Hp + V ) generates a C0 -semigroup on .Lp (Rd ) if .V ∈ Lp,loc (Rd ).

.

Proof of Theorem 1.3 It suffices to show that there is a .λ > 0 such that the operator −(Hp + V + λ I ) is closable and the closure generates a .C0 -semigroup. Hence we may assume that .c0 ≥ 0. But then we can transfer the contribution of .c0 into the potential V . So together we may assume that .c0 = 0. Now .Hp + ω I is accretive by Lemma 2.1. Then also .Hp + V + ω I is accretive. Clearly .Hp + V + ω I is densely defined. We next show that there is a .λ > ω such that .Hp + V + λ I has dense range. Let .λ0 > ω be as in Proposition 2.5 with the choice .p0 = p ∧ 2 and set .λ = λ0 . Let .q ∈ (1, ∞) be the dual exponent of p. Let .f ∈ Lq (Rd ) and suppose that

.

(Hp + V + λ I )ϕ, f Lp ×Lq = 0

.

for all .ϕ ∈ Cc∞ (Rd ). Then .(A# + V + λ I )f = 0 in .D (Rd ). Therefore # # d .−A f = λ f + V f ∈ L1,loc (R ). Moreover, .sgn f A f + V |f | + λ |f | = 0. # # Consequently .V |f | + λ |f | = −Re(sgn f A f ) ≤ −A |f | by the Kato inequality of Theorem 1.2. Hence .(λ I + A# )|f | ≤ −V |f | ≤ 0. This means that (λ I + Bp )u, |f |Lp ×Lq ≤ 0

.

(13)

for all .u ∈ Cc∞ (Rd ) with .u ≥ 0. Then by Proposition 3.1(b) one deduces that (13) is valid for all .u ∈ D(Bp ) ∩ W 1,p (Rd ) with .u ≥ 0. But {(λ I + Bp )u : u ∈ D(Bp ) ∩ W 1,p (Rd ) and u ≥ 0}

.

is dense in .Lp (Rd )+ by Proposition 2.5. So .|f | = 0 and .f = 0. We proved that .Hp + V + λ I has dense range.   We do not know whether Theorem 1.3 extends in the degenerate case to .p = 1. For the Laplacian Kato proved this in [11] and for pure second-order possibly nonsymmetric elliptic operators this was proved in [3]. One can characterise the domain of the closure of .Hp + V .

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Proposition 5.2 Let .p ∈ (1, ∞). Then D(Hp + V ) = {u ∈ Lp (Rd ) : (A + V )u ∈ Lp (Rd ) as distribution}.

.

Proof This follows as in the proof of Theorem 4.1.

 

References 1. W. Arendt, Kato’s inequality: a characterisation of generators of positive semigroups. Proc. Roy. Irish Acad. Sect. A 84, 155–175 (1984) 2. W. Arendt, P. Bénilan, Inegalités de Kato et semi-groups sous-markoviens. Rev. Mat. Univ. Complut. de Madrid 5, 279–308 (1992) 3. W. Arendt, A.F.M. ter Elst, Kato’s inequality, in Analysis and operator theory: Dedicated in Memory of Tosio Kato’s 100th Birthday, ed. by T.M. Rassias, V.A. Zagrebnov, vol. 146 (Springer-Verlag, Berlin, 2019), pp. 47–60. Springer Optimization and Its Applications 4. A. Carbonaro, G. Metafune, C. Spina, Parabolic Schrödinger operators. J. Math. Anal. Appl. 343, 965–974 (2008) 5. A. Devinatz, On an inequality of Tosio Kato for degenerate-elliptic operators. J. Funct. Anal. 32, 312–335 (1979) 6. T.D. Do, Core properties for degenerate elliptic operators with complex bounded coefficients. J. Math. Anal. Appl. 479, 817–854 (2019) 7. T.D. Do, On sectoriality of degenerate elliptic operators. Proc. Edinb. Math. Soc. 64, 689–710 (2021) 8. T.D. Do, A.F.M. ter Elst, One-dimensional degenerate elliptic operators on Lp -spaces with complex coefficients. Semigr. Forum 92, 559–586 (2016) 9. G. Fichera, On a unified theory of boundary value problems for elliptic parabolic equations of second order, in Boundary Problems in Differential Equations (University Wisconsin Press, Madison, 1960), pp. 97–120 10. T. Kato, Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972) 11. T. Kato, Lp -theory of Schrödinger operators with a singular potential, in Aspects of Positivity in Functional Analysis, ed. by R. Nagel, U. Schlotterbeck, M.P.H. Wolf, vol. 122 (Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1986), pp. 63–78. North-Holland Math. Stud. 12. G. Lumer, R.S. Phillips, Dissipative operators in a Banach space. Pacific J. Math. 11, 679–698 (1961) 13. R. Nagel, H. Uhlig, An abstract Kato inequality for generators of positive operators semigroups on Banach lattices. J. Oper. Theory 6, 113–123 (1981) 14. O.A. Ole˘ınik, On hypoellipticity of second order equations, in Partial Differential Equations, ed. by D.C. Spencer, vol. XXXIII of Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, 1973), pp. 145–151 15. E.M. Ouhabaz, Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31 (Princeton University Press, Princeton, 2005) 16. G. Schreieck, J. Voigt, Stability of the Lp -spectrum of Schrödinger operators with form-small negative part of the potential, in Functional Analysis (Essen, 1991) (Dekker, New York, 1994), pp. 95–105 17. B. Simon, An abstract Kato’s inequality for generators of positivity preserving semigroups. Indiana Univ. Math. J. 26, 1067–1073 (1977) 18. A.F.M. ter Elst, J. Rehberg, Consistent operator semigroups and their interpolation. J. Oper. Theory 82, 3–21 (2019)

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19. B. Wong-Dzung, Lp -Theory of degenerate-elliptic and parabolic operators of second order. Proc. Roy. Soc. Edinburgh Sect. A 95, 95–113 (1983) 20. K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, vol. 123, 6th edn. (Springer-Verlag, New York 1980)

Generalized Indefinite Strings with Purely Discrete Spectrum Jonathan Eckhardt and Aleksey Kostenko

Dedicated to the memory of Sergey Nikolaevich Naboko (1950–2020)

Abstract We establish criteria for the spectrum of a generalized indefinite string to be purely discrete and to satisfy Schatten–von Neumann properties. The results can be applied to the isospectral problem associated with the conservative Camassa– Holm flow and to Schrödinger operators with .δ  -interactions. Keywords Generalized indefinite strings · Purely discrete spectrum · Schatten–von Neumann ideals

1 Introduction In this article, we are concerned with the spectral problem .

− f  = z ωf + z2 υf

(1.1)

for a generalized indefinite string .(L, ω, υ). This means that .0 < L ≤ ∞, .ω is −1 a real distribution in .Hloc [0, L) and .υ is a non-negative Borel measure on .[0, L). Spectral problems of this form arise as Lax (isospectral) operators in the study of various nonlinear completely integrable systems (most notably the Camassa–Holm

J. Eckhardt Department of Mathematical Sciences, Loughborough University, Leicestershire, UK e-mail: [email protected] A. Kostenko () Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia Faculty of Mathematics, University of Vienna, Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_16

435

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J. Eckhardt and A. Kostenko

equation [5], where finite time blow-up makes it necessary to allow coefficients of low regularity in (1.1); see [9, 10]). The particular case of (1.1) when the measure .υ vanishes identically and .ω is a non-negative Borel measure on .[0, L) is known as a Krein string and has a venerable history [7, 20, 24]. To the best of our knowledge, spectral problems of the form (1.1) with non-trivial coefficients .υ first appeared in the work of M. G. Krein and H. Langer [25], [26] in their study of indefinite analogues of the moment problem. In the above generality, this spectral problem was introduced quite recently in [12]. Similar to Krein strings, which serve as a canonical model for operators with non-negative simple spectrum (roughly speaking, an arbitrary self-adjoint operator in a separable Hilbert space with non-negative simple spectrum is unitarily equivalent to a Krein string), the spectral problem (1.1) is another canonical model (two other famous models are Jacobi matrices and .2 × 2 canonical systems) for self-adjoint operators with simple spectrum; see [12]. The purpose of this article is to address the question under which conditions on the coefficients the spectrum .σ of (1.1) is purely discrete (that is, consists of isolated eigenvalues without finite accumulation points) or satisfies .

 1 1. In the case of Krein strings, the corresponding results have been known since the late 1950s. The discreteness criterion was first established by I. S. Kac and M. G. Krein in [19], which has been published only in Russian. Due to positivity, one can employ a variational reformulation, which allows to reduce the question about discreteness to the study of the embedding of a form domain into the initial Hilbert space (compare [27]). In this context, the Kac–Krein criterion is related to the Muckenhoupt inequalities [28]. Removing the positivity assumption makes the corresponding considerations much more complicated. For instance, despite its fundamental importance, a discreteness criterion for .2 × 2 canonical systems was found only recently by R. Romanov and H. Woracek [30] (see also [29]). Surprisingly enough (at least to the authors), a discreteness criterion for indefinite strings (the case when the measure .υ vanishes identically and .ω is a real-valued Borel measure on .[0, L)) has essentially been available since the 1970s, when C. A. Stuart [33] established a compactness criterion for integral operators in the Hilbert space .L2 [0, ∞) of the form  ∞ .J : f → q(max( · , t))f (t)dt, (1.3) 0

for a function .q in .L2loc [0, ∞). This is because the operator .J is closely related (see Sect. 3) to the resolvent of the indefinite string when .q is an anti-derivate of .ω. Due

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to this connection, criteria for the spectrum .σ to satisfy (1.2) follow from criteria for the operator .J to belong to the Schatten–von Neumann class .Sp . Such criteria have been established in the work of A. B. Aleksandrov, S. Janson, V. V. Peller, and R. Rochberg [2]. This connection will play a crucial role in our approach to obtain discreteness criteria for generalized indefinite strings. However, let us mention that the discreteness criteria as well as the criteria when the spectrum .σ of a generalized indefinite string satisfies (1.2) can also be obtained by using the recent results of [30], as there is a bijective correspondence between generalized indefinite strings and .2 × 2 canonical systems; see [12, Section 6]. On the other hand, one may look at our results as an alternative way to obtain discreteness criteria for .2 × 2 canonical systems. In conclusion, let us sketch the content of the article. Section 2 is of preliminary character and collects necessary notions and facts on the spectral theory of generalized indefinite strings. In Sect. 3 we are concerned with the resolvent at zero energy of the spectral problem .

− f  = zχf

(1.4)

−1 when .χ is a distribution in .Hloc [0, L). This operator turns out to be closely related to an integral operator of the form (1.3), which allows us to translate several known results from [2, 6, 33]. In the following section, we will then introduce a quadratic operator pencil associated with a generalized indefinite string, which enables us to connect the spectral problem (1.1) with the operators studied in Sect. 3. This connection allows us to derive our main results in Sect. 5; a number of discreteness criteria for generalized indefinite strings. Consequently, in Sect. 6 we apply our findings to the isospectral problem of the conservative Camassa–Holm flow

.

1 − f  + f = z ωf + z2 υf. 4

(1.5)

Section 7 provides another application of our results to one-dimensional Hamiltonians with .δ  -interactions. Finally, in Appendix A we gather a number of known results from [2, 6, 33] about integral operators of the form (1.3) in a way that makes them convenient for us to apply. Throughout this article, we adopt the point of view of linear relations when dealing with linear operators. For the convenience of the reader, we summarize basic notions about linear relations in Appendix B.

Notation Let us first introduce several spaces of functions and distributions. For .L ∈ (0, ∞], we denote with .L2loc [0, L), .L2 [0, L) and .L2c [0, L) the spaces of locally square integrable functions, square integrable functions and square integrable functions

438

J. Eckhardt and A. Kostenko

with compact support in .[0, L), respectively. The space .L˙ 2c [0, L) consists of all functions f in .L2c [0, L) with zero mean, that is, such that  .

L

f (x)dx = 0.

(1.6)

0

When L is finite, the space .L˙ 2 [0, L) can be defined in a similar way. Furthermore, 1 [0, L), .H 1 [0, L) and .H 1 [0, L) the usual Sobolev spaces we denote with .Hloc c 1 Hloc [0, L) = {f ∈ ACloc [0, L) | f  ∈ L2loc [0, L)}, .

(1.7)

1 H 1 [0, L) = {f ∈ Hloc [0, L) | f, f  ∈ L2 [0, L)}, .

(1.8)

Hc1 [0, L) = {f ∈ H 1 [0, L) | supp(f ) compact in [0, L)}.

(1.9)

.

−1 The space of distributions .Hloc [0, L) is the topological dual space of .Hc1 [0, L). We note that the mapping .q → χ , defined by



L

χ (h) = −

.

q(x)h (x)dx,

0

h ∈ Hc1 [0, L),

(1.10)

−1 establishes a one-to-one correspondence between .L2loc [0, L) and .Hloc [0, L). The −1 2 [0, L) unique function .q ∈ Lloc [0, L) corresponding to some distribution .χ ∈ Hloc in this way will be referred to as the normalized anti-derivative of .χ . Finally, a −1 distribution in .Hloc [0, L) is said to be real if its normalized anti-derivative is realvalued almost everywhere on .[0, L). −1 A particular kind of distributions in .Hloc [0, L) arises from Borel measures on the interval .[0, L). More precisely, if .χ is a non-negative Borel measure on .[0, L), −1 [0, L) given by then we will identify it with the distribution in .Hloc

 h →

.

[0,L)

h dχ .

(1.11)

The normalized anti-derivative .q of such a .χ is simply given by the left-continuous distribution function  .q(x) = dχ (1.12) [0,x)

for almost all .x ∈ [0, L), as an integration by parts (use, for example, [4, Exercise 5.8.112] or [17, Theorem 21.67]) shows.

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In order to be able to introduce a self-adjoint realization of the differential equation (1.1) in a suitable Hilbert space later, we also define the function space  ˙1

H [0, L) =

.

1 [0, L) | f  ∈ L2 [0, L), lim {f ∈ Hloc x→L f (x) = 0}, 1 [0, L) | f  Hloc

{f ∈

∈

L2 [0, L)},

L < ∞, L = ∞, (1.13)

as well as its linear subspace H˙ 01 [0, L) = {f ∈ H˙ 1 [0, L) | f (0) = 0},

.

(1.14)

which turns into a Hilbert space when endowed with the scalar product  f, g H˙ 1 [0,L) =

L

.

0

f  (x)g  (x)∗ dx,

0

f, g ∈ H˙ 01 [0, L),

(1.15)

where we use a star to denote complex conjugation. The space .H˙ 01 [0, L) can be viewed as a completion of the space of all smooth functions which have compact support in .(0, L) with respect to the norm induced by (1.15). In particular, the space ˙ 1 [0, L) coincides algebraically and topologically with the usual Sobolev space .H 0 1 ˙ 1 [0, L) are not necessarily bounded, but .H [0, L) when L is finite. Functions in .H 0 0 they satisfy the simple growth estimate  x f 2H˙ 1 [0,L) , .|f (x)| ≤ x 1 − L 0 

2

x ∈ [0, L), f ∈ H˙ 01 [0, L).

(1.16)

Here we employ the convention that whenever an L appears in a denominator, the corresponding fraction has to be interpreted as zero if L is not finite.

2 Generalized Indefinite Strings A generalized indefinite string is a triple .(L, ω, υ) such that .L ∈ (0, ∞], .ω −1 is a real distribution in .Hloc [0, L) and .υ is a non-negative Borel measure on .[0, L). Associated with such a generalized indefinite string is the inhomogeneous differential equation .

− f  = z ωf + z2 υf + χ ,

(2.1)

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−1 where .χ is a distribution in .Hloc [0, L) and z is a complex spectral parameter. Of course, this differential equation has to be understood in a weak sense: A solution 1 [0, L) such that of (2.1) is a function .f ∈ Hloc

f  (0−)h(0) +



L

.

f  (x)h (x)dx = z ω(f h) + z2

0

 [0,L)

f h dυ + χ (h)

(2.2)

for all .h ∈ Hc1 [0, L) and a (unique) constant .f  (0−) ∈ C. All differential equations in this article will be of the form (2.1) and we only refer to [12] for further details. Each generalized indefinite string .(L, ω, υ) gives rise to a self-adjoint linear relation .T in the Hilbert space H = H˙ 01 [0, L) × L2 ([0, L); υ),

(2.3)

.

which is endowed with the scalar product  f, g H =

L

.

0

f1 (x)g1 (x)∗ dx +

 [0,L)

f2 (x)g2 (x)∗ dυ(x),

f, g ∈ H.

(2.4)

Here we denote the respective components of some vector .f ∈ H by adding subscripts, that is, with .f1 and .f2 . Definition 2.1 The linear relation .T in the Hilbert space .H is defined by saying that some pair .(f, g) ∈ H × H belongs to .T if and only if .

− f1 = ωg1 + υg2 ,

υf2 = υg1 .

(2.5)

In order to be precise, we point out that the right-hand side of the first equation −1 in (2.5) has to be understood as the .Hloc [0, L) distribution  h → ω(g1 h) +

.

[0,L)

g2 h dυ

(2.6)

and that the second equation means that .f2 is equal to .g1 almost everywhere on [0, L) with respect to the measure .υ. The linear relation .T defined in this way turns out to be self-adjoint in the Hilbert space .H; see [12, Section 4] for more details. Later on, we will use the following description of the resolvent of .T that can be found in [12, Proposition 4.3]. To this end, we recall that the Wronski determinant .W (ψ, φ) of two solutions .ψ, .φ of the homogeneous differential equation .

.

− f  = z ωf + z2 υf

(2.7)

is defined as the unique number such that W (ψ, φ) = ψ(x)φ  (x) − ψ  (x)φ(x)

.

(2.8)

Generalized Indefinite Strings with Purely Discrete Spectrum

441

for almost all .x ∈ [0, L). It is known that the Wronskian .W (ψ, φ) is non-zero if and only if the functions .ψ and .φ are linearly independent; see [12, Corollary 3.3]. Proposition 2.2 If z belongs to the resolvent set of .T, then one has .

z(T − z)

−1

    1 1 g(x) = g, G(x, · ) H − g1 (x) , z 0 ∗

x ∈ [0, L),

(2.9)

for every .g ∈ H, where the Green’s function .G is given by    1 ψ(x)φ(t), t ∈ [0, x), 1 .G(x, t) = z W (ψ, φ) ψ(t)φ(x), t ∈ [x, L),

(2.10)

and .ψ, .φ are linearly independent solutions of the homogeneous differential equation (2.7) such that .φ vanishes at zero, .ψ lies in .H˙ 1 [0, L) and .zψ lies in 2 .L ([0, L); υ). For the sake of simplicity, we shall always mean the spectrum of the corresponding linear relation .T when we refer to the spectrum .σ of a generalized indefinite string .(L, ω, υ). The same convention also applies to the various spectral types. In particular, we say that the spectrum .σ of a generalized indefinite string .(L, ω, υ) is purely discrete if the linear relation .T has purely discrete spectrum.

3 Some Integral Operators in H˙ 01 [0, L) −1 Throughout this section, let .χ be a distribution in .Hloc [0, L) and denote with .q its normalized anti-derivative. We introduce the linear relation .Kχ in .H˙ 01 [0, L) by defining that a pair .(g, f ) ∈ H˙ 01 [0, L) × H˙ 01 [0, L) belongs to .Kχ if and only if .

− f  = χg,

(3.1)

where equality again has to be understood in a distributional sense. Proposition 3.1 The linear relation .Kχ is (the graph of) a densely defined closed linear operator with core .H˙ 01 [0, L) ∩ Hc1 [0, L) such that Kχ g(x) = χ (δx g),

.

x ∈ [0, L),

(3.2)

for all .g ∈ H˙ 01 [0, L) ∩ Hc1 [0, L), where the kernel function .δx is defined by   max(x, t) , δx (t) = min(x, t) 1 − L

.

x, t ∈ [0, L).

(3.3)

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J. Eckhardt and A. Kostenko

Proof We define the linear operator .Kχ ,0 with domain .H˙ 01 [0, L) ∩ Hc1 [0, L) by 

L

Kχ ,0 h(x) = χ (δx h) = −

.

q(t)(δx h) (t)dt,

x ∈ [0, L),

0

for .h ∈ H˙ 01 [0, L) ∩ Hc1 [0, L). In order to verify that the function .Kχ ,0 h indeed belongs to .H˙ 01 [0, L), one first computes that  χ (δx h) =

x



t

.

0

q(s)h (s)ds dt −

0



x

q(t)h(t)dt 0



L

−x

x L

q(t)h (t)dt +

0



L

q(t)(th (t) + h(t))dt

0

for all .x ∈ [0, L). This shows that the function .x → χ (δx h) is locally absolutely continuous on .[0, L) with square integrable derivative given by  .

L

− q(x)h(x) − x

1 q(t)h (t)dt + L 



L

q(t)(th (t) + h(t))dt

(3.4)

0

for almost every .x ∈ [0, L). Since it also shows that .χ (δx h) → 0 as .x → L when L is finite and that .χ (δ0 h) = 0, we conclude that .Kχ ,0 h belongs to .H˙ 01 [0, L). By using the expression in (3.4) for the derivative of .Kχ ,0 h, one verifies that .

− (Kχ ,0 h) = χ h

in a distributional sense, which implies that (the graph of) .Kχ ,0 is contained in .Kχ . Since the operator .Kχ ,0 is densely defined, its adjoint .K∗χ ,0 is a closed linear operator in .H˙ 01 [0, L). Hence, for the remaining claims it suffices to prove that K∗χ ,0 ⊆ Kχ ∗ ⊆ K∗χ ⊆ K∗χ ,0 ,

.

where the last inclusion is evident as .Kχ ,0 ⊆ Kχ . In order to verify the first inclusion, suppose that .(g, f ) ∈ K∗χ ,0 and let .h ∈ H˙ 01 [0, L) ∩ Hc1 [0, L). From the expression for the derivative of .Kχ ,0 h in (3.4) and an integration by parts, we get 

L

.

0

f  (t)h (t)∗ dt = f, h H˙ 1 [0,L) = g, Kχ ,0 h H˙ 1 [0,L) 0

 = lim

x→L 0

0

x

g  (t)(Kχ ,0 h) (t)∗ dt



x

= lim − x→L

0

q(t)∗ (gh∗ ) (t)dt = χ ∗ (gh∗ ).

Generalized Indefinite Strings with Purely Discrete Spectrum

443

As it does not matter that all our test functions h vanish at zero, this shows that the pair .(g, f ) belongs to .Kχ ∗ . For the second inclusion, we need to prove that f, g∗ H˙ 1 [0,L) = g, f∗ H˙ 1 [0,L)

.

0

0

when .(g∗ , f∗ ) ∈ Kχ ∗ and .(g, f ) ∈ Kχ . To this end, we first note that the respective differential equations that the pairs .(g∗ , f∗ ) and .(g, f ) satisfy entail that  ∗ .f∗ (x) + q(x) g∗ (x)

 = d∗ +

f  (x) + q(x)g(x) = d +



x

0 x

q(t)∗ g∗ (t)dt =: f∗[1] (x),

q(t)g  (t)dt =: f [1] (x),

0

for almost all .x ∈ [0, L) and some constants d, .d∗ ∈ C; see [12, Equation (3.6)]. Integration by parts then gives 

x

f

.

0



(t)g∗ (t)∗ dt



x

− 0

x g  (t)f∗ (t)∗ dt = f [1] (t)g∗ (t)∗ − g(t)f∗[1] (t)∗ t=0

for every .x ∈ [0, L). This clearly implies that f, g∗ H˙ 1 [0,L) − g, f∗ H˙ 1 [0,L) = lim f [1] (x)g∗ (x)∗ − g(x)f∗[1] (x)∗

.

0

x→L

0

and we are left to verify that the limit (which is already known to exist) is zero. However, this follows from the fact that the function

.

|f [1] (x)g∗ (x)∗ − g(x)f∗[1] (x)∗ |2 |f  (x)g∗ (x)∗ − g(x)f∗ (x)∗ |2   = x x 1− L x 1 − Lx  

is integrable near L due to the estimate in (1.16) applied to .g∗ and g. In the course of the proof of Proposition 3.1, we found the adjoint of .Kχ . Corollary 3.2 The adjoint of the operator .Kχ is given by K∗χ = Kχ ∗ .

.

(3.5)

In particular, the operator .Kχ is self-adjoint when the distribution .χ is real. An inspection of the definition of the operator .Kχ also proves the following. Corollary 3.3 For all functions g, .h ∈ H˙ 01 [0, L) ∩ Hc1 [0, L) one has Kχ g, h H˙ 1 [0,L) = χ (gh∗ ).

.

0

(3.6)

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The next result about boundedness of the operator .Kχ when .χ is a non-negative Borel measure on .[0, L) will be useful in Sect. 4. Corollary 3.4 Suppose that .χ is a non-negative Borel measure on .[0, L). The operator .Kχ is bounded if and only if the inclusion .Iχ : H˙ 01 [0, L) → L2 ([0, L); χ ) is bounded. In this case, the adjoint of .Iχ is given by ∗ .Iχ g(x)

 =

x ∈ [0, L),

δx g dχ ,

[0,L)

(3.7)

for all functions .g ∈ L2 ([0, L); χ ) and one has .Kχ = I∗χ Iχ . Proof For every .h ∈ H˙ 01 [0, L) ∩ Hc1 [0, L), we get from Corollary 3.3 that  Iχ h, Iχ h L2 ([0,L);χ ) =

.

[0,L)

|h|2 dχ = χ (hh∗ ) = Kχ h, h H˙ 1 [0,L) . 0

Since .Kχ is self-adjoint, it follows that the operator .Kχ is bounded if and only if the inclusion .Iχ is bounded. In this case, the above equality also shows that .Kχ = I∗χ Iχ and hence it remains to note that the adjoint of .Iχ is given by I∗χ g(x) = I∗χ g, δx H˙ 1 [0,L) = g, Iχ δx L2 ([0,L);χ ) =



.

0

[0,L)

δx g dχ ,

x ∈ [0, L),

for all functions .g ∈ L2 ([0, L); χ ).

 

The operator .Kχ turns out to be unitarily equivalent to a particular integral operator in .L2 [0, L), which allows us to readily translate the boundedness and compactness criteria collected in Appendix A. For simplicity, we will state and prove the cases of an unbounded interval and a bounded interval separately. Theorem 3.5 Suppose that L is not finite. The following assertions hold true: (i) The operator .Kχ is bounded if and only if there is a constant .c ∈ C such that  .

∞

lim sup x x→∞

|q(t) − c|2 dt < ∞.

(3.8)

x

In this case, the constant c is given by 1 x→∞ x



c = lim

.

x

q(t)dt.

(3.9)

0

(ii) The operator .Kχ is compact if and only if there is a constant .c ∈ C such that  .

∞

lim x

x→∞

x

|q(t) − c|2 dt = 0.

(3.10)

Generalized Indefinite Strings with Purely Discrete Spectrum

445

(iii) For each .p > 1, the operator .Kχ belongs to the Schatten–von Neumann class .Sp if and only if there is a constant .c ∈ C such that ∞





∞

x

.

0

p/2 |q(t) − c| dt 2

x

dx < ∞. x

(3.11)

(iv) If the operator .Kχ belongs to the Hilbert–Schmidt class .S2 , then its Hilbert– Schmidt norm is given by  Kχ S = 2

∞

2

.

2

x|q(x) − c|2 dx,

(3.12)

0

where the constant c is given by (3.9). (v) If the operator .Kχ belongs to the trace class .S1 , then ∞





∞

x

.

0

1/2 |q(t) − c| dt 2

x

dx < ∞, x

(3.13)

the function .q − c is integrable and the trace of .Kχ is given by 

∞

tr Kχ =

.

(c − q(x))dx,

(3.14)

0

where the constant c is given by (3.9). Proof We first note that the map .U : f → f  is unitary from .H˙ 01 [0, ∞) to .L2 [0, ∞) with inverse simply given by −1

U

.



x

f (x) =

f (t)dt,

x ∈ [0, ∞).

0

If f belongs to .L˙ 2c [0, ∞), then .U−1 f belongs to .H˙ 01 [0, ∞) ∩ Hc1 [0, ∞) and from the expression in (3.4) for the derivative of .Kχ h when .h ∈ H˙ 01 [0, ∞) ∩ Hc1 [0, ∞) we get  x  UKχ U−1 f (x) = (Kχ U−1 f ) (x) = −q(x) f (t)dt −

∞

q(t)f (t)dt = −Jf (x)

.

0

x

for almost every .x ∈ [0, ∞), where .J is the integral operator in .L2 [0, ∞) defined in Appendix A. Now the claims follow from Theorem A.1.   With the connection established in the proof of Theorem 3.5, the criteria from Theorem A.3 become readily available as well. Theorem 3.6 Suppose that L is not finite and that .χ is a non-negative Borel measure on .[0, ∞). The following assertions hold true:

446

J. Eckhardt and A. Kostenko

(i) The operator .Kχ is bounded if and only if  .

lim sup x

dχ < ∞.

(3.15)

dχ = 0.

(3.16)

[x,∞)

x→∞

(ii) The operator .Kχ is compact if and only if  .

lim x

x→∞

[x,∞)

(iii) For each .p > 1/2, the operator .Kχ belongs to the Schatten–von Neumann class .Sp if and only if 

∞

p

 x

.

0

[x,∞)

dχ

dx < ∞. x

(3.17)

(iv) If the operator .Kχ belongs to the trace class .S1 , then its trace is given by  tr Kχ =

.

[0,∞)

x dχ (x).

(3.18)

(v) If the operator .Kχ belongs to the Schatten–von Neumann class .S1/2 , then the measure .χ is singular with respect to the Lebesgue measure. In a similar way, it is also possible to obtain boundedness and compactness criteria for the operator .Kχ when the interval .[0, L) is bounded. Theorem 3.7 Suppose that L is finite. The following assertions hold true: (i) The operator .Kχ is bounded if and only if  .

x

lim sup (L − x)

|q(t)|2 dt < ∞.

(3.19)

0

x→L

(ii) The operator .Kχ is compact if and only if  .

x

lim (L − x)

x→L

|q(t)|2 dt = 0.

(3.20)

0

(iii) For each .p > 1, the operator .Kχ belongs to the Schatten–von Neumann class .Sp if and only if L





x

(L − x)

.

0

0

p/2 |q(t)| dt 2

dx < ∞. L−x

(3.21)

Generalized Indefinite Strings with Purely Discrete Spectrum

447

(iv) If the operator .Kχ belongs to the Hilbert–Schmidt class .S2 , then its Hilbert– Schmidt norm is bounded by  Kχ 2S ≤ 2

L

.

2

(L − x)|q(x)|2 dx.

(3.22)

0

(v) If the operator .Kχ belongs to the trace class .S1 , then L





(L − x)

.

0

1/2

x

|q(t)| dt 2

0

dx < ∞, L−x

(3.23)

the function .q is integrable and the trace of .Kχ is given by L  2x

 tr Kχ =

.

0

L

 − 1 q(x)dx.

(3.24)

Proof We first note that the map .U : f → f  is unitary from .H˙ 01 [0, L) to .L˙ 2 [0, L) with inverse simply given by U−1 f (x) = −



L

x ∈ [0, L).

f (t)dt,

.

x

If f belongs to .L˙ 2c [0, L), then .U−1 f belongs to .H˙ 01 [0, L) ∩ Hc1 [0, L) and from the expression in (3.4) for the derivative of .Kχ h when .h ∈ H˙ 01 [0, L) ∩ Hc1 [0, L) we get −1

UKχ U

.

 f (x) =

L

x

1 (q(x) − q(t))f (t)dt + L



L

  q(t) tf (t) −

0



L

f (s)ds dt t

for almost every .x ∈ [0, L). The first term on the right-hand side becomes 

L

.



L

(q(x) − q(t))f (t)dt = q(x)

x



x

f (t)dt +



0

x

L

q(t)f (t)dt −

q(t)f (t)dt. 0

Furthermore, if .JL is the integral operator defined in (A.14) with .qL = q, then an integration by parts gives L  t

 JL f, 1 L2 [0,L) =

.

0

 q(s)f (s)ds + q(t)

0



L

=− 0

f (s)ds dt t

  q(t) tf (t) −

L t



L





L

f (s)ds dt + L

q(t)f (t)dt, 0

448

J. Eckhardt and A. Kostenko

so that in combination we conclude that UKχ U−1 f = JL f −

.

1 JL f, 1 L2 [0,L) = PJL Pf, L

(3.25)

where .P : L2 [0, L) → L˙ 2 [0, L) is the orthogonal projection onto .L˙ 2 [0, L). This implies that the operator .Kχ is bounded if and only if the operator .JL is bounded. In fact, boundedness of .JL clearly entails boundedness of .Kχ . On the other side, if the operator .Kχ is bounded, then we may conclude from (3.25) that the linear functional .f → JL f, 1 L2 [0,L) defined on the domain .L˙ 2c [0, L) is closable because the integral operator .JL is closable (see [16, Theorem 3.8] for example). Since this implies that the linear functional is bounded, we infer from (3.25) that the operator ˙ 2c [0, L). Finally, as the subspace .L˙ 2c [0, L) has codimension one .JL is bounded on .L 2 in the domain .Lc [0, L) of .JL , we may conclude that the operator .JL is bounded. Now the claims follow from Theorem A.4. In particular, one has tr Kχ = tr JL −

.

1 JL 1, 1 L2 [0,L) = L



L

q(x)dx −

0

2 L



L

(L − x)q(x)dx,

0

 

which gives the required trace formula in (3.24).

The connection established in the proof of Theorem 3.7 again also makes the boundedness and compactness criteria from Theorem A.5 available. Theorem 3.8 Suppose that L is finite and that .χ is a non-negative Borel measure on .[0, L). The following assertions hold true: (i) The operator .Kχ is bounded if and only if  .

lim sup (L − x)

[0,x)

x→L

dχ < ∞.

(3.26)

(ii) The operator .Kχ is compact if and only if  .

lim (L − x)

x→L

[0,x)

dχ = 0.

(3.27)

(iii) For each .p > 1/2, the operator .Kχ belongs to the Schatten–von Neumann class .Sp if and only if L



(L − x)

.

0

p

 [0,x)

dχ

dx < ∞. L−x

(3.28)

Generalized Indefinite Strings with Purely Discrete Spectrum

449

(iv) If the operator .Kχ belongs to the trace class .S1 , then its trace is given by  tr Kχ =

.

  x dχ (x). x 1− L [0,L)

(3.29)

(v) If the operator .Kχ belongs to the Schatten–von Neumann class .S1/2 , then the measure .χ is singular with respect to the Lebesgue measure.

4 Quadratic Operator Pencils We are now going to establish a connection between the integral operators from the previous section and a generalized indefinite string .(L, ω, υ). To this end, we first introduce an associated quadratic operator pencil .S in the Hilbert space .H˙ 01 [0, L) as follows: For every .z ∈ C, the linear relation .S(z) in .H˙ 01 [0, L) is defined by requiring that a pair .(f, g) ∈ H˙ 01 [0, L) × H˙ 01 [0, L) belongs to .S(z) if and only if .

− f  = z ωf + z2 υf − g  .

(4.1)

−1 In order to be precise, we interpret .−g  here as the .Hloc [0, L) distribution

 h →

.

L

g  (x)h (x)dx.

(4.2)

0

Proposition 4.1 For each .z ∈ C, the linear relation .S(z) is (the graph of) a densely defined closed linear operator with core .H˙ 01 [0, L) ∩ Hc1 [0, L) and S(z) = I − Kzω+z2 υ .

(4.3)

.

Proof By comparing the definition of the linear relation .S(z) with the definition of the operator .Kzω+z2 υ , we see that some pair .(f, g) ∈ H˙ 01 [0, L) × H˙ 01 [0, L) belongs to .S(z) if and only if .(f, f − g) belongs to .Kzω+z2 υ , which shows (4.3). Now the claims follow readily from Proposition 3.1.   We refer to .S as a quadratic operator pencil because one can show that S(z) = I − zKω − z2 Kυ ,

.

z ∈ C.

(4.4)

From the relation (4.3) and Corollary 3.2, we are able to determine the adjoint. Corollary 4.2 For each .z ∈ C, the adjoint of the operator .S(z) is given by S(z)∗ = S(z∗ ).

.

In particular, the operator .S(z) is self-adjoint when z is real.

(4.5)

450

J. Eckhardt and A. Kostenko

Since the operator pencil .S and the linear relation .T both arise from the same differential equation, it is not surprising that they are closely related. Proposition 4.3 If z belongs to the resolvent set of .T, then the operator .S(z) has an everywhere defined bounded inverse and  S(z)−1 = P z(T − z)−1 + I P∗ ,

.

(4.6)

where .P denotes the projection from .H onto .H˙ 01 [0, L). Proof We are going to show first that for every .z ∈ C one has  S(z)−1 ⊇ {(g, f ) ∈ S(z)−1 | f ∈ L2 ([0, L); υ)} ⊇ P z(T − z)−1 + I P∗ ,

.

(4.7)

where the right-hand side should be understood as a product In  of linear relations. fact, if a pair .(g, f ) ∈ H˙ 01 [0, L) × H˙ 01 [0, L) belongs to .P z(T − z)−1 + I P∗ , then there is an .h ∈ H with .P h = f such that .(P∗ g, h) belongs to .z(T − z)−1 + I. This implies that .(h − P∗ g, zh) belongs to .T and hence .

− (h1 − g) = zωh1 + zυh2 ,

υh2 = zυh1 ,

(4.8)

by Definition 2.1, which shows that .(f, g) belongs to .S(z). If z is not zero, then we conclude from the second equation in (4.8) that .f ∈ L2 ([0, L); υ). Otherwise, when z is zero, one notes that .P∗ g = h belongs to the range of .T, which shows again that .f ∈ L2 ([0, L); υ) in view of Definition 2.1. The converse inclusion  {(g, f ) ∈ S(z)−1 | f ∈ L2 ([0, L); υ)} ⊆ P z(T − z)−1 + I P∗

.

only holds for non-zero .z ∈ C in general. In order to prove it, we suppose that a pair .(f, g) belongs to .S(z) such that .f ∈ L2 ([0, L); υ). The definition of .S(z) then shows that (4.8) holds with .h ∈ H given by .h1 = f and .h2 = zf . It follows that ∗ ∗ −1 + I, which .(h − P g, zh) belongs to .T and hence .(zP g, zh) belongs to .z(T − z)  ∗ −1 shows that .(zg, zf ) belongs to .P z(T − z) + I P and because z is not zero, so does the pair .(g, f ). Finally, if z belongs to the resolvent set of .T, then so does .z∗ and we get  S(z)−1 = (S(z∗ )∗ )−1 = (S(z∗ )−1 )∗ ⊆ P z(T − z)−1 + I P∗

.

from (4.7), which yields (4.6) as well as the remaining claims.

 

Remark 4.4 We have seen in the proof of Proposition 4.3 that  S(z)−1 ⊇ {(g, f ) ∈ S(z)−1 | f ∈ L2 ([0, L); υ)} = P z(T − z)−1 + I P∗

.

(4.9)

holds as long as z is not zero. The inclusion is indeed strict in some cases. For example, if the measure .υ is such that .H˙ 01 [0, L) is not contained in .L2 ([0, L); υ)

Generalized Indefinite Strings with Purely Discrete Spectrum

451

and we take .ω = υ, then .S(−1) is simply the identity operator, whereas the middle part in (4.9) becomes its restriction to functions which belong to .L2 ([0, L); υ). This connection with the linear relation .T allows us to find a description of the inverse of .S(z) via the resolvent of .T when z belongs to the resolvent set of .T. Corollary 4.5 If z belongs to the resolvent set of .T, then one has S(z)−1 g(x) = g, G(x, · )∗ H˙ 1 [0,L) ,

.

0

x ∈ [0, L),

(4.10)

for every .g ∈ H˙ 01 [0, L), where the Green’s function G is given by  1 ψ(x)φ(t), t ∈ [0, x), .G(x, t) = W (ψ, φ) ψ(t)φ(x), t ∈ [x, L),

(4.11)

and .ψ, .φ are linearly independent solutions of the homogeneous differential equation (2.7) such that .φ vanishes at zero, .ψ lies in .H˙ 1 [0, L) and .zψ lies in 2 .L ([0, L); υ). Proof The claimed representation for the inverse of .S(z) when z belongs to the   resolvent set of .T follows immediately from Propositions 4.3 and 2.2. Under some additional assumptions on the coefficients, we can say much more about the relation between the operator pencil .S and the linear relation .T. Theorem 4.6 Zero belongs to the resolvent set of .T if and only if the operators .Kω and .Kυ are bounded. In this case, the following assertions hold true: (i) For all .z ∈ C one has S(z) = I − zKω − z2 Kυ .

(4.12)

 S(z)−1 = P z(T − z)−1 + I P∗ ,

(4.13)

.

(ii) For all .z ∈ C one has .

where .P denotes the projection from .H onto .H˙ 01 [0, L). (iii) For each .z ∈ C, the operator .S(z) has an everywhere defined bounded inverse if and only if z belongs to the resolvent set of .T. In this case, one has (T − z)−1 =

.

    S(z)−1 0 Kω I∗υ I 0 I zI∗υ , 0 I Iυ 0 0 I zIυ I



where the inclusion .Iυ : H˙ 01 [0, L) → L2 ([0, L); υ) is bounded.

(4.14)

452

J. Eckhardt and A. Kostenko

(iv) The inverse of .T is given by T−1 =

.

  Kω I∗υ . Iυ 0

(4.15)

Proof Suppose first that zero belongs to the resolvent set of .T and let .f = T−1 g, where .g ∈ H with .g2 = 0. Definition 2.1 implies that .f2 = Iυ g1 ∈ L2 ([0, L); υ) and a comparison with the definition of .Kω reveals that .f1 = Kω g1 . From boundedness of .T−1 we then infer that Kω g1 2H˙ 1 [0,L) + Iυ g1 2L2 ([0,L);υ) = f 2H ≤ T−1 2 g2H

.

0

= T−1 2 g1 2H˙ 1 [0,L) . 0

This shows that the operator .Kω and the inclusion .Iυ are bounded, which also guarantees boundedness of the operator .Kυ in view of Corollary 3.4. On the other side, for .g ∈ H with .g1 = 0, Definition 2.1 implies that .f = T−1 g certainly satisfies −1 is self-adjoint, we conclude that it is given by (4.15). Item (i) .f2 = 0. Because .T follows readily from (4.3) and (3.2), whereas Item (ii) follows from (4.9) when z is not zero and from (4.6) when z is zero (since zero belongs to the resolvent set of .T). For non-zero .z ∈ C, the equivalence in Item (iii) follows from the Frobenius–Schur factorization     S(z) 0 I −zI∗υ I 0 −1 , .I − zT = 0 I 0 I −zIυ I which also yields the identity in (4.14) because −1  (T − z)−1 = I − zT−1 T−1 .

.

It only remains to note that Item (iii) also holds when z is zero. In order to complete the proof, we need to show that boundedness of the operators .Kω and .Kυ implies that zero belongs to the resolvent set of .T. Under these assumptions, the inclusion .Iυ is bounded by Corollary 3.4 so that we may set f =

.

    Kω g1 + I∗υ g2 Kω I∗υ g= Iυ 0 Iυ g1

for a given .g ∈ H. From the definition of .Kω and after computing that 

L

.

0

(I∗υ g2 ) (x)h (x)dx

=

I∗υ g2 , h∗ H˙ 1 [0,L) 0

∗

= g2 , Iυ h L2 ([0,L);υ) =

 [0,L)

g2 h dυ

Generalized Indefinite Strings with Purely Discrete Spectrum

453

for all functions .h ∈ H˙ 01 [0, L) ∩ Hc1 [0, L), we infer that .

− f1 = ωg1 + υg2 ,

υf2 = υg1 ,

which means that .(f, g) belongs to .T. Since g was arbitrary, this implies that the domain of the self-adjoint operator .T−1 is the whole space and thus .T−1 is bounded by the closed graph theorem, which guarantees that zero belongs to the resolvent set of .T.   The operator pencil .S is essentially a Schur complement of the block operator matrix in (4.15) and the factorization (4.14) is essentially a Frobenius–Schur decomposition. Corollary 4.7 If zero belongs to the resolvent set of .T, then the non-zero spectrum of .T−1 coincides with the non-zero spectrum of the block operator matrix1  √  Kυ Kω . √ Kυ 0

(4.16)

and all non-zero eigenvalues of the block operator matrix in (4.16) are simple. Proof The claim about the spectra follows from the Frobenius–Schur factorization 

Kω .I − z √ Kυ

√

Kυ 0





  √  I −z Kυ S(z) 0 I 0 √ = 0 I 0 I −z Kυ I

and Theorem 4.6 (iii). If .λ is a non-zero eigenvalue of the block operator matrix in (4.16) with eigenvector .f ∈ H˙ 01 [0, L) × H˙ 01 [0, L), then one readily finds that S(λ−1 )f1 = 0,

.

f2 = λ−1 Kυ f1 .

However, the kernel of .S(z) is at most one-dimensional as it consists of solutions of the homogeneous differential equation (2.7) that vanish at zero. We conclude that the eigenvector f is unique up to scalar multiples.   In view of the following section, let us point out that zero always belongs to the resolvent set of .T when the spectrum of .T is purely discrete. Remark 4.8 If the resolvent .(T − z)−1 is compact for some (and hence for all) z in the resolvent set of .T, then zero belongs to the resolvent set of .T. Indeed, the spectrum of .T consists only of isolated eigenvalues in this case and since the kernel of .T is trivial, it follows that zero can not be in the spectrum. Of course, compactness of the resolvent can be replaced by the weaker condition that zero does not belong to the essential spectrum of .T.

1 Here

√ and below, . Kυ always denotes the positive square root of the positive operator .Kυ .

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5 Purely Discrete Spectrum The main results of this article are a number of criteria for the spectrum .σ of a generalized indefinite string .(L, ω, υ) to be discrete and to satisfy .

 1 0, the spectrum .σ satisfies (5.1) if and only if the operator .Kω belongs to the Schatten–von Neumann class .Sp and the operator .Kυ belongs to the Schatten–von Neumann class .Sp/2 . (iv) If the spectrum .σ satisfies (5.1) with .p = 2, then .

 1 = Kω 2S + 2 tr Kυ . 2 λ2

(5.2)

λ∈σ

(v) If the spectrum .σ satisfies (5.1) with .p = 1, then .

1 = tr Kω . λ

(5.3)

λ∈σ

Proof Item (i) follows readily from Theorem 4.6 because the spectrum .σ of the generalized indefinite string .(L, ω, υ) is, by definition, the spectrum of the linear relation .T. For Item (ii), suppose first that the spectrum .σ is discrete. Because the kernel of .T is trivial, we infer that zero belongs to the resolvent set of .T. Corollary 4.7 then implies that the self-adjoint block operator matrix in (4.16) is compact and thus the operators .Kω and .Kυ are compact as well. On the other side, if we suppose that .Kω and .Kυ are compact, then zero belongs to the resolvent set of .T in view of (i) and it follows from Corollary 4.7 that the spectrum .σ is discrete because the block operator matrix in (4.16) is compact. In order to prove Item (iii), let us suppose first that the spectrum .σ satisfies (5.1) so that the spectrum .σ is discrete and zero belongs to the resolvent set of .T. Corollary 4.7 then implies that the self-adjoint block operator matrix in (4.16) √ belongs to the Schatten–von Neumann class .Sp and thus the operators .Kω and . Kυ belong to the Schatten–von Neumann class .Sp as well. This clearly entails that the

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455

operator .Kυ belongs to the Schatten–von Neumann class .Sp/2 . For the converse, we suppose that the operator .Kω belongs to the Schatten–von Neumann class .Sp and that the operator .Kυ belongs to the Schatten–von Neumann class .Sp/2 . The block operator matrix in (4.16) then belongs to the Schatten–von Neumann class .Sp because it is a sum  .

K √ω Kυ

√

Kυ 0



   √  Kυ Kω 0 0 = + √ Kυ 0 0 0

of two block operator matrices that belong to the Schatten–von Neumann class .Sp . It follows from Corollary 4.7 that the spectrum .σ satisfies (5.1). Finally, it remains to note that one has

.

  2 √  √ 2  1 Kυ Kω Kω + Kυ Kω Kυ √ √ = tr K2ω + 2 tr Kυ = tr = tr Kυ 0 Kυ Kω Kυ λ2 λ∈σ

if the spectrum .σ satisfies (5.1) with .p = 2 and .

 √  1 Kυ K = tr √ ω = tr Kω Kυ 0 λ λ∈σ

if the spectrum .σ satisfies (5.1) with .p = 1.

 

From Proposition 5.1 and the corresponding boundedness and compactness criteria for the operators .Kω and .Kυ in Sect. 3, we readily derive the remaining theorems in this section with more explicit conditions in terms of the coefficients. The normalized anti-derivative of the distribution .ω will be denoted with .w. Theorem 5.2 Suppose that L is not finite. The following assertions hold true: (i) Zero does not belong to the spectrum .σ if and only if there is a constant .c ∈ R such that  ∞  . lim sup x (w(t) − c)2 dt + x dυ < ∞. (5.4) x→∞

[x,∞)

x

In this case, the constant c is given by 1 .c = lim x→∞ x



x

w(t)dt.

(5.5)

0

(ii) The spectrum .σ is discrete if and only if there is a constant .c ∈ R such that  .

lim x

x→∞

x

∞

 (w(t) − c)2 dt + x

[x,∞)

dυ = 0.

(5.6)

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J. Eckhardt and A. Kostenko

(iii) For each .p > 1, the spectrum .σ satisfies (5.1) if and only if there is a constant .c ∈ R such that ∞





∞

x

.

0

p/2

 (w(t) − c)2 dt + x

x

[x,∞)

dυ

dx < ∞. x

(5.7)

(iv) If the spectrum .σ satisfies (5.1) with .p = 2, then  ∞   1 2 = 2 x(w(x) − c) dx + 2 x dυ(x), λ2 0 [0,∞)

.

(5.8)

λ∈σ

where the constant c is given by (5.5). (v) If the spectrum .σ satisfies (5.1) with .p = 1, then ∞





∞

x

.

0

1/2 (w(t) − c)2 dt

x

dx < ∞, x

(5.9)

where the constant c is given by (5.5), the function .w − c is integrable with 1  ∞ = (c − w(x))dx . λ 0

(5.10)

λ∈σ

and the measure .υ is singular with respect to the Lebesgue measure. One also obtains similar criteria for the case when L is finite. Theorem 5.3 Suppose that L is finite. The following assertions hold true: (i) Zero does not belong to the spectrum .σ if and only if  .

x

lim sup (L − x)

 w(t) dt + (L − x) 2

0

x→L

[0,x)

dυ < ∞.

(5.11)

(ii) The spectrum .σ is discrete if and only if  .

x

lim (L − x)

x→L

 w(t) dt + (L − x) 2

0

[0,x)

dυ = 0.

(5.12)

(iii) For each .p > 1, the spectrum .σ satisfies (5.1) if and only if 

L



x

(L − x)

.

0

0

p/2

 w(t) dt + (L − x) 2

[0,x)

dυ

dx < ∞. L−x

(5.13)

Generalized Indefinite Strings with Purely Discrete Spectrum

457

(iv) If the spectrum .σ satisfies (5.1) with .p = 2, then .

   L   1 x 2 dυ(x). ≤ 2 (L − x)w(x) dx + 2 x 1 − L λ2 0 [0,L)

(5.14)

λ∈σ

(v) If the spectrum .σ satisfies (5.1) with .p = 1, then 

L



(L − x)

.

1/2

x

2

w(t) dt

0

0

dx < ∞, L−x

(5.15)

the function .w is integrable with   1  L  2x = − 1 w(x)dx . λ L 0

(5.16)

λ∈σ

and the measure .υ is singular with respect to the Lebesgue measure. The special cases considered in the remaining two theorems in this section are known as Krein strings [7, 20, 24] in the literature. Theorem 5.4 Suppose that L is not finite, that .ω is a non-negative Borel measure on .[0, ∞) and that the measure .υ vanishes identically. The following assertions hold true: (i) Zero does not belong to the spectrum .σ if and only if  .

lim sup x

dω < ∞.

(5.17)

dω = 0.

(5.18)

[x,∞)

x→∞

(ii) The spectrum .σ is discrete if and only if  .

lim x

x→∞

[x,∞)

(iii) For each .p > 1/2, the spectrum .σ satisfies (5.1) if and only if 

∞

x

.

0

p

 [x,∞)

dω

dx < ∞. x

(5.19)

(iv) If the spectrum .σ satisfies (5.1) with .p = 1, then 1  = x dω(x). . λ [0,∞)

(5.20)

λ∈σ

(v) If the spectrum .σ satisfies (5.1) with .p = 1/2, then the measure .ω is singular with respect to the Lebesgue measure.

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Again, we also get criteria for Krein strings when L is finite. Theorem 5.5 Suppose that L is finite, that .ω is a non-negative Borel measure on [0, L) and that the measure .υ vanishes identically. The following assertions hold true:

.

(i) Zero does not belong to the spectrum .σ if and only if  .

lim sup (L − x)

[0,x)

x→L

dω < ∞.

(5.21)

(ii) The spectrum .σ is discrete if and only if  .

lim (L − x)

x→L

[0,x)

dω = 0.

(5.22)

(iii) For each .p > 1/2, the spectrum .σ satisfies (5.1) if and only if L



(L − x)

.

0

p

 [0,x)

dω

dx < ∞. L−x

(5.23)

(iv) If the spectrum .σ satisfies (5.1) with .p = 1, then   1  x = dω(x). . x 1− λ L [0,L)

(5.24)

λ∈σ

(v) If the spectrum .σ satisfies (5.1) with .p = 1/2, then the measure .ω is singular with respect to the Lebesgue measure. Remark 5.6 Items (i) and (ii) in Theorems 5.4 and 5.5 were first proved by Kac and Krein in [19] (see also [20]). Item (iii) in Theorems 5.4 and 5.5 is due to Kac (see [18, §3.2] for further details). The original approach of Kac and Krein is different from the one in [2] and it also provides quantitative bounds on the bottom of the essential spectrum. Positivity allows to employ variational techniques (for example, via embeddings of weighted .L2 and Sobolev spaces) in order to investigate discreteness of the spectrum and we refer to [27, §1.3.1], [28] for further details. Remark 5.7 It was observed by Krein in the 1950s that for Krein strings 1 #{λ ∈ σ | λ < n2 } = . lim n→∞ n π



L

ρ(x)dx,

(5.25)

0

where .ρ is the square root of the Radon–Nikodým derivative of .ω with respect to the Lebesgue measure. This, in particular, implies that the measure .ω is singular with respect to the Lebesgue measure if (5.1) holds with .p = 1/2. The study of eigenvalue distributions of strings with singular, or, more specifically, with self-

Generalized Indefinite Strings with Purely Discrete Spectrum

459

similar coefficients, has attracted a considerable interest during the last decades and in this respect we only refer to [31, 32, 34, 35] for further results.

6 The Isospectral Problem for the Conservative Camassa–Holm Flow In this section, we are going to demonstrate how our results apply to the isospectral problem of the conservative Camassa–Holm flow. To this end, let u be a real-valued 1 [0, ∞) and .υ be a non-negative Borel measure on .[0, ∞). We define function in .Hloc −1 [0, ∞) by the distribution .ω in .Hloc 

∞

ω(h) =

.

 u(x)h(x)dx +

0

∞

u (x)h (x)dx,

0

h ∈ Hc1 [0, ∞),

(6.1)

so that .ω = u − u in a distributional sense. The isospectral problem of the conservative Camassa–Holm flow is associated with the differential equation .

1 − f  + f = z ωf + z2 υf, 4

(6.2)

where z is a complex spectral parameter. Just like for generalized indefinite strings, this differential equation has to be understood in a weak sense in general (we refer to [9, 11] and [13, Section 7] for further details). The differential equation (6.2) gives rise to a self-adjoint linear relation .T in the Hilbert space H0 = H01 [0, ∞) × L2 ([0, ∞); υ)

(6.3)

.

equipped with the scalar product  f, g H0 = .

0

∞

 1 ∞ f1 (x)g1 (x)∗ dx 4 0  f2 (x)g2 (x)∗ dυ(x), +

f1 (x)g1 (x)∗ dx +

[0,∞)

f, g ∈ H0 , (6.4)

defined by saying that a pair .(f, g) ∈ H0 × H0 belongs to .T if and only if .

1 − f1 + f1 = ωg1 + υg2 , 4

υf2 = υg1 .

More details can be found in [3, 8] and [11, Subsection 4.1] in particular.

(6.5)

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J. Eckhardt and A. Kostenko

We have shown in [13, Section 7] that it is possible to transform the differential equation (6.2) into the differential equation for a generalized indefinite string .(∞, ω, ˜ υ) ˜ defined as follows: With the diffeomorphism .s : [0, ∞) → [0, ∞) given by s(t) = log(1 + t),

t ∈ [0, ∞),

.

(6.6)

˜ to be a real-valued measurable function on .[0, ∞) such that we define .w ˜ w(t) = u(0) −

.

u (s(t)) + u(s(t)) 1+t

(6.7)

for almost all .t ∈ [0, ∞), where we note that the right-hand side is well-defined ˜ is locally square integrable, so that there is almost everywhere. The function .w −1 ˜ as its normalized anti-derivative. a real distribution .ω˜ in .Hloc [0, ∞) that has .w Furthermore, the non-negative Borel measure .υ˜ on .[0, ∞) is defined by setting  υ(B) ˜ =

.

B

1 dυ ◦ s(t) = 1+t



e−x dυ(x)

(6.8)

s(B)

for all Borel sets .B ⊆ [0, ∞). It then follows from [13, Section 7] that the spectrum of the linear relation .T coincides with the spectrum of the generalized indefinite string .(∞, ω, ˜ υ). ˜ From the criteria in Sect. 5, we thus readily obtain similar criteria for the spectrum of .T to be discrete and to satisfy  .

λ∈σ (T)

1 1, the spectrum of .T satisfies (6.9) if and only if there is a constant .c ∈ R such that ∞  ∞

 .

0

 2 ex−t u (t) + u(t) − c et dt +

x

 [x,∞)

p/2 ex−t dυ(t) dx < ∞. (6.13)

(iv) If the spectrum of .T satisfies (6.9) with .p = 1, then ∞  ∞



x−t

e

.

0

  2 u (t) + u(t) − c et dt

1/2 dx < ∞,

(6.14)

x

where the constant c is given by (6.11), the function .u + u − cex is integrable with 1  ∞ = . (c ex − u (x) − u(x))dx (6.15) λ 0 λ∈σ

and the measure .υ is singular with respect to the Lebesgue measure. Proof Since the spectrum of .T coincides with the spectrum of the generalized indefinite string .(∞, ω, ˜ υ), ˜ the claims follow from Theorem 5.2 after noting that .

1 x



x

˜ w(t)dt = u(0) −

0

1 x



log(1+x)

u (t) + u(t) dt

0

for all .x ∈ (0, ∞), as well as that  .

x

y

˜ (w(t) − c) ˜ 2 dt =



 [x,y)



d υ(t) ˜ =

log(1+y)

 2 e−t u (t) + u(t) − (u(0) − c)e ˜ t dt,

log(1+x)

[log(1+x),log(1+y))

e−t dυ(t),

for all x, .y ∈ [0, ∞) with .x < y and constants .c˜ ∈ R.

 

For applications to the conservative Camassa–Holm flow, it is also of interest to consider the isospectral problem for (6.2) on the whole line. Since the corresponding linear relation is a finite rank perturbation of two half-line problems (see the proof of [11, Lemma 5.2] for example), the criteria from Theorem 6.1 can readily be extended to the full line case.

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J. Eckhardt and A. Kostenko

7 Schrödinger Operators with δ  -interactions The results of Sect. 5 also apply to one-dimensional Schrödinger operators with .δ  interactions. To simplify our considerations, we restrict ourselves to the case of the positive semi-axis. More specifically, let .χ be a real-valued Borel measure on .[0, ∞) that is singular with respect to the Lebesgue measure. For the sake of simplicity, we shall also assume that .χ does not have a point mass at zero. The Borel measure .ωχ on .[0, ∞) is then defined as the sum of the measure .χ and the Lebesgue measure, that is,  .ωχ (B) = χ (B) + dx (7.1) B

for every Borel set .B ⊆ [0, ∞). We consider the operator .Hχ in the Hilbert space L2 [0, ∞) associated with the differential expression

.

τχ = −

.

d d dx dωχ (x)

(7.2)

and subject to the Neumann boundary condition at zero. The operator .Hχ can be viewed as a Hamiltonian with .δ  -interactions. Namely, if .χ is a discrete measure, that is,  .χ = β(s)δs , (7.3) s∈X

where X is a discrete subset of .[0, ∞), .β is a real-valued function on X and .δs is the unit Dirac measure centred at s, then the differential expression .τχ can be formally written as (see [14, Example 2.2]) .

−

 d2 + β(s) · , δs δs , 2 dx

(7.4)

s∈X

which is the Hamiltonian with .δ  -interactions on X of strength .β (see [1, 22, 23]). It is known (see [14] and [15]) that under the above assumptions on .χ , the operator 2 .Hχ is self-adjoint in .L [0, ∞). The spectral properties of .Hχ turn out to be closely connected to the generalized indefinite string .Sχ = (∞, ωχ , 0); see [13, Lemma 8.1]. Lemma 7.1 The operator .Hχ is unitarily equivalent to the operator part of the linear relation .Tχ associated with the string .Sχ . Taking this connection into account and applying the results from Sect. 5, we arrive at the following results for Hamiltonians with .δ  -interactions. As usual, we will denote with .q the normalized anti-derivative of .χ , which is given by (1.12).

Generalized Indefinite Strings with Purely Discrete Spectrum

463

Theorem 7.2 The following assertions hold true: (i) Zero does not belong to the spectrum of .Hχ if and only if there is a constant .c ∈ R such that  ∞ . lim sup x (t + q(t) − c)2 dt < ∞. (7.5) x→∞

x

In this case, the constant c is given by x + .c = lim x→∞ 2

  t 1− dχ (t). x [0,x)



(7.6)

(ii) The spectrum of .Hχ is discrete if and only if there is a constant .c ∈ R such that  .

∞

lim x

x→∞

(t + q(t) − c)2 dt = 0.

(7.7)

x

(iii) For each .p > 1, the spectrum of .Hχ satisfies  .

λ∈σ (Hχ )

1 n

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J. Eckhardt and A. Kostenko

for all .x ∈ [xn , xn+1 ) and any .n ≥ 0. Taking into account that the first two summands are dominated by the last two, one easily proves sufficiency.   Remark 7.5 Clearly, discreteness of the spectrum of .Hχ is a rare event (for instance, by Theorem 7.2, existence of the limit in (7.6) is necessary; by employing a completely different approach, some sufficient conditions in the case when the support of .χ is a discrete set were obtained in [21, Section 6.4]). Moreover, in this case the eigenvalues of .Hχ accumulate at .+∞ and at .−∞ (see [23, Proposition 3.1]). Acknowledgments We are grateful to Mark Malamud, Roman Romanov and Harald Woracek for useful discussions and hints with respect to the literature. This research was supported by the Austrian Science Fund (FWF) under Grants No. P30715, I-4600 (A.K.) and by the Slovenian Research Agency (ARRS) under Grant No. N1-0137 (A.K.).

Appendix A: On a Class of Integral Operators Let .q be a function in .L2loc [0, ∞) and consider the integral operator .J in the Hilbert space .L2 [0, ∞) defined by  Jf (x) =

.

∞



x

q(max(x, t))f (t)dt = q(x)

0



∞

f (t)dt +

0

q(t)f (t)dt

(A.1)

x

for functions .f ∈ L˙ 2c [0, ∞). Since the subspace .L˙ 2c [0, ∞) is dense in .L2 [0, ∞), the operator .J is densely defined. The theorems in this appendix gather a number of results from [2] for these kinds of integral operators. Theorem A.1 The following assertions hold true: (i) The operator .J is bounded if and only if there is a constant .c ∈ C such that  .

∞

lim sup x x→∞

|q(t) − c|2 dt < ∞.

(A.2)

x

In this case, the constant c is given by 1 x→∞ x



c = lim

.

x

q(t)dt.

(A.3)

0

(ii) The operator .J is compact if and only if there is a constant .c ∈ C such that  .

∞

lim x

x→∞

x

|q(t) − c|2 dt = 0.

(A.4)

Generalized Indefinite Strings with Purely Discrete Spectrum

467

(iii) For each .p > 1, the operator .J belongs to the Schatten–von Neumann class .Sp if and only if there is a constant .c ∈ C such that ∞





x

.

0

∞

p/2 |q(t) − c|2 dt

x

dx < ∞. x

(A.5)

(iv) If the operator .J belongs to the Hilbert–Schmidt class .S2 , then its Hilbert– Schmidt norm is given by  ∞ 2 .J = 2 x|q(x) − c|2 dx, (A.6) S 2

0

where the constant c is given by (A.3). (v) If the operator .J belongs to the trace class .S1 , then 1/2  ∞  ∞ dx 2 x < ∞, . |q(t) − c| dt x 0 x the function .q − c is integrable and the trace of .J is given by  ∞ .tr J = (q(x) − c)dx,

(A.7)

(A.8)

0

where the constant c is given by (A.3). Proof Sufficiency of the conditions in (i), (ii), and (iii) follows readily from [2, Section 3] upon noticing that one has  ∞  ∞ .Jf (x) = q(max(x, t))f (t)dt = (q(max(x, t)) − c)f (t)dt 0

0

L˙ 2c [0, ∞).

for functions .f ∈ In order to prove that the condition in (i) is also necessary, let us suppose that the operator .J is bounded. For every .n ∈ N, we consider the function fn = 1[0,1) −

.

1 1[1,n+1) , n

where .1I denotes the characteristic function of an interval .I ⊆ [0, ∞). Clearly, the functions .fn belong to .L˙ 2c [0, ∞) and converge to .1[0,1) in .L2 [0, ∞) as .n → ∞. Since the operator .J is bounded, this implies that the functions .Jfn converge in 2 .L [0, ∞). In view of the definition of .J, we then find that ⎧ 1  n+1 ⎪ xq(x) + x q(t)dt − n1 1 q(t)dt, x ∈ [0, 1), ⎪ ⎪ ⎨  n+1 .Jfn (x) = q(x) − n1 q(x)(x − 1) − n1 x q(t)dt, x ∈ [1, n + 1), ⎪ ⎪ ⎪ ⎩ 0, x ∈ [n + 1, ∞),

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J. Eckhardt and A. Kostenko

for almost all .x ∈ [0, ∞). From this we are able to infer that the limit 1 n→∞ n



n+1

c := lim

.

q(t)dt = lim Q − Jfn , 1[0,1) L2 [0,∞) n→∞

1

exists in .C, where the function Q in .L2 [0, ∞) is defined by 



Q(x) = 1[0,1) (x) xq(x) +

1

.

 q(t)dt .

x

Moreover, we see that for almost every .x ∈ [0, ∞) one has  .

lim Jfn (x) =

n→∞

Q(x) − c,

x ∈ [0, 1),

q(x) − c,

x ∈ [1, ∞).

Since, on the other side, it can be readily checked that one also has 

∞

.

0

 (q(max(x, t)) − c)1[0,1) (t)dt =

Q(x) − c,

x ∈ [0, 1),

q(x) − c,

x ∈ [1, ∞),

we may conclude that the bounded extension of .J to .L2c [0, ∞) satisfies  J1[0,1) (x) =

∞

.

0

(q(max(x, t)) − c)1[0,1) (t)dt

and is hence explicitly given by 

∞

Jf (x) =

.

(q(max(x, t)) − c)f (t)dt

0

for all functions .f ∈ L2c [0, ∞). It now remains to apply [2, Theorem 3.1] to deduce that the function .q satisfies (A.2) and the constant c is necessarily given by (A.3). If the operator .J is moreover compact, then [2, Theorem 3.2] yields (A.4) and if it belongs to .Sp for some .p > 1, then [2, Theorem 3.3] yields (A.5). This proves that the conditions in (ii) and (iii) are also necessary. Finally, the formula (A.6) for the Hilbert–Schmidt norm in (iv) follows from [2, Remark in Section 3] and the necessary condition (A.7) for the operator .J to belong to the trace class .S1 as well as the formula (A.8) for the trace in (v) follow from [2, Theorem 6.2].   Remark A.2 Let us stress that boundedness and compactness criteria for the integral operator .J have been established before in [6] and [33], respectively. In Theorem A.1 (iii), the value .p = 1 is a threshold since the condition (A.7) is only necessary for the operator .J to belong to the trace class .S1 ; see [2, Section 6].

Generalized Indefinite Strings with Purely Discrete Spectrum

469

Theorem A.3 Let .χ be a non-negative Borel measure on .[0, ∞) and suppose that  q(x) =

(A.9)

dχ

.

[0,x)

for almost all .x ∈ [0, ∞). The following assertions hold true: (i) The operator .J is bounded if and only if  .

lim sup x

dχ < ∞.

(A.10)

dχ = 0.

(A.11)

[x,∞)

x→∞

(ii) The operator .J is compact if and only if  .

lim x

x→∞

[x,∞)

(iii) For each .p > 1/2, the operator .J belongs to the Schatten–von Neumann class .Sp if and only if 

∞

x

.

p



0

[x,∞)

dχ

dx < ∞. x

(A.12)

(iv) If the operator .J belongs to the trace class .S1 , then its trace is given by  tr J = −

.

[0,∞)

(A.13)

x dχ (x).

(v) If the operator .J belongs to the Schatten–von Neumann class .S1/2 , then the measure .χ is singular with respect to the Lebesgue measure. Proof By means of the connection established in the proof of Theorem A.1, the claims in (i), (ii) and (iii) follow from [2, Theorem 4.6], upon also noting that 1 . lim x→∞ x

 0

x

 q(t)dt = lim

x→∞

   t 1− dχ (t) = dχ x [0,x) [0,∞)

in this case. The claim in (iv) then follows from Theorem A.1 (v) and the claim in (v) follows from [2, Corollary 8.12].   We also want to consider related operators on a finite interval. To this end, let L be a positive number and define the operator .JL in the Hilbert space .L2 [0, L) by 

L

JL f (x) =

.

0



x

qL (min(x, t))f (t)dt = 0



L

qL (t)f (t)dt + qL (x)

f (t)dt x

(A.14)

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for functions .f ∈ L2c [0, L), where .qL is a function in .L2loc [0, L). As a Carleman integral operator, the operator .JL is closable (see [16, Theorem 3.8] for example). Theorem A.4 The following assertions hold true: (i) The operator .JL is bounded if and only if  .

x

lim sup (L − x)

|qL (t)|2 dt < ∞.

(A.15)

0

x→L

(ii) The operator .JL is compact if and only if  .

x

lim (L − x)

x→L

|qL (t)|2 dt = 0.

(A.16)

0

(iii) For each .p > 1, the operator .JL belongs to the Schatten–von Neumann class .Sp if and only if 

L



x

(L − x)

.

0

p/2 |qL (t)| dt 2

0

dx < ∞. L−x

(A.17)

(iv) If the operator .JL belongs to the Hilbert–Schmidt class .S2 , then its Hilbert– Schmidt norm is given by  L 2 .JL  (L − x)|qL (x)|2 dx. (A.18) S =2 2

0

(v) If the operator .JL belongs to the trace class .S1 , then 1/2  x  L dx (L − x) < ∞, . |qL (t)|2 dt L −x 0 0 the function .qL is integrable and the trace of .JL is given by  L .tr JL = qL (x)dx. 0

Proof We first observe that for functions .f ∈ L2c [0, L) one has 

L

JL f (L − x) =

.



L

= 

qL (min(L − x, t))f (t)dt

0

qL (min(L − x, L − t))f (L − t)dt

0 L

= 0

qL (L − max(x, t))f (L − t)dt

(A.19)

(A.20)

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471

for almost all .x ∈ (0, L). Taking into account that the map .f → f (L− · ) is unitary on .L2 [0, L), the claims in (i), (ii), and (iii) follow readily from the corresponding results in [2, Section 3] with the function .ϕ in .L2loc (0, ∞) given by ϕ(x) =

.

 qL (L − x),

x ∈ (0, L), x ∈ [L, ∞).

0,

The claims in (iv) and (v) then follow from [2, Remark in Section 3] and [2,   Theorem 6.2], respectively. The value .p = 1 is again a threshold in Theorem A.4 (iii) because the condition (A.19) is only necessary for the operator .JL to belong to the trace class. Theorem A.5 Let .χ be a non-negative Borel measure on .[0, L) and suppose that  qL (x) =

.

[0,x)

(A.21)

dχ

for almost all .x ∈ [0, L). The following assertions hold true: (i) The operator .JL is bounded if and only if  .

lim sup (L − x)

[0,x)

x→L

dχ < ∞.

(A.22)

(ii) The operator .JL is compact if and only if  .

lim (L − x)

x→L

[0,x)

dχ = 0.

(A.23)

(iii) For each .p > 1/2, the operator .JL belongs to the Schatten–von Neumann class .Sp if and only if L



(L − x)

.

p

 [0,x)

0

dχ

dx < ∞. L−x

(A.24)

(iv) If the operator .JL belongs to the trace class .S1 , then its trace is given by  tr JL =

.

[0,L)

(L − x)dχ (x).

(A.25)

(v) If the operator .JL belongs to the Schatten–von Neumann class .S1/2 , then the measure .χ is singular with respect to the Lebesgue measure.

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Proof By means of the connection established in the proof of Theorem A.4, the claims in (i), (ii) and (iii) follow from [2, Theorem 4.6], the claim in (iv) follows from Theorem A.4 (v) and the claim in (v) follows from [2, Corollary 8.12].  

Appendix B: Linear Relations Let .H be a separable Hilbert space. A (closed) linear relation in .H is a (closed) linear subspace of .H × H. Since every linear operator in .H can be identified with its graph, the set of linear operators can be regarded as a subset of all linear relations in .H. Recall that the domain, the range, the kernel and the multi-valued part of a linear relation . are given, respectively, by dom() = {f ∈ H | ∃g ∈ H such that (f, g) ∈ }, .

(B.1)

ran() = {g ∈ H | ∃f ∈ H such that (f, g) ∈ }, .

(B.2)

ker() = {f ∈ H | (f, 0) ∈ }, .

(B.3)

mul() = {g ∈ H | (0, g) ∈ }.

(B.4)

.

The adjoint linear relation .∗ of a linear relation . is defined by   ∗ = (f˜, g) ˜ ∈ H × H | g, f˜ H = f, g ˜ H for all (f, g) ∈  .

.

(B.5)

The linear relation . is called symmetric if . ⊆ ∗ . It is called self-adjoint if ∗ . =  . Note that .mul() is orthogonal to .dom() if . is symmetric. For a closed symmetric linear relation . satisfying .mul() = mul(∗ ) (the latter is further equivalent to the fact that . is densely defined on .mul()⊥ ), setting Hop = dom() = mul()⊥ ,

.

(B.6)

we obtain the following orthogonal decomposition  = op ⊕ ∞ ,

.

(B.7)

where .∞ = {0} × mul() and .op is the graph of a closed symmetric linear operator in .Hop , called the operator part of .. Notice that for non-closed symmetric linear relations, the decomposition (B.7) may not hold true. If .1 and .2 are linear relations in .H, then their sum .1 + 2 and their product .2 1 are defined by 1 + 2 = {(f, g1 + g2 ) | (f, g1 ) ∈ 1 , (f, g2 ) ∈ 2 }, .

.

2 1 = {(f, g) | (f, h) ∈ 1 , (h, g) ∈ 2 for some h ∈ H}.

(B.8) (B.9)

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The inverse of a linear relation . is given by −1 = {(g, f ) ∈ H × H | (f, g) ∈ }.

.

(B.10)

Consequently, one can consider .( − z)−1 for any .z ∈ C. The set of those .z ∈ C for which .( − z)−1 is the graph of a closed bounded operator on .H is called the resolvent set of . and denoted by .ρ(). Its complement .σ () = C\ρ() is called the spectrum of .. If . is self-adjoint, then taking into account (B.7) we obtain ( − z)−1 = (op − z)−1 ⊕ Omul() .

.

(B.11)

This immediately implies that .ρ() = ρ(op ), .σ () = σ (op ) and, moreover, one can introduce the spectral types of . as those of its operator part .op .

References 1. S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, 2nd edn. (AMS Chelsea Publishing, Providence, RI, 2005) 2. A.B. Aleksandrov, S. Janson, V.V. Peller, R. Rochberg, An interesting class of operators with unusual Schatten–von Neumann behavior, in Function Spaces, Interpolation Theory and Related Topics (Lund, 2000) (de Gruyter, Berlin, 2002), pp. 61–149 3. C. Bennewitz, B.M. Brown, R. Weikard, Inverse spectral and scattering theory for the half-line left-definite Sturm–Liouville problem. SIAM J. Math. Anal. 40(5), 2105–2131 (2008/09) 4. V.I. Bogachev, Measure Theory, vol. I, II (Springer, Berlin, 2007) 5. R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993) 6. R.S. Chisholm, W.N. Everitt, On bounded integral operators in the space of integrable-square functions. Proc. Roy. Soc. Edinburgh Sect. A 69, 199–204 (1970/71) 7. H. Dym, H.P. McKean, Gaussian Processes, Function Theory and the Inverse Spectral Problem. Probability and Mathematical Statistics, vol. 31 (Academic Press, New York-London, 1976) 8. J. Eckhardt, Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators. J. Differ. Equ. 253(2), 604–634 (2012) 9. J. Eckhardt, The inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. Arch. Ration. Mech. Anal. 224(1), 21–52 (2017) 10. J. Eckhardt, A. Kostenko, An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Commun. Math. Phys. 329(3), 893–918 (2014) 11. J. Eckhardt, A. Kostenko, Quadratic operator pencils associated with the conservative Camassa–Holm flow. Bull. Soc. Math. France 145(1), 47–95 (2017) 12. J. Eckhardt, A. Kostenko, The inverse spectral problem for indefinite strings. Invent. Math. 204(3), 939–977 (2016) 13. J. Eckhardt, A. Kostenko, On the absolutely continuous spectrum of generalized indefinite strings. Ann. Henri Poincaré 22(11), 3529–3564 (2021) 14. J. Eckhardt, A. Kostenko, M. Malamud, G. Teschl, One-dimensional Schrödinger operators with δ  -interactions on Cantor-type sets. J. Differential Equations 257, 415–449 (2014) 15. J. Eckhardt, G. Teschl, Sturm–Liouville operators with measure-valued coefficients. J. Anal. Math. 120(1), 151–224 (2013) 16. P.R. Halmos, V.S. Sunder, Bounded Integral Operators on L2 spaces (Springer, Berlin, 1978)

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17. E. Hewitt, K. Stromberg, Real and Abstract Analysis (Springer, New York, 1965) 18. I.S. Kac, Spectral theory of a string, Ukrainian Math. J. 46(3), 159–182 (1994) 19. I.S. Kac, M.G. Krein, Criteria for the discreteness of the spectrum of a singular string, (Russian). Izv. Vysš. Uˇcebn. Zaved. Matematika 2(3), 136–153 (1958) 20. I.S. Kac, M.G. Krein, On the spectral functions of the string. Am. Math. Soc. Transl. Ser. 2 103, 19–102 (1974) 21. A. Kostenko, M. Malamud, 1-D Schrödinger operators with local point interactions on a discrete set. J. Differential Equations 249(2), 253–304 (2010) 22. A. Kostenko, M. Malamud, 1-D Schrödinger operators with local point interactions: a review, in Spectral Analysis, Integrable Systems, and Ordinary Differential Equations, ed. by H. Holden, et al. Proceedings of Symposia in Pure Mathematics, vol. 87 (American Mathematical Society, Providence, 2013), pp. 235–262 23. A. Kostenko, M. Malamud, Spectral theory of semibounded Schrödinger operators with δ  interactions. Ann. Henri Poincaré 15(3), 501–541 (2014) 24. S. Kotani, S. Watanabe, Kre˘ın’s spectral theory of strings and generalized diffusion processes, in Functional analysis in Markov processes (Katata/Kyoto, 1981). Lecture Notes in Mathematics, vol. 923 (Springer, Berlin, 1982), pp. 235–259 25. M.G. Kre˘ın, H. Langer, On some extension problems which are closely connected with the theory of Hermitian operators in a space κ . III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Beiträge Anal. 14, 25–40 (1979); 15, 27–45 (1980) 26. H. Langer, Spektralfunktionen einer Klasse von Differentialoperatoren zweiter Ordnung mit nichtlinearem Eigenwertparameter. Ann. Acad. Sci. Fenn. Ser. A I Math. 2, 269–301 (1976) 27. V.G. Maz’ja, Sobolev Spaces (Springer, Berlin, 1985) 28. B. Muckenhoupt, Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972) 29. C. Remling, K. Scarbrough, The essential spectrum of canonical systems. J. Approx. Theory 254, 105395, 11 pp. (2020) 30. R. Romanov, H. Woracek, Canonical systems with discrete spectrum. J. Funct. Anal. 278(4), 108318, 44 pp. (2020) 31. I.A. Sheipak, Asymptotics of the spectrum of a differential operator with the weight generated by the Minkowski function. Math. Notes 97(2), 289–294 (2015) 32. M. Solomyak, E. Verbitsky, On a spectral problem related to self-similar measures. Bull. London Math. Soc. 27, 242–248 (1995) 33. C.A. Stuart, The measure of non-compactness of some linear integral operators. Proc. Roy. Soc. Edinburgh Sect. A 71, 167–179 (1973) 34. A.A. Vladimirov, I.A. Sheipak, Self-similar functions in L2 [0, 1] and the Sturm–Liouville problem with singular indefinite weight. Sb. Math. 197(11), 1569–1586 (2006) 35. A.A. Vladimirov, I.A. Sheipak, Asymptotics of the eigenvalues of the Sturm–Liouville problem with discrete self-similar weight. (Russian) Mat. Zametki 88(5), 662–672 (2010); translation in: Math. Notes 88(5–6), 637–646 (2010)

Soliton Asymptotics for the KdV Shock Problem of Low Regularity Iryna Egorova, Johanna Michor, and Gerald Teschl

Dedicated to the memory of Sergey Naboko

Abstract We revisit the asymptotic analysis of the KdV shock problem in the soliton region. Our approach is based on the analysis of the associated Riemann– Hilbert problem and we extend the domain of validity of the asymptotic formulas while at the same time requiring less decay and smoothness for the initial data. Keywords KdV equation · Shock problem · Nonlinear steepest descent · Low regularity · Solitons

1 Introduction and Main Results The aim of this note is to revisit the RHP approach (introduced by Deift and Zhou [1] extending ideas of Manakov [13] and Its [10]) for the study of the long-

Research supported by the Austrian Science Fund (FWF) under Grant No. P31651 and the National Academy of Sciences of Ukraine (NASU) under Grant No. 0122U111111. I. Egorova B. Verkin Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine J. Michor Faculty of Mathematics, University of Vienna, Vienna, Austria e-mail: [email protected] G. Teschl () Faculty of Mathematics, University of Vienna, Vienna, Austria Erwin Schrödinger International Institute for Mathematics and Physics, Wien, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_17

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time asymptotics for solutions of the Korteweg–de Vries (KdV) equation qt (x, t) − 6q(x, t)qx (x, t) + qxxx (x, t) = 0

.

(1.1)

with step-like initial data .q(x) = q(x, 0) satisfying the condition .

lim q(x) = 0,

lim q(x) = −c2 ,

x→∞

x→−∞

c > 0.

(1.2)

This is known as the KdV shock problem and the solution will split into a decaying dispersive tail on the background .−c2 , a dispersive shock wave, and a number of solitons. Moreover, it was shown by Khruslov [11] that at the wave front of the dispersive shock, .x = 4c2 t, solitons will emerge which are not associated with points of the discrete spectrum. However, while these principal regions are well understood ([2, 4, 8, 9]), the regions where the corresponding asymptotics are established do not overlap. In particular, it is typically a quite delicate task to improve the domain of validity of these formulas to achieve the aforementioned overlap. In this vein, the aim of the present paper is to refine the Riemann–Hilbert analysis in the soliton region 2 .x > 4c t to both increase the domain of validity as well as weaken the decay and smoothness requirements for the initial data. In particular, the degree of decay will appear in the domain of validity. More specifically, we assume that the initial data are such that .q(x) ∈ C n0 (R) and ˆ ˆ   x m0 −1 |q (i) (x)|dx < ∞, i = 1, . . . , n0 , . |x|m0 |q(x)| + |q(−x) + c2 | dx + R R+ (1.3) where .m0

≥ 4,

n0 ≥ m0 + 3.

(1.4)

In the following we refer to the Cauchy problem (1.1)–(1.4) as the KdV shock problem of low regularity. For comparison, the previously available results for the shock problem (1.1)–(1.4) from [2] in the soliton region .x

≥ (4c2 + ε)t

were established under the assumption of exponential decay: ˆ .

R+

  eρx |q(x)| + |q(−x) + c2 | dx < ∞,

ρ > c.

(1.5)

Decaying (nonsteplike) initial data of low regularity were considered for the KdV equation in [7] (with .c = 0, .m0 = 6, .n0 = 3) and for the mKdV equation in [12].

Soliton Asymptotics for the KdV Shock Problem of Low Regularity

477

Both results are obtained outside a small sector containing the transition region . xt ∼ 0, that is, for .x > εt . In this connection, two interesting questions arise: Is it possible to expand the boundary of the soliton region for the KdV shock problem (and thus narrowing the boundaries of the transition region) using the RHP approach? And is it even possible to achieve this under the low regularity assumptions (1.3)? Using the classical inverse scattering transform the multisoliton asymptotics were recently derived in [4] in the expanded soliton region .x

> 4c2 t +

m0 − 3/2 − ε log t, 2c

m0 ≥ 3.

(1.6)

Namely, assume that the discrete spectrum of the scattering problem associated with (1.1)–(1.4) is given by .−κN2 < · · · < −κ12 , and that the corresponding norming constants of the right eigenfunctions are given by .γj , .j = 1, . . . , N . Then for .t → ∞ uniformly in the domain (1.6) the solution to (1.1)–(1.4) can be represented as  .q(x, t)

=q

sol

(x, t) + O



1

(1.7)

,

3

t m0 − 2 −ε

where .q

sol

(x, t) = −

N  j =1

 cosh2 κj x − 4κj3 t −

2κj2 1 2

γ2

log 2κjj −

N

κ −κ

i j i=j +1 log κi +κj

.

(1.8)

In the present paper we use the RHP approach to show the following result: Theorem 1.1 Assume that the initial datum .q(x) satisfies (1.3)–(1.4), does not have a resonance at the edge of the continuous spectrum .−c2 , and has a nonempty discrete spectrum .−κN2 < · · · < −κ12 . Assume that .x → ∞, .t → ∞ such that .(x, t)

 β ∈ D := x ≥ 4c2 t + log t, c

Then in the domain .D we have   1 sol .q(x, t) = q (x, t) + O ν , t

t  1,

β≥0 .

 1 1 ≥ . ν = min m0 − 3, β + 2 2

(1.9)

(1.10)

The proof is based, among other things, on a new matrix solution of the underlying model problem and will be given in Sects. 2.2, 2.3 and 3. Our restrictions (1.4) on the regularity assumptions (1.3) on the initial datum were made such that one can guarantee the existence of a unique classical solution .q(x, t) of (1.1) remaining within the realm of classical scattering theory, i.e. such that (1.11) below holds for all times, as established in [4]. However, (1.3), (1.4) is

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not necessary for such a solution to exist and if existence of a classical solution satisfying (1.11) is known by other means (see e.g. [6] for results in this direction), the minimal estimates used to prove Theorem 1.1 imply the following: Corollary 1.2 (Largest Possible Class) Assume that for the initial nonresonant data satisfying (1.3) with .m0 = 4 and .n0 = 5, a unique classical solution of (1.1)– (1.3) exists and satisfies

.q(x)

ˆ .

+∞

|x|(|q(x, t)| + |q(−x, t) + c2 |)dx < ∞,

∀t ∈ R.

(1.11)

0

Then the following asymptotic is valid for .t → ∞ uniformly in the domain . x ≥ 4c2 t: .q(x, t)

= q sol (x, t) + O(t −1/2 ).

2 From the Initial RHP to the Pre-model RHP 2.1 Statement of the Initial RH Problem Let .q(x) be as in Theorem 1.1 and let .q(x, t) be the solution of (1.1)–(1.4). Condition (1.4) implies (cf. [4]) that this solution exists, is unique and satisfies (1.11). Let .φ(k, x, t) be the right Jost solution of the associated Schrödinger equation .L(t)y

=−

d2 y + q(x, t)y = k 2 y, dx 2

satisfying .

lim e−ikx φ(k, x, t) = 1,

(2.1)

x→+∞

and let .φ1 (k, x, t) be the corresponding Jost solution associated with the left background, .

lim eik1 x φ1 (k, x, t) = 1,

x→−∞

k1 :=



k 2 + c2 .

(2.2)

Here .k1 > 0 for .k ∈ [0, ic)r . The subscript “r ” in the last notation indicates the right side of the cut along the interval .[0, ic]. Note that the function .φ(k, x, t) is a holomorphic function of k in .C+ := {k ∈ C : Im k > 0} and continuous up to the real axis. It is real-valued for .k ∈ [0, ic], and does not have a discontinuity on this interval. The function .φ1 (k, x, t) is holomorphic in the domain .C+ \ (0, ic] and continuous up to the boundary. On the different sides of .[0, ic] it takes complex

Soliton Asymptotics for the KdV Shock Problem of Low Regularity

479

conjugated values. Denote the Wronskian of the Jost solutions by .W (k)

= φ1 (k, x, 0)φ (k, x, 0) − φ1 (k, x, 0)φ(k, x, 0),

∂ where .f = ∂x f . The conditions of Theorem 1.1 exclude a possible resonance at the point .ic, that is, we assume the condition .W (ic)

= 0.

On .[0, ic] introduce the function .χ(k)

:=

4k [k1 ]r . |W (k)|2

(2.3)

One can verify that .χ(k) = i|χ(k)| and .χ(ic) = χ(0) = 0. Let .R(k) be the right reflection coefficient of the initial data .q(x) and let .γj

:= φ(iκj , ·, 0)−2 L2 (R)

be the right norming constants for .j = 1, . . . , N . The set  .

R(k), k ∈ R;

|χ(k)|, k ∈ [0, ic];

iκj , γj , j = 1, . . . , N ,

(2.4)

constitutes the minimal set of scattering data to uniquely reconstruct the solution of the initial value problem (1.1)–(1.4) (cf. [3, 4]). The Jost solutions (2.2) and (2.1) are connected by the scattering relation .T

(k, t)φ1 (k, x, t) = φ(k, x, t) + R(k, t)φ(k, x, t),

k ∈ R,

where .T (k, t) and .R(k, t) are the right transmission and reflection coefficients. We will use the notation .T (k) = T (k, 0) and .R(k) = R(k, 0). We define a vector-valued function .m(k, x, t) = (m1 (k, x, t), m2 (k, x, t)), meromorphic with respect to the spectral parameter .k ∈ C \ (R ∪ [−ic, ic]) for fixed .x, t , as follows: 

 T (k, t)φ1 (k, x, t)eikx , φ(k, x, t)e−ikx , k ∈ C+ \ (0, ic], .m(k, x, t) = k ∈ C− \ [−ic, 0), m(−k, x, t)σ1 , (2.5) where .σ1 = ( 01 10 ) is the first Pauli matrix. The vector function .m(k, x, t) has at most simple poles at the points .±iκj . For .k → ∞, the following asymptotic formula holds .q(x, t)

  = lim 2k 2 m1 (k, x, t)m2 (k, x, t) − 1 , k→∞

(2.6)

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which we will use to extract our asymptotics. Let .ε > 0 and .δ > 0 be two arbitrary small parameters. We divide the domain .D

 β := (x, t) : x ≥ 4c2 t + log t, c

t  T0  1,

β≥0

into a union of the following regions DN = {(x, t) ∈ D : x ≥ (4κN2 + ε)t}, sol DN = {(x, t) ∈ D : (4κN2 − ε)t ≤ x ≤ (4κN2 + ε)t}, .

Dj = {(x, t) ∈ D : (4κj2 − ε)t ≥ x ≥ (4κj2−1 + ε)t}, Djsol = {(x, t) ∈ D : (4κj2 − ε)t ≤ x ≤ (4κj2 + ε)t}, D0 = {(x, t) : 4c2 t +

β c

j = 1, . . . , N − 1,

log t ≤ x ≤ (4κ12 − ε)t}.

Denote the small nonintersecting circles around the points of the discrete spectrum by .Dj

:= {k : |k − iκj | < δ},

Tj := ∂Dj = {k : |k − iκj | = δ},

j = 1, . . . , N,

with counterclockwise oriented boundaries (see Fig. 1). Let .T∗j = {k : −k ∈ Tj } be small circumferences around the points .−iκj , again with counterclockwise orientation. Introduce the functions .Pj (k)

:=

N  k + iκj l=j

and the matrices ⎛ .Aj (k)

k − iκj

1

k ∈ C+ ,

,

0

⎞

= ⎝ iγj2 e2t (iκj ) ⎠ , − k−iκj 1

j = 1, . . . , N ;

PN+1 (k) = 1,

⎞ ⎛ k−iκj 1 − 2 2t (iκ ) j ⎠ iγj e Bj (k) = ⎝ , 0 1

(2.7)

j = 1, . . . , N,

(2.8) where .Aj (k) = Aj (k, x, t), .Bj (k) = Bj (k, x, t). The phase function . (k) =

(k, x, t) is defined by . (k)

x = 4ik 3 + ik , t

k ∈ C.

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In the domain .(k, x, t) ∈ C+ ×Dj , .j = 1, . . . , N −1, we redefine .m(k) given by (2.5) as .m(k, j )

= m(k, x, t, j ) ⎧ −σ3 ⎪ ⎪ ⎨ m(k)Al (k)[Pj +1 (k)] , k ∈ Dl , 1 ≤ l ≤ j, = m(k)Bl (k)[Pj +1 (k)]−σ3 , k ∈ Dl , N ≥ l > j, ⎪ ⎪ ⎩ k ∈ (C+ \ (0, ic]) \ ∪N m(k)[Pj +1 (k)]−σ3 , l=1 Dl ,

(2.9)

0 ) is the third Pauli matrix. For .(k, x, t) ∈ C+ × D we set where .σ3 = ( 10 −1 N

.m(k, x, t, N )

=

m(k)Al (k), m(k),

k ∈ Dl ,

l ≤ N,

+

k ∈ (C \ (0, ic]) \ ∪N l=1 Dl .

(2.10)

In the domain .(k, x, t) ∈ C+ × Djsol , .j = 1, . . . , N , we set sol

.m

(k, j ) = msol (k, x, t, j ) ⎧ −σ3 ⎪ ⎪ ⎨m(k)Al (k)[Pj +1 (k)] , =

k ∈ Dl , 1 ≤ l < j,

(k)]−σ3 ,

m(k)Bl (k)[Pj +1 k ∈ Dl , N ≥ l > j, ⎪ ⎪ ⎩ + −σ 3 m(k)[Pj +1 (k)] , k ∈ (C \ (0, ic]) \ ∪N l=1 Dl .

(2.11)

We also redefine .m(k, j ) = m(−k, j )σ1 and .msol (k, j ) = msol (−k, j )σ1 for .k ∈ C− . Next we introduce the jump contour .

∗ = R+ ∪ R− ∪ [ic, 0] ∪ [−ic, 0] ∪N l=1 (Tl ∪ Tl )

(2.12)

as depicted in Fig. 1 with the following orientation: left-to-right on .R+ , right-toleft on .R− = R∗+ , top-down on .[ic, 0], bottom-top on .[−ic, 0] = [ic, 0]∗ , and counterclockwise on .Tj and .T∗j . By .I ∗ we refer to the contour consisting of the points .−k : k ∈ I with the following orientation: if k moves in the positive direction of I , then .−k moves in the positive direction of .I ∗ . We observe that .m(k, j ) is a piecewise holomorphic vector function with jumps on . and .msol (k, j ) is a piecewise meromorphic function with simple poles at .iκj and .−iκj and with the same jumps as .m(k, j ) except at .Tj and .T∗j , where it does not have jumps. Note that .m1 (k, x, t, j )m2 (k, x, t, j )

= m1 (k, x, t)m2 (k, x, t),

k → ∞,

(x, t) ∈ Dj ,

(2.13)

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Fig. 1 Part of the jump contour .

and sol

.m1

(k, x, t, j )msol 2 (k, x, t, j ) = m1 (k, x, t)m2 (k, x, t),

k → ∞, (x, t) ∈ Djsol .

(2.14) Theorem 2.1 Let (2.4) be the right scattering data of the initial datum .q(x). Assume that x and t are arbitrary large fixed values such that .(x, t) ∈ Dj (resp. .(x, t) ∈ Djsol ). Then the vector function .m(k, j ) = m(k, x, t, j ) (respectively .msol (k, j ) = msol (k, x, t, j )) defined in (2.9), (2.10) (resp. (2.11)) is the unique solution of the following vector Riemann–Hilbert problem: Find a vector-valued function .m(k, j ), holomorphic (resp. .msol (k, j ), meromorphic) away from . (resp. away from . \ (Tj ∪ T∗j )), satisfying:

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1. The jump condition: sol sol sol (k, j )), where .m+ (k, j ) = m− (k, j )v(k, j ) (resp. .m+ (k, j ) = m− (k, j )v ⎧ ⎪ 1 − |R(k)|2 ⎪ ⎪ ⎪ ⎪ 2t (k) ⎪ Pj−2 ⎪ +1 (k)R(k)e ⎪ ⎪  ⎪ ⎪ ⎪ 1 ⎪ ⎨ −2 2t (k) χ(k)P .v(k, j ) = j +1 (k)e ⎪ ⎪ ⎪ ⎪ Al (k)[Pj +1 (k)]−σ3 , ⎪ ⎪ ⎪ ⎪ −σ3 ⎪ ⎪ ⎪ Bl (k)[Pj +1 (k)] , ⎪ ⎪ ⎩ σ1 v(−k)σ1 ,

 −R(k)Pj2+1 (k)e−2t (k) , k ∈ R+ , 1  0 , k ∈ [ic, 0], 1 k ∈ Tl , l ≤ j ; k ∈ Tl , l > j ; ∗ k ∈ R− ∪ [−ic, 0] ∪N l=1 Tl ;

(resp. .v sol (k, j ) = v(k, j ), k ∈  \ (Tj ∪ T∗j ); 2. the pole conditions:

v sol (k, j ) = I, k ∈ Tj ∪ T∗j )



 0 0 Res m (k, j ) = lim m (k, j ) 2t (iκj ) 0 , k→iκj iγj2 Pj−2 iκj +1 (iκj )e .   0 −iγj2 Pj2+1 (iκj )e2t (iκj ) sol sol , Res m (k, j ) = lim m (k, j ) k→−iκj −iκj 0 0 sol

sol

3. the symmetry conditions: .m(−k, j )

= m(k, j )σ1 ,

k ∈ C \ ,

(resp. .msol (−k, j ) = msol (k, j )σ1 , k ∈ C \ ( \ (Tj ∪ T∗j ))) 4. the normalization condition .

lim m(iκ, j ) = lim msol (iκ, j ) = (1 1).

κ→∞

κ→∞

5. The function .m(k, j ) resp. .msol (k, j ) has continuous limits as k approaches .. Remark 2 The results listed in this theorem are slight modifications of the results obtained in [2, 5], and we omit the proof.

2.2 Properties of the Scattering Data and Their Analytic Continuations First assume that the initial datum .q(x) is smooth and satisfies (1.5). Then the + ± reflection coefficient .R(k) has an analytic continuation to .O− ρ ∪ Oρ , where .Oρ = {k : ± Re k > 0, 0 < Im k < ρ}. In contrast to the case of fast decaying initial

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datum, in the step-like case the analytic continuation of .R(k) has a jump along ∩ {0 < Im k < ρ}, given by (cf. [2, Lemma 3.2])

.[ic, 0]

.R− (k)

− R+ (k) + χ(k) = 0,

k ∈ [ic, 0] ∩ {0 < Im k < ρ}.

(2.15)

On the other hand, if the initial datum satisfies (1.3), which is the case we consider, and .χ(k) are .m0 − 1 times continuously differentiable except for the node point .k = 0, and the following formula is valid ([4, Section 3]) .R(k)

l+1

.(i)

dl χ(ih), h→+0 dhl

R (l) (+0) + (−i)l+1 R (l) (+0) = lim

where .

dl R(k), k→+0 dk l

R (l) (+0) = lim

l = 0, 1, . . . , m0 − 1.

Respectively, m 0 −1 .

l=0

m m 0 −1 0 −1 R (l) (+0) l R (l) (+0) χ (l) (0) l l k − (−k) = k, l! l! l! l=0

for k = ih.

(2.16)

l=0

Thus, this formula agrees with (2.15) for the case (1.3). In fact, the decomposition of .χ at 0 has only odd degrees of k and .R

Set .τ =

c+κ1 2 .

(l)

(+0) = (−1)l R (l) (+0).

(2.17)

In the domain .O+ , where ±

.O

:= {k : ± Re k > 0,

0 < Im k
0,

x ≥ 4c2 t.

On the remaining parts of .C and .C∗ the estimates are literally the same. As a result we get .k

s



 v(k, ˆ j ) − I L1 (C∪C∗ )∩L∞ (C∪C∗ ) ≤ e−Ct ,

C > 0,

x ≥ 4c2 t.

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It remains to estimate .v(k, ˆ j )−I = vˆjsol (k, j )−I on the contour .[ic, 0]. The only nonzero element of this matrix is .v ˆ21 (k, j )

2t (k) = Pj−2 +1 (k)e



χ(k), f (k),

k ∈ [ic, ic2 ], k ∈ [ ic2 , 0],

(2.32)

where the function .f (k) continuous on .[ ic2 , 0] is defined in (2.21) and satisfies (2.22). Note that the analytic continuation of .Ra (k, t) does not have a jump on .[ ic2 , 0]. This was taken into account to get (2.32). Consider first the contour .[ic, ic2 ]. Recall that we study the nonresonant case and therefore .χ(k)

= C(k − ic)1/2 (1 + o(1)),

k → ic,

C = 0,

see (2.2), (2.3). For .x ≥ 4c2 t + ζ , .ζ ≥ 0, an elementary estimate holds for .k = iκ, c . ≤ κ ≤ c, 2 .t (iκ, x, t)

= 4(κ 2 − c2 )κt − κζ < (κ − c)

c 3c2 t − ζ ≤ 0. 4 2

Therefore, .|v ˆ21 (iκ, j )|

≤ C(c − κ)1/2 e−C(c−κ)−cζ .

(2.33)

For all regions under consideration except of .D0 we have .ζ ≥ (4κ12 − 4c2 − ε)t, and therefore .|v ˆ21 (k, j )|

sol + |vˆ21 (k, j )| ≤ Ce−δt ,

k ∈ [ic, ic2 ],

(x, t) ∈ Dj ∪ Djsol ,

where .C = C(j ) > 0, .δ = δ(j ) > 0, .j = 1, . . . , N. In the domain .D0 , which depends on .β, consider first the case .β = 0, that is, .ζ = 0. We observe that the function on the r.h.s. in (2.33) can be estimated from above via .maxy∈R+ u(y, t), where .u(x, t) = 1 √ 1 , we conclude y 1/2 e−Cy . Since . ∂u ∂y = 0 for .y = 2Ct and .maxy∈R+ u(y, t) = 2Cte

that .|vˆ21 (k, 0)| = O(t −1/2 ) uniformly with respect to k and x in the domain under consideration. If .β > 0, that is, .D0 = {(x, t) : x ≥ 4c2 t + βc log t}, then .|v ˆ21 (k, 0)|

1

== O(t −β− 2 ),

k ∈ [ic, ic2 ],

(x, t) ∈ D0 .

Soliton Asymptotics for the KdV Shock Problem of Low Regularity

491

sol (k, j ) on the interval .[ ic , 0]. Taking into It remains to estimate .vˆ21 (k, j ) = vˆ21 2 account (2.32) and (2.22) we conclude that .|v ˆ21 (ih, j )|

3 t−8c2 ht−2 β h log t c

≤ C(j )hm0 −1 e8h

≤ C max hm0 −1 e−4c

2 th

c h∈[0, 2 ]

j = 0, . . . , N.

,

Since .

C(m0 ) max y m0 −1 e−Cyt ≤ m −1 , t 0 y∈R+

then .|v ˆ21 (k, j )|

sol + |vˆ21 (k, j )| ≤

C(m0 , j ) , t m0 −1

k ∈ [ ic2 , 0].

(2.34)

Collecting estimates (2.31)–(2.34) together and taking into account the symmetry property .v(k, ˆ j)

= σ1 v(−k, ˆ j )σ1 ,

ˆ k ∈ ,  

we get (2.30).

Hence we can reformulate our pre-model RHP as follows Theorem 2.7 Assume that .(x, t) ∈ Dj (resp. .(x, t) ∈ Djsol ). Then • the vector function .m(k, ˆ j, x, t) (resp. .m ˆ sol (k, j, x, t)) is the unique piecewise holomorphic (resp. meromorphic) solution of the jump problem .m ˆ + (k, j )

=m ˆ − (k, j )(I + W(k, j )),

ˆ sol (resp. m ˆ sol + (k, j ) = m − (k, j )(I + W(k, j )),

ˆ k ∈ , ˆ k ∈ )

where the jump matrix .W(k, j ) satisfies the symmetry condition .W(k, j )

= σ1 W(−k, j )σ1 ,

ˆ k ∈ ,

(2.35)

and the following estimate: .k

s

−ν W(k, j )L1 ()∩L ∞ () ˆ ˆ = O(t ),

 1 , ν = min m0 − 3, β + 2

s = 0, 1, 2.

(2.36)

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In a vicinity of the point .k = 0, .W(k, j )

= O(k 2 ),

ˆ k ∈ ,

k → 0.

(2.37)

• Both functions .m(k, ˆ j ) and .m ˆ sol (k, j ) satisfy the same symmetry conditions .m(−k, ˆ j)

= m(k, ˆ j )σ1 ,

ˆ k ∈ C \ ,

(2.38)

and normalization conditions .

lim m(iκ, ˆ j ) = lim m ˆ sol (iκ, j ) = (1 1).

κ→∞

κ→∞

(2.39)

• The function .m ˆ sol (k, j ) has simple poles at .±iκj and satisfies the pole conditions 

 0 0 Res m ˆ (k, j ) = lim m ˆ (k, j ) , k→iκj iγj2 (x, t) 0 iκj .   0 −iγj2 (x, t) sol sol Res m ˆ (k, j ) = lim m ˆ (k, j ) , k→−iκj −iκj 0 0 sol

sol

(2.40)

where 2

.γj

3

(x, t) = γj2 e8κj t−2κj x

 N   κl − κj 2 , κl + κj

(x, t) ∈ Djsol .

(2.41)

l=j +1

• In a vicinity of .k = 0, .m ˆ1

sol

2 (k, j ) = m ˆ sol 2 (k, j ) + O(k ),

k → 0.

(2.42)

Proof All propositions of Theorem 2.7 are proven except of (2.37) and (2.42). Estimate (2.37) is straightforward from (2.29), (2.27), (2.32), (2.22), (1.4) (for at least .m0 ≥ 4) and Lemma 2.5. For (2.42), we have the following: (1) the jump matrix .vˆ sol (k, j ) in a vicinity of point .k = 0 satisfies .v ˆ sol (k, j ) = I + O(k 2 ); (2) the vector function .msol (k, j ) is bounded there and has continuous limits when approaching the contour; (3) it satisfies (2.38). Then .m ˆ 1,+ (k, j )

sol

2 2 =m ˆ sol ˆ sol 2,+ (k, j ) + O(k ) = m 1,− (k, j ) + O(k ) 2 =m ˆ sol 2,− (k, j ) + O(k ),

which in turn implies (2.42).

k → 0,  

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493

3 Solution of the Model Problem and Final Asymptotic Analysis The model problem for .m(k, ˆ j ) is trivial: To find a vector function holomorphic in .C and satisfying the symmetry and normalization condition. Its unique solution is the constant vector .mmod (k, j ) = (1, 1), which is the same for all domains .Dj , .j = 0, 1, . . . , N . The second model problem, namely to find a vector function .mmod,sol (k, j ) = S(k, j ) meromorphic in .C and satisfying (2.38)–(2.40), was solved in [7]. The (unique) vector solution .S(k, j ) is given by (cf. [7, Lemma 2.6 and Theorem 4.4]) S(k, j ) = (S1 (k, j ), S2 (k, j )) , S2 (k, j ) = S1 (−k, j ),   . k + iκj 1 −1 2 (2κj ) γj (x, t) , S1 (k, j ) = 1+ k − iκj 1 + (2κj )−1 γj2 (x, t)

where .γj2 (x, t) is defined by (2.41). Verification of the pole, symmetry and normalization conditions is straightforward. To apply the standard final asymptotic analysis for the vector RH problems, we need to construct a matrix solution .M(k, j ) to the model problem in .Djsol , which satisfies the additional symmetry .M(−k, j ) = σ1 M(k, j )σ1 . In .Dj it will evidently be the identity matrix. For .Djsol we cannot expect that a bounded invertible symmetric matrix solution exists. Indeed, we observe that .S1 (0, j )

= S2 (0, j ),

and .S1 (0, j )

= S2 (0, j ) = 0,

for

1 − (2κj )−1 γj2 (x, t) = 0.

The set of pairs .(x, t) satisfying this condition is a line in .Djsol containing arbitrary large x and t . According to [5], for such .(x, t) a bounded symmetric invertible matrix model solution does not exist. When admitting poles for .M(k, j ), one has to ensure ˆ j )M −1 (k, j ) has only removable singularities. that the error vector .m(k, Let us first construct an antisymmetric vector solution for the model problem in sol with a simple pole at .k = 0. We look for a solution of the form .D j   ρj ρj μj μj V(k, x, t, j ) = V(k, j ) = −1 + , 1+ + + , k k − iκj k k + iκj . V(−k, j ) = −V(k, j ),

Im k > 0,

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where the constants .ρj = ρj (x, t) and .μj = μj (x, t) are chosen to satisfy the pole conditions and the condition .V2 (iκj , j ) = S2 (iκj , j ). Then .V2 (iκj , j )

=1+

μj ρj 1 + = = S2 (iκj , j ) iκj 2iκj 1 + (2κj )−1 γj2 (x, t)

and μj = Res V1 (iκj , j ) = iγj2 (x, t)V2 (iκj , j ) iκj

.

= Res S1 (iκj , j ) = iκj

iγj2 (x, t) 1 + (2κj )−1 γj2 (x, t)

.

We get .ρj (x, t)

=−

iγj2 (x, t) 1 + (2κj )−1 γj2 (x, t)

,

μj (x, t) =

iγj2 (x, t) 1 + (2κj )−1 γj2 (x, t)

,

ρj = −μj ;

(3.1) 

 iκj μj (x, t) iκj μj (x, t) .V(k, j ) = −1 + . , 1− k(k − iκj ) k(k + iκj )

In terms of .μj the solution .S(k, j ) = mmod,sol (k, j ) has the representation 

 μj (x, t) μj (x, t) .S(k, j ) = 1+ , 1− . k − iκj k + iκj

(3.2)

For .(x, t) ∈ Djsol introduce the matrix   1 S1 (k, j ) − V1 (k, j ) S2 (k, j ) − V2 (k, j ) .M(k, j ) = M(k, j, x, t) = . 2 S1 (k, j ) + V1 (k, j ) S2 (k, j ) + V2 (k, j )

(3.3)

From (2.41)–(3.3) it follows that ⎛ .M(k, j )

=

μj (x,t) 2k ⎝ μ (x,t) k+iκ j j 2k k−iκj

1+

μj (x,t) k−iκj 2k k+iκj μ (x,t) 1 − j 2k

−

⎞ ⎠,

μj (x, t) =

iγj2 (x, t) 1 + (2κj )−1 γj2 (x, t)

.

(3.4) Its evident properties are listed in the following

Soliton Asymptotics for the KdV Shock Problem of Low Regularity

495

Lemma 3.1 The matrix .M(k, j ) = M(k, j, x, t) given by (2.41), (3.4) for .(x, t) ∈ Djsol satisfies • • • • •

symmetry: .M(−k, j ) = σ1 M(k, j )σ1 ; normalization: .M(k, j ) → I as .k → ∞; the matrix .M(k, j ) is meromorphic in .C with simple poles at .±iκj and .k = 0; .det M(k, j ) = 1; mod,sol (k, j ) = (1, 1)M(k, j ); .m

Theorem 3.2 Let .m ˆ sol (k, j ) be the solution of RHP (2.35)–(2.41) and let err

.m

(k, j ) := m ˆ sol (k, j )M −1 (k, j ),

(x, t) ∈ Djsol .

Then 1. the vector function .merr (k, j ) does not have singularities at .±iκj ; 2. .merr (k, j ) does not have a singularity at .k = 0; 3. the following estimate is valid uniformly for .(x, t) ∈ Djsol : .k

s

−ν M(k, j )W(k, j )[M(k, j )]−1 L1 ()∩L ∞ () ˆ ˆ = O(t ),

where 

1 .ν = min m0 − 3, β + , 2

s = 0, 1, 2.

Proof Consider first the point .iκj . By definition, ⎞ ⎛ μj (x,t) μj (x,t) k−iκj  1 − 2k k+iκj ⎠ err ⎝ μ (x,t)2kk+iκ , .m (k, j ) = m ˆ sol ˆ sol μ (x,t) 1 (k, j ), m 2 (k, j ) − j 2k k−iκjj 1 + j 2k 

(3.5)

2 lim (k − iκj )m ˆ sol ˆ sol 1 (k, j ) = iγj (x, t)m 2 (iκj , j ).

k→iκj

err Since .[M(·, j )]−1 12 = O(k − iκj ), then .m2 (iκj , j ) is well defined. For the first element of this vector we have using (3.1) .

lim (k − iκj )merr 1 (iκj , j ) =

k→iκj

  μj (x, t) − μj (x, t)m ˆ sol (iκ , j ) 1 − ˆ sol iγj2 (x, t)m j 2 2 (iκj , j ) = 0. 2iκj

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The arguments for the point .−iκj are the same due to symmetry, .merr (k, j ) = merr (−k, j )σ1 . To prove item 2, note that by (2.42) and (3.5), err

.m

(k, j ) = (K, K)M −1 (k, j ) + O(k)   k + iκj  k − iκj  μj  μj  1+ 1+ , 1+ + O(k) = K 1− 2k k − iκj 2k k + iκj   μj (x, t) μj (x, t) , 1+ = K 1− + O(k), K = merr 1 (+0 + i0, j ). k − iκj k + iκj

To prove item 3, we observe that .μj (x, t) defined by (3.1), (3.4), (2.41) is uniformly bounded in .Djsol . Therefore, .M(k, j ) and .M −1 (k, j ) admit an estimate from above of C ˆ So outside a small the form .1 + |k| , where C does not depend on .(x, t) and .k ∈ . vicinity of .k = 0 the estimate in item 3 is fulfilled because of (2.36). Near .k = 0 we   apply (2.37).

The rest of the final asymptotic analysis is a trivial modification of the standard “small norm” arguments for symmetric vector RH problems. Indeed, let .m(k, ˆ j ) and .m ˆ sol (k, j ) be as in Theorem 2.7 and let .M(k, j ) be defined by (3.3) in .Djsol and by the identity matrix in .Dj . Set

er

.m

(k) =

m(k, ˆ j ),

(x, t) ∈ Dj ,

m ˆ sol (k, j )[M(k, j )]−1 ,

(x, t) ∈ Djsol ,

j = 0, . . . , N,

and define .W

er

(k) = M(k, j )W(k, j )[M(k, j )]−1 ,

j = 0, . . . , N.

Then .mer (k) = mer (−k)σ1 is the unique piecewise holomorphic solution of the jump problem er

.m+ (k)

er = mer − (k)(I + W (k)),

ˆ k ∈ ,

where .W

er

(k) = σ1 Wer (−k)σ1 ,

ˆ k ∈ ,

and .k

s

W (k)L1 ()∩L ∞ () ˆ ˆ = O(t er

−ν



),

1 ν = min m0 − 3, β + 2

≥

1 , 2

s = 0, 1, 2.

(3.6)

Soliton Asymptotics for the KdV Shock Problem of Low Regularity

497

This vector function is continuous up to the boundaries and satisfies the normalization condition .

lim mer (iκ) = (1, 1).

κ→∞

Let .C denote the Cauchy operator associated with .ˆ , .(Ch)(k)

=

1 2πi

ˆ ˆ 

h(s)

ds , s−k

ˆ k ∈ C \ ,

ˆ . Let .C+ f and .C− f be its non-tangential limit values where .h = (h1 , h2 ) ∈ L2 () from the left and right sides of .ˆ , respectively. ˆ ∪ L∞ () ˆ → L2 () ˆ by .CW f = As usual, we introduce the operator .CW : L2 () er er C− (f W ), where .W is our error matrix. Then .CW  2 ˆ ˆ L ()→L2 ()

−ν ≤ CWer L∞ () ˆ ≤ O(t )

as well as .(I

− CW er )−1 L2 ()→L 2 () ˆ ˆ ≤

1 1 − O(t −ν )

(3.7)

for sufficiently large t . Consequently, for .t  1, we may define a vector function .w(k)

  = (1, 1) + (I − CW )−1 CW (1, 1) (k).

By (3.7), .w(k)

−1 err − (1, 1)L2 () 2 () 2 () ˆ ≤ (I − CW ) L2 ()→L ˆ ˆ C− L2 ()→L ˆ ˆ W L∞ () ˆ

= O(t −ν ).

(3.8)

With the help of w, the function .mer (k) can be represented as er

.m

(k) = (1, 1) +

1 2πi

ˆ ˆ 

w(z)Wer (z)dz , z−k

and in virtue of (3.8) and (3.6) we obtain as .k → +i∞, er

.m

(k) = (1, 1) +

1 2πi

ˆ ˆ 

(1, 1)Wer (z) dz + E(k), z−k

where .|E(k)|

≤

C O(t −ν−1 ) Wer L2 () , ˆ w(z) − (1, 1)L2 () ˆ ≤ |k| |k|

k → ∞.

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The term .O(t −ν−1 ) is uniformly bounded with respect to .(x, t) ∈ D. In the regime .Re k = 0, .Im k → +∞, we have .

1 2πi

ˆ ˆ 

(1, 1)Wer (z) f0 (x, t) f1 (x, t) dz = (1, −1) + (1, 1) k−z 2ikt ν 2k 2 t ν + O(t −ν )O(k −3 ) + O(t −ν−1 )O(k −1 ),

where .f0,1 (x, t) are scalar functions uniformly bounded in .D. Furthermore, .O(k −s ) are vector functions depending only on k and .O(t −ν ), .O(t −ν−1 ) are as above. Hence, m ˆ sol (k, j ) = mer (k, j )M(k, j ) = S(k, j ) + +

.

f0 (x, t) (1, −1)M(k, j ) 2ikt ν

f1 (x, t) S(k, j ) + O(t −ν )O(k −3 ) + O(t −ν−1 )O(k −1 ), 2k 2 t ν (x, t) ∈ Djsol ;

j = 1, . . . , N ;

and m(k, ˆ j ) = mer (k, j ) = (1, 1) + .

f0 (x, t) f1 (x, t) (1, −1) + (1, 1) 2ikt ν 2k 2 t ν

+ O(t −ν )O(k −3 ) + O(t −ν−1 )O(k −1 ),

(x, t) ∈ Dj ;

j = 0, . . . , N.

(3.9) From (3.4) it follows that .(1, −1)M(k, j ) = (1, −1) + O(k 2 ). Therefore, (2.14) implies that for .(x, t) ∈ Djsol , .m1 (k)m2 (k)

=m ˆ sol ˆ sol 1 (k, j )m 2 (k, j ) = S1 (k, j )S2 (k, j ) + O(t −ν )O(k −2 ), k → ∞.

(3.10)

By (3.2), .S1 (k, j )S2 (k, j )

−1= =

2iκj μj (x, t) − μ2j (x, t) (k − iκj )(k + iκj ) −2κj γj2 (x, t)

2 (1 + o(1)). k 2 1 + (2κj )−1 γj2 (x, t) 

Comparing this with (3.10) and (2.6) we conclude that in the region .Djsol , .q(x, t)

= qjsol (x, t) + O(t −ν );

−4κj γj2 (x, t) qjsol (x, t) :=  2 . 1 + (2κj )−1 γj2 (x, t)

(3.11)

Soliton Asymptotics for the KdV Shock Problem of Low Regularity

499

On the other hand, sol

.qj

8κj2

(x, t) = −  √

2κj γj (x,t)

8κj2 2 = −  2 , 3 3 γj (x,t) eκj x−4κj t+j + e−κj x+4κj t−j + √ 2κj

where .j

N  γj2 κi − κj 1 = − log − log . 2 2κj κi + κj i=j +1

Thus, sol

.qj

(x, t) = −

 cosh2 κj x − 4κj3 t −

2κj2 1 2

log

γj2 2κj

−

N

i=j +1 log

κi −κj κi +κj

.

Note that in .D \ Djsol , this function admits the estimate .O(e−C(ε)t ), and taking into account the weaker estimate (3.11), we conclude that .q(x, t)

=

N 

qjsol (x, t) + O(t −ν ),

(x, t) ∈

j =1

N 

Djsol .

j =1

On the other hand, (3.9) implies .q(x, t)

= O(t −ν ),

(x, t) ∈

N 

Dj .

j =0

This proves Theorem 1.1.

References 1. P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Ann. of Math. (2) 137, 295–368 (1993). https://doi.org/10.2307/2946540 2. I. Egorova, Z. Gladka, V. Kotlyarov, G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation with steplike initial data. Nonlinearity 26, 1839–1864 (2013). https://doi.org/ 10.1088/0951-7715/26/7/1839 3. I. Egorova, Z. Gladka, T.-L. Lange, G. Teschl, Inverse scattering theory for Schrödinger operators with steplike potentials. Zh. Mat. Fiz. Anal. Geom. 11, 123–158 (2015). https://doi. org/10.15407/mag11.02.123 4. I. Egorova, J. Michor, G. Teschl, Soliton asymptotics for the KdV shock problem via classical inverse scattering. J. Math. Anal. Appl. 514, 126251 (2022). https://doi.org/10.1016/j.jmaa. 2022.126251

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5. I. Egorova, M. Piorkowski, G. Teschl, Asymptotics of KdV shock waves via the Riemann– Hilbert approach. Preprint. arXiv:1907.09792 6. S. Grudsky, A. Rybkin, On classical solutions of the KdV equation. Proc. Lond. Math. Soc. (3) 121(2), 354–371 (2020). https://doi.org/10.1112/plms.12326 7. K. Grunert, G. Teschl, Long-time asymptotics for the Korteweg–de Vries equation via nonlinear steepest descent. Math. Phys. Anal. Geom. 12, 287–324 (2009). https://doi.org/10. 1007/s11040-009-9062-2 8. A.V. Gurevich, L.P. Pitaevskii, Decay of initial discontinuity in the Korteweg–de Vries equation. JETP Lett. 17(5), 193–195 (1973) 9. A.V. Gurevich, L.P. Pitaevskii, Nonstationary structure of a collisionless shock wave. Soviet Phys. JETP 38(2), 291–297 (1974) 10. A.R. Its, Asymptotic behavior of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations. Soviet. Math. Dokl. 24(3), 452–456 (1981) 11. E.Y. Khruslov, Asymptotics of the Cauchy problem solution to the KdV equation with step-like initial data. Matem. sborn. 99, 261–281 (1976) 12. J. Lenells, The nonlinear steepest descent method for Riemann–Hilbert problems of low regularity. Indiana Univ. Math. J. 66(4), 1287–1332 (2017). https://doi.org/10.1512/iumj.2017. 66.6078 13. S.V. Manakov, Nonlinear Frauenhofer diffraction. Sov. Phys. JETP 38(4), 693–696 (1974)

Realizations of Meromorphic Functions of Bounded Type Christian Emmel and Annemarie Luger

In respectful memory of Sergey Naboko, who knew Herglotz-Nevanlinna functions so well

Abstract In this article it is shown that every function meromorphic in the upper halfplane and of bounded type does have a realization with the resolvent of a selfadjoint relation in a Krein space. Keywords Realizations · Functions of bounded type · Herglotz-Nevanlinna functions · Quasi-Herglotz functions · Krein spaces · Self-adjoint relations

1 Introduction Realizations of locally analytic functions are basically a way of writing the function in terms of the resolvent of an operator. Let us consider an example: Given an operator one can define a function, e.g., if A is a self-adjoint operator in a Hilbert space .H and .u ∈ H, then   f (ζ ) := (A − ζ )−1 u, u H

.

is analytic (at least) in the resolvent set of A, and—due to the positive definiteness of the inner product of the Hilbert space—maps the complex upper half plane into itself. Conversely, given a function g, one might ask if it can be written in the above form. In this example the answer is that this is possible if and only if g is analytic in the upper half plane, with non-negative imaginary part there, and satisfying

C. Emmel · A. Luger () Department of Mathematics, Stockholms Universitet, Stockholm, Sweden e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_18

501

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limy→∞ y|g(iy)| < ∞. One immediate consequence is that such functions can only have simple poles on the real line, since eigenvalues of self-adjoint operators in Hilbert spaces cannot have algebraic multiplicity larger than one. With other classes of operators and a bit more general representations also other large classes of analytic functions can be realized. These realizations can be a very useful tool for investigating functions and, conversely, also the analytic properties of the function can be used in order to derive properties of the corresponding operators. For example, in system theory functions with a realization are usually called transfer functions, from the vast literature let us just name for instance [25], the text book [5] or a recent result that uses transfer functions [16]. Another example are Weyl-functions (or Q-functions) in connection with extension theory of symmetric operators (see the classic book [1] or recently [3], and as a more concrete recent application e.g., [14]). Examples for such correspondences are Herglotz-Nevanlinna functions, which are exactly those functions, which admit an realization of the form (2.1) in a Hilbert space, whereas Generalized Nevanlinna functions are characterized by realizations in a Pontryagin space (i.e., an indefinite inner product space with finitely many negative squares), see [18]. Definitizable functions are defined via realizations with definitizable operators in Krein spaces, [17]. In the current paper we use in some sense an opposite approach and start with a particular rather general class of functions, namely, analytic functions that are meromorphic in the upper halfplane and of bounded type. This class also includes so called real complex functions and, in particular, Koebe inner functions, see e.g., [13]. For these functions we show that there exists a realization in a Krein space, see Theorem 5.1. In such a generality the only results that we are aware of are of the type given in [8] and also [2] , where, however, the authors have to restrict the realization to a smaller, simply connected, domain than the one where the function originally is given, or alternatively continuity on the boundary has to be assumed. Note, that results in this area often are obtained for functions in a certain domain, but then can be transferred into other domains, e.g., from the upper half-plane to the unit disk. In this text we have chosen to adopt an upper half plane point of view. This article is organized as follows. After this introduction we collect necessary preliminaries in Sect. 2. For the proof of the main result, namely, the existence of realizations in Krein spaces for meromorphic functions of bounded type, we need two tools, which are provided in Sects. 3 and 4, respectively. First, we show in Sect. 3 that if a function admits a (minimal) realization of the form (2.1) in a Krein space, then also every Moebius transformation (with real coefficients) admits a (minimal) realization in the same space. This result is known for Generalized Nevanlinna function, but here it is given in full generality and the proof has been simplified substantially. In Sect. 4 we construct realizations for real Quasi-Herglotz functions, which have been introduced in [21]. These two tools are then put together in Sect. 5 in order to obtain the main result, which is also illustrated by examples.

.

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2 Preliminaries 2.1 Realizations This article is concerned with resolvent type realizations in Krein spaces and we start with specifying what we mean by this. Recall that a Krein space .K is a complex vector space with an indefinite inner product .[·, ·]K such that .K is decomposable into a direct orthogonal sum K = K+ [+]K− ,

.

where .(K+ , [·, ·]K ) and .(K− , −[·, ·]K ) are Hilbert spaces. If either .K+ or .K− is finite dimensional then .K is called Pontragin space. We denote by .B(K1 , K2 ) the set of bounded linear operators between Krein spaces .K1 and .K2 and as usual .B(K) = .B(K, K). For a linear relation T (i.e., multivalued operator) the adjoint is denoted by .T + (or also .T ∗ in case that the underlying spaces are Hilbert spaces), for details see e.g., [9]. In what follows we will mainly deal with scalar functions. But since some results hold for matrix- or operator valued functions as well, we give here a general definition of what we mean by a realization. Here and in the following we denote by .D the given domain of the function, and we tacitly assume that .D is open. Definition 2.1 Let .H0 be a Hilbert space and .q : D ⊂ C → B(H0 ) an analytic function. We say that q admits a realization if there exists a Krein space .K, a selfadjoint relation A on .K with .(A) = ∅, a point .ζ0 ∈ (A) ∩ D and a mapping .Γζ0 ∈ B(H0 , K) such that for all .ζ ∈ (A) ∩ D it holds   q(ζ ) = q(ζ0 )∗ + (ζ − ζ0 )Γζ+0 I + (ζ − ζ0 )(A − ζ )−1 Γζ0 .

.

(2.1)

The pair .(A, Γζ0 ) is called realization of q with point of reference .ζ0 . Sometimes also the notion representation is used instead of realization. Note that the left side of (2.1) is defined for .ζ ∈ D, whereas the right side makes sense for .ζ ∈ (A), and hence it is required for this identity to hold on the intersection of these sets. Of most interest are of course realizations for which .D = (A). In what follows we want to illustrate different extreme situations. In the first example we have .(A)  D, whereas in the second .(A)  D. In order to avoid trivial situations we will have additional assumptions on .D and .(A) later in this text. 1 Example 2.2 It is easily checked that the function .q1 : C \ {1} → C; ζ → 1−ζ has 2 a realization in the space .K = C (with the usual Euclidean inner product) and with

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the operators A and .Γ := Γi be given by the matrices 

  1 0 .A = 0 −1

and

Γ =

1−i 2

0

.

Here the function .q1 is defined on the maximal domain of holomorphy, whereas the representing relation has an extra spectral point .z = −1, that is not a singularity of the right hand side of (2.1). 1 Example 2.3 The function .q2 : C+ → C; ζ → − ζ +i satisfies .Im q2 (ζ ) ≥ 0 for + .ζ ∈ C and hence it is known, cf. Proposition 2.13), that it admits a realization with a self-adjoint relation A in a Hilbert space and therefore .(A) ⊂ C \ R, whereas .q2 is defined in the upper half plane only.

Note that the self-adjointness of A in .K implies that .(A) is symmetric with respect to the real line. Hence if we define a function by the right handside of (2.1) then this function q will satisfy .q(ζ ) = q(ζ )∗ . Consequently, realizations of the form (2.1) are usually used for functions that either satisfy this symmetry property or are defined in a subset of the upper half-plane. In the latter case we therefore extend a function in the following way: Let .q : D ⊂ C+ → B(H0 ) be given, then we define the extension (which again is denoted by q) as q(ζ ) :=

.

⎧ ⎨q(ζ ),

if ζ ∈ D

⎩q(ζ ) , if ζ ∈ D∗ , ∗

(2.2)

where .D∗ := {ζ ∈ C : ζ ∈ D}. Example 2.4 In the above example the realization is extending the function .q2 as

q2 (ζ ) =

.

1 − ζ +i ζ ∈ C+ 1 − ζ −i ζ ∈ C−

,

and hence the extended function cannot be analytically extended to .R. Remark 2.5 In case that q is a scalar function, i.e., .B(H0 ) C, then (2.1) becomes q(ζ ) = q(ζ0 ) + (ζ − ζ0 )

.

   I + (ζ − ζ0 )(A − ζ )−1 vζ0 , vζ0 ,

(2.3)

since then the mapping .Γζ0 acts as .1 → vζ0 with some element .vζ0 ∈ K, and hence Γζ+0 k = [k, vζ0 ] for .k ∈ K.

.

Note that the example in the introduction is indeed a special case of such a realization, since under the assumption there we have .vζ0 ∈ dom(A) and hence (2.3) simplifies when introducing .u := (A − ζ0 )vζ0 .

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In connection with Definition 2.1 also the notion of .Γ -fields plays an important role. Definition 2.6 Let A be a self-adjoint relation on a Krein space .K with .(A) = ∅, and .H0 a Hilbert space. A family .{Γζ }ζ ∈(A) , with .Γζ ∈ B(H0 , K), is called .Γ -field (induced by A) if it holds that   Γw = I + (w − ζ )(A − w)−1 Γζ

.

∀ζ, w ∈ (A).

The following proposition shows how .Γ -fields are related to realizations, it is well known and follows from the resolvent identity, see e.g., [4]. Proposition 2.7 Let the function .q : D ⊂ C+ → B(H0 ) be analytic and .(A, Γζ0 ) a realization of q. Then   Γw := I + (w − ζ0 )(A − w)−1 Γζ0

.

∀w ∈ (A)

constitutes a .Γ - field (called associated .Γ - field) and it holds (extending q to .D∗ as usual) .

q(ζ ) − q(w)∗ = Γw+ Γζ ζ −w

∀ζ, w ∈ (A) ∩ (D ∪ D∗ ) with ζ = w.

(2.4)

Remark 2.8 In particular, we see from (2.4) that q(ζ ) = q(w)∗ + (ζ − w) · Γw+ Γζ   = q(w)∗ + (ζ − w) · Γw+ I + (ζ − w)(A − ζ )−1 Γw .

.

Consequently, for every .w ∈ (A) ∩ D also .(A, Γw ) is a realization of q. Thus, given a realization .(A, Γζ0 ) we can “vary” the point of reference .ζ0 within .(A) ∩ D by fixing the relation A and adjusting the operator .Γζ . In connection with e.g., Generalized Nevanlinna functions minimality is a crucial property of a realization, since there it makes the realization essentially unique and guarantees that domain of holomorphy of the function is contained in the resolvent set of the representing relation. In the current more general situation we cannot expect such results in full generality, but minimality is still relevant. Definition 2.9 Let .q : D ⊂ C+ → B(H0 ) be analytic and .(A, Γζ0 ) a realization of q. We call .(A, Γζ0 ) minimal if for the associated .Γ -field it holds that .K = span{Γζ : ζ ∈ (A)}. 1 in Example 2.2 is not minimal, but Example 2.10 The realization of .q1 (ζ ) = 1−ζ there obviously exists a minimal realization in the one-dimensional space .C.

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Finally, we want to mention continuity properties for later reference. These follow directly by invoking that the resolvent of a relation A is a continuous function on .(A), see e.g. [3]. Proposition 2.11 Let .{Γζ }ζ ∈(A) be a .Γ -field induced by A. Let .ζ ∈ (A), {ζn }n∈N ⊂ (A) with .ζn → ζ . Then .Γζn → Γζ in norm.

.

Proposition 2.12 Let .H0 be a Hilbert space, .K a Krein space, A a self-adjoint relation on .K with .(A) = ∅, .ζ0 ∈ (A) and finally .Γζ0 ∈ B(H0 , K). Then the function .q : (A) → B(H0 ) defined by   q(ζ ) = (ζ − ζ0 )Γζ+0 I + (ζ − ζ0 )(A − ζ )−1 Γζ0

.

is analytic.

2.2 Functions of Bounded Type The most general class of functions in this text, for which also the main result in Theorem 5.1 holds, are scalar (i.e., complex valued) meromorphic functions of bounded type in the upper half plane. More precisely, these are functions q that are meromorphic in .C+ and there exist bounded analytic functions .g1 and .g2 such that g1 (ζ ) .q(ζ ) = g2 (ζ ) . In what follows we recall some subclasses, which are important for us here.

2.2.1

Herglotz-Nevanlinna Functions

A function that maps the open upper halfplane .C+ analytically into the closed upper half plane is called Herglotz-Nevanlinna function. Using for instance a Möbius transform .ϕ from .C+ into the unit disc .D, it follows that if q is a Herglotz-Nevanlinna function then .ϕ(q) is bounded and analytic in .C+ . Since   −1 .q = ϕ ϕ(q)) , Herglotz-Nevanlinna functions are of bounded type. As mentioned in the introduction these functions also can be characterized by realizations of the form (2.3), but also via integral representations. More precisely, the following proposition holds. Proposition 2.13 For .q : C+ → C analytic, the following are equivalent: 1. The function q is Herglotz-Nevanlinna, i.e., .Im q(ζ ) ≥ 0 for .Im ζ ≥ 0. 2. The function q admits a minimal realization (2.3) in a Hilbert space .K. In this case .C+ ⊂ ρ(A).

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3. There exist numbers .a ∈ R and .b ≥ 0 and a positive finite Borel measure .σ such that  1 + tζ 1 (2.5) .q(ζ ) = a + bz + dν π R t −ζ for all .ζ ∈ C+ . 4. The Nevanlinna kernel .Nq (ζ, w) :=

q(ζ )−q(w) ζ −w

is positive definite.

Note that the integral representation (2.5) can be seen as a concrete instance of the realization (2.3). Indeed, if .b = 0 then A can be chosen to be the multiplication operator .f → t · f (t) in the space .L2 (R, ν). If, however, .b > 0 then A is a relation with one-dimensional multivalued part. 1 from Example 2.3 has a minimal Example 2.14 The function .q2 (ζ ) = − ζ +i 2 realization in the space .L (R, ν), where .ν is absolutely continuous with density 1 . , in particular, this space is infinite dimensional. 1+t 2

For more details about Herglotz-Nevanlinna functions, in particular, in connection with operator theory see e.g., [1], an overview with applications can be found in the recent review articles [22, 23].

2.2.2

Generalized Nevanlinna Functions

In the view of realizations the class of Generalized Nevanlinna functions is a natural extension of Herglotz-Nevanlinna functions, see e.g., [18]. Proposition 2.15 Let .q : D ⊂ C+ → C be meromorphic in .C+ , then the following are equivalent: 1. The function q admits a minimal realization (2.3) in a Pontryagin space .K with + .κ negative squares. In this case .C ∩ hol(q) ⊂ ρ(A). 2. The Nevanlinna kernel .Nq (ζ, w) := q(ζζ)−q(w) has .κ negative squares. −w In this case the function is called a Generalized Nevanlinna function, .q ∈ Nκ . There also exists an integral representation and a distributional representation, but these are not of importance for us here. However, there is yet another equivalent representation via factorizations, see [7, 10] and for a more recent alternative proof [26]. Proposition 2.16 A function q belongs to the Generalized Nevanlinna class .Nκ if and only if there exist .q0 ∈ N0 and a rational function r of degree .κ such that q(ζ ) = r(ζ ) · q0 (ζ ) · r(ζ ).

.

(2.6)

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In particular, this shows that also Generalized Nevanlinna functions are of bounded type. Moreover, we would like to mention that this factorization result can be utilized in order to construct a concrete realization, where the realizations of the factors .q0 and r are used as building bricks, see e.g., [11, 15, 19].

2.2.3

Extended Nevanlinna Class

Generalized Nevanlinna functions are comparably well studied, however, in applications, they do not always suffice. Hence an even larger class has been introduced in [6], which we adapt here slightly for our purpose. Definition 2.17 Let .q : D ⊂ C+ → C be meromorphic in .C+ and of bounded type there. Then q has boundary values .q(x) := lim q(x + iy) for almost all .x ∈ R. The y→0

function q is said to belong to the extended Nevanlinna class if Im(q(x)) ≥ 0 for almost all x ∈ R.

.

Moreover, its index .I (q) ∈ N0 ∪ {∞} is defined as the number of negative squares of the associated Nevanlinna kernel Nq (ζ, w) :=

.

q(ζ ) − q(w) , ζ −w

ζ, w ∈ D.

Obviously the extended Nevanlinna class contains the Generalized Nevanlinna class and hence also all Herglotz-Nevanlinna functions. Finally we want to mention that in the recent past meromorphic functions of bounded type with real boundary values have attracted substantial interest, see for example [12], [24] and the survey [13]. A very important subclass is constituted by the so called Koebe inner functions which are generated by inner functions, cf., [13]. Recall that an analytic function with boundary values of modulus one almost everywhere is called inner. Definition 2.18 A Koebe inner function is a function of the form h1 :=

.

V (V + 1)2

or

h2 :=

(V + 1)2 , V

where V is an inner function on the upper half plane. ζ Note that the map .ζ → (ζ +1) 2 maps the unit circle on the positive real axis and hence .h1 and .h2 have real (even positive) boundary values almost everywhere.

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3 Realizations of Möbius Transforms The main purpose of this section is to derive realizations for real Möbius transforms of functions with given realizations. In the case of Generalized Nevanlinna functions similar questions have been considered e.g., in [20]. Here, however, we work with more general realizations and we also provide considerably simpler formulas and proofs. Theorem 3.1 Let .q : D ⊂ C+ → B(H0 ) be analytic such that q is boundedly invertible for some points in .D and denote the inverse function by .

qˆ : Dˆ := {ζ ∈ D : q(ζ ) is boundedly invertible} → B(H0 ),

q(ζ ˆ ) := −q(ζ )−1 .

If q has a realization .(A, Γζ0 ) (on .K) then for suitable .ζ also the inverse function .qˆ can be represented in the form (2.1). More precisely, we choose .ζ0 ∈ Dˆ ∩ (A), set .q0 = q(ζ0 ), and .q ˆ0 = q(ζ ˆ 0 ), and consider the operator .Γˆζ0 = Γζ0 qˆ0 ∈ B(H0 , K) and the linear relation .Aˆ given by   (Aˆ − ζ0 )−1 := (A − ζ0 )−1 − Γζ0 q0−1 Γζ+0 I + (ζ0 − ζ0 )(A − ζ0 )−1 .

.

ˆ ∩ Dˆ the inverse function .qˆ can be written as Then for every .ζ ∈ (A) ∩ (A)   q(ζ ˆ ) = q(ζ ˆ 0 )∗ + (ζ − ζ0 )Γˆζ+0 I + (ζ − ζ0 )(Aˆ − ζ )−1 Γˆζ0 .

.

(3.1)

ˆ and for every .ζ ∈ (A) ∩ (A) ˆ ∩ Dˆ it Moreover, .Aˆ is self-adjoint with .ζ0 ∈ (A) holds that (Aˆ − ζ )−1 − (A − ζ )−1 = Γζ q(ζ ˆ )Γζ+ .

.

(3.2)

Remark 3.2 The minus sign in the definition of .qˆ has its origin in the fact that the classes of Herglotz-Nevanlinna functions and Generalized Nevanlinna functions are closed under this transformation. If we make mild assumptions on the realization of q then we can guarantee that ˆ which means that we ˆ ∩ D, the representation (3.1) holds even for all .ζ ∈ (A) ˆ For simplicity we formulate this result only in case indeed obtain a realization of .q. of scalar functions, since the notion of zeros of operator valued functions is more involved and not needed for our purpose here. For a scalar analytic function q we denote by .Z(q) the set of its zeros. Corollary 3.3 Let .0 ≡ q : D ⊂ C+ → C be analytic. Assume that q has a ˆ Then the realization .(A, Γζ0 ) for which .(A) is dense in .C and with .ζ0 ∈ (A) ∩ D.

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ˆ Γˆζ0 ) from Theorem 3.1 is a realization of the inverse function pair .(A, .

qˆ : Dˆ = D \ Z(q) → C,

q(ζ ˆ ) := −q(ζ )−1 .

Proof (Theorem 3.1) In order to simplify notation throughout the proof we introduce for a linear relation T Tww12 := I + (w1 − w2 )(T − w1 )−1

.

∀w1 ∈ (T ), w2 ∈ C

and we note that then the resolvent identity yields Tww21 Tww10 = Tww20

(Tww12 )−1 = Tww21

and

.

∀w1 , w2 ∈ (T ), w0 ∈ C.

Using this we may rewrite the induced .Γ -field as .Γζ = Aζ0 Γζ0 , and also .Γζ+ = ζ

Γζ+0 Aζ0 . and the realization becomes ζ

q(ζ ) = q0∗ + (ζ − ζ0 )Γζ+0 Aζ0 Γζ0 = q0 + (ζ − ζ0 )Γζ+0 Aζ0 Γζ0 , ζ

ζ

.

∀ζ ∈ (A) ∩ D (3.3)

where the latter equality follows from .q(ζ ) = q(ζ )∗ (see Remark after Example 2.3). From the definition of .Aˆ follows ζ ζ ζ Aˆ ζ0 = Aζ0 + (ζ − ζ0 )Γζ0 q0−1 Γζ+0 Aζ00

.

(3.4)

In terms of this notation we show that .

q(ζ ˆ ) = qˆ0∗ + (ζ − ζ0 )Γˆζ+0 Aˆ ζ0 Γˆζ0 ζ

ˆ ˆ ∩ D. ∀ζ ∈ (A) ∩ (A)

(3.5)

To this end, we first note the following identity, which follows directly from (2.4) q0 − q0∗ = (ζ0 − ζ0 )Γζ+0 Γζ0 .

.

(3.6)

and infer using this identity .

    I − (ζ0 − ζ0 )Γζ0 q0−1 Γζ+0 Γζ0 = Γζ0 I − (ζ0 − ζ0 )q0−1 Γζ+0 Γζ0 = Γζ0 q0−1 q0∗ . (3.7)

Now we are ready to verify that (3.5) holds by checking that the right hand side is ˆ We use the definition of .Γˆζ0 in ˆ ∩ D. the negative inverse of q for .ζ ∈ (A) ∩ (A)

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line 2, and identity (3.7) in line 4 to calculate  .

   ζ ζ qˆ0∗ + (ζ − ζ0 )Γˆζ+0 Aˆ ζ0 Γˆζ0 · q0∗ + (ζ − ζ0 )Γζ+0 Aζ0 Γζ0     ζ ζ = −q0−∗ + (ζ − ζ0 )q0−∗ Γζ+0 Aˆ ζ0 Γζ0 q0−1 · q0∗ + (ζ − ζ0 )Γζ+0 Aζ0 Γζ0  ζ ζ = −I + (ζ − ζ0 ) − q0−∗ Γζ+0 Aζ0 Γζ0 + q0−∗ Γζ+0 Aˆ ζ0 Γζ0 q0−1 q0∗ + (ζ − ζ0 )q0−∗ Γζ+0 Aˆ ζ0 Γζ0 q0−1 Γζ+0 Aζ0 Γζ0 ζ



ζ

ζ = −I + (ζ − ζ0 )q0−∗ Γζ+0 Aˆ ζ0   ζ ζ ζ −1 + −1 + ˆ − Aζ0 + (I − (ζ0 − ζ0 )Γζ0 q0 Γζ0 )Aζ0 + (ζ − ζ0 )Γζ0 q0 Γζ0 Aζ0 Γζ0

= −I, where the last equality follows from the following calculation (using (3.4) in line 2)   ζ ζ − Aˆ ζ0 + I − (ζ0 − ζ0 )Γζ0 q0−1 Γζ+0 Aζ0 + (ζ − ζ0 )Γζ0 q0−1 Γζ+0 = − Aζ0 − (ζ − ζ0 )Γζ0 q0−1 Γζ+0 Aζ00 + Aζ0 ζ

.

ζ

ζ

− (ζ0 − ζ0 )Γζ0 q0−1 Γζ+0 Aζ0 + (ζ − ζ0 )Γζ0 q0−1 Γζ+0   ζ ζ =Γζ0 q0−1 Γζ+0 −(ζ − ζ0 )Aζ00 − (ζ0 − ζ0 )Aζ0 + (ζ − ζ0 )I   =Γζ0 q0−1 Γζ+0 −ζ + ζ0 − ζ0 + ζ0 + ζ − ζ0 I = 0. ζ

ˆ For .ζ = ζ0 Next we verify (3.2), which is clear for .ζ = ζ0 from the definition of .A. we verify using (3.4) in line 4 and (3.3) in the last equality   (ζ − ζ0 ) (Aˆ − ζ )−1 − (A − ζ )−1 − Γζ q(ζ ˆ )Γζ+

.

=Aˆ ζ0 − Aζ0 − (ζ − ζ0 )Aζ0 Γζ0 q(ζ ˆ )Γζ+0 Aζ0   ζ0 ζ ζ ζ0 ζ ζ0 ζ + ˆ ˆ ˆ =Aζ Aζ − Aζ0 Aζ − (ζ − ζ0 )Aζ0 Aζ Γζ0 q(ζ ˆ )Γζ0 Aζ0 ζ

ζ

0

ζ

ζ

0

   ζ ζ0 ζ −1 + ζ0 ˆ =Aζ Aζ Aζ0 + (ζ − ζ0 )Γζ0 q0 Γζ0 Aζ0 0

  ζ ζ ζ Aζ0 + (ζ − ζ0 )Aζ0 Γζ0 q(ζ Aζ0 ˆ )Γζ+0 0

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  ζ ζ0 ζ ˆ ˆ )Γζ+0 + Γζ0 q0−1 Γζ+0 =Aζ Aζ − Aζ − (ζ − ζ0 ) Γζ0 q(ζ 0

0

ˆ )Γζ+0 + (ζ − ζ0 )Γζ0 q0−1 Γζ+0 Aζ0 Γζ0 q(ζ ζ

 ζ Aζ0

  ζ ζ ζ ˆ )Γζ+0 Aζ0 = 0. = − (ζ − ζ0 )Aˆ ζ0 Γζ0 q0−1 q0 − q(ζ ) + (ζ − ζ0 )Γζ+0 Aζ0 Γζ0 q(ζ Lastly, .Aˆ is self-adjoint if and only if the Cayley transform .Aˆ ζ00 is unitary (see [9]). We calculate, using (3.4) ζ

ζ ζ ζ ζ Aˆ ζ00 = Aζ00 + (ζ0 − ζ0 )Γζ0 q0−1 Γζ+0 Aζ00 = (I + (ζ0 − ζ0 )Γζ0 q0−1 Γζ+0 )Aζ00 .

.

Since A is self-adjoint, .Aζ00 is unitary and .(I + (ζ0 − ζ0 )Γζ0 q0−1 Γζ+0 ) is also unitary, which is straightforward to verify using (3.6).   ζ

ˆ ∩ Dˆ the Proof (Corollary 3.3) Theorem 3.1 states that for all .ζ ∈ (A) ∩ (A) inverse function .qˆ can be written as   q(ζ ˆ ) = q(ζ ˆ 0 )∗ + (ζ − ζ0 )Γˆζ+0 I + (ζ − ζ0 )(Aˆ − ζ )−1 Γˆζ0 .

.

ˆ Γˆζ0 ) is indeed a realization of .qˆ we must show In order to show that the pair .(A, ˆ Since .(A) is dense in .C and ˆ ∩ D. that this equality extends to the whole of .(A) ˆ ∩ Dˆ is open it follows that .(A) ˆ ∩ Dˆ ⊂ (A) ˆ ∩ Dˆ (A) ∩ (A)

.

is a dense inclusion. Furthermore, recall that the right hand side in the above equality ˆ (see Proposition 2.12) and that .qˆ is analytic on defines an analytic function on .(A) ˆ by definition. Thus, since the equality holds on a dense subset of .(A) ˆ it ˆ ∩ D, .D ˆ ∩ Dˆ by continuity.   holds on the whole of .(A) In Theorem 3.1 the construction seems to depend on the point .ζ0 . However, we prove now that under suitable assumptions the constructed relation .Aˆ does not depend on this choice. More precisely, recall that if .(A, Γζ0 ) is a realization for q and .ζ ∈ (A) ∩ D, then also .(A, Γζ ) is a realization for q (see Proposition 2.7 and Remark 2.8). Consequently, we can apply Theorem 3.1 to every realization .(A, Γζ ) where .ζ ∈ (A) ∩ Dˆ ⊂ (A) ∩ D is arbitrary. Lemma 3.4 Let .0 ≡ q : D ⊂ C+ → C be analytic. Assume that q has a ˆ Then realization .(A, Γζ ) for which .(A) ∩ D is connected and .ζ ∈ (A) ∩ D. ˆ the obtained relation .A from applying Theorem 3.1 does not depend on the chosen point of reference .ζ .

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Proof Within this proof we use the notation .Aˆ (ζ ) in order to indicate the chosen point of reference in Theorem 3.1. First, note that since .(A) ∩ D is assumed to be connected and .Z(q) has no accumulation points in .(A) ∩ D (q is analytic on ˆ is also .(A) ∩ D) we consequently conclude that .((A) ∩ D) \ Z(q) = (A) ∩ D connected. Therefore, it is sufficient to show that the function (A) ∩ Dˆ  ζ → Aˆ (ζ )

.

ˆ Then .ζ ∈ (Aˆ (ζ ) ) by construction and we is locally constant. Let .ζ ∈ (A) ∩ D. can consider an open neighbourhood U ˆ ζ ∈ U ⊂ (Aˆ (ζ ) ) ∩ (A) ∩ D.

.

Let .w ∈ U be arbitrary. Since .w ∈ U and .w ∈ (Aˆ (w) ) by construction, it follows that w ∈ (Aˆ (ζ ) ) ∩ (A) ∩ Dˆ and w ∈ (Aˆ (w) ) ∩ (A) ∩ Dˆ

.

and we infer by (3.2) that + −1 (Aˆ (ζ ) − w)−1 = Γw q(w)Γ ˆ = (Aˆ (w) − w)−1 . w + (A − w)

.

Therefore, .Aˆ (ζ ) = Aˆ (w) and the function is constant on U , which finishes the proof.   ˆ ∩ Dˆ we can express .q(ζ Corollary 3.3 states that whenever .ζ ∈ (A) ˆ ) as in (2.1). ˆ This is the subject of the Now we want to say more about the resolvent set .(A). following Corollary. Corollary 3.5 Let .0 ≡ q : D ⊂ C+ → C be analytic. Assume that q has a realization .(A, Γζ0 ) for which .(A) is dense in .C, .(A) ∩ D is connected and .ζ0 ∈ ˆ Let .(A, ˆ Γˆζ0 ) be the realization from Theorem 3.1. Then (A) ∩ D. ˆ (A) ∩ Dˆ ⊂ (A).

.

Proof We have seen in Lemma 3.4 that .Aˆ does not depend on the point of reference ˆ Therefore, since by construction the point of reference is in the ζ ∈ (A) ∩ D. ˆ resolvent set of the relation .Aˆ it follows that .(A) ∩ Dˆ ⊂ (A).  

.

In addition, we also determine the structure of the Gamma field .{Γˆζ }ζ ∈(A) ˆ (recall Definition 2.6 ) in relation to the transformation of Corollary 3.3. Lemma 3.6 Let .0 ≡ q : D ⊂ C+ → C be analytic. Assume that q has a realization ˆ .(A, Γζ0 ) for which .(A) is dense in .C, .(A) ∩ D is connected and .ζ0 ∈ (A) ∩ D.

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ˆ Γˆζ0 ) be the constructed realization from Corollary 3.3. Then it holds that Let .(A, .

  −1 Γˆw = Γw · (−q(w)−1 ) and Γˆw = Γw · −q(w)

ˆ for all w ∈ (A) ∩ D.

ˆ (see Corollary 3.5). We calculate using Proof Consider .w ∈ (A) ∩ Dˆ ⊂ (A) ˆ the definition of .Γζ0 (see Theorem 3.1) in line 2, Eq. (3.2) in line 3 and Eq. .(2.4) in line 5   ˆ − w)−1 Γˆζ0 .Γˆw = I + (w − ζ0 )(A   = I + (w − ζ0 )(Aˆ − w)−1 Γζ0 · (−q(ζ0 )−1 )   = − I + (w − ζ0 )(A − w)−1 + (w − ζ0 )Γw (−q −1 (w))Γw+ Γζ0 q(ζ0 )−1   = − I + (w − ζ0 )(A − w)−1 Γζ0 q(ζ0 )−1   + Γw q −1 (w) (w − ζ0 )Γw+ Γζ0 q(ζ0 )−1 = − Γw q(ζ0 )−1 + Γw q(w)−1 (q(w) − q(ζ0 ))q(ζ0 )−1 = Γw · (−q(w)−1 ). ˆ By taking adjoints   in (3.2) it can be verified analogously that we also have .Γw = −1 Γw · −q(w) .   Concerning minimality we have the following result: Proposition 3.7 Let .0 ≡ q : D ⊂ C+ → C be analytic. Assume that q has a realization .(A, Γζ0 ) for which .(A) ∩ (D ∪ D∗ ) is dense in .C, .(A) ∩ D is connected and .ζ0 ∈ Dˆ ∩ (A). If the realization .(A, Γζ0 ) is minimal, then the realization ˆ Γˆζ0 ) of .qˆ from Corollary 3.3 is minimal. .(A, Proof We have to show that ˆ = K. span{Γˆζ : ζ ∈ (A)}

.

Since the relations A and .Aˆ are self-adjoint we have that the resolvent sets .(A) and ˆ are symmetric with respect to .R. Let .w ∈ (A) ∩ Dˆ ⊂ (A). ˆ We have seen in (A) Lemma 3.6 that   −1 −1 .Γˆw = Γw · (−q(w) ) and Γˆw = Γw · −q(w) .

.

Consequently, it holds that span{Γζ : ζ ∈ (A) ∩ (Dˆ ∪ Dˆ ∗ )} = span{Γˆζ : ζ ∈ (A) ∩ (Dˆ ∪ Dˆ ∗ )}.

.

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Secondly, since .(A) ∩ (D ∪ D∗ ) is open and dense in .C and .Z(q) ∪ Z(q)∗ has no accumulation point in .(A) ∩ (D ∪ D∗ ), it follows that (A) ∩ (Dˆ ∪ Dˆ ∗ ) = ((A) ∩ (D ∪ D∗ )) \ (Z(q) ∪ Z(q)∗ ) ⊂ (A)

.

is a dense inclusion. Consequently, we conclude by the continuous dependence of Γζ on .ζ (see Proposition 2.11) that

.

span{Γζ : ζ ∈ (A)} = span{Γζ : ζ ∈ (A) ∩ (Dˆ ∪ Dˆ ∗ )}.

.

Using the assumed minimality we arrive at K = span{Γζ : ζ ∈ (A)} = span{Γζ : ζ ∈ (A) ∩ (Dˆ ∪ Dˆ ∗ )}

.

ˆ = span{Γˆζ : ζ ∈ (A) ∩ (Dˆ ∪ Dˆ ∗ )} ⊂ span{Γˆζ : ζ ∈ (A)},  

which finishes the proof. So far we have only considered the Möbius transform .m(z) = generalize our result in the following way:

1 −z . However,

we can

Corollary 3.8 Let .0 ≡ q : D ⊂ C+ → C be analytic. Consider a real Möbius transform m(z) =

.

az + b , cz + d

a, b, c, d ∈ R : bc − ad = 0.

Assume that q has a realization .(A, Γζ0 ) for which .(A) ∩ (D ∪ D∗ ) is dense in .C, (A) ∩ D is connected and .ζ0 ∈ (A) ∩ D \ Z(cq + d). Then .m(q) has a realization .(Aˆ m , Γˆζm0 ) with

.

((A) ∩ D) \ Z(cq + d) ⊂ (Aˆ m ).

.

Furthermore, if .(A, Γζ0 ) is minimal, then .(Aˆ m , Γˆζm0 ) is minimal as well. Proof First, note that adding a real constant is incorporated in a realization by changing the constant term in (2.1) and multiplying by a real nonzero constant is incorporated by rescaling the inner product. Therefore, these operations do not change A or minimality. Consequently, we may assume without loss of generality that .c = 0 and decompose  .m = z → −cz → −cz − d →

u a u 1

→

→ + cz + d cz + d c cz + d with u :=



bc − ad . c

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1 In this decomposition, only the negative inversion .−cz − d → cz+d is neither an addition with a real constant nor a multiplication by a real nonzero constant. Consequently, it is the only operation left to treat. We have proven Corollary 3.3 for the existence of a realization, Corollary 3.5 for the description of the resolvent   set and finally Proposition 3.7 for the statement about minimality.

4 Quasi-Herglotz Functions In the proof of the main result we will make use of realizations of so-called (real) Quasi-Herglotz functions. These are basically (real) linear combinations of Herglotz-Nevanlinna functions, and have been introduced and investigated in [21]. We repeat the definition and relevant results here for convenience. Definition 4.1 A function .q : C+ → C is called real Quasi-Herglotz function if there are two Herglotz-Nevanlinna functions .q1 , .q2 such that .q = q1 − q2 . Note that the class of real Quasi-Herglotz functions is closed under addition, subtraction and also multiplication by real scalars. Using the integral representation for Herglotz-Nevanlinna functions in Proposition 2.13 one obtains the following characterization, see [21]. Proposition 4.2 A function .q : C+ → C is a real Quasi-Herglotz function if and only if there exist .a, b ∈ R and a signed Borel measure v on .R such that q(ζ ) = a + bz +

.

1 π

 R

1 + tζ dv t −ζ

∀ζ ∈ C+ .

(4.1)

The parameters .a, b, v are unique and we call .(a, b, v) the representing triple of q. Hence there is a unique way to decompose a Quasi-Herglotz function q as in Definition 4.1. Indeed, we can associate two Herglotz-Nevanlinna functions .h1 and .h2 with the following triples h1 ↔ (a, max(b, 0), v+ )

.

h2 ↔ (0, min(b, 0), v− )

where .v = v+ − v− is the Jordan decomposition. From the realizations of .h1 and h2 , which exist by Proposition 2.13, a realization for the Quasi-Herglotz function q can be constructed.

.

Theorem 4.3 Let .q be a real Quasi-Herglotz function and .ζ0 ∈ C+ be arbitrary. Consider .h1 and .h2 as defined above and their minimal realizations .(A1 , Γζ10 ) and

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(A2 , Γζ20 ) on Hilbert spaces .H1 and .H2 respectively. Then .(A, Γζ0 ) defined on the Krein space .K

.

K := H1 ⊕ H2 ,

.

[(x1 , x2 ), (y1 , y2 )]K := [x1 , y1 ]H1 − [x2 , y2 ]H2 ∀(x1 , x2 ), (y1 , y2 ) ∈ K with A given by (A − ζ0 )

.

−1

 0 (A1 − ζ0 )−1 , = 0 (A2 − ζ0 )−1 

and

Γ ζ0 =

 Γζ10 Γζ20

is a minimal realization for q with .C \ R ⊂ (A). Proof It follows by straightforward computation that this is indeed a realization. Minimality follows from the fact that .h1 and .h2 were chosen in such a way that their associated measures .v+ and .v− are mutually singular. More precisely, assume that .b = 0, the proof for .b = 0 is essentially the same. Consider the minimal realizations .(A1 , Γζ10 ) and .(A2 , Γζ20 ) on the Hilbert spaces 2 2 .H1 = L (R, v+ ) and .H2 = L (R, v− ), where .A1 and .A2 are the multiplication operators by the independent variable. Note that minimal realizations of HerglotzNevanlinna functions are unique up to isometric isomorphisms. Note also that the closure in the definition of minimality is with respect to the positive inner product in the Krein space .K induced by the fundamental symmetry. In this case this means we consider the space .L2 (R, v+ ) ⊕ L2 (R, v− ). Since .v+ and .v− are mutually singular we have the isometric isomorphism (of Hilbert spaces) K = L2 (R, v+ ) ⊕ L2 (R, v− ) ∼ = L2 (R, v+ + v− ),

.

(f1 , f2 ) → f1 + f2

Consequently, we only have to show that   L2 (R, v+ + v− ) = span Γζ10 + Γζ20 : z ∈ (A1 ) ∩ (A2 )

.

Now, consider the Herglotz-Nevanlinna function .h = h1 + h2 , which has the representing measure .v+ + v− and a minimal realization .(A3 , Γζ30 ) on .L2 (R, v+ + v− ), where .A3 is also the multiplication operator by the independent variable. By standard theory on realizations of Herglotz-Nevanlinna functions (see e.g.,[19]) we infer that (A3 ) = (A1 ) ∩ (A2 )

.

and

Γζ10 + Γζ20 = Γζ30 ,

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where .Γζ10 + Γζ20 in the latter equality is to be understood as a mapping into 2 .L (R, v+ + v− ). Since the realization of .h3 was minimal it follows that   L2 (R, v+ + v− ) = span Γζ30 : z ∈ (A3 )   = span Γζ10 + Γζ20 : z ∈ (A1 ) ∩ (A2 ) ,

.

 

which finishes the proof.

Note that in the theorem above also any other decomposition of q (as in Definition 4.1) can be used, however, then in general the resulting realization is not minimal again. The class of real Quasi-Herglotz functions is analytically characterized in [21]. However, for our purpose here the following is sufficient. Proposition 4.4 Let q be a bounded analytic function in the upper halfplane, i.e., q ∈ H ∞ (C+ ). Then q is a real Quasi-Herglotz function.

.

Proof Rewriting .q = (q + i · q∞ ) − i · q∞ provides a decomposition as in Definition 4.1.   We now apply the theory of Sect. 3 to obtain: Corollary 4.5 Let .0 ≡ q be a real Quasi-Herglotz function and consider a real Möbius transform m(z) =

.

az + b , cz + d

a, b, c, d ∈ R : bc − ad = 0.

Let .ζ0 ∈ C+ \ Z(cq + d) be arbitrary. Then .m(q) has a minimal realization ˆ m , Γˆ m ) admitting .(A ζ0

C+ \ Z(cq + d) ⊂ (Aˆ m ).

.

Proof By Theorem 4.3 the function q has a minimal realization .(A, Γζ0 ) with .C+ ⊂ (A). We apply Corollary 3.8 to q noting that .D = C+ and .C \ R ⊂ (A) and end up with the desired realization.  

5 Main Theorem In this section we finally prove the main result, which guarantees the existence of a realization for every meromorphic function of bounded type. Theorem 5.1 Let .f : D ⊂ C+ → C be a meromorphic function of bounded type in the upper half plane and .ζ0 ∈ D. Then f has a realization .(A, Γζ0 ) admitting

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(A) ∩ C+ = D. In other words, f can be expressed by formula (2.1), i.e.

.

  f (ζ ) = f (ζ0 ) + (ζ − ζ0 )Γζ+0 I + (ζ − ζ0 )(A − ζ )−1 Γζ0

.

∀ ζ ∈ D.

Proof Since .f is of bounded type we can write .f = ff12 , where .f1 and .f2 are in ∞ (C+ ) and do not have common zeros. We denote by .P(f ) the poles of f and .H note that .P(f ) = Z(f2 ). We rewrite f =

.

f1 f1 + 2f1 ∞ − 2f1 ∞ f1 + 2f1 ∞ −2f1 ∞ = = + f2 f2 f2 f2

and define g1 :=

.

f2 , f1 + 2f1 ∞

g2 :=

−f2 . 2f1 ∞

It is obvious that .g2 ∈ H ∞ (C+ ) and that .Z(g2 ) = Z(f2 ) = P(f ). For .g1 we note that     .f1 (z) + 2f1 ∞  ≥ f1 ∞ ∀z ∈ C+ . Thus, .g1 ∈ H ∞ (C+ ) and .Z(g1 ) = Z(f2 ) = P(f ). Consequently, the functions .g1 and .g2 are real Quasi-Herglotz functions (Proposition 4.4) with f =

.

1 1 + g1 g2

and

Z(g1 ) = Z(g2 ) = P(f ).

Since .ζ0 ∈ D = C+ \ P(f ) we have by Corollary 4.5 minimal realizations 1 g1

and

(A2 , Γζ20 ) for

C+ \ P(f ) ⊂ (A1 )

and

C+ \ P(f ) ⊂ (A2 ).

(A1 , Γζ10 ) for

.

1 g2

with the resolvent sets admitting .

It is straightforward to verify that .(A, Γζ0 ) defined on the Krein space .K given by K := K1 ⊕ K2 ,

.

[(x1 , x2 ), (y1 , y2 )]K := [x1 , y1 ]K1 + [x2 , y2 ]K2 ∀(x1 , x2 ), (y1 , y2 ) ∈ K    1 Γζ10 0 (A − ζ0 )−1 −1 , Γ ζ0 = (A − ζ0 ) = 2 −1 Γζ20 0 (A − ζ0 )

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is a realization for f . Since .(A) = (A1 ) ∩ (A2 ) holds it follows in addition that + .D = C \ P(f ) ⊂ (A).   In particular, the above theorem can be applied to the extended Nevanlinna class. Corollary 5.2 Let .f : D ⊂ C+ → C be in the extended Nevanlinna class and .ζ0 ∈ D. Then f has a (in general non-minimal) realization .(A, Γζ0 ) admitting + = D. .(A) ∩ C Of particular interest are functions of the extended Nevanlinna class with infinite index. These also provide examples of functions, for which so far no realizations have been known. Example 5.3 Consider an infinite Blaschke product B. Then the Koebe inner function f :=

.

(B + 1)2 B

is in the extended Nevanlinna class, however it is not a Generalized Nevanlinna function (nor a definitizable function) because it has infinitely many poles in the upper half plane. Consequently, it holds .I (f ) = ∞. 2 More generally, it can be shown that the Koebe inner function . (V +1) has infinite V index if and only if either V has a non trivial singular part or infinitely many zeros. It also has to be noted that the construction in Theorem 3.1 provides a realization, which in general is not minimal, even if we know that in the upper halfplane the poles of the function do coincide with the spectrum of the representing relation. In the following example we demonstrate the construction with a simple example. 1 Example 5.4 Let .q1 (ζ ) = 1−ζ be the Herglotz-Nevanlinna function in Examples 2.2 and 2.10. First we rewrite .q1 as a fraction of bounded analytic functions, e.g. as

q1 (ζ ) =

.

Here for the nominator .f1 (ζ ) =

1 ζ +i

.

we have .f1 ∞ = 1 and hence

q1 (ζ ) =

.

1 ζ +i 1−ζ ζ +i

1 1−ζ ζ +i 1 ζ +i +2

+

1 1−ζ ζ +i

.

−2

1−ζ For the Quasi-Herglotz functions .g1 (ζ ) = 2ζ +1+2i and .g2 (ζ ) = − 12 · 1−ζ ζ +i there is a minimal realization. Let us for instance look at the second term, which reminds of the function in Example 2.3. It can be checked that the signed measures for .g2

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is absolutely continuous .dν = ρdλR with density .ρ(t) = 2(t1−t 2 +1) . Without writing down the detailed formulas we can see that the Krein spaces for the representations of both .g1 and .g2 (and hence also their orthogonal sum) are infinite dimensional, even if we know that the function .q1 does admit a minimal representation in a onedimensional space. The above example shows that the construction itself in the proof of the existence result in Theorem 5.1 is not yet optimal. However, it has to be pointed out that it can be applied to a much larger class of functions than results as in [2, 8]. The main significant difference is, that we can allow both domains with holes (in particular, meromorphic functions) and do not need to assume continuity on the boundary.

References 1. N. Akhiezer, and I. Glazman, in Theory of Linear Operators in Hilbert Space (Dover Publications, New York, 1993). Translated from the Russian and with a preface by Merlynd Nestell, Reprint of the 1961 and 1963 translations 2. D. Alpay, F. Colombo, I. Sabadini, Realizations of holomorphic and slice hyperholomorphic functions: the Krein space case. Indag. Math. 31, 607–628 (2020) 3. J. Behrndt, S. Hassi, H. de Snoo, in Boundary Value Problems, Weyl Functions, and Differential Operators (Birkhäuser, Cham, 2020) 4. M. Borogovac, A. Luger, Analytic characterizations of Jordan chains by pole cancellation functions of higher order. J. Funct. Anal. 267, 4499–4518 (2014) 5. R. Curtain, H. Zwart, in Introduction to Infinite-Dimensional Systems Theory: A State-Space Approach (Springer Nature, 2020) 6. P. Delsarte, Y. Genin, Y. Kamp, Pseudo-Caratheodory functions and Hermitian Toeplitz matrices. Philips J. Res. 41, 1–54 (1986) 7. V. Derkach, S. Hassi, H. de Snoo, Operator models associated with Kac subclasses of generalized Nevanlinna functions. Methods Funct. Anal. Topol. 5, 65–87 (1999) 8. A. Dijksma, H. Langer, H. de Snoo, Representations of holomorphic operator functions by means of resolvents of unitary or selfadjoint operators in Krein spaces. Oper. Theory Adv. Appl. 24, 123–143 (1987) 9. A. Dijksma, H. de Snoo, Symmetric and selfadjoint relations in Krein spaces. I. Oper. Theory Adv. Appl. 24, 145–166 (1987) 10. A. Dijksma, H. Langer, A. Luger, Y. Shondin, A factorization result for generalized Nevanlinna functions of the class Nκ . Integral Equ. Oper. Theory. 36, 121–125 (2000) 11. A. Dijksma, H. Langer, A. Luger, Y. Shondin, Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization. Oper. Theory Adv. Appl. 154, 69– 90 (2004) 12. S. Garcia, A ∗-closed subalgebra of the Smirnov class. Proc. Amer. Math. Soc. 133, 2051–2059 (2005) 13. S. Garcia, J. Mashreghi, W. Ross, Real complex functions. Contemp. Math. 679, 91–128 (2016) 14. F. Gesztesy, S. Naboko, R. Weikard, M. Zinchenko, Donoghue-type m-functions for Schrödinger operators with operator-valued potentials. J. Anal. Math. 137, 373–427 (2019) 15. S. Hassi, H. Wietsma, Minimal realizations of generalized Nevanlinna functions. Opuscula Math. 36, 749–768 (2016) 16. B. Jacob, S. Möller, C. Wyss, Stability radius for infinite-dimensional interconnected systems. Syst. Control Lett. 138, 104662 (2020)

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17. P. Jonas, On operator representations of locally definitizable functions. Oper. Theory Adv. Appl. 162, 41–72 (2006) 18. M. Krein, H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77, 187–236 (1977) 19. M. Langer, A. Luger, Scalar generalized Nevanlinna functions: realizations with block operator matrices. Oper. Theory Adv. Appl. 162, 253–276 (2006) 20. A. Luger, A factorization of regular generalized Nevanlinna functions. Integr. Equ. Oper. Theory 43, 326–345 (2002) 21. A. Luger, M. Nedic, On quasi-Herglotz functions in one variable. C. R. Sér. Math., 360, 937– 970 (2022) 22. A. Luger, M.-J.Y. Ou, On applications of Herglotz-Nevanlinna functions in material sciences, I: extended applications and generalized theory, in Research in the Mathematics of Materials Science. Association for Women in Mathematics Series, vol 31 (Springer, Cham, 2022) 23. A. Luger, M.-J.Y. Ou, On applications of Herglotz-Nevanlinna functions in material sciences, II: extended applications and generalized theory, in Research in the Mathematics of Materials Science. Association for Women in Mathematics Series, vol 31 (Springer, Cham, 2022) 24. G. Ramon, S. Donald, Real Outer Functions. Indiana Univ. Math. J. 52, 1397–1412 (2003) 25. G. Weiss, Transfer Functions of Regular Linear Systems. Part I: characterizations of Regularity. Trans. Amer. Math. Soc. 342, 827–854 (1994) 26. H. Wietsma, The factorization of generalized Nevanlinna functions and the invariant subspace property. Indag. Math. (N.S.). 30, 26–38 (2019)

Spectral Transition Model with the General Contact Interaction Pavel Exner and Jiˇrí Lipovský

Dedicated to the memory of Sergei Naboko

Abstract Using a technique introduced by Sergei Naboko, we analyze a generalization of the parameter-controlled model of spectral transition, originally proposed by Smilansky and Solomyak, to the situation where the singular interaction responsible for the effect is characterized by the ‘full’ family of four real numbers with the ‘diagonal’ part of the coupling being position-dependent. Keywords Smilansky-Solomyak model · Irreversible quantum graph · Four-parameter interaction

1 Introduction Many memories appear when thinking of Sergei Naboko, his charming personality and sharp mathematical mind. Here we choose as a starting point one of his papers [10] written with Michael Solomyak, in which he used his deep knowledge of Jacobi operators to enrich our understanding of a model of spectral transition originally proposed by Smilansky and Solomyak [11, 12]. This model has different physical

P. Exner () Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Prague, Czechia Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, Prague, Czechia e-mail: [email protected] J. Lipovský Department of Physics, Faculty of Science, University of Hradec Králové, Hradec Králové, Czechia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_19

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interpretations, either as a model of an irreversible behavior on a graph coupled to a caricature heat bath [11] or as the two-dimensional Schrödinger operator with a singular interaction of a position-dependent strength [13, 14]. It contains a parameter that controls its spectrum; it has a critical value at which the spectral nature abruptly changes from a below bounded and partly discrete to an absolutely continuous one covering the whole real axis. The original model was modified in various ways. For instance, the harmonic confining potential in (2.1a) below can be replaced by a more general function [15] and the line on which the coupling is imposed can be replaced by a more general graph [16]. Another modification consists of replacing the singular interaction by a regular potential channel [2, 3]; the advantage is that one is able in this setting to compare the quantum dynamics with its classical counterpart which exhibits an interesting irregular scattering behavior [9]. We note also that even the basic model still poses open questions, for example, having in addition to the discrete spectrum in the subcritical case also an infinite family of resonances the behavior of which is not fully understood [8]. A generalization going, so to say, in the opposite direction to [2, 3] consists of replacing the .δ interaction of the original model by a more singular coupling. In [7] we did that with the interaction commonly known as .δ  [1]. The aim of the present paper is to extend the conclusions to the situation where such a singular interaction is of the most general type depending on four real parameters; the mentioned .δ and  .δ coupling are now included as particular cases. We are going to show that the effect of abrupt spectral transition is robust, however, it occurs now on a hypersurface in the parameter space. The key element in our analysis of the spectral transition is the link to spectral properties of a specific Jacobi operator introduced in [10]. The paper is structured as follows. In the next section, we introduce our 4parameter model. Section 3 is devoted to the study of the quadratic form associated with the Hamiltonian; we prove a lower bound for it. In Sect. 4, we obtain the recurrence system which defines the Jacobi operator associated to the problem. Section 5 includes the proof of self-adjointness of the Hamiltonian. In Sect. 6, the absolutely continuous spectrum of the Hamiltonian is obtained, using the known properties of another Jacobi operator, which differs from the Jacobi operator associated with the system by a compact operator. Theorem 6.2 similarly to the previous results on .δ and .δ  -coupling shows the abrupt change of the spectra of the Jacobi operator from purely absolutely continuous to discrete depending on a real parameter depending on the coupling parameters. Using this result, Theorem 6.3 studies the absolutely continuous spectrum of the Hamiltonian. Finally, in Sect. 7 we obtain results on the discrete spectrum, in particular, we compare the number of eigenvalues of the Hamiltonian and the Jacobi operator (see Theorem 7.2) and find the asymptotic formulæ for the number of eigenvalues (Theorem 7.3). The Appendix in Sect. 8 inclused some technical results needed in Sect. 5.

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2 The Model The model we are going to discuss describes a quantum system the Hamiltonian of which is the operator of the form   2 ∂ 2 1 ∂  2 .Hα,β,γ (x, y) = − (x, y) + − 2 (x, y) + y (x, y) 2 ∂x 2 ∂y

(2.1a)

with the general contact interaction with position-dependent coefficients supported by the axis .x = 0, characterized by choosing the operator domain as the family of functions in the Sobolev space . ∈ H 2 ((0, ∞) × R) ⊕ H 2 ((−∞, 0) × R) satisfying the boundary conditions  ∂ ∂ α  . (0+, y) − (0−, y) = y (0+, y) + (0−, y) ∂x ∂x 2  ∂ γ  ∂ (0+, y) + (0−, y) , . + 2 ∂x ∂x  γ¯  (0+, y) + (0−, y) 2  β  ∂ ∂ + (0+, y) + (0−, y) 2y ∂x ∂x

(2.1b)

(0+, y) − (0−, y) = −

(2.1c)

with the parameters .α, β ∈ R and .γ ∈ C. We note that the conditions defining the general contact interaction are written in different ways [1, Appendix K.1]; here we choose the form proposed in [6] which has the advantage that one easily singles out the particular cases of .δ- and .δ  -interactions: the choice .β = γ = 0 leads to the original Smilansky-Solomyak model with the .δ-interaction on the x axis [11–14] and .α = γ = 0 yields its .δ  -modification discussed in [7]. One of the features of those models was that the spectrum was independent of the coupling constant sign. In the general case the mirror transformation, .y → −y can be compensated by a simultaneous change of the ‘diagonal’ parameters, .α → −α and .β → −β which thus leaves the spectrum invariant.

3 The Quadratic Form A convenient way to deal with the operator (2.1) is to use the quadratic form method. Theorem 3.1 The operator .Hα,β,γ is associated with the quadratic form given by aα,β,γ [] = a0 [] +

.

 1 b1 [] + b2 [] + b3 [] for β

aα,0,γ [] = a0 [] + b4 [] for

β = 0,

β = 0,

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where   ∂ 2 1  ∂ 2 1   +   + y 2 ||2 dxdy,   ∂x  2  ∂y  2 R2 ˆ y |(0+, y) − (0−, y)|2 dy, b1 [] = R ˆ αβ + |γ |2 b2 [] = y |(0+, y) + (0−, y)|2 dy, 4 R ˆ

¯ ¯ y Re γ ((0+, y) + (0−, y))((0+, y) − (0−, y)) dy, b3 [] = R ˆ α b4 [] = y |(0+, y) + (0−, y)|2 dy R4 ˆ

a0 [] =

.

and the domain .D = dom a0 of the form .a0 is  1 1 .D =  ∈ H ((0, ∞) × R) ⊕ H ((−∞, 0) × R) : a0 [] < ∞ . Proof The quadratic form is given by the integral over .R2 of the expression ¯ y)(H )(x, y) where H is the symbol given by the right-hand side .(x, of (2.1a). By integration by parts in both variables .x, y and using the fact that ¯ y) ∂ .(x, ∂x (x, y) vanishes as .x, y → ±∞ we obtain ˆ   ∂ ∂ ¯ ¯ (0+, y) (0+, y) − (0−, y) (0−, y) dy. .aα,β,γ [] = a0 [] + ∂x ∂x R Introducing the shorthands f+ := (0+, y) + (0−, y) ,

f− := (0+, y) − (0−, y),

f+ :=   (0+, y) +   (0−, y) ,

f− :=   (0+, y) −   (0−, y)

.

one can rewrite the last integral as ˆ 1 (f¯+ f− + f¯− f+ ) dy. . 2 R We start with the case .β = 0. Rewrite the matching conditions (2.1b) and (2.1c) as .

f+ =

2y γ¯ (f− + f+ ), β 2

f− =

γy y |γ |2 f− + (α + )f+ . β 2 β

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and substituting it into the above equation we arrive at ˆ aα,β,γ [] − a0 [] =

.

=

y (αβ + |γ |2 )|f+ |2 + 4|f− |2 + 4Re (γ f¯+ f− ) dy R 4β

1 (b1 [] + b2 [] + b3 []). β

For .β = 0 the matching conditions (2.1b) and (2.1c) yield γ¯ f− = − f+ , 2

f− =

.

αy γ f+ + f+ , 2 2

so that .

 αy 1 ¯  1  αy γ (f+ f− + f¯− f+ ) = |f+ |2 + f¯+ f+ + f¯− f+ = |f+ |2 , 2 2 2 2 4

and as a result, we get .aα,0,γ [] − a0 [] = b4 [] which completes the proof.



Next we state a simple lemma; if a proof is needed it can be found, for instance in [7] as Lemma 5. Lemma 3.2 For all .c, d ∈ C we have .2|Re (cd)| ¯ ≤ |c|2 + |d|2 . The second simple lemma generalizes Lemma 3.2: Lemma 3.3 Let .σj , .j = 1, 2, 3 be the Pauli matrices and .σ0 the .2 × 2 identity matrix. Let further .u, v be complex two-component column vectors. Then for any .ωj ∈ R we have  

 3    1 Re u¯ T · ωj σj · v ≤ u2 + v2 |ω0 | + ω12 + ω22 + ω32 2

.

j =0

Proof Since the real part of a complex number is bounded by its modulus, we have  3      

  3 3   T      ¯ u = uv · v Re u¯ T · ωj σj · v ≤  · ω σ ω σ j j j j ,   

.

j =0

j =0

j =0

where the matrix norm is the operator norm. Using .2uv ≤ u2 + v2 we complete the proof noting that the eigenvalues of the matrix 3  .

j =0

are .ω0 ±



ω12 + ω22 + ω32 .



ω0 + ω3 ω1 − iω2 ωj σj = ω1 + iω2 ω0 − ω3





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For the following particular choice of parameters .ωj , .j = 0, . . . , 3 ω0 = 4 + αβ + |γ |2 ,

.

ω1 = αβ + |γ |2 − 4 ,

ω2 = 4 Im γ ,

ω3 = 4 Re γ (3.1)

 we denote the matrix . 3j =0 ωj σj by .. This matrix appears later in the text. (j ) 1 Let the subspaces of .Dj , .j = 1, 2 consist  H ((−∞, 0)) ⊕  offunctions.ψ ∈  (j )  (j ) (j ) K+ ψ (0+) K+ H 1 ((0, ∞)) with . (j ) = Span (j ) , where . (j ) are eigenvectors ψ (0−) K− K− of .. Simple calculation yields up to a constant (1)

K+. = ω3 −



ω12 + ω22 + ω33 = 4Re γ −

 (αβ + |γ |2 − 4)2 + 16|γ |2 , . (3.2a)

(1)

K− = ω1 + iω2 = αβ + |γ |2 − 4 + 4iIm γ , . (3.2b)   (2) K+ = ω3 + ω12 + ω22 + ω33 = 4Re γ + (αβ + |γ |2 − 4)2 + 16|γ |2 , . (3.2c) (2)

K− = ω1 + iω2 = αβ + |γ |2 − 4 + 4iIm γ .

(3.2d)

The following lemma generalizes [7, Lemma 6] Lemma 3.4 For all .δ > 0 and .ψ ∈ H 1 ((−∞, 0)) ⊕ H 1 ((0, ∞)) it holds ˆ δ|ψ(0±)|2 ≤ ±

±∞ 

 |ψ  (x)|2 + δ 2 |ψ(x)|2 dx .

.

0±

On the subspaces of .Dj these inequalities are saturated for

˜ (j ) (x) := .ψ δ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(j )

K− eδx

(j ) (j ) δ(|K+ |2 +|K− |2 ) (j ) −δx K+ e  (j ) (j ) δ(|K+ |2 +|K− |2 )

for

x0

,

(3.3)

(j )

with .j = 1, 2 and the constants .K± given by (3.2). Proof The proof is a minor modification of the one in [7, Lemma 6]. The ´0 result follows from the positivity of the integrals . −∞ |ψ  (x) − δψ(x)|2 dx and ´∞  2 . 0 |ψ (x) + δψ(x)| dx. The second part of the lemma can be verified by a direct  inspection.

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With these preliminaries we can derive upper and lower bounds to the forms involved: Theorem 3.5 We have the following inequalities: .b1 []

+ b2 [] + b3 [] ≤

|4 + αβ + |γ |2 | +

 (αβ + |γ |2 − 4)2 + 16|γ |2 a0 [], √ 2 2β

α b4 [] ≤ √ a0 []. 2

For .β = 0 

.aα,β,γ []



 (αβ + |γ |2 − 4)2 + 16|γ |2 a0 [] √ 2 2β    1 |4 + αβ + |γ |2 | + (αβ + |γ |2 − 4)2 + 16|γ |2 ≥ 2 . 1− √ 2 2 2β

≥ 1−

|4 + αβ + |γ |2 | +

For .β = 0 .aα,0,γ []

    α 1 α ≥ 1 − √ a0 [] ≥ 1 − √ 2 . 2 2 2

Proof We proceed a way similar to [7], where the bound for .b1 [] was obtained, and to [14], where bounds for .b2 and .b4 were found, cf. [14, Lemma 2.1] or [7, Theorem 4] for more details. To get the first inequality, one has to estimate the expression .b1 [] + b2 [] + b3 []. In view of the mentioned results, we can skip a part of the computation related to the first two terms and focus on the form .b3 [] only. As in [7, 14], we use the expansion of the function in terms of the ‘transverse’ basis,  .(x, y) = ψn (x)χn (y), (3.4) n∈N0 with the coefficients .ψn depending on the variable x only, where .χn (y) is the n-th Hermite function, that is, the normalized harmonic oscillator eigenfunction referring to the eigenvalue .n + 12 , and .N0 denotes the set of non-negative integers. We note that the Ansatz (3.4) can be also used to analyze the model numerically as it was done in [8] for the eigenvalues and resonances appearing in the subcritical case. The quadratic form .a0 can be in terms of the coefficient functions .ψn written as a0 [] =

.

  ˆ    |ψn (x)|2 + n + 12 |ψn (x)|2 dx, R n∈N0

(3.5)

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giving the bound a0 [] ≥

.

ˆ 1  1 |ψn (x)|2 dx = 2 . 2 2 R n∈N0

Moreover, one can use the well-known recurrent relation for Hermite functions, √ .

n + 1χn+1 (y) −

√ √ 2yχn (y) + nχn−1 (y) = 0.

(3.6)

Using the Ansatz (3.4) in the definition of .b3 [] substituting subsequently for yχn (y) from (3.6), we get

.

b3 [] =

 ˆ

.

n,m∈N0

R

  y Re γ ψ¯ m (0+) + ψm (0−) χ¯ m (y)



 × ψn (0+) − ψn (0−) χn (y) dy ˆ    1  χ¯ m (y) Re γ ψ¯ m (0+) + ψm (0−) =√ 2 R n,m∈N0  √  √ × ψn (0+) − ψn (0−) n + 1χn+1 (y) + nχn−1 (y) dy. ´ Using the orthogonality of Hermite functions, . R χ¯ m (y)χn (y) dy = δmn , we further obtain  √   1  b3 [] = √ Re γ n + 1 ψ¯ n+1 (0+) + ψ¯ n+1 (0−) 2 n∈N0 

√  + n ψ¯ n−1 (0+) + ψ¯ n−1 (0−)  × (ψn (0+) − ψn (0−)

.

  1   √  Re γ n ψ¯ n (0+) + ψ¯ n (0−) ψn−1 (0+) − ψn−1 (0−) =√ 2 n∈N    √  + n ψ¯ n−1 (0+) + ψ¯ n−1 (0−) ψn (0+) − ψn (0−) √ 

2n Re Re γ ψ¯ n (0+)ψn−1 (0+) − ψ¯ n (0−)ψn−1 (0−) = n∈N

 + i Im γ ψ¯ n (0−)ψn−1 (0+) − ψ¯ n (0+)ψn−1 (0−) .

Spectral Transition Model with the General Contact Interaction

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In the first term of the second equality, we have changed the summation index from n to .n − 1; notice that the sum now runs over .N, not .N0 . In an analogous way, the other two terms entering the form .aα,β,γ [] can be written as b1 [] + b2 [] =

√

.

2n Re

   ψ¯ n (0+) − ψ¯ n (0−) ψn−1 (0+) − ψn−1 (0−)

n∈N

  αβ + |γ |2  ψ¯ n (0+) + ψ¯ n (0−) ψn−1 (0+) + ψn−1 (0−) 4  4 + αβ + |γ |2 √ = 2n Re 4 n∈N   × ψ¯ n (0+)ψn−1 (0+) + ψ¯ n (0−)ψn−1 (0−) +

+

 αβ + |γ |2 − 4  ψ¯ n (0+)ψn−1 (0−) + ψ¯ n (0−)ψn−1 (0+) . 4

The sum of all the three forms can be written elegantly using the Pauli matrices and the identity matrix, b1 [] + b2 [] + b3 [] =

.

   √n √ Re u¯ T ·  · v , 2 2 n∈N

(3.7)

  ψn (0+) where the notation is the same as in Lemma 3.3, .u = and .v = ψn (0−)   ψn−1 (0+) , and the numbers .ωj are given by (3.1). Applying now Lemma 3.3, ψn−1 (0−) we obtain  √n  .b1 [] + b2 [] + b3 [] ≤ √ |ψn (0+)|2 + |ψn (0−)|2 4 2 n∈N  + |ψn−1 (0+)|2 + |ψn−1 (0−)|2 

× |4 + αβ + |γ |2 | + (αβ + |γ |2 − 4)2 + 16|γ |2 . We divide the sum into two parts and in the part containing .ψn−1 we raise the summation index by one; in this way we get √  √n + n + 1   |ψn (0+)|2 + |ψn (0−)|2 .b1 [] + b2 [] + b3 [] ≤ √ 4 2 n∈N0 

× |4 + αβ + |γ |2 | + (αβ + |γ |2 − 4)2 + 16|γ |2 .

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 √ √ Furthermore, we employ the inequality . n + n + 1 ≤ 2 n + b1 [] + b2 [] + b3 [] ≤

.

1 2

which yields



(αβ + |γ |2 − 4)2 + 16|γ |2 √ 2 2     n + 12 |ψn (0+)|2 + |ψn (0−)|2 . × n∈N0 |4 + αβ + |γ |2 | +

Applying now Lemma 3.4 with .δ =



n+

1 2

we arrive at

|4 + αβ + |γ |2 | +

b1 [] + b2 [] + b3 [] ≤

.



(αβ + |γ |2 − 4)2 + 16|γ |2 √ 2 2

 ˆ  

|ψ  (x)|2 + n + 12 |ψ(x)|2 dx × R n∈N0  |4 + αβ + |γ |2 | + (αβ + |γ |2 − 4)2 + 16|γ |2 = √ 2 2 × a0 []. The obtained upper bound for .b1 [] + b2 [] + b3 [] leads to the lower bound for .aα,β,γ []. The second inequality follows from .a0 [] ≥ 12 2 which is a consequence of (3.5). The last inequality, the lower bound to .aα,0,γ [] for .β = 0, follows from an upper bound to .b4 [] that can be obtained in a way similar to [14, Lemma 2.1] or [7, Theorem 4] with the use of Lemma 3.2.  The form .aα,β,γ [] is closed on D. First, the form .a0 [] is closed, as proven in [4]. The closedness of the full form can be obtained by the same reasoning as in [4, Proposition 2.2] using the bounds in Theorem 3.5; as in the said paper, the argument applies only if the form is semibounded from below.

4 The Jacobi Operator The second main step of our argument is the construction of the Jacobi operator associated with .Hα,β,γ . Let us begin with the case .β = 0. First of all, we rewrite the matching conditions (2.1b) and (2.1c) in an equivalent form [6], .

∂ y (αβ + |γ |2 + 4 + 4 Re γ )(0+, y) (0+, y) = ∂x 4β

+ (αβ + |γ |2 − 4 − 4i Im γ )(0−, y) ,

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y ∂ (0−, y) = (−αβ − |γ |2 + 4 − 4i Im γ )(0+, y) ∂x 4β + (−αβ − |γ |2 − 4 + 4 Re γ )(0−, y)] Mimicking the construction in [7, 10] we use the Ansatz (3.4) and the recurrent relation (3.6), multiply both equations from the left by .χ¯ m (y) and integrate over .R obtaining thus conditions for the coefficient functions, .

∂ψm 1 (αβ + |γ |2 + 4 + 4Re γ ) (0+) = √ ∂x 4 2β √ √ × (ψm−1 (0+) m + ψm+1 (0+) m + 1) + (αβ + |γ |2 − 4 − 4iIm γ ) √

√ × (ψm−1 (0−) m + ψm+1 (0−) m + 1) , .

(4.1a)

1 ∂ψm (0−) = √ (−αβ − |γ |2 + 4 − 4iIm γ ) ∂x 4 2β √ √ × (ψm−1 (0+) m + ψm+1 (0+) m + 1) + (−αβ − |γ |2 − 4 + 4Re γ ) √

√ × (ψm−1 (0−) m + ψm+1 (0−) m + 1) .

(4.1b)

We seek solutions of the form

where .ζn () :=



.

φn (x, ) = k1 (, n) e−ζn ()x ,

x > 0, .

(4.2a)

φn (x, ) = k2 (, n) eζn ()x ,

x < 0,

(4.2b)

n+

1 2

− , to the equation

 . − φn (x) +

  1 n + −  φn (x) = 0 , 2

n ∈ N0 ,

(4.3)

with .φn ∈ H 2 (−∞, 0) ⊕ H 2 (0, ∞) satisfying the conditions (4.1). In the definition of .ζn () we take the branch of the square root which is analytic in .C\[n + 12 , ∞) and for a number . from this set it holds Re ζn () > 0 ,

.

Im ζn () · Im  < 0.

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P. Exner and J. Lipovský

Substituting the Ansätze (4.2) into (4.1) we get .

√ 1 (αβ + |γ |2 + 4 + 4Re γ )(k1 (, n − 1) n − ζn ()k1 (, n) = √ 4 2β √ + k1 (, n + 1) n + 1) + (αβ + |γ |2 − 4 − 4iIm γ ) √

√ × (k2 (, n − 1) n + k2 (, n + 1) n + 1) , 1 (−αβ − |γ |2 + 4 − 4iIm γ ) ζn ()k2 (, n) = √ 4 2β √ √ × (k1 (, n − 1) n n + 1) + k1 (, n + 1) + (−αβ − |γ |2 − 4 + 4Re γ ) √ × (k2 (, n − 1) n √

+ k2 (, n + 1) n + 1) .

Adding and subtracting the previous two equations we get √ 2 ζn () k2 (, n) − k1 (, n)] = √ [(k1 (, n − 1) − k2 (, n − 1)) n 2β √

+ (k1 (, n + 1) − k2 (, n + 1)) n + 1

.

√ γ¯ +√ (k1 (, n − 1) + k2 (, n − 1)) n 2β √

+ (k1 (, n + 1) + k2 (, n + 1)) n + 1 , √ γ −ζn () k1 (, n) + k2 (, n)] = √ [(k1 (, n − 1) − k2 (, n − 1)) n 2β √

+ (k1 (, n + 1) − k2 (, n + 1)) n + 1 √ αβ + |γ |2 (k1 (, n − 1) + k2 (, n − 1)) n √ 2 2β √

+ (k1 (, n + 1) + k2 (, n + 1)) n + 1 . +

To simplify these relations, we define C± (n) := k1 (, n) ± k2 (, n),

.

Spectral Transition Model with the General Contact Interaction

535

then the previous two equations can be rewritten as ⎛   √2 C− (n) 2β ⎝ = . − ζn () C+ (n) √γ

⎞

√   √ √γ¯ 2β ⎠ C− (n − 1) n + C− (n + 1) n + 1 √ √ 2 αβ+|γ C+ (n − 1) n + C+ (n + 1) n + 1 √ | 2β 2 2β

Computing the eigenvalues and eigenspaces of the matrix involved in the previous equation and putting −1/4   1 − 4γ C− (n) + 4 − αβ − |γ |2 Cn,1 . := n + 2 

 (4.4a) + (αβ + |γ |2 − 4)2 + 16|γ |2 C+ (n) , . −1/4   4γ C− (n) + − 4 + αβ + |γ |2 Cn,2 := n + 12 

 + (αβ + |γ |2 − 4)2 + 16|γ |2 C+ (n) ,

(4.4b)

we find that .Cn,1 satisfies the same equation as, for instance, in [7], namely .(n

1/4 1/4  1/4   Cn+1,1 + 2μ n + 12 ζn ()Cn,1 + n1/2 n − 12 Cn−1,1 = 0 + 1)1/2 n + 32

(4.5) with .μ = μ1 given by .μ1

:=

4 + αβ + |γ |2 −



√ 2 2β (αβ + |γ |2 − 4)2 + 16|γ |2

(4.6a)

.

Similarly, .Cn,2 satisfies the same equation, the only difference is that with .μ equals now to .μ2

:=

4 + αβ + |γ |2 +



√ 2 2β (αβ + |γ |2 − 4)2 + 16|γ |2

(4.6b)

.

With the above mentioned symmetry .(α, β) → (−α, −β) in mind, we can without the loss of generality take .β > 0. Then we easily find that for .α > 0 it holds .μ1 ≥ μ2 > 0, for .α < 0 it holds .μ1 < 0 < μ2 and for .α = 0 the parameter .μ1 diverges. The general solution of the Eq. (4.3) can be written in the form of linear combination of functions corresponding to the above-mentioned eigenspaces .φn = (1) (2) Cn,1 ηn + Cn,2 ηn , where (1) .ηn

=

ηn(2) =

 " (n + 1/2)1/4 (ω3 − ω12 + ω22 + ω32 ) e−ζn ()x x > 0 x 0.

.

(3.53)

4 The Weyl–Titchmarsh Function in the ϕ-Periodic Case in Terms of the Green’s Function and Its Nevanlinna–Herglotz Property In our final and principal section, we now introduce the .2 × 2 matrix-valued Weyl–Titchmarsh M-function associated with .Tϕ , .ϕ ∈ [0, 2π ), and prove it is a Nevanlinna–Herglotz function. For its precise connection to the spectral theory of .Tϕ , along the lines of Theorem A.2, we refer to [11]. To get started, and in complete analogy to the case of separated boundary conditions treated in detail in Appendix A, see, in particular, (A.15) and (A.24), we now introduce the .2 × 2 matrix-valued Weyl–Titchmarsh M-function .M0,ϕ (z, a), associated with .Tϕ , .ϕ ∈ [0, 2π ), expressed in terms of the Green’s function .Gϕ (z, · , · ), .z ∈ C\σ (Tϕ ), by

 2−1 G[1] ϕ (z, a, a) .M0,ϕ (z, a) = (4.1)  [1] [1]  2−1 G[1] ϕ (z, a, a) ∂1 ∂2 Gϕ (z, a, a)



0 (z,b,a)  (z,b,a)− 2−1 φ θ0 (z,b,a) φ 1 0

 = , θ0 (z,b,a) − θ0 (z,b,a) 2[D(z) − cos(ϕ)] 2−1 φ 0 (z,b,a)− Gϕ (z, a, a)

z ∈ C\σ (Tϕ ).

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595

Here we employed the notation,   [1]     ∂1 + ∂2[1] Gϕ (z, a, a) = [p(x1 )∂x1 + p(x2 )∂x2 ]Gϕ (z, x1 , x2 )x =a, x =a , 1 2    = p(x)(d/dx)Gϕ (z, x, x)x=a

.

:= G[1] ϕ (z, a, a),   [1] [1]    ∂1 ∂2 Gϕ (z, a, a) = p(x1 )∂x1 p(x2 )∂x2 Gϕ (z, x1 , x2 )x =a,x =a 1 2  [1] [1]  = ∂2 ∂1 Gϕ (z, a, a).

(4.2)

The principal aim of this section is to show that .M0,ϕ ( · , a) is a Nevanlinna– Herglotz function. The proof of this fact relies on some basic abstract facts from the theory of matrix-valued Nevanlinna–Herglotz functions summarized below in Lemma 4.1 and Remark 4.2. One recalls that the matrix-valued function .M( · ) is a matrix-valued Nevanlinna– Herglotz function if it is analytic on .C+ with .Im(M( · )) ≥ 0 on .C+ . One then extends .M( · ) to .C− by reflection, that is, M(z) = M(z)∗ ,

.

z ∈ C− ,

(4.3)

so that .M( · ) is analytic on .C\R. However, .M|C+ and .M|C− , generally, are not analytic continuations of each other. Any matrix-valued Nevanlinna–Herglotz function permits the measure representation,   ˆ 1 λ dΩ(λ) , z ∈ C\R, . − .M(z) = C + Dz + (4.4) λ − z 1 + λ2 R C = C∗,

D ≥ 0, . ˆ dΩ(λ) Im(M(i)) − D = exists. R 1 + λ2

(4.5) (4.6)

Conversely, any .M( · ) satisfying (4.4)–(4.6) is a matrix-valued Nevanlinna– Herglotz function. Moreover, for .λ1 , λ2 ∈ R, .λ1 < λ2 , the Stieltjes inversion formula, Ω((λ1 , λ2 ]) = π −1 lim lim

ˆ

λ2 +δ

.

δ↓0 ε↓0

holds.

λ1 +δ

dλ Im(M(λ + iε)),

(4.7)

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Lemma 4.1 .M( · ) represents a meromorphic Nevanlinna–Herglotz function with poles at the discrete set .{λj }j ∈J ⊂ R, .J ⊆ Z a finite or infinite index set, and .M( · ) self-adjoint on .R\{λj }j ∈J , if and only if .M( · ) is of the form M(z) = C + Dz +

%

.

 Aj

j ∈J

 λj 1 − , λj − z 1 + λ2j

z ∈ C\R, .

C = C∗,

D ≥ 0, . % Im(M(i)) − D = Aj j ∈J

(4.8) (4.9)

1 exists, . 1 + λ2j

(4.10)

Aj = limε↓0 ε Im(M(λj + iε)) = −i limε↓0 εM(λj + iε) = Ω({λj }) ≥ 0, Aj = 0,

j ∈ J, (4.11)

where .Ω( · ) denotes the .(matrix-valued .) measure in the Nevanlinna–Herglotz representation in (4.4) of .M( · ). Proof It is clear from the representation of .M( · ) in (4.4) and the Stieltjes inversion formula (4.7) that the support of .Ω( · ) equals .{λj }j ∈J and that .Ω({λj }) = Aj = 0, .j ∈ J .     Remark 4.2 Since for any .N × N matrix .B = B ,m 1≤ ,m≤N , .N ∈ N, with entries in .C, satisfying .B ≥ 0, one has B , ≥ 0,

.

1 ≤ ≤ N, . 1/2

(4.12)

1/2

|B ,m | ≤ B , Bm,m ≤ [B , + Bm,m ]/2,

1 ≤ , m ≤ N,

(4.13)

(in particular, if .B 0 , 0 = 0 then the . 0 th row and . 0 th column of B vanishes), condition (4.10) is satisfied if and only it holds for all diagonal elements, that is, if and only if .

% (Aj ) , j ∈J

1 + λ2j

< ∞,

1 ≤ ≤ N.

(4.14)

Put differently, if .M( · ) is meromorphic and permits a representation of the type (4.8) such that .Aj ≥ 0, .j ∈ J , .C = C ∗ , .D ≥ 0, and each diagonal term of .M( · ) is either a real constant or a scalar-valued Nevanlinna–Herglotz function, then .M( · ) is a matrix-valued Nevanlinna–Herglotz function. . Given these preparations, the main result of this paper may be stated as follows.

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597

Theorem 4.3 Assume Hypothesis 3.1, and suppose that .ϕ ∈ [0, 2π ). Then the .2×2 Weyl–Titchmarsh M-function .M0,ϕ ( · , a) associated to .Tϕ , and defined by (4.1), is a Nevanlinna–Herglotz function. Proof Let .ϕ ∈ [0, 2π ). By definition, .M0,ϕ ( · , a) is meromorphic with poles at most at the discrete set .σ (Tϕ ) = {λn (ϕ)}∞ n=1 . In addition, .M0,ϕ ( · , a) is self-adjoint on .R\σ (Tϕ ). The singularity structure of .M0,ϕ ( · , a) at each eigenvalue .λn (ϕ), .n ∈ N, of .Tϕ can be investigated by considering the following four cases .(i)–.(iv). (i) .ϕ ∈ (0, 2π )\{π }. In this case, all eigenvalues .λn (ϕ), .n ∈ N, of .Tϕ are simple, ˙ n (ϕ)) = 0 for each .n ∈ N by (3.33) and (3.41), and one obtains .D(λ

.

M0,ϕ (z, a)

=

.

⎛ ×⎝

z→λn (ϕ)

1 ˙ n (ϕ)) 2[z − λn (ϕ)]D(λ

(4.15)

 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)

0 (λn (ϕ),b,a) φ

 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)





⎞ ⎠

−θ 0 (λn (ϕ),b,a)

+ O(1). Since



detC2

.

0 (λn (ϕ),b,a) φ

 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)



 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)

 



−θ 0 (λn (ϕ),b,a)

= 1 − D(λn (ϕ))2 > 0,

(4.16)

where the inequality follows from (3.33), the two eigenvalues of the .2 × 2 matrix on the right-hand side in (4.15) have the same sign. By (3.40)

.

0 (λn (ϕ), b, a) φ < 0, ˙ n (ϕ)) D(λ



−θ 0 (λn (ϕ), b, a) < 0, ˙ n (ϕ)) D(λ

(4.17)

and hence 1 . ˙ n (ϕ)) D(λ





0 (λn (ϕ),b,a) φ

 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)



 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)





−θ 0 (λn (ϕ),b,a)

< 0.

(4.18)

˙ n (ϕ)) = 0, and either .φ 0 (λn (ϕ), b, a) = 0 or . θ0 (λn (ϕ), b, a) (ii) .ϕ ∈ {0, π }, .D(λ . = 0. By (3.33) and the equality in (4.16),

.



detC2

.

= 0,

0 (λn (ϕ),b,a) φ

 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)



 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a)

 



−θ 0 (λn (ϕ),b,a)

(4.19)

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and one eigenvalue of the .2 × 2 matrix on the right-hand in (4.15) is equal to zero so that

 

0 (λn (ϕ),b,a) φ 2−1 φ 0 (λn (ϕ),b,a)− θ0 (λn (ϕ),b,a) 1 .   ˙ n (ϕ)) 2−1 φ  (λn (ϕ),b,a)− D(λ θ0 (λn (ϕ),b,a) −θ 0 (λn (ϕ),b,a) 0 ≤ 0.

(4.20)

Apart from (4.20), the remainder of case .(ii) equals case .(i). In particular, (4.15) holds. ˙ n (ϕ)) = 0, and .φ 0 (λn (ϕ), b, a) = .(iii) .ϕ ∈ {0, π }, .D(λ θ0 (λn (ϕ), b, a) = 0. By (3.44) and (3.45),  θ0 (λn (ϕ), b, a) = φ 0 (λn (ϕ), b, a) = eiϕ ∈ {1, −1},

.

(4.21)

and the calculation in (4.15) yields M0,ϕ (z, a)

.

=

z→λn (ϕ)

(4.22)

O(1).

This is the exceptional case where a zero of all matrix elements on the righthand side of (4.1) cancel the pole term in (4.15) produced by .[D( · ) − cos(ϕ)] at .z = λn (ϕ). ˙ n (ϕ)) = 0. Invoking (3.42), (3.43), (3.44), (3.45), (3.51), and .(iv) .ϕ ∈ {0, π }, .D(λ (3.53), one obtains .

0 (λn (ϕ), b, a) = φ θ0 (λn (ϕ), b, a) = 0, .

(4.23)

0 (λn (ϕ), b, a) = eiϕ ∈ {1, −1}, . θ0 (λn (ϕ), b, a) = φ

(4.24)

¨ n (ϕ)) = 0, D(λ

(4.25)

and one concludes for the analog of (4.15) in this case M0,ϕ (z, a)

=

.

⎛ ⎜ ×⎝

z→λn (ϕ)

1 ¨ n (ϕ)) [z − λn (ϕ)]D(λ ˙ 0 (λn (ϕ),b,a) φ

2−1

  φ˙ 0

˙0 (λn (ϕ),b,a) (λn (ϕ),b,a)−θ

2−1

!

 

˙0 (λn (ϕ),b,a) φ˙ 0 (λn (ϕ),b,a)−θ

 

− θ˙0 (λn (ϕ),b,a)

!⎞ ⎟ ⎠

+ O(1)



˙0 (λn (ϕ), b, a) φ˙ 0 (λn (ϕ), b, a) −θ 1 = 

 ¨ n (ϕ))  [z − λn (ϕ)]D(λ φ˙ 0 (λn (ϕ), b, a) −  θ˙0 (λn (ϕ), b, a) + O(1),

(4.26)

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599

employing (3.47). The relation in (3.23) implies 

˙0 (λn (ϕ), b, a) φ˙ 0 (λn (ϕ), b, a) −θ > 0, .det 2

 

 C φ˙ 0 (λn (ϕ), b, a) −  θ˙0 (λn (ϕ), b, a)

(4.27)

so the eigenvalues of the .2 × 2 matrix in (4.27) have the same sign. By (3.18), (3.20), (4.23), and (4.24), ˙ 0 (λn (ϕ), b, a) = eiϕ φ

ˆ

b

.

r(x  )dx  φ0 (λn (ϕ), x  , a)2 , .

(4.28)

r(x  )dx  θ0 (λn (ϕ), x  , a)2 .

(4.29)

a

 − θ˙0 (λn (ϕ), b, a) = eiϕ

ˆ

b

a

Combining the observations in (3.51) and (3.52) with (4.28) and (4.29) then yields ˙ 0 (λn (ϕ), b, a) φ . < 0, ¨ n (ϕ)) D(λ

 − θ˙0 (λn (ϕ), b, a) < 0. ¨ n (ϕ)) D(λ

(4.30)

The inequalities in (4.30) imply 

˙0 (λn (ϕ), b, a) φ˙ 0 (λn (ϕ), b, a) −θ 1 . < 0. 

 ¨ n (ϕ))  D(λ φ˙ 0 (λn (ϕ), b, a) −  θ˙0 (λn (ϕ), b, a)

(4.31)

Finally, applying of Lemma 4.1 and Remark 4.2 proves that .M0,ϕ ( · , a) is a .2 × 2 Nevanlinna–Herglotz function in all cases .(i)–.(iv).   Remark 4.4 (i) For the precise connection between .M0,ϕ ( · , a) and the spectral theory of .Tϕ , along the lines of Theorem A.2 in the special case of separated boundary conditions, we refer to [11]. The fact that a can be replaced by .x0 ∈ (a, b) is also shown in [11]. The latter reference also contains the proper extension of Theorem 4.3 to all self-adjoint extensions of .Tmin with general (coupled) boundary conditions. .(ii) In contrast to (A.16) in the case of separated boundary conditions, where .det 2 (Mα,γ ,δ (z, x0 )) = −1/4, .z ∈ ρ(Tγ ,δ ), one now computes, employing C (3.13) and hence .

0 (z, · , a) W (θ0 (z, · , a), φ0 (z, · , a)) = 1 = θ0 (z, · , a)φ

.

− θ0 (z,

(4.32)

0 (z, · , a), z ∈ C, x0 ∈ (a, b), · , a)φ

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and (4.1), that detC2 (M0,ϕ (z, a))

.

= 4−1 [D(z) − cos(ϕ)]−2

 2 ! 0 (z, b, a) − 0 (z, b, a) × −φ θ0 (z, b, a) − 4−1 φ θ0 (z, b, a)

 ! 0 (z, b, a) 2 θ0 (z, b, a) + φ = 4−1 [D(z) − cos(ϕ)]−2 1 − 4−1

 = 4−1 [D(z) − cos(ϕ)]−2 1 − D(z)2 , z ∈ ρ(Tϕ ). (4.33) 

.

Appendix A: Classical Weyl–Titchmarsh Theory in the Case of Separated Boundary Conditions In this appendix we review the classical Weyl–Titchmarsh approach in connection with separated boundary conditions as summarized, for instance, in [8, App. J], [12, Ch. 6]. Moreover, we summarize the basic formulas in the case where the fundamental system .θ0 ( · , x, x0 ), .φ0 ( · , x, x0 ) is replaced by .θα ( · , x, x0 ), .φα ( · , x, x0 ), .α ∈ [0, π ). Throughout this appendix we assume Hypothesis 3.1 and fix .α ∈ [0, π ). Associated with the differential expression .τ we consider the self-adjoint operator .Tγ ,δ in .L2 ((a, b); rdx) corresponding to separated boundary conditions (if any) indexed by .γ , δ ∈ [0, π ), and the usual fundamental system of solutions .φα (z, · , x0 ) and .θα (z, · , x0 ), .z ∈ C, of .τ u = zu, with respect to a fixed reference point .x0 ∈ (a, b), satisfying the initial conditions φα (z, x0 , x0 ) = −θα[1] (z, x0 , x0 ) = − sin(α), .

φα[1] (z, x0 , x0 ) = θα (z, x0 , x0 ) = cos(α),

α ∈ [0, π ), z ∈ C, x0 ∈ (a, b). (A.1)

Again we note that for any fixed .x, x0 ∈ (a, b), .φα (z, x, x0 ) and .θα (z, x, x0 ) are entire with respect to z and that W (θα (z, · , x0 ), φα (z, · , x0 ))(x) = 1,

.

z ∈ C, x0 ∈ (a, b).

(A.2)

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601

Particularly important solutions of .τ u = zu are the Weyl–Titchmarsh solutions .ψα,b (z, · , x0 ) or .ψα,b,δ (z, · , x0 ) at b (and similarly, .ψα,a (z, · , x0 ) or .ψα,a,γ (z, · , x0 ) at a) of .τ u = zu, uniquely characterized as follows: (i) If .τ is in the limit point case at b (resp., a), one introduces .ψα,b (z, · , x0 ) (resp., .ψα,a (z, · , x0 )) via the requirement

.

  ψα,b (z, · , x0 ) ∈ L2 ([x0 , b); rdx), resp., ψα,a (z, · , x0 ) ∈ L2 ((a, x0 ]; rdx) ,

.

[1] sin(α)ψα,b (z, x0 , x0 ) + cos(α)ψα,b (z, x0 , x0 ) = 1 [1] (resp., sin(α)ψα,a (z, x0 , x0 ) + cos(α)ψα,a (z, x0 , x0 ) = 1),

(A.3) z ∈ C\R.

The crucial condition in (A.3) is the .L2 -property at b (resp., a), which uniquely determines .ψα,b (z, · , x0 ) (resp., .ψα,a (z, · , x0 )) up to constant (possibly, zdependent) multiples by the limit point case hypothesis of .τ at a and b. In particular, for .α, β ∈ [0, π ), ψα,b (z, · , x0 ) = Cb (z, α, β, x0 )ψβ,b (z, · , x0 )

.

(resp., ψα,a (z, · , x0 ) = Ca (z, α, β, x0 )ψβ,a (z, · , x0 ))

(A.4)

for some coefficients Cb (z, α, β, x0 ) ∈ C, (resp., Ca (z, α, β, x0 ) ∈ C). (ii) If .τ is in the limit circle case at b (resp., a), one introduces .ψα,b,δ (z, · , x0 ) (resp., .ψα,a,γ (z, · , x0 )) by requiring that

.

ψα,b,δ (z, · , x0 ) (resp., ψα,a,γ (z, · , x0 )) satisfies the separated boundary

.

condition at b (resp., a) of the form,  α,b,δ α,b,δ (z, b, x0 ) = 0 sin(δ)ψ (z, b, x0 ) + cos(δ)ψ  α,b,γ (z, a, x0 ) = 0), α,b,γ (z, a, x0 ) + cos(γ )ψ (resp., sin(γ )ψ

(A.5)

[1] sin(α)ψα,b,δ (z, x0 , x0 ) + cos(α)ψα,b,δ (z, x0 , x0 ) = 1 [1] (resp., sin(α)ψα,a,γ (z, x0 , x0 ) + cos(α)ψα,a,γ (z, x0 , x0 ) = 1),

z ∈ C\R.

Notational Convention To minimize the case distinctions to be made in the following, we will adopt the notation of case .(ii) and should the limit point case of .τ be present at b or a we simply ignore the extra .δ- or .γ -dependence. In either case .(i) or .(ii), the normalizations employed in (A.3) and (A.5) show that .ψα, b,δ (z, · , x0 ) are of the type a,γ

ψα, b,δ (z, x, x0 ) = θα (z, x, x0 ) + mα, b,δ (z, x0 )φα (z, x, x0 ), .

a,γ

a,γ

z ∈ C\R, x ∈ (a, b),

(A.6)

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for some coefficients .mα, b,δ (z, x0 ), .z ∈ C\R, the Weyl–Titchmarsh m-functions a,γ

associated with .τ , .α, γ , δ, and .x0 , mα, b,δ (z, x0 ) = cos(α)ψ [1]b,δ (z, x0 , x0 ) − sin(α)ψα, b,δ (z, x0 , x0 ),

.

α,a,γ

a,γ

a,γ

z ∈ C\R. (A.7)

One recalls the fundamental identities ˆ

b

r(x)dx ψα,b,δ (z1 , x, x0 )ψα,b,δ (z2 , x, x0 ) =

.

x0

ˆ

mα,b,δ (z1 , x0 ) − mα,b,δ (z2 , x0 ) , z 1 − z2

x0

(A.8)

r(x)dx ψα,a,γ (z1 , x, x0 )ψα,a,γ (z2 , x, x0 ) a

=−

mα,a,γ (z1 , x0 ) − mα,a,γ (z2 , x0 ) , z 1 − z2

z1 , z2 ∈ C\R, z1 = z2 ,

and concludes mα, b,δ (z, x0 ) = mα, b,δ (z, x0 ),

.

a,γ

a,γ

z ∈ C\R.

(A.9)

Choosing .z1 = z, .z2 = z in (A.8), one infers ˆ

b

r(x)dx |ψα,b,δ (z, x, x0 )|2 =

x0

ˆ

.

x0

Im(mα,b,δ (z, x0 )) > 0, Im(z)

r(x)dx |ψα,a,γ (z, x, x0 )|2 = −

a

z ∈ C\R,

Im(mα,a,γ (z, x0 )) > 0, Im(z)

z ∈ C\R. (A.10)

In addition, since .mα, b,δ ( · , x0 ) are known to be analytic on .C\R, one obtains that a,γ

±mα, b,δ ( · , x0 ) are Nevanlinna–Herglotz functions.

.

a,γ

The Green’s function .Gγ ,δ (z, x, x  ), .z ∈ ρ(Tγ ,δ ), .x, x  ∈ (a, b), of .Tγ ,δ then reads Gγ ,δ (z, x, x  ) =

.

 ×

1 W (ψα,b,δ (z, · , x0 ), ψα,a,γ (z, · , x0 ))

ψα,a,γ (z, x, x0 )ψα,b,δ (z, x  , x0 ), a < x ≤ x  < b, ψα,b,δ (z, x, x0 )ψα,a,γ (z, x  , x0 ), a < x  ≤ x < b,

(A.11) z ∈ C\R,

with W (ψα,b,δ (z, · , x0 ), ψα,a,γ (z, · , x0 )) = mα,a,γ (z, x0 ) − mα,b,δ (z, x0 ),

.

z ∈ C\R. (A.12)

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603

Thus, ((Tγ ,δ − zI )−1 f )(x) =

ˆ

b

r(x  )dx  Gγ ,δ (z, x, x  )f (x  ),

a

.

(A.13)

z ∈ C\R, x ∈ (a, b), f ∈ L ((a, b); rdx). 2

For each .x ∈ R, the diagonal Green’s function of .Tγ ,δ , denoted by .gγ ,δ (z, x), has the Nevanlinna–Herglotz property, that is, gγ ,δ ( · , x) = Gγ ,δ ( · , x, x),

.

x ∈ (a, b), is a Nevanlinna–Herglotz function. (A.14)

Given .mα, b,δ (z, x0 ), one also introduces the .2 × 2 matrix-valued Weyl– a,γ

Titchmarsh function   Mα,γ ,δ (z, x0 ) = Mα,γ ,δ, ,  (z, x0 ) ,  =1,2 ⎛ .

=

⎞

1 mα,a,γ (z,x0 )+mα,b,δ (z,x0 ) 1 0 )−mα,b,δ (z,x0 ) 2 mα,a,γ (z,x0 )−mα,b,δ (z,x0 ) ⎠ ⎝ mmα,a,γ (z,x , mα,a,γ (z,x0 )mα,b,δ (z,x0 ) 1 α,a,γ (z,x0 )+mα,b,δ (z,x0 ) 2 mα,a,γ (z,x0 )−mα,b,δ (z,x0 ) mα,a,γ (z,x0 )−mα,b,δ (z,x0 )

z ∈ C\R,

(A.15)

and notes that detC2 (Mα,γ ,δ (z, x0 )) = −1/4,

.

z ∈ C\R.

(A.16)

By inspection, .Mα,γ ,δ (z, x0 ) is a Nevanlinna–Herglotz matrix and hence possesses a representation of the type Mα,γ ,δ (z, x0 ) = Cα,β,γ (x0 ) + Dα,γ ,δ (x0 )z   ˆ . λ 1 dΩα,γ ,δ (λ, x0 ) , − + λ − z 1 + λ2 R

z ∈ C\R,

(A.17)

where Cα,γ ,δ (x0 ) = Cα,γ ,δ (x0 )∗ = Re(Mα,γ ,δ,x0 (i)), .

Dα,γ ,δ (x0 ) = lim ˆ

η↑∞

Mα,γ ,δ (iη, x0 )) ≥ 0, iη

dΩα,γ ,δ (λ, x0 ) < ∞. 1 + λ2 R

(A.18)

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The Stieltjes inversion formula for the .2 × 2 nonnegative matrix-valued measure dΩα,γ ,δ ( · , x0 ) then reads

.

Ωα,γ ,δ ((λ1 , λ2 ], x0 ) = π

−1

.

ˆ lim lim δ↓0 ε↓0

λ2 +δ

λ1 +δ

dλ Im(Mα,γ ,δ (λ + iε, x0 )),

(A.19)

λ1 , λ2 ∈ R, λ1 < λ2 . Moreover, since the diagonal entries of .Mα,γ ,δ are Nevanlinna–Herglotz functions, the diagonal entries of the measure .dΩα,γ ,δ are nonnegative measures. The offdiagonal entries of the measure .dΩα,γ ,δ then equal a complex measure (whose real and imaginary parts naturally consist of differences of two finite nonnegative measures). In particular, the entries .dΩα,γ ,δ, ,  , . ,  = 1, 2, of the matrix-valued measure .dΩα,γ ,δ are scalar-valued measures. Thus, Mα,γ ,δ,1,1 ( · , x0 ) =

.

1 mα,a,γ ( · , x0 ) − mα,b,δ ( · , x0 )

and Mα,γ ,δ,2,2 ( · , x0 ) =

mα,a,γ ( · , x0 )mα,b,δ ( · , x0 ) mα,a,γ ( · , x0 ) − mα,b,δ ( · , x0 )

are (scalar-valued) Nevanlinna–Herglotz functions.

(A.20)

In the special context of Sturm–Liouville operators one can actually show that Dα,γ ,δ (x0 ) = 0.

.

(A.21)

Next, we turn to the connection between the .2 × 2 Weyl–Titchmarsh matrix Mα,γ ,δ (z, x0 ) and the Green’s function .Gγ ,δ (z, · , · ) of .Tγ ,δ . Introducing

.

 [1]   ∂1 + ∂2[1] Gγ ,δ (z, x0 , x0 )    = [p(x1 )∂x1 + p(x2 )∂x2 ]Gγ ,δ (z, x1 , x2 )x =x , x =x , 1 0 2 0     = p(x)(d/dx)Gγ ,δ (z, x, x) x=x

.

0

:=

G[1] γ ,δ (z, x0 , x0 ),

(A.22)   [1] [1]    ∂1 ∂2 Gγ ,δ (z, x0 , x0 ) = p(x1 )∂x1 p(x2 )∂x2 Gγ ,δ (z, x1 , x2 )x =x ,x =x 1 0 2 0   [1] [1]  = ∂2 ∂1 Gγ ,δ (z, x0 , x0 ); x0 , x, x ∈ (a, b), the expression (A.15) for .Mα,γ ,δ (z, x0 ) then can be rewritten as Mα,γ ,δ,1,1 (z, x0 ) = cos2 (α)Gγ ,δ (z, x0 , x0 ) + sin(α) cos(α)G[1] γ ,δ (z, x0 , x0 )   + sin2 (α) ∂1[1] ∂2[1] Gγ ,δ (z, x0 , x0 ),

.

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605

Mα,γ ,δ,1,2 (z, x0 ) = Mα,γ ,δ,2,1 (z, x0 )

 = − sin(α) cos(α)G(z, x0 , x0 ) + 2−1 cos2 (α) − sin2 (α) G[1] γ ,δ (z, x0 , x0 )   + sin(α) cos(α) ∂1[1] ∂2[1] Gγ ,δ (z, x0 , x0 ), (A.23)

Mα,γ ,δ,2,2 (z, x0 ) = sin2 (α)Gγ ,δ (z, x0 , x0 ) − sin(α) cos(α)G[1] γ ,δ (z, x0 , x0 )   + cos2 (α) ∂1[1] ∂2[1] Gγ ,δ (z, x0 , x0 ); z ∈ C\R, x0 ∈ (a, b). In particular, in the case .α = 0, one obtains the simple formula ⎛ M0,γ ,δ (z, x0 ) = ⎝ .

Gγ ,δ (z, x0 , x0 ) 2−1 G[1] γ ,δ (z, x0 , x0 )

⎞ 2−1 G[1] (z, x , x ) 0 0 γ ,δ ⎠,   [1] [1] ∂1 ∂2 Gγ ,δ (z, x0 , x0 )

(A.24)

z ∈ C\R. Relations (A.23) can be expressed as the following similarity relation,  Mα,γ ,δ (z, x0 ) = .

   cos(α) sin(α) cos(α) − sin(α) M0,γ ,δ (z, x0 ) , − sin(α) cos(α) sin(α) cos(α)

(A.25)

z ∈ C\R, x0 ∈ (a, b). Thus, .Gγ ,δ ( · , x0 , x0 ) and appropriate first quasi-derivatives of .Gγ ,δ ( · , x, x  ) taking x and .x  at .x0 , uniquely determine .Mα,γ ,δ (z, x0 ) in a straightforward fashion (see also [8, App. J]). Remark A.1 (i) We note that in formulas (A.3)–(A.17) and (A.22)–(A.24) one can of course replace .z ∈ C\R by .z ∈ ρ(Tγ ,δ ). .(ii) It is possible to take the limit .x0 ↓ a and/or .x0 ↑ b in (A.23), (A.24). . .

Given the Nevanlinna–Herglotz representation of .Mα,γ ,δ ( · , x0 ) in (A.17), (A.18), one introduces the unitary maps ⎧ ⎨L2 ((a, b); rdx) → L2 (R; dΩα,γ ,δ ( · , x0 )) Fα,γ ,δ (x0 ) :   ⎩h →  h (·,x ) =  h ( · , x ),  h (·,x ) , α

.

0

0

α,1

α,2

0

 

´ d  r(x)dx θα ( · , x, x0 )h(x) hα,1 ( · , x0 ) c  hα ( · , x0 ) = , = s-lim ´ d  c↓a,d↑b hα,2 ( · , x0 ) r(x)dx φα ( · , x, x0 )h(x) c

(A.26)

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where .s-lim in (A.26) refers to the .L2 (R; dΩα,γ ,δ ( · , x0 ))-limit. The associated inverse operator is then given by Fα,γ ,δ (x0 )−1 :

.

hα ( · ) =

⎧ ⎨L2 (R; dΩα,γ ,δ ( · , x0 )) → L2 ((a, b); rdx)

(A.27) ⎩ h → hα , ˆ μ2 s-lim (θα (λ, · , x0 ), φα (λ, · , x0 )) dΩα,γ ,δ (λ, x0 )  h(λ),

μ1 ↓−∞,μ2 ↑∞ μ1

where .s-lim in (A.27) now refers to the .L2 ((a, b); rdx)-limit. One then obtains the following result, which we state for brevity without proof (see, e.g., [12, Chs. 4, 13], [13]): Theorem A.2 Assume Hypothesis 3.1. In addition, let .F ∈ C(R), .α ∈ [0, π ), and x0 ∈ (a, b). Then,

.

Fα,γ ,δ (x0 )F (Tγ ,δ )Fα,γ ,δ (x0 )−1 = MF

.

(A.28)

in .L2 (R; dΩα,γ ,δ ( · , x0 )) .(cf. (A.32).). Moreover,  tr  σ (Tγ ,δ ) = supp (dΩα,γ ,δ ( · , x0 )) = supp dΩα,γ ,δ ( · , x0 ) .

.

(A.29)

Here dΩ tr = dΩ1,1 + dΩ2,2

.

(A.30)

denotes the trace measure of a .2 × 2 matrix-valued nonnegative measure   dΩ = dΩ ,  ,  =1,2

.

(A.31)

on .R, and .MG represents the maximally defined operator of multiplication by the dΩ tr -measurable function G in the Hilbert space .L2 (R; dΩ),

.

    MG h2 (λ) for dΩ tr -a.e. λ ∈ R, h (λ) = G(λ) h(λ) = G(λ) h1 (λ), G(λ)     k ∈ L2 (R; dΩ) . h ∈ dom(MG ) =  k ∈ L2 (R; dΩ)  G (A.32)

.

This summarizes the traditional approach to .2×2 Weyl–Titchmarsh theory which focuses on separated boundary conditions at a and b (if any).

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References 1. N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space, Volume II (Pitman, Boston, 1981) 2. J. Behrndt, S. Hassi, H. de Snoo, Boundary Value Problems, Weyl Functions, and Differential Operators. Monographs in Mathematics, vol. 108 (Springer, Birkhäuser, 2020) 3. S. Clark, F. Gesztesy, R. Nichols, Principal solutions revisited, in Stochastic and Infinite Dimensional Analysis, ed. by C.C. Bernido, M.V. Carpio-Bernido, M. Grothaus, T. Kuna, M.J. Oliveira, J.L. da Silva. Trends in Mathematics (Springer, Birkhäuser, 2016), pp. 85–117 4. E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (Krieger Publication, Malabar, FL, 1985) 5. N. Dunford, J.T. Schwartz, Linear Operators. Part II: Spectral Theory (Wiley Interscience, New York, 1988) 6. M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic Press, Edinburgh, 1973) 7. J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl, Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional potentials. Opuscula Math. 33, 467–563 (2013) 8. F. Gesztesy, H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol. I: (1 + 1)-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics, vol. 79 (Cambridge University Press, Cambridge, 2003) 9. F. Gesztesy, L.L. Littlejohn, R. Nichols, On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below. J. Differ. Equ. 269, 6448–6491 (2020) 10. F. Gesztesy, L.L. Littlejohn, R. Nichols, M. Piorkowski, J. Stanfill, Donoghue m-functions for singular Sturm–Liouville operators, arXiv:2107.09832. St. Petersburg Math. J. (to appear) 11. F. Gesztesy, R. Nichols, Sturm–Liouville M-functions in terms of Green’s functions, preprint, 2022 12. F. Gesztesy, R. Nichols, M. Zinchenko, Sturm–Liouville Operators, Their Spectral Theory, and Some Applications, book in preparation 13. F. Gesztesy, M. Zinchenko, On spectral theory for Schrödinger operators with strongly singular potentials. Math. Nachr. 279, 1041–1082 (2006) 14. P. Hartman, Ordinary Differential Equations (SIAM, Philadelphia, 2002) 15. P. Hartman, A. Wintner, On the assignment of asymptotic values for the solutions of linear differential equations of second order. Am. J. Math. 77, 475–483 (1955) 16. E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956) 17. K. Jörgens, F. Rellich, Eigenwerttheorie Gewöhnlicher Differentialgleichungen (Springer, Berlin, 1976) 18. H. Kalf, A characterization of the Friedrichs extension of Sturm–Liouville operators. J. Lond. Math. Soc. (2) 17, 511–521 (1978) 19. P. Kuchment, Floquet theory for partial differential equations, in Operator Theory: Advances and Applications, vol. 60 (Birkhäuser, Basel, 1993) 20. W. Leighton, M. Morse, Singular quadratic functionals. Trans. Am. Math. Soc. 40, 252–286 (1936) 21. W. Magnus, S. Winkler, Hill’s Equation (Dover, New York, 1979) 22. M.A. Naimark, Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, Transl. by E.R. Dawson, English translation edited by W.N. Everitt (Ungar Publishing, New York, 1968) 23. H.-D. Niessen, A. Zettl, Singular Sturm–Liouville problems: the Friedrichs extension and comparison of eigenvalues. Proc. London Math. Soc. (3) 64, 545–578 (1992) 24. D.B. Pearson, Quantum Scattering and Spectral Theory (Academic Press, London, 1988) 25. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators (Academic Press, New York, 1978)

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26. F. Rellich, Die zulässigen Randbedingungen bei den singulären Eigenwertproblemen der mathematischen Physik. (Gewöhnliche Differentialgleichungen zweiter Ordnung.). Math. Z. 49, 702–723 (1943/44). (German) 27. F. Rellich, Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung. Math. Ann. 122, 343–368 (1951). (German) 28. R. Rosenberger, A new characterization of the Friedrichs extension of semibounded Sturm– Liouville operators. J. London Math. Soc. (2) 31, 501–510 (1985) 29. G. Teschl, Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators, 2nd edn. Graduate Studies in Mathematics, vol. 157 (American Mathematical Society, Providence, RI, 2014) 30. J. Weidmann, Linear operators in Hilbert spaces, in Graduate Texts in Mathematics, vol. 68 (Springer, New York, 1980) 31. J. Weidmann, Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, vol. 1258 (Springer, Berlin, 1987) 32. J. Weidmann, Lineare Operatoren in Hilberträumen. Teil II: Anwendungen (Teubner, Stuttgart, 2003) 33. A. Zettl, Sturm–Liouville Theory. Mathematical Surveys and Monographs, vol. 121 (American Mathematical Society, Providence, RI, 2005)

On Discrete Spectra of Bergman–Toeplitz Operators with Harmonic Symbols L. Golinskii, S. Kupin , J. Leblond, and M. Nemaire

Dedicated to the memory of Sergey Naboko, our dear friend and colleague.

Abstract In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using the methods of classical perturbaton theory and recent results by Borichev–Golinskii–Kupin and Favorov–Golinskii, we obtain a quantitative result on the distribution of the discrete spectrum of the operator in the unbounded (outer) component of its Fredholm set. Keywords Discrete spectrum of a perturbed operator · Lieb–Thirring type inequality · Blaschke type inequality · Hardy–Toeplitz operator · Bergman–Toeplitz operator

L. Golinskii B. Verkin Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine e-mail: [email protected] S. Kupin () IMB, CNRS, Université de Bordeaux, Talence Cedex, France e-mail: [email protected] J. Leblond FACTAS, INRIA Sophia Antipolis-Méditerranée, Sophia Antipolis Cedex, France e-mail: [email protected] M. Nemaire IMB, CNRS, Université de Bordeaux, Talence Cedex, France FACTAS, INRIA Sophia Antipolis-Méditerranée, Sophia Antipolis Cedex, France e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_22

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1 Introduction Various problems of modern analysis require the study of certain classes of “model” operators. One of the important families is the class of Toeplitz operators and operators related to them. Probably, the most classical objects of this kind are Toeplitz operators on Hardy spaces of analytic functions, see Nikolski [9] for a complete account on the subject. The applications of operators of this class can be found in Nikolski [10]. Another “similar” class is the family of Toeplitz operators on Bergman spaces. Their study started in late 80’s of the last century, see Zhu [16] for a nice overview of the topic. We proceed with some definitions. Let .D = {z : |z| < 1} and .T = {z : |z| = 1} be the unit disk and the unit circle in the complex plane, respectively. For a function ∞ 2 2 .ϕ ∈ L (T), the Hardy–Toeplitz operator .Tϕ : H (T) → H (T) is defined as Tϕ h = P+ (ϕh),

.

h ∈ H 2 (T),

(1)

where .P+ is the well-known Riesz othogonal projection from .L2 (T) to .H 2 (T), see Garnett [5]. The function .ϕ is called a symbol of the operator. For the sake of brevity, we call operator .Tϕ (1) an HT-operator. The definition of a Bergman–Toeplitz operator .Tψ (a BT-operator, for short), is rather similar to the above one. Indeed, let .L2a (D) be the closed subspace in .L2 (D) of analytic on .D functions, see [16, Sect. 3.4, 4.3] for notation. Given .ψ ∈ L∞ (D), set Tψ : L2a (D) → L2a (D),

.

Tψ h = Pˆ+ (ψh),

(2)

where .Pˆ+ is the orthogonal projection acting from .L2 (D) to .L2a (D), see Zhu [16, Chapter 7]. For a function .ϕ ∈ L∞ (T), it is sometimes convenient to consider a harmonic function .ϕˆ on .D, the harmonic extension of .ϕ to .D, given by ˆ ϕ(z) ˆ =

.

1 − |z|2 ϕ(t) m(dt), T |t − z|2

z ∈ D.

(3)

Here, m is the probability Lebesgue measure on .T. Certainly, .ϕˆ ∈ L∞ (D) and .ϕ ˆ ∞ ≤ ϕ∞ . Given an HT-operator .Tϕ (with its symbol defined on .T), we consider its associated BT-operator .Tϕˆ with symbol .ϕˆ given by (3). Notice that even though we use “similar-looking” notation for an HT-operator .Tϕ and a BT-operator .Tϕˆ , the confusion is not possible since the functions .ϕ and .ϕˆ are defined on .T and .D, respectively. The domains of definitions of corresponding symbols will be always clear from the context of the discussion.

Discrete Spectra of Bergman–Toeplitz Operators

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We shall be concerned with Toeplitz operators .Tϕ , Tϕˆ having symbols defined as follows. Set ϕ(t) := g(t) ¯ + f (t),

.

t ∈ T,

f, g ∈ H ∞ (T).

(4)

Clearly, we have ψ(z) := ϕ(z) ˆ = g(z) ¯ + f (z),

.

z ∈ D.

(5)

It is plain that .ϕˆ has the non-tangential boundary values on the unit circle ϕ(t) ˆ = lim ϕ(rt), ˆ

.

r→1−

for a. e. t ∈ T,

and these boundary values coincide with .ϕ a.e. on .T. Despite the similarity of definitions (1), (2), the BT- operators exhibit considerably reacher spectral behavior as compared to HT-operators. For instance, the essential spectrum of HT-operator .Tϕ is connected, see Widom [14], while, in general, the essential spectrum of BT-operator .Tψ is not. There are non-trivial compact BT-operators with quite simple (even radial) symbols [16, Sections 7.2, 7.3], while a compact HT-operator is necessarily zero [9, Part B, Chapter 4]. Sundberg–Zheng [13] showed that there are BT-operators with harmonic symbols having isolated eigenvalues in their spectrum. In subsequent papers, Zhao– Zheng [15], Guan–Zhao [6] and Guo–Zhao–Zheng [7] presented a class of BToperators with harmonic symbols posessing “rather big” discrete spectrum, that is, the set of isolated eigenvalues of finite algebraic multiplicity. So, in contrast to HT-operators, the notion of the discrete spectrum of a BToperator with harmonic symbol makes sense. The study of the properties of the discrete spectrum for BT-operators with symbols (5) is the core of the present paper. Unlike the articles [6, 7, 15], our results are essentially based on the perturbation techniques from operator theory and function-theoretic results of Borichev–Golinskii–Kupin [2, 3] and Favorov–Golinskii [4]. We also need the definition of the Sobolev space .W 1,2 (T) of absolutely continuous functions on the unit circle .T with derivative in .L2 : W 1,2 (T) := {h : T → C, h ∈ AC, h ∈ L2 (T)}.

.

Quite naturally, the norm of a function .h ∈ W 1,2 (T) is defined as ||h||W 1,2 (T) = ||h||L2 (T) + ||h ||L2 (T) .

.

For further purposes, we would like to introduce two closely related characteristics of compact sets on the complex plane .C. The following definitions are borrowed from Perkal [12, Section 2] and Peller [11, Section 4], respectively.

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Definition 1.1 Let .r > 0. A closed set .E ⊂ C is called r -convex if   .C\E = B(x, r) : B(x, r) ⊂ C\E , that is, the complement to E can be covered with open disks of a fixed radius .r > 0, which lie in that complement. Definition 1.2 A compact set .K ⊂ C is called circularly convex, if there is .r > 0 such that for each .λ ∈ C\K with .dist(λ, K) < r there are points .μ ∈ ∂K and .ν ∈ C\K so that |μ − ν| = r,

.

λ ∈ (μ, ν],

{ζ : |ν − ζ | < r} ⊂ C\K.

For example, if K is a convex set, or the boundary .∂K is of .C 2 -class (without intersections and cusps), then K is an circularly convex set. Note that the later definition is a bit more stringent than the former one. When K is a (closed) Jordan curve (a rectifiable continuous curve with no self-intersections), one can also see that the above definitions are “equivalent”; that is, K is circularly convex whenever there is .r > 0 so that K is r-convex. A short reminder on standard notions and notations from operator theory is given in Sect. 2.1 below. For instance, see (3) for the notion of the unbounded (outer) open component of the Fredholm domain .F0 (T ). The main result of this note is the following theorem. Theorem 1.3 Let .Tϕ be an HT-operator with the symbol .ϕ (4) from .W 1,2 (T), .ϕˆ its harmonic extension (3), and .Tϕˆ be the BT-operator associated to .Tϕ . Assume that the spectrum .σ (Tϕ ) is a circularly convex set. Then, for each .ε > 0  .

dist3+ε (λ, σ (Tϕ )) ≤ C(ϕ, ε) ϕ  22 .

(6)

λ∈σd (Tϕˆ )∩F0 (Tϕ )

Corollary 1.4 Let q and p be algebraic polynomials, .ϕ = q + p be a harmonic polynomial, and assume that the image .ϕ(T) is a Jordan curve without cusps. Then .(6) holds for the discrete spectrum of BT-operator .Tϕˆ .

2 Some Preliminaries 2.1 Generalities from Operator Theory In this section, we recall some well-known notions of the classical operator theory, see Kato [8, Section IV.5]. Let T be a bounded linear operator on a (separable) Hilbert space H . As usual, the resolvent set of T is ρ(T ) := {λ ∈ C : (T − λ) : H → H is bijective }.

.

(1)

Discrete Spectra of Bergman–Toeplitz Operators

613

It follows that .(T − λ)−1 is bounded for .λ ∈ ρ(T ). The spectrum of T is defined as σ (T ) = C\ρ(T ).

.

(2)

Furthermore, we say that a bounded operator T is Fredholm, if its kernel and co-kernel are of finite dimension. The essential spectrum of T is defined as σess (T ) = {λ ∈ C : (T − λ) is not Fredholm}.

.

One can see that .σess (T ) is a closed subset of .σ (T ). One considers also the Fredholm domain of T , .F(T ) = C\σess (T ). Clearly, .ρ(T ) ⊂ F(T ). We represent .F(T ) as F(T ) =

N 

.

Fj (T ),

(3)

j =0

where .Fj (T ) are disjoint (open) connected components of the set. Of course, the number of the components .Fj (T ) is .N + 1, and it can be finite or infinite. We agree that .F0 (T ) stays for the unbounded connected component of .F(T ). The discrete spectrum .σd (T ) of T is the set of all isolated eigenvalues of T of finite algebraic multiplicity. For convenience, we put σ0 (T ) := σd (T ) ∩ F0 (T ) ⊂ σd (T ).

.

(4)

Let .A0 , A be bounded operators on a Hilbert space such that .A − A0 is compact. The operators A and .A0 are called compact perturbations of each other. The celebrated Weyl’s theorem states that σess (A) = σess (A0 ),

.

(5)

see Kato [8, Section IV.5.6]. We shall be interested in the situation when .σ0 (A) is at most countable set, .σ0 (A) = {λj }j ≥1 and it accumulates to the essential spectrum .σess (A0 ) only.

2.2 Reminder on Hilbert-Schmidt Operators In this subsection, we recall briefly the notion of a Hilbert-Schmidt operator and its simplest properties, see Birman-Solomyak [1, Section 11.3]. Let A be a compact operator. The sequence of singular values .{sj (A)}j ≥1 is defined as sj (A) = λj (A∗ A)1/2 ,

.

sj (A) ≥ 0,

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where .λj (A∗ A) are eigenvalues of the compact operator .A∗ A. Without loss of generality one can suppose that .{sj (A)}j ≥1 forms a decreasing sequence, and, moreover .

lim sj (A) = 0.

j →+∞

One says that .A ∈ S2 , the Hilbert-Schmidt class of compact operators, iff A2S := 2



.

sj (A)2 < ∞.

j ≥1

Equivalently, .A ∈ S2 if and only if .{sj (A)}j ≥1 ∈ 2 . Alternatively, the .S2 -norm of the operator can be computed as A2S = 2



.

|(Aej , ek )|2 ,

j,k≥1

where .{ej }j ≥1 is an arbitrary orthonormal basis in the given Hilbert space.

2.3 On the Discrete Spectrum of a Perturbed Operator: A Result of Favorov–Golinskii Some useful quantitative bounds for the rate of convergence of the discrete spectrum of a perturbed operator are given in Favorov–Golinskii [4, Section 5]. A special case of [4, Theorem 5.1] (cf. a remark right after its proof and formula (5.8)) looks as follows. Theorem 2.1 ([4]) Let .A0 be a bounded linear operator on a Hilbert space, which satisfies the conditions: 1. There is .r > 0 such that spectrum .σ (A0 ) is an r-convex set. 2. The resolvent .R(z, A0 ) = (A0 − z)−1 is subject to the bound R(z, A0 ) ≤

.

C(A0 ) , distp (z, σ (A0 ))

p > 0,

z ∈ F0 (A0 ).

(6)

Let B be a Hilbert–Schmidt operator, and .A = A0 + B. Then for each .ε > 0  .

λ∈σd (A)∩F0 (A0 )

dist2p+1+ε (λ, σ (A0 )) ≤ C(σ (A0 ), p, ε) B22 .

(7)

Discrete Spectra of Bergman–Toeplitz Operators

615

If .σess (A0 ) does not split the plane, and (6) holds for all .λ ∈ C\σ (A0 ), then (7) is true for the whole discrete spectrum .σd (A). For the class of (non-selfadjoint) HT-operators .A0 = Tϕ , .ϕ in (4), the essential spectrum, in general, splits the plane.

3 Proof of the Main Result Let .ϕ be as in (4). Consider the HT-operator .Tϕ (1) and the associated BT-operator Tϕˆ (2). The technical way to compare these operators is to look at their matrices in appropriately chosen bases. Namely, define √ n (1) .eH,n (t) = t , eB,n (z) = n + 1 zn , n ≥ 0.

.

It is plain that the systems .{eH,n }n≥0 and .{eB,n }n≥0 are the orthonormal bases in H 2 (T) and .L2a (D), respectively. Set

.

Tϕ = [(Tϕ eH,i , eH,j )H 2 (T) ]i,j ≥0 ,

.

Tϕˆ = [(Tϕˆ eB,i , eB,j )L2a (T) ]i,j ≥0 .

(2)

The operators (or, infinite matrices) .Tϕ and .Tϕˆ are unitarily equivalent to original operators .Tϕ and .Tϕˆ . For this reason, it will be convenient to “identify” .Tϕ and .Tϕ , as well as .Tϕˆ and .Tϕˆ , respectively. By definition .Tϕ and .Tϕˆ both act on .2 (Z+ ). So, one can argue on the operators .Tϕ and .Tϕˆ being “close” in a certain sense. Rewriting relation (4) in more detailed form, we have ϕ(z) ˆ = g(z) + f (z), f (z) = .

∞ 

∞

fk z ∈ H (T), k

g(z) =

k=0

ϕ(eiθ ) = ϕ(e ˆ iθ ) =



 bj eij θ ,

bj =

j ∈Z

∞ 

gk zk ∈ H ∞ (T), (3)

k=0

fj ,

j ≥ 0,

g−j ,

j < 0.

Proposition 3.1 Assume that the symbol .ϕ (4) belongs to .W 1,2 (T). Then .Tϕˆ − Tϕ is a Hilbert-Schmidt operator and Tϕˆ − Tϕ 2S ≤

.

2

π2  2 ϕ 2 . 24

(4)

Proof The matrix representation of .Tϕ is obvious: .Tϕ = [bi−j ]i,j ≥0 . So let us compute the matrix .Tϕˆ = [τi,j ]i,j ≥0 in the orthonormal basis .{eB,n }n≥0 (1). For .l, k ≥ 0 ˆ 1 ϕ(z)e ˆ τk,k+l = (Tϕˆ eB,k+l , eB,k )L2a (D) = B,k+l (z) eB,k (z) dxdy π D . √ ˆ (k + l + 1)(k + 1) k+l k ϕ(z)z ˆ = z dxdy, π D

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or, in polar coordinates, τk,k+l

.

√ ˆ ˆ (k + l + 1)(k + 1) 1 2π  bn r |n|+2k+l+1 ei(n+l)θ drdθ. = π 0 0 n∈Z

Finally,  τk,k+l =

.

k+1 b−l , k+l+1

k, l ≥ 0.

The same formula holds for .τk+l,k , .k, l ≥ 0, and so  τi,j =

.

min(i, j ) + 1 bi−j , max(i, j ) + 1

i, j ≥ 0.

(5)

Let us estimate the Hilbert-Schmidt norm Tϕˆ − Tϕ 2S =



.

2

|τi,j − bi−j |2 =

i,j ≥0

 i,j ≥0

 2 min(i, j ) + 1 2 |bi−j | 1 − . max(i, j ) + 1

As  .

aij =

∞ ∞  

i,j ≥0

ak,k+l +

l=0 k=0

∞ ∞  

ak+l,k ,

l=1 k=0

we have Tϕˆ − Tϕ 2S =

∞ 

∞

 l 2 |bl |2 + |b−l |2

l=0

k=0

.

2

=



|l bl |2

l∈Z

⎛ ≤⎝

=

k=0

⎞ ∞  |lbl |2 ⎠

l∈Z

k=0



|lbl |2

∞  k=0

1 k+l+1+

2 √ k+1

1

2 √ √ (k + |l| + 1) k + |l| + 1 + k + 1



l∈Z

as claimed.

∞ 

(k + l + 1)

√

1 √ (k + 1) (2 k + 1)2



1 π2 π2  2  2 ϕ ϕ 2 . =  = 2 4·6 24 4 (k + 1)2 

Discrete Spectra of Bergman–Toeplitz Operators

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Weyl’s theorem concerning spectra of compact perturbations, mentioned above, leads to the following Corollary 3.2 Let the symbol .ϕ satisfy hypothesis of the above Proposition. Then the essential spectrum of BT-operator .Tϕˆ is σess (Tϕˆ ) = σess (Tϕ ) = ϕ(T) =: Γ,

.

and the discrete spectrum .σd (Tϕˆ ) is at most countable set of eigenvalues of finite algebraic multiplicity with all its accumulation points on .Γ . We go on with the proof of the quantitative version of the above Corollary. Proof of Theorem 1.3 Let .A0 = Tϕ , .A = Tϕˆ , see (2). We only have to ensure that the conditions of Theorem 2.1 are met. It is clear that .ϕ ∈ W 1,2 (T) implies that .ϕ ∈ W , the Wiener algebra of absolutely convergent Fourier series. By [11, Theorem 4], the resolvent .(A0 − z)−1 admits the linear growth, that is, (6) holds with .p = 1. Next, by Proposition 3.1, the difference .A − A0 is the Hilbert-Schmidt operator with the norm bound (4). The proof is complete. 

Acknowledgments SK, JL, MN are partially supported by the project ANR-18-CE40-0035.

References 1. M. Birman, M. Solomjak, in Spectral theory of Selfadjoint Operators in Hilbert Space. Mathematics and Its Applications (Soviet Series) (D. Reidel Publishing, Dordrecht, 1987) 2. A. Borichev, L. Golinskii, S. Kupin, A Blaschke-type condition and its application to complex Jacobi matrices. Bull. Lond. Math. Soc. 41(1), 117–123 (2009) 3. A. Borichev, L. Golinskii, S. Kupin, On zeros of analytic functions satisfying non-radial growth conditions. Rev. Mat. Iberoam. 34(3), 1153–1176 (2018) 4. S. Favorov, L. Golinskii, Blaschke-type conditions on unbounded domains, generalized convexity, and applications in perturbation theory. Rev. Mat. Iberoam. 31(1), 1–32 (2015) 5. J. Garnett, in Bounded Analytic Functions. Graduate Texts in Mathematics, vol. 236, Revised first edn. (Springer, New York, 2007) 6. N. Guan, X. Zhao, Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols. Sci. China Math. 63(5), 965–978 (2020) 7. K. Guo, X. Zhao, D. Zheng, The spectral picture of Bergman Toeplitz operators with harmonic polynomial symbols, submitted. https://arxiv.org/abs/2007.07532 8. T. Kato, Perturbation theory for linear operators. Classics in Mathematics (Springer-Verlag, Berlin, 1995). Reprint of the 1980 edition 9. N. Nikolski, in Operators, Functions, and Systems: An Easy Reading, I. Hardy, Hankel, and Toeplitz. Mathematical Surveys and Monographs, vol. 92 (American Mathematical Society, Providence, 2002) 10. N. Nikolski, in Operators, Functions, and Systems: An Easy Reading, II. Model Operators and systems. Mathematical Surveys and Monographs, vol. 93 (American Mathematical Society, Providence, 2002)

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11. V. Peller, Spectrum, similarity, and invariant subspaces of Toeplitz operators. Math. USSR-Izv. 29(1), 133–144 (1987) 12. J. Perkal, Sur les ensembles ε-convexes. Colloq. Mat. 4, 1–10 (1956) 13. C. Sundberg, D. Zheng, The spectrum and essential spectrum of Toeplitz operators with harmonic symbols. Indiana Univ. Math. J. 59(1), 385–394 (2010) 14. H. Widom, On the spectrum of a Toeplitz operator. Pacific J. Math. 14, 365–375 (1964) 15. X. Zhao, D. Zheng, The spectrum of Bergman Toeplitz operators with some harmonic symbols. Sci. China Math. 59(4), 731–740 (2016) 16. K. Zhu, in Operator Theory in Function spaces. Mathematical Surveys and Monographs, vol. 138, 2nd edn. (American Mathematical Society, Providence, 2007)

One Dimensional Discrete Schrödinger Operators with Resonant Embedded Eigenvalues Wencai Liu and Kang Lyu

Dedicated to the memory of Sergey Nikolaevich Naboko (1950–2020)

Abstract In this paper, we introduce a new family of functions to construct Schrödinger operators with embedded eigenvalues. This particularly allows us to construct discrete Schrödinger operators with arbitrary prescribed sets of eigenvalues. Keywords Schrödinger operators · Embedded eigenvalues · Resonant eigenvalues

1 Introduction In this paper, we study the discrete Schrödinger operator, H u(n) = ( + V )u(n) = u(n + 1) + u(n − 1) + V (n)u(n),

.

and the eigen-equation .H u = Eu, where . is the free discrete Schrödinger operator, and V is the potential.

W. Liu Department of Mathematics, Texas A&M University, College Station, TX, USA e-mail: [email protected] K. Lyu () School of Mathematics and Statistics, Nanjing University of Science and Technology, Jiangsu, People’s Republic of China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_23

619

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W. Liu and K. Lyu

For simplicity, we only consider the eigen-equation on the half line, H u(n) = u(n + 1) + u(n − 1) + V (n)u(n) = Eu(n), n ≥ 1,

.

(1.1)

with the boundary condition .

u(1) = tan θ, θ ∈ [0, π ). u(0)

(1.2)

By Weyl’s criterion for the essential spectrum, when .V (n) = o(1) as .n → ∞, the essential spectrum of H σess (H ) = σess () = [−2, 2].

.

We are interested in constructions of potentials V such that .H =  + V have eigenvalues embedded into the essential spectrum .[−2, 2]. The study of problems of embedded eigenvalues has a long history. The classical K sin(2π kx + φ) could provide Wigner-von Neumann functions .Vk (x) = 1+x one embedded eigenvalue. Naboko [20] and Simon [24] constructed continuous Schrödinger operators with countably many (dense) eigenvalues. Recently there are remarkable developments in the area of embedded eigenvalues for both continuous and discrete Schrödinger operators [1, 4–6, 8, 9, 11, 14, 16–19, 23, 25]. The previous results about perturbed discrete Schrödinger operators can be summarized as follows. For any .E ∈ (−2, 2), let .E = 2 cos π k(E) with .k(E) ∈ (0, 1) (.k(E) is referred to as quasimomentum). Naboko and Yakovlev [22] constructed potentials V such that . + V have any given eigenvalues that their quasimomenta are rationally independent (see [8, 20, 24] for continuous cases and [12, 21] for Stark type potentials). In [3], Jitomirskaya and Liu introduced the piecewise constructions and gluing techniques to construct embedded eigenvalues of Laplacians on noncompact manifolds. Those techniques have been further developed by Liu and Ong in [15]. In particular, they constructed potentials of any prescribed embedded ˜ = 1 eigenvalues when their quasimomenta are non-resonant, namely .k(E) + k(E) ˜ See [13] for an alternative construction based on explicit (in other words, .E = −E). Wigner-von Neumann type functions. It is natural to ask if we could relax the non-resonance assumption in [13, 15], namely construct discrete Schrödinger operators with arbitrary prescribed sets of eigenvalues. ˜ When .k(E) = 1 − k(E), their corresponding Wigner-von Neumann functions K K ˜ + φ) are the same up to .Vk (x) = sin(2π kx + φ) and .Vk˜ (x) = 1+x sin(2π kx 1+x a phase .φ. This is a big obstacle to construct potentials such that both E and .E˜ are eigenvalues. We also want to mention that non-resonant conditions appear naturally in the study of Schrödinger operators with Wigner-von Neumann type or bounded variation potentials [2, 19].

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

621

The main goal of this paper is to construct arbitrary embedded eigenvalues (without non-resonance restriction). The proof (contains two steps) follows the general scheme of Jitomirskaya and Liu [3]. The first step is to construct a “nice” function (we call it a generating function) attached to one chosen eigenvalue E. The second step is to glue all pieces together by adding eigenvalues slowly (the idea is inspired by Naboko [20] and Simon [24]). With the recent development [12, 13, 15], the second step is rather standard. So following the plan of Jitomirskaya and Liu [3], the difficulty lies in finding the generating function. The Wigner-von K Neumann function .Vk (x) = 1+x sin(2π kx + φ) (k is the quasimomentum of E) is a natural candidate, which has been used to do the constructions in [3] and [13]. In order to study perturbed periodic operators, inspired by [8], Liu and Ong [15] chose the generating function by solving an equation involving Prüfer angles and potentials. Both generating functions fail in the case of resonant eigenvalues, since the generating function attached to E completely breaks the oscillatory estimate to ˜ (for example, see Lemma 4.2). the eigen-equation .H u = Eu In the present work, we propose a new family of functions as our generating functions. There are two novelties in our constructions of generating functions. We first obtain a function .V˜k + V˜k˜ attached to a pair of resonant eigenvalues E and .E˜ by solving a system of two equations. Both functions .V˜k and .V˜k˜ have similar properties of (but not) Wigner-von Neumann functions. However, .V˜k + V˜k˜ can not directly serve as a generating function since the Prüfer angle .θk˜ (n) + θk (n) may stay fixed up to an integer by the fact that .k + k˜ = 1. In order to overcome the difficulty, we 1 modify the function .V˜k + V˜k˜ by adding an extra function . 1+n . In other words, our 1 1 ˜ ˜ generating functions have the format .Vk +Vk˜ + 1+n . The additional function . 1+n will play a significant role in our constructions. We believe the novel family of functions ˜k + V˜ ˜ + 1 introduced in this paper will have wider applications, in particular .V k 1+n problems of perturbed periodic operators with resonant embedded eigenvalues. Theorem 1.1 Suppose .{Ej }m j =1 are a finite set of distinct points in .(−2, 2). Then

O(1) for any given .{θj }m j =1 ⊂ [0, π ), there exist potentials .V (n) = 1+n such that for any .j = 1, 2, · · · , m, the eigen-equation .H u = ( + V )u = Ej u has a solution 2 .u ∈ l (N) satisfying the boundary condition

.

u(1) = tan θj . u(0)

Theorem 1.2 Suppose .{Ej }∞ j =1 are a countable set of distinct points in .(−2, 2). ⊂ [0, π ) and any non-decreasing positive function .h(n) Then for any given .{θj }∞ j =1 with .limn→∞ h(n) = ∞, there exist potentials V satisfying .

|V (n)| ≤

h(n) , 1+n

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such that for any .j = 1, 2, · · · , the eigen-equation .H u = ( + V )u = Ej u has a solution .u ∈ l 2 (N) satisfying the boundary condition .

u(1) = tan θj . u(0)

2 Some Basic Lemmas The basic tool we use in this paper is the modified Prüfer transformation. We refer readers to [7] and [23] for more details. For any .E ∈ (−2, 2), denote by .E = 2 cos π k with .k = k(E) ∈ (0, 1). Suppose .u(n) = u(n, E) is a solution of (1.1). Let Y (n) =

.

1 sin π k



sin π k 0 − cos π k 1



 u(n − 1) . u(n)

Define the Prüfer variables .R(n) and .θ (n) as  Y (n) = R(n)

.

 sin(π θ (n) − π k) . cos(π θ (n) − π k)

Then .R(n) and .θ (n) obey .

V (n)2 R(n + 1)2 V (n) sin 2π θ (n) + =1− sin2 π θ (n) 2 sin π k R(n) sin2 π k

(2.1)

and .

cot(π θ (n + 1) − π k) = cot π θ (n) −

V (n) When .| sin πk | ≤

1 10 ,

V (n) . sin π k

by (2.1), one has that

V (n) sin 2π θ (n) + O . ln R(n + 1) − ln R(n) = − sin π k 2

(2.2)

2



V (n)2 sin2 π k

 (2.3)

Now let us introduce some useful lemmas.    V (n)  1 Lemma 2.1 ([7, Proposition 2.4]) Suppose . sin π k  < 2 , then the .θ (n) defined by (2.2) satisfies    V (n)  .  |θ (n + 1) − θ (n) − k| ≤  . sin π k 

(2.4)

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

   V (n)  Lemma 2.2 Suppose . sin πk 
0, there exists .N ∈ N (depending on k and .ε > 0) such that for any .θ ∈ R and .ν ∈ {2, 4}, we have   −1  1 N    . cos(θ ± νπ kl) ≤ ε, N  l=0

(2.7)

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  −1  1 N    sin(θ ± νπ kl) ≤ ε. . N 

(2.8)

l=0

Proof Clearly, (2.8) follows from (2.7). In order to avoid repetition, we only prove (2.7) for .ν = 2. Case 1: k is rational In this case, let .k = pq , and p and q are coprime. Direct computations imply that / 2π Z, for .x ∈ .

cos θ + cos(θ + x) + · · · + cos(θ + (N − 1)x)   sin θ − x2 + N x − sin θ − x2 = . 2 sin x2

Therefore, one has that for .N = q,  N −1  1     . cos(θ + 2π kl) = 0. N  l=0

This obviously implies (2.7). Case 2: k is irrational We note that the map .T : x → x + 2kπ mod 2π is ergodic. By the ergodicity, one has that for large N,  N −1   N −1  ˆ 2π 1   1       . cos(θ + 2π kl) =  cos(θ + 2π kl) − cos tdt  ≤ ε. N  N  0 l=0

l=0

We finish the proof.

3 Technical Preparations Let .N ∈ N, which will be determined later. Lemma 3.1 Let .k ∈ (0, 1), .L ∈ N, .n0 ∈ N and .b ∈ N. Assume that .F = F (n) and ϕ = ϕ(n) satisfy for any .m ∈ N,

.

.

ln F (n0 + (m + 1)N)2 = ln F (n0 + mN )2 −

(3.1)

K (1 − cos 2π ϕ(n0 + mN ) + δ(m)) 2(n0 + mN − b) sin π k

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

625

and ϕ(n0 + (m + 1)N ) = ϕ(n0 + mN ) + L

.

+

K (100 + ε(m)) , (n0 + mN − b)π sin π k

where .K > 108 π N , .|δ(m)| ≤ enough .n0 − b > 0,

sin π k 104

(3.2)

and .|ε(m)| < 1. Then we have that for large 

F (n0 + mN ) ≤ 1.5F (n0 )

.

n0 − b + mN n0 − b

−100 .

(3.3)

Proof Denote by .ϕ(n ˜ 0 + mN ) = ϕ(n0 + mN) − mL. Then by (3.1) and (3.2) one has that .

ln F (n0 + (m + 1)N )2 = ln F (n0 + mN)2 −

(3.4)

K (1 − cos 2π ϕ(n ˜ 0 + mN ) + δ(m)) 2(n0 + mN − b) sin π k

and ϕ(n ˜ 0 + (m + 1)N ) = ϕ(n ˜ 0 + mN ) +

.

K (100 + ε(m)) . (n0 + mN − b)π sin π k (3.5)

Inequality .1 − cos 2π ϕ(n ˜ 0 + mN ) + δ(m) ≥ − sin10π4 k and (3.4) imply that .

ln F (n0 + (m + 1)N )2 ≤ ln F (n0 + mN )2 + In the case that .ϕ(n ˜ 0 + mN ) mod Z ∈



1 2 3, 3

1 K . 2 × 104 n0 + mN − b

(3.6)

 , one has that

1 − cos 2π ϕ(n ˜ 0 + mN ) + δ(m) ≥ 1,

.

and hence (by (3.4)), .

ln F (n0 + (m + 1)N )2 ≤ ln F (n0 + mN )2 −

1 K . 2 sin π k n0 + mN − b

(3.7)



Without loss of generality, we assume that .ϕ(n ˜ 0 ) ∈ − 43 , − 13 . By (3.5) we have that there exists an increasing sequence .{ml }∞ l=1 ⊂ N such that for any .l ∈ N, ϕ(n ˜ 0 + m2l N ) ≤

.

3l − 1 3l − 1 , ϕ(n ˜ 0 + (m2l + 1)N ) > , 3 3

(3.8)

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and ϕ(n ˜ 0 + m2l+1 N ) ≤

.

3l + 1 3l + 1 , ϕ(n ˜ 0 + (m2l+1 + 1)N ) > . 3 3

(3.9)

By (3.5), one has that ϕ(n ˜ 0 + (m2l+1 + 1)N )

.



m2l+1

=ϕ(n ˜ 0 + m2l N ) +

m=m2l

K (100 + ε(m)). (n0 + mN − b)π sin π k

(3.10)

Inequalities (3.8) and (3.9) imply that ϕ(n ˜ 0 + (m2l+1 + 1)N ) − ϕ(n ˜ 0 + m2l N ) ≥

.

2 . 3

(3.11)

By (3.10) and (3.11), we have that

.

m2l+1  K 2 ≤ (100 + ε(m)) 3 m=m (n0 + mN − b)π sin π k 2l

m2l+1  K ≤101 N π sin π k m=m

2l

≤102Ck ln

1 n0 −b N

+m

m2l+1 N + n0 − b , m2l N + n0 − b

(3.12)

8 where .Ck = N π K sin π k ≥ 10 . Replacing (3.11) with the inequality

ϕ(n ˜ 0 + m2l+1 N ) − ϕ(n ˜ 0 + (m2l + 1))N ) ≤

.

2 3

and following the proof of (3.12), we have 98Ck ln

.

2 m2l+1 N + n0 − b ≤ . m2l N + n0 − b 3

Therefore, one concludes that for any .l ∈ N, 98Ck ln

.

2 m2l+1 N + n0 − b m2l+1 N + n0 − b ≤ ≤ 102Ck ln . m2l N + n0 − b 3 m2l N + n0 − b

(3.13)

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

627

Similar to the proof of (3.13), we have that for any .l ∈ N, 98Ck ln

.

m2l+2 N + n0 − b m2l+2 N + n0 − b 1 ≤ ≤ 102Ck ln . m2l+1 N + n0 − b 3 m2l+1 N + n0 − b

(3.14)

Rewrite (3.13) and (3.14) as .

1 1 m2l+1 N + n0 − b ≤ ≤ ln , 153Ck m2l N + n0 − b 147Ck

(3.15)

.

1 1 m2l+2 N + n0 − b ≤ ≤ ln . 306Ck m2l+1 N + n0 − b 294Ck

(3.16)

and

By (3.6) and (3.7), we obtain for any .l ∈ N, m2l+1 N + n0 − b K ln m2l N + n0 − b 104 N   1 +O , (3.17) m2l N + n0 − b

ln F (n0 + m2l+1 N )2 ≤ ln F (n0 + m2l N )2 +

.

and .

m2l+2 N + n0 − b K ln 2N sin π k m2l+1 N + n0 − b   1 . +O (3.18) m2l+1 N + n0 − b

ln F (n0 + m2l+2 N )2 ≤ ln F (n0 + m2l+1 N )2 −

By (3.15) and (3.17), one has for large .n0 − b, K

1

F (n0 + m2l+1 N )2 ≤ F (n0 + m2l N )2 e 146NCk ×104 ≤ F (n0 + m2l N )2 e 105 . (3.19)

.

By (3.16) and (3.18), one has for large .n0 − b, F (n0 + m2l+2 N)2 ≤ F (n0 + m2l+1 N )2 e

.

− 613NCK sin π k k

≤ F (n0 + m2l N )2 e

−

1 103

. (3.20)

Moreover, it is not difficult to see that (similar to the proof of (3.19)) for any l ∈ N and .m ∈ [m2l , m2l+2 ],

.

1

F (n0 + mN)2 ≤ F (n0 + m2l N )2 e 105 .

.

(3.21)

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Iterating (3.20) (l times) and by (3.21), one has that for any .l ∈ N and .m ∈ [m2l , m2l+2 ], F (n0 + mN)2 ≤ F (n0 + m0 N )2 e

.

−

l + 1 103 105

(3.22)

.

Similar to the proof of (3.13) or (3.14), one has that .

ln

1 m0 N + n0 − b ≤ . n0 − b 98Ck

(3.23)

Similar to the proof of (3.19) (also see (3.21)), one has that 1

F (n0 + m0 N )2 ≤ e 105 F (n0 )2 .

(3.24)

.

By (3.22) and (3.24), one has that for any .l ∈ N and .m ∈ [m2l , m2l+2 ], F (n0 + mN )2 ≤ F (n0 )2 e

.

−

l + 2 103 105

(3.25)

.

By (3.15) and (3.16), one has .

l+1 l+1 m2l+2 N + n0 − b ≤ ≤ ln . 102Ck m0 N + n0 − b 98Ck

(3.26)

By (3.23) and (3.26), one has that .

ln

m2l+2 N + n0 − b m0 N + n0 − b l+2 m2l+2 N + n0 − b = ln + ln ≤ . n0 − b m0 N + n0 − b n0 − b 98Ck

This implies that l ≥ 98Ck ln

.

m2l+2 N + n0 − b mN + n0 − b − 2 ≥ 98Ck ln − 2. n0 − b n0 − b

By (3.25), (3.27) and the fact that .Ck ≥ 108 , one has 2

F (n0 + mN )2 ≤F (n0 )2 e 105

.

≤F (n0 ) e 2

This implies (3.3).

3 103

+

2 103



e

−

98Ck 103

ln

mN+n0 −b n0 −b

mN + n0 − b n0 − b

−200 .

(3.27)

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

629

4 Constructions of Potentials and Proof of Theorems 1.1 and 1.2 In this section, we always assume that .E = 2 cos π k, Ej = 2 cos π kj , .E˜ = −E = ˜ ˜ we also denote by .(R(n), θ (n)), (Rj (n), θj (n)), (R(n), θ˜ (n)) the Prüfer 2 cos π k, ˜ We remark that .k = 1 − k˜ and .sin π k = variables of .E, Ej , E˜ (or say .k, kj , k). ˜ sin π k. Proposition 4.1 Let .A = {Ej }m j =1 ⊂ (−2, 2). Let .E ∈ (−2, 2) be such that both ˜ E and .E = −E are not in A. Suppose .θ0 , θ˜0 ∈ [0, π ). Then there exist constants .K1 (E, A), K2 (E, A) and a function .V (E, A, n0 , b, θ0 , θ˜0 ) such that the following holds for .n0 − b ≥ K2 (E, A): Perturbation:

supp(V ) ⊂ [n0 , ∞), and for any .n ≥ n0 ,

.

.

  K (E, A) 1   ; V (E, A, n0 , b, θ0 , θ˜0 ) ≤ n−b

(4.1)

Solution for E: the solution of .( + V )u = Eu with the boundary condition .θ (n0 ) = θ0 satisfies for any n with .n > n0 ,  R(n) ≤ 2

.

n−b n0 − b

−100 R(n0 );

(4.2)

˜ with the boundary Solution for .E˜ (if .E = 0): the solution of .( + V )u = Eu condition .θ˜ (n0 ) = θ˜0 satisfies for any .n > n0 , ˜ R(n) ≤2

.

Solution for .Ej :



n−b n0 − b

−100

˜ 0 ); R(n

(4.3)

any solution of .( + V )u = Ej u satisfies for any .n > n0 , Rj (n) ≤ 2Rj (n0 ).

.

(4.4)

The proof of the following lemma is similar to that of [10, Lemma 3.1]. For readers’ convenience, we include a proof. Lemma 4.2 Let E and .A = {Ej }m j =1 satisfy the same assumptions as in Proposition 4.1. Assume for .n > b, .

|V (n)| ≤

K . n−b

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Then we have for any .n > n0 > b,      n sin 2π θ (l)  C(E, K)  ≤ , .   n0 − b l=n0 l − b 

(4.5)

and for any .j = 1, 2, · · · , m      n sin 2π θ (l) sin 2π θj (l)  C(E, A, K) ≤ . .   l−b n0 − b l=n0 

(4.6)

Proof Let us first show (4.5). It suffices to prove      n e2πiθ(l)  C(E, K)  ≤ . .   n0 − b l=n0 l − b  Indeed,        n 2π i(θ(l)+k)  n n 2πiθ(l) 2πiθ(l)      2π ki   e e e =  − . (e − 1)    l−b   l−b l − b   l=n0 l=n0 l=n0    n 2π i(θ(l)+k)  n n n 2πiθ(l+1) 2πiθ(l+1) 2πiθ(l)      e e e e  =  − + − l−b l−b l−b l − b   l=n0

l=n0

l=n0

l=n0

     n 2π i(θ(l)+k)   n 2πiθ(l+1)  n n 2πiθ(l+1)  2πiθ(l)     e  e e e     ≤ + − − l−b l − b   l−b l − b  l=n0 l=n0 l=n0 l=n0     n  n−1  2πiθ(l+1) 2π i(θ(l+1)−θ(l)−k)    2 e2πiθ(l+1)  e 1 − e  ≤ + −   +   l−b   n0 − b l−b l +1−b l=n0

l=n0

l=n0

l=n0

  n  n−1 2π i(θ(l+1)−θ(l)−k)    2 1  1 − e = + . +   n0 − b  l−b (l − b)(l + 1 − b)

(4.7)

By (2.4), one has that θ (l + 1) − θ (l) − k =

.

O(K) . (l − b) sin π k

(4.8)

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

631

By (4.7) and (4.8), we have       ∞ ∞ n 2πiθ(l)     2π ki 2 K 1 1 e   ≤ + O + . (e − 1)   n −b 2 l − b sin π k (l − b) (l − b)2 0   l=n0 l=n0 l=n0 (4.9) ≤

C(E, K) . n0 − b

Clearly, .

sin 2π θ (l) sin 2π θj (l) =

cos 2π(θ (l) − θj (l)) − cos 2π(θ (l) + θj (l)) . 2

By changing .k, θ (l) in the proof of (4.5) to .k ± kj , θ (l) ± θj (l), we have (4.6). For simplicity, denote by .K2 = K2 (E, A) and .K1 = K1 (E, A), we require that K2 K1 > 0.

.

For .E = 0, we solve the following system of two equations for .θ (n) and .θ˜ (n) on .[n0 , ∞) with the initial condition .(θ (n0 ), θ˜ (n0 )) = (θ0 , θ˜0 ) .

V (n) sin π k , V (n) , sin π k˜

(4.10)

sin 2π θ (n) + sin 2π θ˜ (n) + 100 . n−b

(4.11)

cot(π θ (n + 1) − π k) = cot π θ (n) − ˜ = cot π θ˜ (n) − ˜ + 1) − π k) cot(π θ(n

where V (n) = K1 (E, A)

.

For .E = 0, instead of solving (4.10), we solve .

cot(π θ (n + 1) − π k) = cot π θ (n) −

V (n) sin π k

(4.12)

with the initial condition .θ (n0 ) = θ0 , where V (n) = K1 (E, A)

.

sin 2π θ (n) + 100 . n−b

(4.13)

Proof (Proposition 4.1) Substitute (4.11) into (4.10) (or (4.13) into (4.12)). Then by solving (4.10) (or (4.12)), we obtain .θ (n) and .θ˜ (n) (or .θ (n)) for .n ≥ n0 . For .n ≥ n0 , define .V (n) as (4.11) (or (4.13) ). We finish the construction of .V (n). Now we are in the position to prove (4.2), (4.3) and (4.4).

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Case 1.

E = 0, namely .k =

.

1 2

Choose a fixed large enough N, which will be determined later. By (2.3) and (4.11), one has .

ln R(n + N )2

= ln R(n)2 −

N −1  l=0

= ln R(n)2 − K1

O(1) V (n + l) sin 2π θ (n + l) + sin π k (n − b)2

N −1  l=0

+

sin 2π θ (n + l) + sin 2π θ˜ (n + l) + 100 sin 2π θ (n + l) (n − b) sin π k

O(1) . (n − b)2

(4.14)

By (2.4) and (4.11), one has that for any l with .0 ≤ l ≤ N − 1, .

 sin 2π θ (n + l) = sin 2π θ (n) + kl +

O(1) 1+n−b

= sin(2π θ (n) + 2π kl) +



O(1) , 1+n−b

(4.15)

˜ and (.k = 1 − k)  ˜ + ˜ . sin 2π θ (n + l) = sin 2π θ˜ (n) + kl

O(1) 1+n−b

= sin(2π θ˜ (n) − 2π kl) +



O(1) . 1+n−b

By trigonometric identities, (4.14), (4.15) and (4.16), we have that .

ln R(n + N)2

= ln R(n)2 − K1

N −1  l=0

− 100K1

N −1  l=0

1 − cos (4π θ (n) + 4π kl) 2(n − b) sin π k

sin (2π θ (n) + 2π kl) (n − b) sin π k

(4.16)

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

− K1 +

633

N −1 

    cos 2π θ˜ (n) − 2π θ (n) − 4π kl − cos 2π θ˜ (n) + 2π θ (n)

l=0

2(n − b) sin π k

O(1) , (n − b)2

where .O(1) depends on E and A through N and .K1 (E, A). Applying Lemma 2.3 with large enough N , and using the fact that .n − b ≥ K2 , one has that there exists .|δ(n)| ≤ sin10π4 k such that .

ln R(n + N )2 = ln R(n)2 −

K1 N (1 − cos 2π ϕ(n) + δ(n)), 2(n − b) sin π k

(4.17)

where .ϕ(n) = θ˜ (n) + θ (n). By (2.5), (4.11) and Lemma 2.3 (with trigonometric identities), we have that there exists .|ε(n)| < 1 such that ϕ(n + N )

.

=ϕ(n) + N +

N −1 

(sin2 π θ (n + l) + sin2 π θ˜ (n + l))

l=0

=ϕ(n) + N + + K1

N −1 

V (n + l) O(1) + π sin π k (n − b)2

O(1) (n − b)2

˜ + l)) (sin2 π θ (n + l) + sin2 π θ(n

l=0

×

sin 2π θ (n + l) + sin 2π θ˜ (n + l) + 100 (n − b)π sin π k

=ϕ(n) + N +

N −1  1 O(1) + 100K 1 2 (n − b)π sin π k (n − b) l=0

+ K1

N −1  l=0

−

sin 2π θ (n + l) + sin 2π θ˜ (n + l) (n − b)π sin π k

N −1 K1  (cos 2π θ (n + l) + cos 2π θ˜ (n + l)) 2 l=0

×

sin 2π θ (n + l) + sin 2π θ˜ (n + l) + 100 (n − b)π sin π k

=ϕ(n) + N +

K1 N (100 + ε(n)) . (n − b)π sin π k

(4.18)

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By (4.17), (4.18) and applying Lemma 3.1 with .L = N, one has that for .n = n0 + mN, .m = 1, 2, · · · , 

n−b .R(n) ≤ 1.5 n0 − b

−100 (4.19)

R(n0 )

By (2.3) and the fact that .n0 − b is large enough, (4.19) implies for any l with 0 ≤ l ≤ N − 1 and .n = n0 + mN + l,

.



n−b .R(n) ≤ 2 n0 − b

−100 (4.20)

R(n0 ).

This implies (4.2). The proof of (4.3) follows that of (4.2) step by step. We omit the details. By (2.3) and (4.11), one obtains

.

ln Rj (n)2 ≤ ln Rj (n0 )2 −

n−1 n−1   V (l) O(1) sin 2π θj (l) + sin π kj (l − b)2

l=n0

= ln Rj (n0 )2 + −

l=n0

O(1) n0 − b

n−1 K1  sin 2π θ (l) + sin 2π θ˜ (l) + 100 sin 2π θj (l). sin π kj l−b l=n0

= ln Rj (n0 )2 +

O(1) , n0 − b

(4.21)

where the last inequality holds by Lemma 4.2. Therefore, for large .n0 − b, we have (4.4). Case 2.

k=

.

1 2

(.E = 0)

Since .sin π k = 1, by (2.3), (2.5) and (4.13), one has

.

ln R(n + 2)2 = ln R(n)2 −

1  j =0

V (n + j ) sin 2π θ (n + j ) +

O(1) (n − b)2

= ln R(n)2 −

O(1) K1 (1 − cos 4π θ (n)) + , n−b (n − b)2

= ln R(n)2 −

O(1) K1 (1 − cos 4π θ (n) + ) n−b n−b

(4.22)

Discrete Schrödinger Operators with Resonant Embedded Eigenvalues

635

and   1 O(1) K1 100 − sin 4π θ (n) + . .θ (n + 2) = θ (n) + 1 + π(n − b) 2 n−b

(4.23)

By (4.22), (4.23) and applying Lemma 3.1 with .ϕ(n) = θ (n) and .L = 1, we have (4.2). In this case, the proof of (4.4) is similar to the proof in Case 1. Proof (Theorems 1.1 and 1.2) As we already mentioned in the introduction, once we have the generating functions (Proposition 4.1) at hand, proofs of Theorems 1.1 and 1.2 follow from the standard construction first introduced in [3] and further developed in [12, 15]. We only give an outline of the proof here. Let .{Nr }r∈N be a non-decreasing sequence. In the construction of Theorem 1.1, .Nr = M for sufficiently large r, where M is the number of all non-resonant eigenvalues being constructed. In Theorem 1.2, .Nr goes to infinity arbitrarily slowly. We further assume .Nr+1 = Nr +1 when .Nr+1 > Nr . The construction is proceeded by inductions. Suppose we construct the potential on .[0, Jr−1 ] before .r − 1 steps. At the rth step, we take .Nr (non-resonant) eigenvalues into consideration. Our strategy r is to construct the potential on .[Jr−1 , Jr ] = ∪N l=1 [Jr−1 + (l − 1)Tr , Jr−1 + lTr ] in a piecewise manner (namely at rth step, we construct .Nr pieces of the potential of size .Tr starting at .Jr−1 and ending at .Jr = Jr−1 + Nr Tr ). For each piece .[Jr−1 +(l −1)Tr , Jr−1 +lTr ], we apply Proposition 4.1 with E being one eigenvalue (in total we have .Nr choices so we obtain .Nr pieces) and A being the rest of eigenvalues. The main difficulty is to control the size of each piece .Tr . The construction in [3, 12, 15] only uses inequalities (4.1), (4.2), (4.3) and (4.4) to obtain appropriate .Tr and .Nr . Therefore Theorems 1.1 and 1.2 follow from Proposition 4.1. Acknowledgments The authors wish to express their gratitude to the anonymous referee, whose comments helped the exposition of this paper. W. Liu was supported by NSF DMS-2000345, DMS2052572 and DMS-2246031. K. Lyu was supported by the National Natural Science Foundation of China (11871031).

References 1. S.A. Denisov, A. Kiselev, Spectral properties of Schrödinger operators with decaying potentials, in Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, vol. 76 of Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, 2007), pp. 565–589 2. J. Janas, S. Simonov, A Weyl-Titchmarsh type formula for a discrete Schrödinger operator with Wigner–von Neumann potential. Studia Math. 201(2), 167–189 (2010) 3. S. Jitomirskaya, W. Liu, Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian. Geom. Funct. Anal. 29(1), 238–257 (2019)

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4. E. Judge, S. Naboko, I. Wood, Eigenvalues for perturbed periodic Jacobi matrices by the Wigner–von Neumann approach. Integr. Equ. Oper. Theory 85(3), 427–450 (2016) 5. E. Judge, S. Naboko, I. Wood, Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach. J. Differ. Equ. Appl. 24(8), 1247–1272 (2018) 6. E. Judge, S. Naboko, I. Wood, Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique. Studia Math. 242(2), 179–215 (2018) 7. A. Kiselev, Y. Last, B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Comm. Math. Phys. 194 (1), 1–45 (1998) 8. A. Kiselev, C. Remling, B. Simon, Effective perturbation methods for one-dimensional Schrödinger operators. J. Differ. Equ. 151(2), 290–312 (1999) 9. H. Krüger, On the existence of embedded eigenvalues. J. Math. Anal. Appl. 395(2), 776–787 (2012) 10. W. Liu, Absence of singular continuous spectrum for perturbed discrete Schrödinger operators. J. Math. Anal. Appl. 472(2), 1420–1429 (2019) 11. W. Liu, The asymptotical behaviour of embedded eigenvalues for perturbed periodic operators. Pure Appl. Funct. Anal. 4(3), 589–602 (2019) 12. W. Liu, Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators. J. Funct. Anal. 276(9), 2936–2967 (2019) 13. W. Liu, Criteria for embedded eigenvalues for discrete Schrödinger operators. Int. Math. Res. Not. IMRN 2021(20), 15803–15832 (2021) 14. W. Liu, Revisiting the Christ-Kiselev’s multi-linear operator technique and its applications to Schrödinger operators. Nonlinearity 34(3), 1288–1315 (2021) 15. W. Liu, D.C. Ong, Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators. J. Anal. Math. 141(2), 625–661 (2020) 16. V. Lotoreichik, S. Simonov, Spectral analysis of the half-line Kronig-Penney model with Wigner–Von Neumann perturbations. Rep. Math. Phys. 74(1), 45–72 (2014) 17. M. Lukic, Schrödinger operators with slowly decaying Wigner-von Neumann type potentials. J. Spectr. Theory 3(2), 147–169 (2013) 18. M. Lukic, A class of Schrödinger operators with decaying oscillatory potentials. Comm. Math. Phys. 326(2), 441–458 (2014) 19. M. Lukic, D.C. Ong, Wigner-von Neumann type perturbations of periodic Schrödinger operators. Trans. Amer. Math. Soc. 367(1), 707–724 (2015) 20. S.N. Naboko, On the dense point spectrum of Schrödinger and Dirac operators. Teoret. Mat. Fiz. 68(1), 18–28 (1986) 21. S.N. Naboko, A.B. Pushnitski˘ı, A point spectrum, lying on a continuous spectrum, for weakly perturbed operators of Stark type. Funktsional. Anal. i Prilozhen. 29(4), 31–44, 95 (1995) 22. S.N. Naboko, S.I. Yakovlev, The point spectrum of a discrete Schrödinger operator. Funktsional. Anal. i Prilozhen. 26(2), 85–88 (1992) 23. C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Comm. Math. Phys. 193 (1), 151–170 (1998) 24. B. Simon, Some Schrödinger operators with dense point spectrum. Proc. Amer. Math. Soc. 125(1), 203–208 (1997) 25. S. Simonov, Zeroes of the spectral density of discrete Schrödinger operator with Wigner-von Neumann potential. Integr. Equ. Oper. Theory 73(3), 351–364 (2012)

On the Invariance Principle for a Characteristic Function Konstantin A. Makarov and Eduard Tsekanovskii

In respectful memory of Sergei Nikolaevich Naboko

Abstract We extend the invariance principle for a characteristic function of a dissipative operator with respect to the group of linear automorphisms of the upper half-plane to the case of general .SL2 (R) transformations. Keywords Dissipative operators · Characteristic function · Deficiency indices · Quasi-self-adjoint extensions · Krein-von Neumann extension

1 Introduction In [18] we introduced the concept of a characteristic function associated with a triple of operators consisting of (i) a densely defined symmetric operator with deficiency indices .(1, 1), (ii) its quasi-self-adjoint dissipative extension and (iii) a (reference) self-adjoint extension. This concept turns out to be convenient in two respects. On the one hand, the issue of choosing an undetermined constant phase factor in the definition of the characteristic function of an unbounded dissipative operator (a quasi-self-adjoint extension), which is due to M. S. Livšic [14] (also see [1]), is cleared. On the other hand, the characteristic function of a triple determines the whole triple up to mutual unitary equivalence, provided that the underlying symmetric operator is prime (see the corresponding uniqueness theorem in [18]).

K. A. Makarov () Department of Mathematics, University of Missouri, Columbia, MO, USA e-mail: [email protected] E. Tsekanovskii Department of Mathematics, Niagara University, Lewiston, NY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_24

637

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K. A. Makarov and E. Tsekanovskii

From a technical point of view, the characteristic function of a triple has shown itself as an adequate tool for solving certain problems in operator theory. For instance, the solution of the Jørgensen-Muhly problem [12] presented in [19, 20] (in the particular case of the deficiency indices .(1, 1)) led to the complete classification of simplest solutions of the classical commutation relations in the form  + tI  t∗ = A Ut AU

.

on

 Dom(A),

t ∈ R,

(1)

 is a dissipative quasi-self-adjoint extension of a symmetric operator .A˙ with where .A deficiency indices .(1, 1) and .Ut is a strongly continuous one-parameter group of unitary operators. Recall that the Jørgensen-Muhly problem is to provide an intrinsic characterization of symmetric operators .A˙ satisfying the commutation relation (1). In this context it is worth mentioning that the solution of the problem was based on the study of the transformation properties of the characteristic function of a triple of operators (generated by the symmetric and dissipative solutions of (1) ˙ with respect to affine and augmented by a reference self-adjoint extension of .A) transformations of the operators of the triple [19, 20]. As opposed to the familiar transformation law for the characteristic function  introduced by M. S. Livšic in [15] (also .SA (z) of a bounded dissipative operator .A see [17]) Sf (A)   ◦ f = SA

.

(2)

valid with respect to affine transformations .f (z) = az + b, .a > 0, .b ∈ R, the extension of the law (2) to the case of triples of unbounded operators requires appropriate modifications related to alignment of relevant phase factors (see [19, 20, Theorem F.1, Appendix F]). The goal of this Note is to extend the transformation law to the case of general automorphisms .f ∈ Aut (C+ ) of the upper half-plane .C+ . The corresponding main result of this Note is as follows, see Theorem 4.2. ˙ A,  A), with .A˙ a prime symmetric Given .f ∈ Aut (C+ ) and a triple .A = (A, ˙  operator, .A a maximal dissipative extension of .A and A its (reference) self-adjoint extension, we show that if the preimage .ω = f −1 (∞) belongs to the spectrum of  then the triple .A, ˙ f (A),  f (A)) f (A) = (f (A),

.

is well defined. Moreover, for an appropriately normalized . SA (z) characteristic function of the triple .A the invariance relation  SA Sf (A) ◦ f = 

.

holds (see (29) for the definition of the normalized characteristic function . SA (z)).

On the Invariance Principle for a Characteristic Function

639

 we If, instead, .ω = f −1 (∞) is a regular point of the dissipative operator .A,  relate the characteristic function .Sf (A) (z) of the bounded dissipative operator .f (A)  to the characteristic function .SA (z) of the triple as Sf (A)  ◦ f = f SA (z),

.

with .f = SA (ω + i0) a unimodular constant factor. As an application of the extended invariance principle we obtain identities relating the Friedrichs and Krein-von Neumann extensions of model homogeneous non-negative symmetric operators and their inverses, see Theorem 7.5.

2 Preliminaries and Basic Definitions Let .A˙ be a densely defined symmetric operator with deficiency indices .(1, 1) and .A its self-adjoint (reference) extension. Following [8, 11, 14, 18] recall the concept of the Weyl-Titchmarsh and Livšic ˙ A). functions associated with the pair .(A, Suppose that (normalized) deficiency elements .g± , g± ∈ Ker(A˙ ∗ ∓ iI ),

.

g±  = 1,

(3)

are chosen in such a way that g+ − g− ∈ Dom(A).

(4)

.

Consider the Weyl-Titchmarsh function1 M(z) = ((Az + I )(A − zI )−1 g+ , g+ ),

.

z ∈ C+ ,

(5)

˙ A) and also the Livšic function associated with the pair .(A, z − i (gz , g− ) · , z + i (gz , g+ )

z ∈ C+ ,

(6)

0 = gz ∈ Ker(A˙ ∗ − zI ),

z ∈ C+ .

(7)

s(z) =

.

.

1 Paying

tribute to historical justice it is worth mentioning that the function .M(z) has been introduced by Donoghue in [8]. However, as one can see from [9, eq. (5.42)], it is elementary to express .M(z) in terms of the classical Weyl-Titchmarsh function which explains the terminology we use.

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K. A. Makarov and E. Tsekanovskii

Recall the important relationship that links the Weyl-Titchmarsh and Livšic functions [18], s(z) =

.

M(z) − i , M(z) + i

z ∈ C+ .

(8)

˙  = (A)  ∗ is a maximal dissipative extension of .A, If .A  f ) ≥ 0, Im(Af,

.

 f ∈ Dom(A),

 is automatically quasi-self-adjoint [2, 18, 21, 24] and therefore then .A  for some || < 1. g+ − g− ∈ Dom(A)

.

(9)

˙ A,  A). By definition, we call . the von Neumann parameter of the triple .A = (A, Given (4) and (9), define the characteristic function .SA (z) associated with the ˙ A,  A) as follows (see [18], cf. [15]) triple .A = (A, s(z) −  , SA (z) =  s(z) − 1

.

z ∈ C+ ,

(10)

˙ A). where .s(z) = s(A,A) (z) is the Livšic function associated with the pair .(A, ˙ We remark that the von Neumann parameter . can explicitly be evaluated in ˙ A,  A) as terms of the characteristic function of the triple .(A,  = S(A, ˙ A,A)  (i).

(11)

.

˙ A) Moreover, as it follows from (10), the Livšic function associated with the pair .(A, admits the representation s(z) = s(A,A) (z) = ˙

.

SA (z) −  ,  SA (z) − 1

z ∈ C+ .

(12)

Summing up, the calculation of the main characteristics (unitary invariants) of the ˙ A,  A) can be performed in accordance with the following algorithm: triple .A = (A, on the first step, deficiency elements .g± that satisfy the relation (3) are to be found, then one calculates the Weyl-Titchmarsh (5) or/and Livšic (6) functions depending of whether the resolvent of the reference self-adjoint operator A or the “deficiency" field .C+ z → gz (7) is available. On the next step, the von Neumann parameter . of the triple (9) is to be determined and finally, one arrives at the characteristic function .SA (z) of the triple given by (10). One can take a different point of view as presented in [18], and in this way we come to a functional model of a prime dissipative triple in which the characteristic functions is considered as a parameter of the model.

On the Invariance Principle for a Characteristic Function

641

3 A Functional Model of a Triple Given a contractive analytic map S, S(z) =

.

s(z) −  ,  s(z) − 1

z ∈ C+ ,

(13)

where .|| < 1 and .s(z) is an analytic, contractive function in .C+ satisfying the Livšic criterion [14] (also see [18, Theorem 1.2]), that is, s(i) = 0

and

.

lim z(s(z) − e2iα ) = ∞ for all

z→∞

α ∈ [0, π ),

(14)

0 < ε ≤ arg(z) ≤ π − ε,

.

introduce the function 1 s(z) + 1 · , i s(z) − 1

M(z) =

.

z ∈ C+ .

(15)

In this case, the function .M(z) admits the representation, ˆ  M(z) =

.

R

1 λ − λ − z 1 + λ2

 dμ(λ),

z ∈ C+ ,

(16)

for some infinite Borel measure .μ(dλ), .

μ(R) = ∞,

(17)

dμ(λ) = 1. 1 R + λ2

(18)

such that ˆ .

In the Hilbert space .L2 (R; dμ) introduce the (self-adjoint) operator .B of multiplication by independent variable on ˆ    2 2 2  λ |f (λ)| dμ(λ) < ∞ .Dom(B) = f ∈ L (R; dμ)  R

(19)

˙ its symmetric restriction on and denote by .B ˆ     ˙ f (λ)dμ(λ) = 0 . .Dom(B) = f ∈ Dom(B)  R

(20)

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K. A. Makarov and E. Tsekanovskii

B as the dissipative quasi-self-adjoint extension of the symmetric Next, introduce . ˙ on the domain given by operator .B ˙ +lin ˙ .Dom( B) = Dom(B) span



 1 1 , − λ−i λ+i

(21)

˙  where the von Neumann parameter . of the triple .(B, B, B) is given by  = S(i).

.

Notice that in this case, ˙ +lin ˙ .Dom(B) = Dom(B) span



 1 1 − . λ−i λ+i

(22)

˙  We will refer to the triple .(B, B, B) as the model triple in the Hilbert space L2 (R; dμ) parameterized by the characteristic functions .SB (z) = S(z).

.

˙ ˙ of the symmetric operator .B Remark 3.1 Notice that the core of the spectrum . σ (B) can be characterized as ˆ    dμ(λ) ˙ = s ∈ R = ∞ and μ({s}) = 0 . (23) . σ (B)  R (λ − s)2 ˙ B) from the model Here .μ(dλ) is the measure (16) associated with the pair .(B, representation of the triple .B. ˙ is ˙ of the spectrum of a prime2 symmetric operator .B Recall that the core . σ (B) just the complement of the set of its quasi-regular points, ˙ =R\ρ ˙  σ (B) (B).

.

For the convenience of the reader we provide a short proof of (23). Proof As in [23], introduce the set ˆ   dμ(λ) .P = s ∈ R  < ∞ or  R (λ − s)2

 μ({s}) > 0 ,

which apparently is the complement of the right hand side of (23). ˙ of the We claim that .P coincides with the set of quasi-regular points .ρ (B) ˙ symmetric (prime) operator .B.

symmetric operator .A˙ is called a prime operator if there is no (non-trivial) subspace invariant under .A˙ such that the restriction of .A˙ to this subspace is self-adjoint.

2A

On the Invariance Principle for a Characteristic Function

643

Indeed, it is well known (see, e.g., [10, 23]) that the set .P does not depend on ˙ the choice of the self-adjoint (reference) extension .B of the symmetric operator .B. Therefore, given .s ∈ P, without loss we may assume that .Ker(B − sI ) = {0}. ˙ which proves the inclusion (B), Hence, .s ∈ ρ(B) ⊆ ρ ˙ P⊆ρ (B).

.

˙ one can always choose a self-adjoint extension On the other hand, if .t ∈ ρ (B),    ˙ .B of .B such that t is an eigenvalue of .B . If .dμ (dλ) is the representing measure ˙ B ) in the associated with the corresponding model representation for the pair .(B, 2  space .L (R, dμ ), then μ ({t}) > 0

.

and therefore, .t ∈ P (here we have again used the independence of the set .P from ˙ ⊆ P and hence the choice of the (reference) extension .B ). Thus, .ρ (B) ˙ P=ρ (B),

.

as stated and (23) follows.

(24)  

a Let .A˙ be a densely defined symmetric operator with deficiency indices .(1, 1), .A maximal non-selfadjoint dissipative extension of .A˙ and .A its self-adjoint (reference) ˙ A, A),  recall that if the symmetric operator .A˙ extension. Given the triple .A = (A, is prime, then both the Weyl-Titchmarsh function .M(z) and the Livšic function .s(z) ˙ A), while the characteristic function are complete unitary invariants of the pair .(A, .S (z) is a complete unitary invariant of the triple .A (see [18]). In particular, the von A ˙ A,  A), not a complete Neumann parameter . is a unitary invariant of the triple .(A, unitary invariant though. Also notice that if the symmetric operator .A˙ from a triple in the Hilbert space .H is not prime and .A˙  is the prime part of .A˙ in a reducing ˙ A, ˙  , A|  A) and .(A|   , A|  ) have the subspace .H ⊂ H, then the triples .(A, H H H same characteristic function, which in many cases allows one to focus on the case where .A˙ is a prime operator. Theorem 3.2 ([18, Theorems 1.4, 4.1]) Suppose that .A˙ and .B˙ are prime, closed, densely defined symmetric operators with deficiency indices .(1, 1). Assume, in addition, that A and B are some self and .B  are maximal dissipative extensions adjoint extensions of .A˙ and .B˙ and that .A ∗ ˙ ˙     ∗ ). of .A and .B, respectively .(A = (A) and .B = (B) Then, ˙ A, ˙ B,  A) and .(B,  B) are mutually unitarily equivalent3 if, and (i) the triples .(A, only if, the corresponding characteristic functions of the triples coincide; ˙ A, ˙ B,  A) and .(B,  B) in Hilbert spaces .HA and .HB are say that triples of operators .(A, mutually unitarily equivalent if there is a unitary map .U from .HA onto .HB such that .B˙ = ˙ −1 , .B  = UAU  −1 , and .B = UAU−1 . UAU

3 We

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K. A. Makarov and E. Tsekanovskii

˙ A,  A) is mutually unitarily equivalent to the model triple (ii) the triple .(A, ˙  .(B, B, B) in the Hilbert space .L2 (R; dμ), where .μ(dλ) is the representing (z) associated with measure for the Weyl-Titchmarsh function .M(z) = M(A,A) ˙ ˙ the pair .(A, A). In particular, ˙ A) and .(B, ˙ B) are mutually unitary equivalent if and only if (iii) the pairs .(A, .M(A,A) (z) = M(B,B) (z). ˙ ˙ For the further reference recall the following resolvent formula [18]. ˙  Theorem 3.3 Suppose that .B = (B, B, B) is the model triple in the Hilbert space 2 .L (R; dμ) given by (19)–(21). B in .L2 (R; dμ) has the form Then the resolvent of the model dissipative operator . ( B − zI )−1 = (B − zI )−1 − p(z)(· , gz )gz ,

(25)

   + 1 −1 p(z) = M(B˙ ,B) (z) + i ,  −1

(26)

.

with .

z ∈ ρ( B) ∩ ρ(B).

.

˙ B) Here .M(B˙ ,B) (z) is the Weyl-Titchmarsh function associated with the pair .(B, continued to the lower half-plane by the Schwarz reflection principle, . is the von Neumann parameter of the triple .B, and the deficiency elements .gz , ˙ ∗ − zI ), gz ∈ Ker((B)

.

z ∈ C \ R,

are given by gz (λ) =

.

1 λ−z

for μ-almost all λ ∈ R.

(27)

Remark 3.4 Notice that if .z = 0 is a quasi-regular point for the symmetric operator ˙ .0 ∈ ρ ˙ and therefore B, (B),

.

0 ∈ ρ(B) ∩ ρ( B),

.

−1 B is a rank-one perturbation of the bounded self-adjoint operator then the inverse . −1 .B and the following resolvent formula −1  B = B−1 − pQ

.

On the Invariance Principle for a Characteristic Function

645

holds. Here    + 1 −1 , p = M(0) + i  −1

.

(28)

Q is a rank-one self-adjoint operator given by ˆ f (s) 1 dμ(s), .(Qf )(λ) = λ R s

μ -a.e. λ ∈ R,

and .M(0) = M(0 + i0) is the boundary value of the Weyl-Titchmarsh function ˙ B) at the point zero. associated with the pair .(B,

4 The Invariance Principle Let .A˙ be a densely defined symmetric operator with deficiency indices .(1, 1), .A  a maximal non-self-adjoint dissipative its self-adjoint (reference) extension and .A ˙  is automatically a quasi-selfextension of .A. In this case, the dissipative operator .A ˙ adjoint extension of .A (see, e.g., [18]), ˙ ∗  ⊂ (A) A˙ ⊂ A

.

and  Dom(A)∩Dom( A˙ = A|  ∗ ) . A

.

˙ A,  A) will be called regular. Throughout this Note such kind of triples .A = (A, First, recall the concept of an invariance principle for the characteristic function ˙ A, ˙  A) with respect to affine transformations of the operators .A, of a triple .A = (A,  and A from .A. .A Given an affine transformation of the upper half-plane, .f (z) = az + b, .a, b ∈ R, .a > 0, introduce the triple ˙ f (A),  f (A)), f (A) = (f (A),

.

where f (X) = aX + bI

.

on

Dom(X),

˙ A,  A. X = A,

˙ is a symmetric operator with deficiency indices .(1, 1), .f (A) is Clearly, .f (A)  is a quasi-self-adjoint dissipative extension of its self-adjoint extension, and .f (A) ˙ Therefore, the triple .f (A)  is regular and hence the characteristic function .f (A). .S (z) of the triple .f (A) is well defined. A

646

K. A. Makarov and E. Tsekanovskii

Along with the characteristic function .SA (z) introduce the normalized characteristic function . SA (z) of the triple .A as 1 − SA (i)  SA (z) = · S (z), 1 − SA (i) A

.

z ∈ C+ .

(29)

˙ A,  A) be a regular triple. Theorem 4.1 ([19, 20, Theorem F.1]) Let .A = (A, Suppose that .f (z) = az + b with .a, b ∈ R, .a > 0, is an affine transformation. Then for the normalized characteristic functions of the triples .A and .f (A) the following invariance principle  SA Sf (A) ◦ f = 

.

(30)

holds. If .A is a regular triple and f is a linear-fractional automorphism of the upper half-plane .C+ , .f ∈ SL2 (R), one can also introduce the triple ˙ f (A),  f (A)), f (A) = (f (A),

.

where the function .f (X) of a linear operator X is understood as f (X) = (aX + cI )(cX + dI )−1

.

on

Ran ((cX + dI )) ,

˙ A,  A, X = A,

provided that .Ker(cX + dI ) = {0}. However, if the preimage .f −1 (∞) is a quasi˙ is not densely regular point for the symmetric operator A, then the operator .f (A) defined and therefore the triple .f (A) is not regular. In fact, if .A˙ is a prime symmetric operator and .f ∈ SL2 (R) is a linear-fractional automorphism, then we have the following alternative: either the triple ˙ f (A),  f (A)) f (A) = (f (A),

.

 is a bounded dissipative operator. is regular or .f (A) The following main result of this Note takes care of the two possible outcomes in the alternative mentioned above. ˙ A,  A) is a regular triple. Assume, in addition, Theorem 4.2 Suppose that .A = (A, ˙ that .A is a prime symmetric operator. Suppose that .f ∈ SL2 (R) is a linearfractional automorphism of the upper half-plane. Then,  (i) if .ω = f −1 (∞) belongs to the spectrum of the dissipative operator .A, ˙ then equivalently, to the core of the spectrum of the symmetric operator .A, the triple ˙ f (A),  f (A)) f (A) = (f (A),

.

is regular.

On the Invariance Principle for a Characteristic Function

647

In this case, for the normalized characteristic functions . SA (z) and . Sf (A) (z) of the triples .A and .f (A) the following invariance principle  SA Sf (A) ◦ f = 

.

(31)

holds;  equivalently, (ii) if .ω = f −1 (∞) is a regular point of the dissipative operator .A, ˙ then the operator .f (A)  is a quasi-regular point of the symmetric operator .A, well defined as a bounded dissipative operator. In this case, for the normalized characteristic function . SA (z) of the triple .A and the characteristic function .Sf (A) (z) of the bounded dissipative operator   .f (A) the following invariance principle Sf (A)  ◦f =

.

1 ·S SA (ω + i0) A

(32)

holds. Remark 4.3 Notice that the technical requirement that .A˙ is a prime operator can easily be relaxed with obvious modifications in the formulation. We defer the proof of Theorem 4.2 to Sect. 6.

5 Invariance Principe for Model Triples Taking into account that each automorphism of the upper half-plane is either a linear transformation of .C+ or a composition of linear transformations of .C+ and the automorphism 1 f (z) = − , z

.

z ∈ C+ ,

(33)

we will concentrate first on the case where the relevant mapping is given by (33). Under this assumption we will establish the corresponding invariance principle for ˙  the model triple of operators .B = (B, B, B) given by (19)–(21). ˙ is automatiNotice, that in the situation in question, the symmetric operator .B cally a prime operator and hence we have the following alternative: either the point ˙ or zero is a quaisi-regular point for .z = 0 belongs to the core of the spectrum of .B that operator (see [1]). We start with the case where the point .z = 0 belongs to the core of the spectrum ˙ of the model symmetric operator .B.

648

K. A. Makarov and E. Tsekanovskii

˙  Theorem 5.1 Suppose that .B = (B, B, B) is the model triple in .L2 (R; dμ) given ˙ of the σ (B) by (19)–(21). Assume that 0 belongs to the core of the spectrum . ˙ symmetric operator .B. Then, (i) (ii) (iii)

˙ −1 is a symmetric operator with deficiency indices .(1, 1); (B) −1 ˙ −1 ; .B is a self-adjoint extension of .(B) −1 ˙ −1 ;  .(B) is a quasi-self-adjoint extension of .(B) In particular, the triple .

˙ −1 , −( C = (−(B) B)−1 , −B−1 )

(34)

.

is well defined and regular. Moreover, (iv) the von Neumann parameters of the triples .B and .C coincide; (v) the Livšic functions .s(−(B˙ )−1 ,−B−1 ) (z) and .s(B˙ ,B) (z) associated with the pairs ˙ −1 , −B−1 ) and .(B, ˙ B) are related as .(−(B)   1 = s(B˙ ,B) (z), s(−(B˙ )−1 ,−B−1 ) − z

z ∈ C+ ;

.

(35)

(vi) the Weyl-Titchmarsh functions .M(−(B˙ )−1 ,−B−1 ) (z) and .M(B˙ ,B) (z) associated ˙ −1 , −B−1 ) and .(B, ˙ B) are related as with the pairs .(−(B)  M(−(B˙ )−1 ,−B−1 )

.

1 − z

 = M(B˙ ,B) (z),

z ∈ C+ ;

(36)

(vii) the characteristic functions .SC (z) and .SB (z) of the triples ˙ −1 , −( C = (−(B) B)−1 , −B−1 )

.

and

˙  B = (B, B, B)

satisfy the invariance identity 

1 .S C −z

 = SB (z),

z ∈ C+ .

(37)

˙ is a prime symmetric operator and the point .z = 0 is not a quasiProof Since .B ˙ that is .0 ∈  ˙ the subspace .Ker((B) ˙ ∗ ) is trivial. For the σ (B), regular point of .B, convenience of the reader, we present the corresponding argument. ˙ ∗ ) = {0}. Then the restriction .B of Indeed, suppose on the contrary that .Ker((B) ∗ ˙ on .(B) ˙ +Ker(( ˙ ∗) ˙ Dom(B ) = Dom(B) B)

.

On the Invariance Principle for a Characteristic Function

649

is a self-adjoint operator. Let .μ (dλ) be the measure from the representation for the ˙ B ), Weyl-Titshmarsh function .M(B˙ ,B ) (z) associated with the pair .(B, ˆ  M(B˙ ,B ) (z) =

.

R

1 λ − λ − z 1 + λ2



dμ (λ),

z ∈ C+ .

˙ ∗ ) = {0}, we see that zero is an Taking into account that .Ker(B ) = Ker((B)  eigenvalue of .B and hence μ ({0}) = 0.

.

However, since the right hand side of (23) after replacing .μ(dλ) with .μ (dλ) ˙ implies remains invariant (see, e.g., [10], cf. Remark 3.1), the membership .0 ∈  σ (B) μ({0}) = μ ({0}) = 0.

.

The obtained contradiction shows that ˙ ∗ ) = {0}. Ker((B)

(38)

.

˙ is dense in .L2 (R; dμ). In particular, .B˙ = (B) ˙ −1 From (38) it follows that .Ran(B) is well defined as a densely-defined unbounded symmetric operator. Also, ˙ ∗ ) = {0} Ker(B) ⊆ Ker((B)

.

and therefore the inverse .B = B−1 of the self-adjoint (multiplication) operator .B is well defined as a self-adjoint operator. Clearly, .B˙ coincides with the restriction of the unbounded self-adjoint operator .B = B−1 on ˆ    f (λ) ˙ = f ∈ Dom(B)  dμ(λ) = 0 . Dom(B)  R λ

.

(39)

˙ −1 ) ⊂ Dom((B)−1 ) ˙ = Dom((B) Using (39), we see that for any .f ∈ Dom(B) we have ˙ − zI )f, hz ) = ((B−1 − zI )f, hz ) = .(B ˆ

ˆ  R

 1 1 dμ(λ) − z f (λ) 1 − λz λ

f (λ) dμ(λ) = 0. = R λ Therefore, the functions hz (λ) =

.

1 , 1 − λz

Im(z) = 0,

(40)

650

K. A. Makarov and E. Tsekanovskii

˙ That is, are deficiency elements of .B. ˙ ∗ − zI ), hz ∈ Ker((B)

.

Im(z) = 0.

(41)

In particular, the symmetric operator .B˙ has deficiency indices .(m, n) with .m, n ≥ 1. To show that .m = n = 1, observe that for any .f ∈ Dom(B) = Dom(B−1 ) the function ˆ f (s) λ (λ) = f (λ) − .f (42) dμ(s) · 2 λ +1 R s has the property that ˆ ˆ ˆ f (λ) f (s) dμ(λ) f(λ) dμ(λ) = dμ(λ) − dμ(s) = 0, R λ R s R λ2 + 1 R λ

ˆ .

so that ˙ f ∈ Dom(B).

.

(43)

Here we have used the normalization condition (18). From (40) and (41) it follows that the function r(λ) =

.

λ2

λ +1

˙ ∗ − iI )+Ker(( ˙ ∗ + iI ). In particular, (42) along ˙ belongs to the subspace .Ker((B) B) ˙ is one-dimensional and with (43) show that the quotient space .Dom(B)/Dom(B) hence m = n = 1.

.

Thus, .B˙ is a symmetric restriction with deficiency indices .(1, 1) of the self-adjoint operator B and the proof of (i) and (ii) is complete.  = ( To check (iii), first observe that .B B)−1 is well-defined, since ˙ ∗ ) = {0}. Ker( B) ⊆ Ker((B)

.

 = ( Next we claim that the operator .B B)−1 is the quasi-self-adjoint extension of .B˙ on   1 1 ˙ + span  = Dom(B) + , .Dom(B) (44) λ−i λ+i ˙  where . is the von Neumann parameter of the model triple .B = (B, B, B).

On the Invariance Principle for a Characteristic Function

651

Indeed, since (see (21)) ˙ + span Dom( B) = Dom(B)



.

1 1 − λ−i λ+i

 (45)

˙ −1 , it suffices to check the following two (mapping) properties for the and .B˙ = (B)   operators .B and .B,  B

.





1 1 − λ−i λ+i

˙ ∗ = (B)



1 1 − λ−i λ+i

 =

i i + λ−i λ+i

and  B

.



i i + λ−i λ+i



˙ ∗ = (B) =



i i + λ−i λ+i



 =i

(−i) i + λ−i λ+i



1 1 − . λ−i λ+i

Here we used that by (27), ˙ ∗ ∓ iI ) = span {h± } , Ker((B)

.

(46)

and also that (see (40) and (41)) ˙ ∗ ∓ iI ) = span {h∓ } , Ker((B)

.

(47)

with h± (λ) =

.

1 . λ∓i

The proof of (iii) is complete. Finally, to calculate the von Neumann parameter of the triple .C given by (34) we proceed as follows. Observing that h+ (λ) + h− (λ) =

.

1 λ 1 + =2 2 , λ−i λ+i λ +1

we see that h+ + h− ∈ Dom(B) = Dom(−B).

.

(48)

Moreover, from (44) it follows that  = Dom(−B).  h+ + h− ∈ Dom(B)

.

(49)

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K. A. Makarov and E. Tsekanovskii

Since ˙ ∗ ∓ iI ) = Ker((B) ˙ ∗ ± iI ) = span{h± }, Ker(−(B)

.

and by (47) ˙ ∗ ± iI ) = span{h± }, Ker((B)

.

the memberships (48) and (49) ensure that . is the von Neumann parameter of the triple ˙ −1 , −( ˙ B,  B) = (−(B) C = (B, B)−1 , −B−1 ),

.

which completes the proof of (iv). ˙ B) and Next, we will evaluate the Livšic functions associated with the pairs .(B, ˙ .(−B, −B). Recall that by (27), ˙ ∗ − zI ) = span {gz } , Ker((B)

.

Im(z) = 0,

where gz (λ) =

.

1 , λ−z

Im(z) = 0.

Since h+ − h− ∈ Dom(B),

.

˙ ∗ ∓ iI ), h± ∈ Ker((B)

˙ B) we obtain the for the Livšic function .s(B˙ ,B) (z) associated with the pair .(B, representation ´ z−i R z − i (gz , h+ ) · = ·´ .s ˙ (z) = (B,B) z + i (gz , h− ) z+i R

dμ(λ) (λ−z)(λ−i) dμ(λ) (λ−z)(λ+i)

From (40) and (41) it follows that  ˙ ∗ − zI ) = span hˆ z , Ker(−(B)

.

Im(z) = 0,

where hˆ z (λ) =

.

1 , 1 + λz

Im(z) = 0.

.

(50)

On the Invariance Principle for a Characteristic Function

653

Observing that h+ + h− = h+ − (−1)h− ∈ Dom(B),

.

˙ ∗ ∓ iI ), h± ∈ Ker(−(B)

in accordance with the definition of the Livšic function .s(−B,−B) (z) associated with ˙ ˙ −B) we obtain the pair .(−B, ´ dμ(λ) (hˆ z , h+ ) z−i z − i R (1+λz)(λ−i) · ·´ .s(−B,−B) (z) = , z ∈ C+ . =− ˙ dμ(λ) z + i (hˆ z , (−1)h− ) z+i R (1+λz)(λ+i) (51) A simple computation using (50) and (51) shows that   1 − = s(B˙ ,B) (z), .s(−B,−B) ˙ z

z ∈ C+ ,

thus proving (35). The proof of (v) is complete. The assertion (vi) is a direct consequence of (v) and the relationship (15) linking the Livšic and Weyl-Titchmarsh functions. The last assertion (vii) is a consequence of (iv) and (v) and the definition of the characteristic function of a triple.   Remark 5.2 Since the point .z0 = i is a fixed point of the automorphism 1 f (z) = − , z

.

z ∈ C+ ,

using (37) we see that   1 .S (i) = S C C − i = SB (i). Therefore, for the normalized characteristic functions  SC (z) = Sf (B) (z) = 

.

1 − SC (i) · S (z), 1 − SC (i) C

z ∈ C+ ,

and 1 − SB (i)  · S (z), SB (z) = 1 − SB (i) B

.

z ∈ C+ ,

associated with the triples 

˙ −1 , −( f (B) = C = −(B) B)−1 , −B−1

.

and

˙  B = (B, B, B),

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K. A. Makarov and E. Tsekanovskii

respectively, we also have the invariance equality  SB . Sf (B) ◦ f = 

.

Notice that if the point .z = 0 is a quasi-regular point of the symmetric operator ˙ then the operator .−(B) ˙ −1 is well defined as a bounded dissipative operator, while B, ∗ ˙ ˙ −1 ˙ −1 is not densely defined (although .(B) .Ker(B) is non-trivial and therefore .(B) ˙ is continuous on .Ran(B)). Therefore, the triple 

˙ −1 , −( .f (B) = −(B) B)−1 , −B−1 .

is not regular in this case. However, we have the following result (cf. [2, Theorem 8.4.4], [25] in system theory that relates the transfer functions of L-systems under the transformation .z → 1z of the spectral parameter). Lemma 5.3 (cf. [2, Theorem 8.4.4], [25]) ˙  Suppose that .B = (B, B, B) is the model triple in .L2 (R; dμ) given by (19)–(21). ˙ Assume that the point .z = 0 is a quasi-regular point of the symmetric operator .B. B has a bounded inverse and the characteristic function In this case, the operator . .S B)−1 and the characteristic function  )−1 (z) of the dissipative operator .−( −(B .S (z) of the triple .B are related as B   SB (z) 1 − = , z ∈ C+ . (52) .S  −(B)−1 z SB (0 + i0) Proof Assume temporarily that .Ker(B) = {0}. Then under the hypothesis that 0 is ˙ we have a quasi-regular point of the symmetric operator .B, 0 ∈ ρ(B) ∩ ρ( B).

.

−1 B is a rank-one perturbation of the self-adjoint By Remark 3.4, the inverse . −1 operator .B , −1  B = B−1 − p Q.

(53)

.

Here    + 1 −1 p = M(0) + i ,  −1

.

(54)

M(0) = M(0+i0) is the value of the Weyl-Titchmarsh function .M(z) = M(B˙ ,B) (z) ˙ B) at the point zero, . is the von Neumann parameter associated with the pair .(B, ˙  of the triple .B = (B, B, B), and Q is a rank-one self-adjoint operator given by

.

ˆ h(λ) 1 .(Qh)(λ) = dμ(λ), λ R λ

μ -a.e. λ.

On the Invariance Principle for a Characteristic Function

655

In accordance with the definition [15] of the characteristic function .S−(B  )−1 (z) −1

of the bounded dissipative operator .− B

we have

−1 −1 ∗  .S (z) = 1 + 2i Im(p) tr −( B ) − zI Q  )−1 −(B



−1 −1 = 1 + 2i Im(p) tr −B + pQ − zI Q ,

where we have used (53) in the last step. Since .pQ is a rank-one perturbation of the self-adjoint operator .−B−1 , from the first resolvent identity it follows that (see, e.g., [22])

−1 −1 Q = .tr −B + pQ − zI

 −1  −B−1 − zI Q  −1  , Q 1 + p tr −B−1 − zI tr

and hence  −1  ´ −B−1 − zI Q 1−p R  .S ´  )−1 (z) = −1  = −(B 1−p R 1 + p tr −B−1 − zI Q 1 + p tr

dμ(λ) λ(1+λz) dμ(λ) λ(1+λz)

,

z ∈ C+ . (55)

On the other hand, using (8) and (10), for the characteristic function .SB (z) of the triple .B one obtains 1+ 1 −  M (z) − i 1− · , SB (z) = − 1 −  M (z) + i 1+ 1−

.

z ∈ C+ ,

and therefore

   M − 1z − i 1+ 1− 1− 1 .S , B −z = −1 −  · 1 M − z + i 1+ 1−

z ∈ C+ .

From (54) it also follows that p−1 − M(0) = −i

.

 +1 1−

and hence

   M − 1z − M(0) + p−1 1− 1 .S , B −z = −1 −  · 1 M − z − M(0) + p−1

z ∈ C+ .

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K. A. Makarov and E. Tsekanovskii

Here we used that .M(0) is real for .0 ∈ ρ(B). Since

 ´ ´ dμ(λ) M − 1z − M(0) + p−1 − R λ(1+λz) + p−1 p 1−p R

 . = ´ dμ(λ) = · ´ p 1−p − R λ(1+λz) + p−1 M − 1z − M(0) + p−1 R

dμ(λ) λ(1+λz) dμ(λ) λ(1+λz)

,

one concludes that ´   1− p 1−p R 1 .S ´ B −z = −1 −  · p · 1−p R

dμ(λ) λ(1+λz) dμ(λ) λ(1+λz)

z ∈ C+ .

,

Thus, taking into account (55), one obtains   1− p 1 .S  −1 (z), B − z = − 1 −  · p · S−(B)

z ∈ C+ .

(56)

Using (54) we get

.

−

+1 1− p M(0) − i − (M(0) + i) 1 −  M(0) + i −1 = · =− · 1− p 1 −  M(0) + i +1 M(0) + i − (M(0) − i) −1

=

s(B˙ ,B) (0) − 

s(B˙ ,B) (0 + i0) − 1

(57)

= SB (0 + i0).

Combining (56) and (57) shows that    −1 1 = SB (0 + i0) SB (z), S−(B  )−1 − z

.

z ∈ C+ ,

which proves the claim provided that .Ker(B) = {0}. To relax the requirement that .Ker(B) = {0}, suppose that .B is a self-adjoint ˙ such that .Ker(B ) = {0}. Such an extension is alway available since extension of .B ˙ However, .0 ∈ ρ (B). .

SB (z) SB (z) = , SB (0 + i0) SB (0 + i0)

z ∈ C+ ,

˙  where .B = (B, B, B ) and hence (52) holds regardless of whether .Ker(B) = {0} or not.  

On the Invariance Principle for a Characteristic Function

657

6 Proof of Theorem 4.2 Now we are ready to establish the invariance principle also for the characteristic function of a triple for general linear-fractional automorphisms of the upper halfplane. Proof Any linear-fractional automorphism C+ z → f (z) =

.

az + b , cz + d

a, b, c, d ∈ R,

ad − bc > 0, .c > 0 can be represented as the composition

.

f = h ◦ ι ◦ g,

.

where h is a linear automorphism of .C+ , ι(z) = −

.

1 z

and

g(z) = z − f −1 (∞),

z ∈ C+ .

˙ then the point 0 belongs If .ω = f −1 (∞) belongs to the core of the spectrum of .A, ˙ = A˙ − ωI . Therefore, to the core of the spectrum of the symmetric operator .g(A) in the model representation of the triple .g(A) the hypotheses of Theorem 5.1 are satisfied. In particular, the triple .ι ◦ g(A) is regular, so is .f (A), since .f = h ◦ ι ◦ g and h is a linear isomorphism. By Theorem 4.1,  Sι◦g(A) ◦ (ι ◦ g). Sf (A) ◦ f = 

.

Applying Theorem 5.1, we get  Sg(A) ◦ g =  SA , Sι◦g(A) ◦ (ι ◦ g) = 

.

where we have used Theorem 4.1 one more time in the last step, thus proving (31). ˙ both .f (A)  and .(ι ◦ g)(A)  are If .ω = f −1 (∞) is a quasi-regular point of .A, bounded dissipative operators. By the invariance principle in the bounded case (see (2)), we have Sf (A)  ◦ f = Sι◦g(A)  ◦ (ι ◦ g).

.

Applying Lemma 5.3 we get Sι◦g(A)  ◦ (ι ◦ g) =

.

1 ◦ g. · S  Sg(A) (0 + i0) g(A)

658

K. A. Makarov and E. Tsekanovskii

By Theorem 4.1, .

1 1 1 · Sg(A) ◦ g = · SA = ·S .   SA (ω + i0) A Sg(A) (0 + i0) SA (ω + i0)

Therefore, Sf (A)  ◦f =

.

1 ·S , SA (ω + i0) A  

completing the proof (32).

7 Applications to the Krein-von Neumann Extensions Theory Recall that if .A˙ is a densely defined (closed) nonnegative operator, then the set of ˙ K , the all nonnegative self-adjoint extensions of .A˙ has the minimal element .[A] Krein-von Neumann extension (different authors refer to the minimal extension by ˙ F , the Friedrichs using different names, see, e.g., [3–6]), and the maximal one .[A] extension. This means, in particular, that for any nonnegative self-adjoint extension A of .A˙ the following operator inequality holds [13]: ˙ F + λI )−1 ≤ (A + λI )−1 ≤ ([A] ˙ K + λI )−1 , ([A]

for all

.

λ > 0.

Hypothesis 7.1 Suppose that .ν ∈ (−1, 1). Assume that .A(ν) is the self-adjoint multiplication operator by independent variable in the Hilbert space H+ = L2 ((0, ∞); λν dλ)

.

˙ and .A(ν) is its restriction on  ˆ   ˙ .Dom(A(ν)) = f ∈ Dom(A(ν))

∞

 f (λ)λ dλ = 0 . ν

0

Analogously, suppose that .B(ν) is the self-adjoint multiplication operator by independent variable in the Hilbert space H− = L2 ((−∞, 0); |λ|ν dλ)

.

˙ and .B(ν) is its restriction on  ˆ   ˙ .Dom(B(ν)) = f ∈ Dom(B(ν))

0

−∞

 f (λ)|λ| dλ = 0 . ν

On the Invariance Principle for a Characteristic Function

659

Lemma 7.2 Assume Hypothesis 7.1. Then the Weyl-Titchmarsh function .Mν (z) ˙ associated with the pair .(A(ν), A(ν)) admits the representation ⎧ ⎨ i − cot π ν   z ν + cot π ν, 2

2 i .Mν (z) = ⎩ π2 log − 1z ,

ν = 0 ν=0

,

z ∈ C+ .

(58)

Analogously, the Weyl-Titchmarsh function .Nν (z) associated with the pair ˙ (B(ν), B(ν)) admits the representation

.

Nν (z) =

.

   ν i + cot π2 ν zi − cot π2 ν, 2 π

ν = 0 ν=0

log z,

,

z ∈ C+ .

(59)

Here .log z denotes the principal branch of the logarithmic function with the cut on the negative semi-axis and

z ν .

i



π  , = exp ν log z − i 2

z ∈ C+ .

Proof Suppose that .ν = 0. We have Mν (z) =

.

1 g+ 2

 (zA(ν) + I )(A(ν) − zI )−1 g+ , g+ ,

z ∈ C+ ,

∗ − iI ). One can choose (see, ˙ where .g+ is a deficiency element from .Ker((A(ν)) e.g., [18])

g+ (z) =

.

1 , λ−z

z ∈ C+ ,

λ ∈ (0, ∞).

We have .

 ˆ (zA(ν) + I )(A(ν) − zI )−1 g+ , g+ =

∞

0

(zλ + 1)λν dλ (λ − z)(1 + λ2 )

and ˆ g+ 2 =

.

0

∞

λν dλ. 1 + λ2

Let . ε , .ε > 0, denote the anti-clockwise oriented contour in the complex plane

ε = {z ∈ C | dist(z, [0, ∞)) = ε}  ∪ z ∈ C | |z| = ε−1 , arg(z) ∈ [arcsin(ε2 ), 2π − arcsin(ε2 )] ,

.

660

K. A. Makarov and E. Tsekanovskii

which consists of “an infinitely distant circle and an indentation round the cut along the positive real axis” [16, § 129, Fig. 46] (as .ε → 0). Given .z ∈ C+ and .ε < min(|z|, 1), by the Residue theorem we get ˛ .

ε

(zλ + 1)λν dλ = 2π i (λ − z)(1 + λ2 )



Res(F, ζ ),

ζ ∈{z,i,−i}

where F (λ) =

.

(zλ + 1)λν . (λ − z)(1 + λ2 )

Going to the limit as .ε → 0 and taking into account that (λ − i0)ν = ei2π ν λν ,

λ > 0,

.

we arrive at the representation ˆ (1 − e

.

i2π ν

∞

) 0

(zλ + 1)λν dλ = lim ε↓0 (λ − z)(1 + λ2 )

˛

ε

= 2π i

(zλ + 1)λν dλ (λ − z)(1 + λ2 ) 

Res(F, ζ ).

ζ ∈{z,i,−i}

Analogously, ˆ

∞

(1 − ei2π ν )

.

0

 λν dλ = 2π i Res(G, ζ ), 2 1+λ ζ ∈{i,−i}

where G(λ) =

.

λν . 1 + λ2

A direct computation shows that 

ζ ∈{z,i,−i} Res(F, ζ )

Mν (z) = 

.

ζ ∈{i,−i} Res(G, ζ )

=

i ν +(−i)ν 2 i ν −(−i)ν 2i

zν −

z ν 2i 1 + eiπ ν − i 1 − eiπ ν i 1 − eiπ ν

π  z ν π = i − cot ν + cot ν, 2 i 2 =

z ∈ C+ .

On the Invariance Principle for a Characteristic Function

661

The case of .ν = 0 can be justified by taking the limit M0 (z) = lim Mν+ (z) =

.

ν→0

  1 2 log − , π z

z ∈ C+ ,

which competes the proof of (58). The proof of (59) is analogous.

(60)

 

Lemma 7.3 Assume Hypothesis 7.1. ˙ Then, if .ν ∈ [0, 1), then .A(ν) is the Friedrichs extension of .A(ν). If .ν ∈ (−1, 0], ˙ then .A(ν) is the Krein-von Neumann extension of .A(ν). ˙ In particular, the Friedrichs and Krein-von Neumann extensions of .A(0) coincide. Proof By Lemma 7.2, the Weyl-Titchmarsh function associated with the pair ˙ (A(ν), A(ν)) can be evaluated as

.

M(A(ν),A(ν)) (z) = Mν (z), ˙

.

z ∈ C+ .

To complete the proof it remains to observe that lim Mν (λ) = −∞,

.

λ↓−∞

0 ≤ ν < 1,

and .

lim Mν (λ) = ∞, λ↑0

−1 < ν ≤ 0,

(61)

and then apply [11, Theorem 4.4], a result that characterizes the Weyl-Titchmarsh function threshold behavior for the Friedrichs and Krein-von Neumann extensions, respectively.   Remark 7.4 Notice that the Cayley transforms of .Mν (z) and .M−ν (z) coincide up to a constant unimodular factor, that is,    ν  z ν i − cot π2 ν zi + cot π2 ν − i −1 Mν (z) − i   z ν = , . =  z iν π π Mν (z) + i − eiπ ν i − cot 2 ν i + cot 2 ν + i i    −ν  z −ν i + cot π2 ν zi − cot π2 ν − i −1 M−ν (z) − i i = . =  −ν   z −ν π π z M−ν (z) + i i + cot 2 ν i − cot 2 ν + i − e−iπ ν i  z ν −1 iπ ν = e  z iν , z ∈ C+ . − eiπ ν i

662

K. A. Makarov and E. Tsekanovskii

Therefore, by (8), the Livšic functions .sν (z) and .s−ν (z) associated with the pairs ˙ ˙ (A(ν), A(ν)) and .(A(−ν), A(−ν)), respectfully, are related as

.

sν (z) = eiπ ν s−ν (z),

.

z ∈ C+ .

Since the knowledge (up to a unimodular factor) of the Livšic function .s(A,A) (z) of a ˙ ˙ A), where .A˙ is a prime symmetric operator and A its self-adjoint extension, pair .(A, determines the symmetric operator .A˙ up to a unitary equivalence (see [1, 14, 19, ˙ ˙ 20]), and, moreover, .A(ν) and .A(−ν) are prime symmetric operators, we conclude ˙ ˙ that .A(ν) and .A(−ν) are unitarily equivalent. The following theorem addresses “intertwining” properties of the Friedrichs and Krein-von Neumann extensions of an operator with respect to the inverse operation. ˙ Theorem 7.5 Assume Hypothesis 7.1 and set .A˙ = A(ν) and .A = A(ν). Then  .

˙ F [A]

−1

  ˙ −1 = (A)

K

(62)

and .

    ˙ K −1 = (A) ˙ −1 . [A] F

(63)

˙ Remark 7.6 Notice that the operators .A(ν), .A(ν), etc., referred to in Hypothesis 7.1 are essentially coincide with the model multiplication operators in the weighted Hilbert space .L2 ((0, ∞); dμ) given by (19)–(21) (after an appropriate renormalization of the weight .dμ(λ) = λν dλ), so that Theorem 5.1 applies. For instance, the ˙ is well defined as a prime symmetric operator with deficiency inverse of .A˙ = A(ν) indices .(1, 1). Proof Comparing (58) and (59) one observes that for .ν ∈ (−1, 1) we have   1 − = N−ν (z), .Mν z

z ∈ C+ .

(64)

By Theorem 5.1 (vi), the left hand side of (64) is the Weyl-Titchmarsh function −1 , −(A(ν))−1 ) and therefore ˙ associated with the pair .(−(A(ν)) −1 −1 ˙ ˙ (−(A(ν)) , −(A(ν))−1 ) ∼ , (B(−ν))−1 ). = ((B(−ν))

.

Here the symbol .∼ = denotes the mutual unitary equivalence of the corresponding pairs. From the definition of the operators .A(ν) and .B(ν) it follows that −1 ˙ ˙ ((B(−ν)) , (B(−ν))−1 ) ∼ −A(−ν)) = (−A(−ν),

.

On the Invariance Principle for a Characteristic Function

663

and therefore −1 ˙ ˙ ((A(ν)) , (A(ν))−1 ) ∼ A(−ν)). = (A(−ν),

.

Suppose that .ν ≥ 0. By Lemma 7.3, ˙ A(ν) = [A(ν)] F.

.

So that −1 −1 −1 ˙ ˙ ˙ ((A(ν)) , (A(ν))−1 ) = ((A(ν)) , ([A(ν)] F) )

.

and hence −1 −1 ∼ ˙ ˙ ˙ ((A(ν)) , ([A(ν)] A(−ν)). F ) ) = (A(−ν),

.

(65)

Again, by Lemma 7.3, the operator .A(−ν) is the Krein-von Neumann extension of ˙ From the mutual unitary equivalence of the pairs (65) it follows that the A(−ν). second operator from the left pair is the Krein-von Neumann extension of the first one. That is,

.

−1 −1 ˙ ˙ ([A(ν)] = [(A(ν)) ]K F)

.

or, equivalently, ˙ F )−1 = [(A) ˙ −1 ]K , ([A]

.

which proves (62) (for .ν ∈ [0, 1)). One also has that −1 ˙ ˙ ((A(−ν)) , (A(−ν))−1 ) ∼ A(ν)). = (A(ν),

.

The same reasoning shows that −1 −1 ˙ ˙ [(A(−ν)) ]F = ([A(−ν)] K) .

.

˙ ˙ However, by Remark 7.4, the operators .A(−ν) and .A(ν) are unitarily equivalent which implies that −1 −1 ˙ ˙ [(A(ν)) ]F = ([A(ν)] K)

.

or, equivalently, ˙ −1 ]F = ([A] ˙ K )−1 , [(A)

.

and (63) follows (for .ν ∈ [0, 1)).

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K. A. Makarov and E. Tsekanovskii

The proof for .ν ∈ (−1, 0) is analogous.

 

Remark 7.7 In view of Remark 7.4, from (65) it also follows that the symmetric ˙ −1 referred to in Theorem 7.5 are unitarily equivalent. operators .A˙ and .(A) Remark 7.8 In connection with (63), it is worth mentioning that in a more general context of multivalued symmetric operators discussed in [7], the Krein-von Neumann extension .SN of a positive subspace S, called the von Neumann extension in [7], has been defined as SN = ([S −1 ]F )−1 ,

.

where .[S −1 ]F is the Friedrichs extension of the relation .S −1 (see also [4] and [5]). Acknowledgments We are very grateful to the referee for valuable comments and suggestions. The first author was partially supported by the Simons collaboration grant 00061759 while preparing this article.

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14. M.S. Livšic, On a class of linear operators in Hilbert space. Mat. Sbornik, 19(2), 239–262 (1946) 15. M. Livšic, On spectral decomposition of linear non-self-adjoint operators. Mat. Sbornik 34(76), 145–198 (1954). (Russian); English transl.: Amer. Math. Soc. Transl. 5(2), 67–114 (1957) 16. L.D. Landau, E.M. Lifshitz, in Quantum Mechanics. Non-relativistic Theory (Pergamon Press, New York, 1965) 17. M.S. Livšic, in Operators, Oscillations, Waves (Open Systems). Translated from the Russian by Scripta Technica, Ltd. English translation ed. by R. Herden. Translations of Mathematical Monographs, vol. 34 (American Mathematical Society, Providence, 1973) 18. K.A. Makarov, E. Tsekanovski˘i, in On the Weyl-Titchmarsh and Livšic Functions. Proceedings of Symposia in Pure Mathematics, vol. 87 (American Mathematical Society, 2013), pp. 291– 313 19. K.A. Makarov, E. Tsekanovskii, Representations of commutation relations in dissipative quantum mechanics. Preprint arXiv:2101.10923v2 [math-ph]) 20. K.A. Makarov, E. Tsekanovskii, in The Mathematics of Open Quantum Systems (World Scientific, Singapore, 2022) 21. R.S. Phillips, in On Dissipative Operators, ed. by A.K. Aziz. Lecture Series in Differential Equations, vol. II (von Nostrand, New York, 1969), pp. 65–113 22. B. Simon, in Spectral Analysis of Rank One Perturbations and Applications. Mathematical Quantum Theory. II. Schrödinger Operators (Vancouver, BC, 1993), pp. 109–149. CRM Proc. Lecture Notes, vol. 8 (Amer. Math. Soc., Providence, 1995) 23. B. Simon, T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Comm. Pure Appl. Math. 39, 75–90 (1986) 24. A.V. Shtraus, On the extensions and the characteristic function of a symmetric operator. Izv. Akad. Nauk SSR, Ser. Mat. 32, 186–207 (1968) 25. E.R. Tsekanovskii, in Characteristic Operator-Functions of Unbounded Operators (in Russian), vol. 11 (Transactions of Mining Institute, Kharkov, 1962), pp. 95–100

A Trace Formula and Classical Solutions to the KdV Equation Alexei Rybkin

Dedicated to the memory of Sergey Naboko, my teacher and friend

Abstract We show that if the initial profile .q (x)´for the Korteweg-de Vries (KdV) ∞ equation is supported on .(a, ∞) , a > −∞, and . a x 7/4 |q (x)| dx < ∞, then the time evolved .q (x, t) is a classical solution of the KdV equation. Keywords KdV equation · Trace formula

1 Introduction We are concerned with the Cauchy problem for the Korteweg-de Vries (KdV) equation  .

∂t u − 6u∂x u + ∂x3 u = 0, x ∈ R, t ≥ 0 u(x, 0) = q(x).

(1.1)

As is well-known, (1.1) is the first nonlinear evolution PDE solved in the seminal 1967 Gardner-Greene-Kruskal-Miura paper [7] by the method which is now referred to as the inverse scattering transform (IST). Much of the original work was done under generous assumptions on initial data q (typically from the Schwarz class) for

A. Rybkin () Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_25

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which the well-posedness of (1.1) was not an issue even in the classical sense.1 But well-posedness in less nice function classes becomes a problem. From the harmonic analysis point of view, the natural function class for well-posedness results is the .L2 based Sobolev space .H s (R). Before 1993 well-posedness was proven for .s > 0. The .s = 0 bar was reached in 1993 in the seminal papers by Bourgain [1], where, among others, he proved that (1.1) is well-posed in .H 0 (R) = L2 (R). Moreover his trademark harmonic analysis techniques could be pushed below .s = 0. We refer the interested reader to the influential [3] for the extensive literature prior to 2003. Until very recently, the best well-posedness Sobolev space for (1.1) remained .H −3/4 (R). Note that harmonic analysis methods break down while crossing .s = −3/4 in an irreparable way. Further improvements required utilizing complete integrability of the KdV equation. The breakthrough occurred in 2019 in Killip-Visan [10] where .s = −1 was reached. That is, (1.1) is well-posed for initial data of the form .q = v + w  where .v, w ∈ L2 (R). For .s < −1 the KdV equation is ill-posed in .H s (R) scale (see [10] for relevant discussions and the literature cited therein). While [10] relies on the complete integrability of the KdV equation it does not utilize the IST transform as all Sobolev spaces allow for the rate of decay that is slower than  what the IST requires. As it was shown by Naboko [12] slower than .q (x) = O |x|−1 may produce dense singular spectrum filling .(0, ∞) leaving any hope that a suitable IST can include such a situation. Well-posedness in .H s (R) with .s ≤ 0 does not of course provide any regularity of KdV solutions and hence solutions can not possibly be understood in the classical sense which requires continuity of at least three spatial and one temporal derivatives of .q (x, t). On the other hand, from the physical point of view, we want to solve (1.1) by the IST and want our solution to satisfy (1.1) in the classical sense. Wellposedness results in the .H s scale do not help. This issue drew much of attention once (1.1) became in the spot light. For the earlier literature account we refer the reader to the substantial 1987 paper [2] by Cohen-Kappeler. The main result of [2] says that if2 ˆ

∞

.

ˆ

−∞ ∞

(1 + |x|) |q (x)| dx < ∞,

.

(1 + |x|)N |q (x)| dx < ∞, N ≥ 11/4,

(1.2) (1.3)

then (1.1) can be solved by the IST and the solution .q (x, t) satisfies the KdV equation in the classical sense, the initial condition being satisfied in .H −1 (R). Note that uniqueness was not proven in [2] and in fact it was stated as an open problem. The best known uniqueness result back then was available for .H 3/2 (R) which of course assumes some smoothness whereas the conditions (1.2)–(1.3) do 1 Here and below by a classical solution to the KdV equation we mean a solution at least three times continuously differentiable in x and once in t for any .t > 0 (but not necessarily .t = 0). ´ ´ 2 . ∞ |f (x)| dx < ∞ means that . ∞ |f (x)| dx < ∞ for all finite a. a

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not. Since any function subject to (1.2)–(1.3) can be properly included in .H s (R) with .s < −1/2, a well-posedness statement in .H s (R) , s < −1/2, would turn the Cohen-Kappeler. existence result into a well-posedness but not in the class (1.2)– (1.3) where initial data belong to. A peculiar consequence of [2] is that a rough initial profile subject to (1.2)–(1.3) turns instantaneously into a three times differentiable function showing a strong smoothing effect of the KdV flow. However, the payoff for the gain of regularity is a slower decay at .+∞ (a smaller N in (1.3) for .t > 0). In [13] we show that essentially any (non-decaying) initial profile supported on .(−∞, a) turns into a spatially meromorphic function suggesting that it is the decay at .+∞ that affects smoothness of the solution. In the recent [8] we improve (1.3) to .N ≥ 5/2. The current note is devoted to improving the power even further. More specifically we prove Theorem 1.1 (Main Theorem) Suppose that a real locally integrable initial profile q in (1.1) satisfies for some .a > −∞ q (x) = 0, x ≤ a (restricted support);

.

ˆ .

∞

(1 + |x|)N |q (x)| dx < ∞, N ≥ 7/4 (decay at + ∞).

(1.4) (1.5)

a

If the unique solution .q (x, t) of the initial-value problem (1.1) admits representation (4.1) then it is the classical one (i.e. three times continuously differentiable in x and once in t), the initial condition being satisfied in the sense .

lim q(x, t) = q (x) in H −1 (R) .

t→+0

(1.6)

The condition (1.4) seems to be very restrictive but it can actually be replaced with ˆ . Sup max (−q (x) , 0) dx < ∞, |I |=1 I

which does not require any decay at .−∞. The main reason why we assume (1.4) is that we rely on a recent result from [6] which is proven under the condition (1.4). Note, that the validity of the trace formula (4.1) is assumed to avoid some technical complications and can be removed. We also believe that the initial condition (1.6) can be understood in a much more specific sense. We plan to come back to these and other interesting questions elsewhere. Let us briefly discuss our arguments. Recall that in [8] we improve N in (1.3) by .1/4. While we rely in [8] on the Faddeev-Marchenko inverse scattering theory, we do it within the Hankel operator approach, which we develop in [9]. Note that the Cohen-Kappeler approach [2] is also based upon the Marchenko integral equation in its classical form, which does not offer a transparent way of relaxation of the condition (1.3), while ours does. Besides, the treatment is quite involved. Our

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approach in [8] rests on the Dyson (second log determinant) formula for .q (x, t) that is written in terms of Hankel operators. Its analysis requires the membership in the trace class of our Hankel operator and its x-derivatives of order up to five. Here we rely on the trace formula introduced in [5] which allows us to get by with three x-derivatives saving extra .3/4. The Hankel operator however still works behind the scene as it is crucially used in [6] to prove Proposition 2. The paper is organized as follows. The short Sect. 2 is devoted to our agreement on notation. In Sect. 3.1 we present some background information on scattering theory and give an important asymptotic formula. Section 4 is devoted to the proof of Theorem 1.1.

2 Notations We follow standard notation accepted in Analysis. .R is the real line, .R± = (0, ±∞), C is the complex plane, .C± = {z ∈ C : ± Im z > 0}. .z is the complex conjugate of n n n .z. As always, .∂x := ∂ /∂x . p As usual, .L (S) , 0 < p ≤ ∞, is the Lebesgue space on a set S. We will also deal with the weighted .L1 spaces .L1N (S) , N > 0, of functions summable with the Nth order moments ˆ   1 + |x|N |f (x)| dx < ∞. . .

S

This function class is basic for direct/inverse scattering theory for Schrödinger operators on the line. Besides, we use Hardy spaces. We recall that a function f analytic in .C± is in the Hardy space .H±2 if ˆ .

sup

∞

y>0 −∞

|f (x ± iy)|2 dx < ∞.

We will also need .H±∞ , the algebra of analytic functions uniformly bounded in .C± . We set .H 2 = H+2 , .H ∞ = H+∞ . We occasionally write .f (x) ∼ g (x) , x → x0 (finite or infinite) if .f (x) − g (x) → 0, x → x0 .

3 Our Framework and Main Ingredients In this section we briefly review the necessary material and introduce our main ingredients.

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3.1 Scattering Data Through this section we assume that q is short-range, i.e. .q ∈ L11 (R). Associate with q the full line Schrödinger operator .Lq = −∂x2 + q(x). As is well-known, 2 .Lq is self-adjoint on .L (R) and its spectrum consists of a finite number of simple negative eigenvalues .{−κn2 }, called bound states, and two fold absolutely continuous component filling .R+ . There is no singular continuous spectrum. Two linearly independent (generalized) eigenfunctions of the a.c. spectrum .ψ± (x, k), k ∈ R, can be chosen to satisfy ψ± (x, k) = e±ikx + o(1), ∂x ψ± (x, k) ∓ ikψ± (x, k) = o(1), x → ±∞.

(3.1)

.

The functions .ψ± are referred to as Jost solutions of the Schrödinger equation Lq ψ = k 2 ψ.

(3.2)

.

Since q is real, .ψ± also solves (3.2) and one can easily see that the pairs .{ψ+ , ψ+ } and .{ψ− , ψ− } form fundamental sets for (3.2). Hence .ψ∓ is a linear combination of .{ψ± , ψ± }. We write this fact as follows (.k ∈ R) T (k)ψ− (x, k) = ψ+ (x, k) + R(k)ψ+ (x, k),

(3.3)

.

where T and R are called transmission and (right) reflection coefficients respectively. The identity (3.3) is totally elementary but serves as a basis for inverse scattering theory and for this reason it is commonly referred to as the basic scattering relation. As is well-known (see, e.g. [11]), the triple .{R, (κn , cn )}, where −1 .cn = ψ+ (·, iκn )

, determines q uniquely and is called the scattering data for .Lq . We will need the following statement from [8]. Proposition 1 (On Reflection Coefficient) Suppose q is real and in .L11 (R), 3 . q|R = 0, and .T (0) = 0. Then − 

1 .R (k) = T (k) 2ik

ˆ

∞

e 0

−2ikx

q (x) dx +

1 (2ik)2

ˆ

∞

e

−2ikx





Q (x) dx ,

0

(3.4) where Q is an absolutely continuous function subject to ˆ ∞  |q| , x ≥ 0, . Q (x) ≤ C1 |q (x)| + C2

(3.5)

x

with some (finite) constants .C1 , C2 dependent on . q L1 and . q L1 only. 1

fact, .T (0) = 0 happens generically and is not a real restriction. Recall that the transmission coefficient doesn’t vanish at .k = 0 only for the so-called exceptional potentials but an arbitrarily small perturbation turns such a potential into generic. In our case it can be achieved by merely shifting the data q.

3 In

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Note that (3.5) implies q ∈ L1N ⇒ Q ∈ L1N −1 .

(3.6)

.

Note that for q supported on the full line, it was proven in [5] that R (k) =

.

T (k) 2ik

ˆ

∞

−∞

e−2ikx g (x) dx,

where g satisfies .

|g (x)| ≤ |q (x)| + const

 ´∞ |q| , x ≥ 0 ´ xx , −∞ |q| , x < 0

(3.7)

and nothing better can be said about g in general. In the case of q supported on (0, ∞) this statement can be improved. Indeed, (3.4) implies that

.

g (x) = q (x) + Q (x)

.

with some absolutely continuous on .(0, ∞) function which derivative .Q satisfies (3.5). We will rely on the following statement from [6, Theorem 8.2]. Proposition 2 (On Jost Solution) Let y (x, t, k) = e−ikx ψ+ (x, t, k) − 1,

.

where .ψ+ (x, t, k) is the right Jost solution corresponding to the solution of (1.1).4 Suppose that .q (x) is subject to the conditions of Theorem 1.1. Then5 ∂xn y (x, t, k) , ∂t y (x, t, k) ∈ H 2 ∩ H ∞ , n = 0, 1, 2, 3.

.

We emphasize that this proposition holds only for .t > 0 in general. For .t = 0 it fails unless we assume extra regularity conditions on q. Note that the existence of .ψ+ (x, t, k) is guaranteed by a much more general result from [4, Theorem 1.4]. In particular, it holds even for .q (x) ∈ L1 . We however believe that it was known already in the 1970s for .q (x) ∈ L11 . At least it follows from [2] for .q (x) ∈ L111/4 . Finally, recall that uniformly in the upper half-plane 1 .T (k) = 1 + 2ik

ˆ

∞

−∞

q (x) dx + O

1 k2

 , k → ∞.

(3.8)

4 Such a solution is also commonly referred to as the time evolved Jost solution under the KdV flow. 5 The Hardy spaces are with respect to k.

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3.2 An Oscillatory Integral The following lemma is an immediate consequence of the method of steepest descent [14]. Lemma 1 Let .Γ be a deformation of .R providing the absolute convergence of the integral ˆ Fn (x, t) :=

.

  λn exp iλ3 t − iλx dλ.

(3.9)

Γ

Then for a fixed .t > 0 as .x → ∞ : Fn (x, t) ∼

.

 12 −(2n+1)/4 (2n−1)/4 −i sin φ (x, t) , n is odd t x , cos φ (x, t) , n is even π

(3.10)

where π 2x 3/2 φ (x, t) := √ − . 1/2 4 3 3t

.

Proof Rescaling in (3.9) .λ → (x/3t)1/2 λ and setting .ω = x (x/3t)1/2 yield Fn (x, t) =

 ω (n+1)/3 ˆ

.

3t

λn eiωS(λ) dλ,

Γ

where S (λ) := λ3 /3 − λ.

.

The phase .S (λ) has two stationary points .λ = ±1 and hence by the steepest descent

ˆ λn eiωS(λ) dλ =

.

Γ

 π −2iω/3+iπ/4 e + (−1)n e2iω/3−iπ/4 ω

  +O ω−3/2 , ω → ∞. Therefore

Fn (x, t)     3 ω (2n−1)/6 −2iω/3+iπ/4 e = + (−1)n e2iω/3−iπ/4 + o ω−1/2 , π t 3t

.

ω→∞ Switching back from .ω to x we arrive at (3.10).

 

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4 Prove of the Main Theorem Since the KdV equation is translation invariant without loss of generality we may assume that .a = 0. Existence, uniqueness, and (1.6) follow from the general results (see e.g. [10]). Our proof is based on the trace formula [5] q (x, t) = qc (x, t) + qd (x, t) ,

(4.1)

.

where qc (x, t) =

.

2i π

ˆ

∞

−∞

qd (x, t) = −4



.

kR (k) ξx,t (k) (1 + y (x, t; k))2 dk,    2 κj cj2 ξx,t iκj 1 + y x, t; iκj .

j

Here .y (x, t; k) (refereed sometimes to as the Faddeev function) is as in Proposition 2 and   3 .ξx,t (k) := exp 8ik t + 2ikx . The integral in (4.1) is understood in the sense of Cesàro means. We however use a more convenient way to regularize this integral. Consider .qc first. By the CauchyGreen formula applied to the strip .0 ≤ Im λ ≤ 1 we have (.λ = u + iv) 2i .qc (x, t) = π +

ˆ ˆ

  2iλξx,t (λ) (1 + y (x, t; λ))2 ∂R λ, λ dudv

0≤Im λ≤1

  1 2iλξx,t (λ) R λ, λ (1 + y (x, t; λ))2 dλ π R+i

=: qc(0) (x, t) + qc(1) (x, t) ,   where .R λ, λ is the (not analytic) continuation of the reflection coefficient .R (k) given by   T (λ) .R λ, λ = 2iλ

ˆ

∞

e 0

−2iλs

1 g (s) ds + 2iλ

ˆ

∞

e

−2iλs





Q (x) dx .

(4.2)

0

  Note that such continuation guaranties .R λ, λ = o (1/ |λ|) in the closed upper half-plane. (1) Consider .qc (x, t) first. Since .ξx,t (λ) rapidly decays along any line in the upper (1) half-plane parallel to the real line, .qc (x, t) is differentiable in x as many times

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675

(1)

as .y (x, t; λ). By Proposition 2 then .qc (x, t) is at least three times continuously (0) differentiable in x. Turn now to .qc (x, t). Since by Proposition 2 ∂xn y (x, t; λ) ∈ H 2 ∩ H ∞ , for n = 0, 1, 2, 3,

.

it follows from  2 ∂x3 ξx,t (λ) (1 + y (x, t; λ))

.

= (2iλ)3 ξx,t (λ)



+ (2iλ)3 ξx,t (λ) 2y (x, t; λ) + y (x, t; λ)2 + 6 (2iλ)2 ξx,t (λ) [1 + y (x, t; λ)] ∂x y (x, t; λ)   + 3 (2iλ) ξx,t (λ) [1 + y (x, t; λ)] ∂x2 y (x, t; λ) + [∂x y (x, t; λ)]2   + ξx,t (λ) 3∂x y (x, t; λ) ∂x2 y (x, t; λ) + [1 + y (x, t; λ)] ∂x2 y (x, t; λ) that for a fixed real x and a positive t, uniformly in the strip .0 ≤ Im λ ≤ 1  2 ∂x3 ξx,t (λ) (1 + y (x, t; λ)) = (2iλ)3 ξx,t (λ) [1 + o (1/λ)] , λ → ∞.

.

Thus, if the integral .

2i π

ˆ

  (2iλ)n ξx,t (λ) ∂R λ, λ dudv,

0≤Im λ≤1 (0)

converges for .n = 1, 2, 3, 4, so do the integrals representing .∂xn qc (x, t) for .n = 0, 1, 2, 3. Apparently it is enough to consider the largest derivative .∂x3 only. Due to (4.2), we have ˆ   2i . (2iλ)4 ξx,t (λ) ∂R λ, λ dudv π 0≤Im λ≤1 ˆ ∞  ˆ 1 = (2iλ)3 ξx,t (λ) T (λ) (−2is) e−2iλs q (s) ds dudv π 0≤Im λ≤1 0 ˆ ∞  ˆ 1 + (2iλ)2 ξx,t (λ) T (λ) (−2is) e−2iλs Q (x) dx dudv π 0≤Im λ≤1 0 = I0 (x, t) + I1 (x, t) + I2 (x, t) ,

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where −2i .I0 (x, t) = π I1 (x, t) =

.

−2i π

ˆ

ˆ

∞

3

(2iλ) ξx,t (λ)

e

−2iλs

 sq (s) ds dudv,

0

0≤Im λ≤1

ˆ

ˆ

∞

(2iλ)2 ξx,t (λ) 0≤Im λ≤1

 e−2iλs sQ (x) dx dudv,

0

 ˆ ∞ ˆ −2i e−2iλs sq (s) ds dudv (2iλ)3 ξx,t (λ) (T (λ) − 1) π 0≤Im λ≤1 0 ˆ ∞  ˆ 2i e−2iλs sQ (x) dx dudv. − (2iλ)2 ξx,t (λ) (T (λ) − 1) π 0≤Im λ≤1 0

I2 (x, t) =

.

Due to (3.8), it is enough to study .I0 (x, t) and .I1 (x, t) . For .I0 (x, t) one has I0 (x, t) =

.

=

−2i π −2i π

ˆ

ˆ

∞

(2iλ)3 ξx−s,t (λ) e−4vs dudv

ds q (s) 0≤Im λ≤1

0

ˆ

ˆ

∞

ds q (s) 0

1

dv e−4vs

ˆ

0

R+iv

(2iλ)3 ξx−s,t (λ) dλ

Here we have used that .λ = u − i = λ − 2i and .du = dλ for each fixed v. Since (2iλ)3 ξx−s,t (λ) is analytic and rapidly decaying on .R + iv we can deform .R + iv to some contour .Γ independent of v, to be chosen later. Thus we have arrived at

.

I0 (x, t) =

.

−i 2π

∞

ˆ  ds q (s) 1 − e−4s (2iλ)3 ξx−s,t (λ) dλ

ˆ

∞

ˆ

1 =− 4π

0

Γ

ˆ  −4s ds q (s) 1 − e λ3 ξx−s,t (λ/2) dλ

0

Γ

Applying Lemma 1 to the inner integral we conclude that the integral .I0 (x, t) converges if ˆ .

∞

s 5/4 |q (s)| ds < ∞.

Turn now to .I1 (x, t). Similarly, one gets 1 .I1 (x, t) = − 2π

ˆ 0

∞





ds Q (s) 1 − e

−4s

ˆ Γ

λ2 ξx−s,t (λ/2) dλ

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677

and hence, as above, .I1 (x, t) converges if ˆ .

∞

s 3/4 Q (s) ds < ∞.

(4.3)

Due to (3.6) we see that if .q ∈ L17/4 then .Q ∈ L13/4 , and hence (4.3) holds.   The term .qd (x, t) is easy. It is differentiable in x as many as .y x, t; iκj and thus we have at least three continuous derivatives. Continuous differentiability of .q (x, t) in .t > 0 follows from .∂t ξx,t = −∂x3 ξx,t . Acknowledgments The author is supported in part by the NSF grant DMS-2009980.

References 1. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation. Geom. Funct. Anal. 3(3), 209–262 (1993) 2. A. Cohen, T. Kappeler, Solutions to the Korteweg-de Vries equation with initial profile in L11 (R) ∩ L1N (R + ). SIAM J. Math. Anal. 18(4), 991–1025 (1987) 3. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T . J. Amer. Math. Soc. 16(3), 705–49 (2003) 4. D. Damanik, C. Remling, Schrödinger operators with many bound states. Duke Math. J. 136(1), 51–80 (2007) 5. P. Deift, E. Trubowitz, Inverse scattering on the line. Comm. Pure Appl. Math. 32(2), 121–251 (1979) 6. S.M. Grudsky, V. Kravchenko, S.M. Torba, Realization of the inverse scattering transform method for the Korteweg–de Vries equation (2022). Preprint. https://doi.org/10.22541/au. 165424504.42620980/v1 7. C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Method for solving the KortewegdeVries equation. Phys. Rev. Lett. 19, 1095–1097 (1967) 8. S. Grudsky, A. Rybkin, On classical solution to the KdV equation. Proc. London Math. Soc. (3) 121, 354–371 (2020) 9. S. Grudsky, A. Rybkin, Soliton theory and Hakel operators. SIAM J. Math. Anal. 47(3), 2283– 2323 (2015) 10. R. Killip, M. Visan, KdV is wellposed in H −1 . Ann. of Math. (2) 190(1), 249–305 (2019) 11. V.A. Marchenko, in Sturm-Liouville Operators and Applications, Revised edn. (AMS Chelsea Publishing, Providence, 2011), p. xiv+396 12. S.N. Naboko, Dense point spectra of Schrödinger and Dirac operators. Theor. Math. Phys. 68, 646–653 (1986) 13. A. Rybkin, Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line. Nonlinearity 23, 1143–1167 (2010) 14. R.S.C. Wong, in Asymptotic Approximations of Integrals. (Academic Press, London, 2001), p. xiv + 540. ISBN: 978-0-89871-497-5

Semiclassical Analysis in the Limit Circle Case Dimitri R. Yafaev

To the memory of Serezha Naboko

Abstract We consider second order differential equations with real coefficients that are in the limit circle case at infinity. Using the semiclassical Ansatz, we construct solutions (the Jost solutions) of such equations with a prescribed asymptotic behavior for .x → ∞. It turns out that in the limit circle case, this Ansatz can be chosen common for all values of the spectral parameter z. This leads to asymptotic formulas for all solutions of considered differential equations, both homogeneous and non-homogeneous. We also efficiently describe all self-adjoint realizations of the corresponding differential operators in terms of boundary conditions at infinity and find a representation for their resolvents. Keywords Second order differential equations · Minimal and maximal differential operators · Self-adjoint extensions · Quasiresolvents and resolvents

D. R. Yafaev () University Rennes, CNRS, Rennes, France SPGU, Saint Petersburg, Russia NTU Sirius, Sochi, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Brown et al. (eds.), From Complex Analysis to Operator Theory: A Panorama, Operator Theory: Advances and Applications 291, https://doi.org/10.1007/978-3-031-31139-0_26

679

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D. R. Yafaev

1 Introduction 1.1 Setting the Problem We consider a second order differential equation .

− (p(x)u (x)) − q(x)u(x) = zu(x),

p(x) > 0, q(x) = q(x), ¯

x ∈ R+ , z ∈ C,

(1.1)

under some minimal regularity assumptions on coefficients .p(x) and .q(x) guaranteeing that its solutions, as well as their derivatives, are continuous on .[0, ∞). We are interested in the limit circle (LC) case at infinity where all solutions are in .L2 (R+ ) for all .z ∈ C. Equation (1.1) is known as the Schrödinger equation if .p(x) = 1, but we keep the same term in the general case. We write .−q(x) because under our assumptions .q(x) → +∞ as .x → ∞. Our analysis relies on a construction of solutions .fz (x) of Eq. (1.1) distinguished by their asymptotics   fz (x) = (p(x)q(x))−1/4 ei(x) 1 + o(1) ,

.

z ∈ C,

x → ∞,

(1.2)

where  (x) =

x

.

 1/2 dy q(y)/p(y)

x0

(.x0 is an arbitrary fixed number). Solutions .fz (x) are known as the Jost solutions of Eq. (1.1). Relation (1.2) shows that the leading terms of asymptotics of the Jost solutions do not depend on .z ∈ C. This fact is specific for the LC case. The functions .fz (x) and .f¯z¯ (x) satisfy the same Eq. (1.1) and are linearly independent so that an arbitrary solution of Eq. (1.1) is their linear combination. We require that 

∞

.

(p(x)q(x))−1/2 dx < ∞

x0

whence .fz ∈ L2 (R+ ) for all .z ∈ C. Thus, according to (1.2), we are in the LC case.

1.2 Limit Point versus Limit Circle The Weyl limit point/circle theory states that differential equation (1.1) always has a non-trivial solution in .L2 (R+ ) for .Im z = 0. This solution is either unique (up to a constant factor) or all solutions of (1.1) belong to .L2 (R+ ). The first instance is known as the limit point (LP) case and the second one—as the limit circle (LC)

Semiclassical Analysis in the Limit Circle Case

681

case. Consistent presentations of the Weyl theory can be found, for example, in the books [1], Chapter IX, [2], Chapter XIII and [6], Chapter X.1. Our goal is to study self-adjoint operators in the space .L2 (R+ ) associated with a differential operator   (Hu)(x) = − p(x)u (x) − q(x)u(x).

(1.3)

.

The operator .H defined on a domain .C0∞ (R+ ) is symmetric in .L2 (R+ ), but to make it self-adjoint, one has to add boundary conditions at .x = 0 and, eventually, for .x → ∞. The boundary condition at the point .x = 0 looks as u (0) = αu(0)

.

where

α = α. ¯

(1.4)

The value .α = ∞ is not excluded. In this case (1.4) should be understood as the equality .u(0) = 0. We always require condition (1.4), fix .α and do not keep track of .α in notation. Let us define a symmetric operator .H00 by an equality .H00 u = Hu on a domain .D(H00 ) consisting of smooth functions .u(x) satisfying boundary condition (1.4) and such that .u(x) = 0 for sufficiently large x. The operator .H00 is essentially self-adjoint if and only if the LP case occurs (see, e.g., Theorem X.7 in the book [6]). In the LC case, .H00 has a one parameter family .Hω (here .ω is a point on the unit circle .T ⊂ C) of self-adjoint extensions distinguished by some conditions for .x → ∞. Their description can be performed in different terms. Our analysis relies on asymptotic formula (1.2). Alternative possibilities are briefly discussed in Sect. 4.5.

1.3 Plan of the Paper The existence of the Jost solutions is proven in Sect. 2. Another important element of our approach is a construction of an operator .R(z) (we call it the quasiresolvent) ∗ (the adjoint playing the role of the resolvent for the maximal operator .Hmax := H00 of .H00 ). We emphasize that .R(z) is an analytic function of .z ∈ C. The operator .R(z) is constructed in Sect. 3. This section is close to Sect. 2 of paper [12]. In Sect. 4 we show (see Theorem 4.4) that all functions .u ∈ D(Hmax ) have asymptotic behavior   u(x) = (p(x)q(x))−1/4 s+ ei(x) + s− e−i(x) + o(1) ,

.

x → ∞,

(1.5)

with some coefficients .s± = s± (u) ∈ C. We distinguish a set .D(Hω ) ⊂ D(Hmax ) by the condition s+ (u) = ωs− (u)

.

where

|ω| = 1.

(1.6)

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D. R. Yafaev

Asymptotic coefficients .s± (u) in formula (1.5) play the role of boundary values .u(0) and .u (0) for functions u in the local Sobolev space .H2loc , and equality (1.6) plays the role of boundary condition (1.4). Theorem 4.5 shows that the restriction .Hω of the operator .Hmax on .D(Hω ) is self-adjoint, and all self-adjoint extensions of the operator .Hmin coincide with one of the operators .Hω . Our proofs of these results are independent of the von Neumann formulas. Finally, we construct the resolvents of the operators .Hω in Theorem 4.6. The construction of this paper is similar to the approach used in Sect. 3 of [11] in the case of Jacobi operators. Conditions on coefficients look completely differently for Jacobi and Schrödinger operators, but, in both cases, we are in the LC case and the leading terms of asymptotics of the corresponding Jost solutions do not depend on the spectral parameter. This implies that spectral properties of these two classes of operators are similar.

2 The Semiclassical Ansatz We here construct solutions of the Schrödinger equation (1.1) with asymptotics (1.2) for .x → ∞.

2.1 Regular Solutions To avoid inessential technical complications, we always suppose that .p ∈ C 1 (R+ ), .q ∈ C(R+ ) and the functions .p(x), .q(x) have finite limits as .x → 0. We assume that .p(x) > 0 for .x ≥ 0. The solutions of Eq. (1.1) exist, belong to .C 2 (R+ ) and they have limits .u(+0) =: u(0), .u (+0) =: u (0). A solution .u(x) is distinguished uniquely by boundary conditions .u(0) = u0 , .u (0) = u1 . Recall that for arbitrary solutions u and v of Eq. (1.1) their Wronskian {u, v} := p(x)(u (x)v(x) − u(x)v  (x))

.

does not depend on .x ∈ R+ . Clearly, the Wronskian .{u, v} = 0 if and only if the solutions u and v are proportional. We introduce a couple of regular solutions of Eq. (1.1) by boundary conditions at the point .x = 0:

.

 ϕz (0) = 1, θz (0) = 0,

ϕz (0) = α, θz (0) = −p(0)−1 ,

if α ∈ R

(2.1)

Semiclassical Analysis in the Limit Circle Case

683

and .

 ϕz (0) = 0,

ϕz (0) = 1,

θz (0) = p(0)−1 ,

if α = ∞.

θz (0) = 0,

(2.2)

Obviously, .ϕz (x) (but not .θz (x)) satisfy boundary condition (1.4) and the Wronskian {ϕz , θz } = 1. In the LC case all solutions of Eq. (1.1) are in .L2 (R+ ). In particular,

.

ϕz ∈ L2 (R+ ),

.

θz ∈ L2 (R+ )

for all

z ∈ C.

(2.3)

2.2 Jost Solutions The Jost solutions .fz (x) of the differential equation (1.1) are distinguished by their asymptotics (1.2) for .x → ∞. They will be constructed in Theorem 2.1. Let us set  a(x) = (p(x)q(x))

.

−1/4

ξ(x) =

,

q(x) p(x)

 and

x

(x) =

ξ(y)dy

(2.4)

x0

so that p(x)ξ(x)a 2 (x) = 1.

(2.5)

.

Notation (2.4) will be used throughout the whole paper. Theorem 2.1 Suppose that a ∈ L2 (x0 , ∞)

.

and

a(pa  ) ∈ L1 (x0 , ∞)

(2.6)

for some .x0 > 0. Then for all .z ∈ C, Eq. (1.1) has a solution .fz (x) with asymptotics fz (x) = a(x)ei(x) (1 + o(1))

(2.7)

.

as .x → ∞. If, additionally, .

lim p(x)a  (x)a(x) = 0,

(2.8)

x→∞

then fz (x) = iξ(x)a(x)ei(x) (1 + o(1)),

.

x → ∞.

(2.9)

684

D. R. Yafaev

Remark 1 In the leading particular case .p(x) = 1 conditions (2.6) mean that 

∞

.

q(x)−1/2 dx < ∞ and

x0



∞

q(x)−1/4 |(q(x)−1/4 ) |dx < ∞.

x0

Condition (2.8) reduces to .q(x)−3/2 q  (x) = o(1) for .x → ∞. In a detailed notation, formula (2.7) coincides with (1.2). We call .fz (x) the Jost solution. We emphasize that the leading term of asymptotics of .fz (x) does not depend on z. This is specific for the LC case. Otherwise a proof of Theorem 2.1 is relatively standard (cf. [5], Chapter 6). It relies on the fact that the Liouville-Green Ansatz A(x) = a(x)ei(x)

.

(2.10)

satisfies Eq. (1.1) with a sufficiently good accuracy. We start a proof of Theorem 2.1 with a multiplicative change of variables fz (x) = A(x)ψz (x).

.

(2.11)

The following result will be obtained by a direct computation. Lemma 1 Set  ρz (x) = a(x) p(x)a  (x)) + za 2 (x).

(2.12)

(p(x)fz (x)) + q(x)fz (x) + zfz (x) = 0

(2.13)

.

Then the equation .

is equivalent to an equation  ξ  (x)   ψ (x) + ρz (x)ξ(x)ψz (x) = 0 ψz (x) + 2iξ(x) − ξ(x) z

.

(2.14)

for the function .ψz (x) defined by formula (2.11). Proof Differentiating (2.11) twice, we find that (pfz ) = pAψz + (2pA + p A)ψz + (pA ) ψz .

.

Substituting this expression into (2.13) and dividing by pA, we rewrite (2.13) as   A p   1  (pA ) ψz + + q + z ψz = 0. ψz + 2 + A p p A

.

(2.15)

Semiclassical Analysis in the Limit Circle Case

685

By definitions (2.4) and (2.10) the coefficient at .ψz here equals 2

.

A p a p ξ + = 2 + 2iξ + = 2iξ − . A p a p ξ

(2.16)

Next, we compute the coefficient at .ψz in (2.15). Using (2.5), we see that pA = (pa  + ia −1 )ei

.

and   (pA ) = (pa  ) − a −1 ξ ei

.

whence A−1 (pA ) + q = a −1 (pa  ) .

(2.17)

.

Substituting now expressions (2.16) and (2.17) into (2.15), we obtain Eq. (2.14) with the coefficient   ρz (x) = (pξ )−1 a −1 (pa  ) + z .

.



In view of (2.5) this coincides with definition (2.12). Next, we reduce the differential equation (2.14) to a Volterra integral equation −1



ψz (x) = 1 + (2i)

.

∞

 1 − e−2i(x) e2i(y) ρz (y)ψz (y)dy,

x ≥ x0 .

(2.18)

x

Lemma 2 Let assumptions (2.6) be satisfied. Then Eq. (2.18) has a unique solution ψz (x). This solution satisfies Eq. (2.14) and

.

ψz (x) → 1,

.

ψz (x) = o(ξ(x))

as

x → ∞.

(2.19)

Proof Note that .ρz ∈ L1 (x0 , ∞) according to conditions (2.6). Therefore a bounded solution .ψz (x) of Eq. (2.18) can be standardly constructed by iterations. Let us check that .ψz (x) satisfies Eq. (2.14). Differentiating (2.18), we see that ψz (x) = ξ(x)e−2i(x)



∞

.

e2i(y) ρz (y)ψz (y)dy

(2.20)

x

and ψz (x) = −ξ(x)ρz (x)ψz (x)

.



+(ξ (x) − 2iξ (x))e 2

−2i(x)



∞ x

e2i(y) ρz (y)ψz (y)dy.

(2.21)

686

D. R. Yafaev

Substituting expressions (2.20) and (2.21) into the left-hand side of (2.14), we see that it equals zero. Relations (2.19) are direct consequences of (2.18) and (2.20). 

Now it is easy to conclude the proof of Theorem 2.1. Proof Define the function .fz (x) by formula (2.11). According to Lemma 1 it satisfies differential equation (1.1), and according to Lemma 2 it has asymptotics (2.7). Moreover, differentiating (2.11), we obtain fz (x) = iξ(x)fz (x) + a(x)ei(x) ψz (x) + a  (x)ei(x) ψz (x).

.

As we have already seen, the first term on the right has asymptotics (2.9). It follows from relations (2.19) that the second and third terms are .o(ξ(x)a(x)) and .o(a  (x)), respectively. So, it remains to observe that o(a  (x)) = o(p(x)−1 a(x)−1 ) = o(ξ(x)a(x))

.

according to condition (2.8) and identity (2.5).



Theorem 2.1 defines the Jost solutions for .x > x0 . Then the functions .fz (x) are extended to all .x ≥ 0 as solutions of the differential equation (1.1). Let us introduce the second solution .f¯z¯ (x) of differential equation (1.1). It follows from asymptotics (2.7), (2.9) and identity (2.5) that the Wronskian   {fz , f¯z¯ } = 2i lim p(x)ξ(x)a 2 (x) = 2i

.

x→∞

(2.22)

so that these solutions are linearly independent. We also observe that a solution .fz (x) of Eq. (1.1) is determined uniquely by conditions (2.7) and (2.9). Indeed, if .f˜z (x) is another solution of Eq. (1.1) satisfying these conditions, then the Wronskian .{fz , f˜z } = 0 so that .f˜z (x) = cfz (x) for some .c ∈ C. This constant equals 1 according again to (2.7). Thus, the following result is a direct consequence of Theorem 2.1. Proposition 1 Under the assumptions (2.6) and (2.8) Eq. (1.1) is in the LC case .(at infinity.). This result is not really new; cf. Theorem XIII.6.20 in the book [2].

2.3 Arbitrary Solutions of the Homogeneous Equation An arbitrary solution of the Schrödinger equation (1.1) is a linear combination of the Jost solutions .fz (x) and .f¯z¯ (x). In particular, this is true for regular solutions .ϕz (x) and .θz (x) distinguished by the boundary conditions (2.1) or (2.2): ϕz (x) = σ+ (z)fz (x) + σ− (z)f¯z¯ (x)

.

(2.23)

Semiclassical Analysis in the Limit Circle Case

687

and θz (x) = τ+ (z)fz (x) + τ− (z)f¯z¯ (x),

.

(2.24)

where the coefficients .σ± (z) and .τ± (z) can be expressed via the Wronskians:

.

2iσ+ (z) = {ϕz , f¯z¯ },

2iσ− (z) = −{ϕz , fz },

2iτ+ (z) = {θz , f¯z¯ },

2iτ− (z) = −{θz , fz }.

σ− (z) = σ+ (¯z)

τ− (z) = τ+ (¯z)

(2.25)

Observe that and

.

(2.26)

because .ϕz (x) = ϕz¯ (x) and .θz (x) = θz¯ (x). Of course, all coefficients .σ± (z) and τ± (z) are entire functions of z. According to (2.23) and (2.24) the following result is a direct consequence of Theorem 2.1.

.

Theorem 2.2 Under the assumptions of Theorem 2.1 the solutions .ϕz (x) and .θz (x) of Eq. (1.1) have asymptotics   ϕz (x) = a(x) σ+ (z)ei(x) + σ− (z)e−i(x) + o(1)

(2.27)

  θz (x) = a(x) τ+ (z)ei(x) + τ− (z)e−i(x) + o(1)

(2.28)

.

and .

as .x → ∞. These asymptotic formulas can be differentiated in x; in particular,   ϕz (x) = iξ(x)a(x) σ+ (z)ei(x) − σ− (z)e−i(x) + o(1) .

.

(2.29)

In view of conditions (2.1) or (2.2) the Wronskian .{ϕz , θz } = 1. On the other hand, we can calculate this Wronskian using relations (2.22) and (2.23), (2.24). This yields an identity   2i σ+ (z)τ− (z) − σ− (z)τ+ (z) = 1,

.

∀z ∈ C.

(2.30)

Below, we need also the following fact. Proposition 2 Under the assumptions of Theorem 2.1 we have an identity  |σ− (z)| − |σ+ (z)| = Im z

.

2

2

0

∞

|ϕz (x)|2 dx.

(2.31)

688

D. R. Yafaev

Proof Multiplying Eq. (1.1) for .ϕz (x) by .ϕ¯z (x), integrating over .(0, ∞) and taking the imaginary part, we see that  .

− Im 0

x

  p(y)ϕz (y) ϕ¯z (y)dy = Im z



x

|ϕz (y)|2 dy.

0

Next, we integrate by parts on the left and take into account the boundary condition (1.4) whence .

  − Im p(x)ϕz (x)ϕ¯z (x) = Im z



x

|ϕz (y)|2 dy.

(2.32)

0

It follows from asymptotic formulas (2.27) and (2.29) that   p(x)ϕz (x)ϕ¯z (x) =i |σ+ (z)|2 − |σ− (z)|2   + i σ+ (z)σ¯ − (z)e2i(x) − σ− (z)σ¯ + (z)e−2i(x) + o(1)

.

where equality (2.5) has been used. Let us take the imaginary part of this expression. Then the second term on the right disappears. Substituting this expression into (2.32) and passing to the limit .x → ∞, we arrive at (2.31). 

2.4 Conditions on the Coefficients Let us discuss the assumptions of Theorem 2.1. The main condition is .a ∈ L2 (R+ ). It requires that the product .p(x)q(x) → ∞ sufficiently rapidly, roughly speaking, faster than .x 2 . The second inclusion (2.6) as well as condition (2.8) play auxiliary roles. They mean that .p(x) does not grow too rapidly compared to .a(x) and exclude too wild oscillations of the functions .p(x) and .a(x). For example, for the functions β γ .p(x) = x , .q(x) = x (for large x) conditions (2.6) and (2.8) are satisfied if β + γ > 2 and

.

β − γ < 2.

(2.33)

This implies that, necessarily, .γ > 0, but it may be an arbitrary small number. Observe that .β → 2 if .γ → 0. It is noteworthy that (2.33) allows negative .β provided .γ > 2 + |β|. In particular, according to Proposition 1 for such .β and .γ the operator .Hmin is in the LC case. Condition on .γ is very important here. Indeed, the results of [10] show that if .q(x) = 0 and .p(x) → 0 very rapidly, then the corresponding Schrödinger operator is selfadjoint, its spectrum is absolutely continuous and coincides with .[0, ∞).

Semiclassical Analysis in the Limit Circle Case

689

3 Schrödinger Operators and Their Quasiresolvents We refer to the books [3], §17, and [6], Sect. X.1, for background information on the theory of symmetric differential operators.

3.1 Minimal and Maximal Operators We here consider differential operators (1.3) in the space .L2 (R+ ). The scalar product in this space is denoted . ·, ·; I is the identity operator. Let us first define a minimal operator .H00 by the equality .H00 u = Hu on domain 2 .D(H00 ) that consists of functions .u ∈ C (R+ ) such that .u(x) = 0 for sufficiently  large x, limits .u(+0) =: u(0), .u (+0) =: u (0) exist and condition (1.4) is satisfied. Thus, the boundary condition (1.4) at .x = 0 is included in the definition of the operator .H00 so that its self-adjoint extensions are determined by conditions for .x → ∞ only. The closure of .H00 will be denoted .Hmin . This operator is symmetric in the space 2 .L (R+ ), and under assumptions of this paper its domain .D(Hmin ) can be described ∗ =: H efficiently (see Proposition 4). The adjoint operator .Hmin max is again given by the formula .Hmax u = Hu on the set .D(Hmax ) consisting of functions u in the local Sobolev space .H2loc , satisfying boundary condition (1.4) and such that .u ∈ L2 (R+ ) and .Hu ∈ L2 (R+ ). In the LC case, the operator .Hmax is not symmetric. Integrating by parts, we see that for all .u, v ∈ D(Hmax )  

Hu, v − u, Hv = lim p(x) u(x)v¯  (x) − u (x)v(x) ¯

.

x→∞

(3.1)

where the limit in the right-hand side exists but is not necessarily zero. Recall that ∗∗ ∗ Hmin = Hmin = Hmax .

.

The operator .Hmin is self-adjoint if and only if the LP case occurs. In this paper we are interested in the LC case when ∗ Hmin = Hmax = Hmin .

.

Self-adjoint extensions H of the operator .Hmin satisfy the condition ∗ Hmin ⊂ H = H ∗ ⊂ Hmin =: Hmax .

.

In the LC case the operators .Hmax are not symmetric.

690

D. R. Yafaev

Since the operator .Hmin commutes with the complex conjugation, its deficiency indices d± := dim ker(Hmax − zI ),

.

± Im z > 0,

are equal, i.e. .d+ = d− =: d, and, so, .Hmin admits self-adjoint extensions. For an arbitrary .z ∈ C, all solutions of Eq. (1.1) with boundary condition (1.4) are given by the formula .u(x) = ϕz (x) for some . ∈ C. They belong to .D(Hmax ) if and only if .ϕz ∈ L2 (R+ ). Therefore .d = 0 if .ϕz ∈ L2 (R+ ) for .Im z = 0; otherwise .d = 1.

3.2 Quasiresolvent of the Maximal Operator Recall that in the LC case inclusions (2.3) are satisfied. Following [12], let us define, for all .z ∈ C, a bounded operator .R(z) in the space .L2 (R+ ) by the equality 

x

(R(z)h)(x) = θz (x)

.



∞

ϕz (y)h(y)dy + ϕz (x)

0

θz (y)h(y)dy.

(3.2)

x

Actually, the operator .R(z) belongs to the Hilbert-Schmidt class. It depends analytically on .z ∈ C and .R(z)∗ = R(¯z). We prove (see Theorem 3.3) that, in a natural sense, .R(z) plays the role of the resolvent of the operator .Hmax . We call it the quasiresolvent of the operator .Hmax . Let us enumerate some simple properties of the operator .R(z). Differentiating definition (3.2), we see that 

(R(z)h) (x) =

.

θz (x)



x

ϕz (y)h(y)dy 0

+ ϕz (x)



∞

θz (y)h(y)dy

(3.3)

x

for all .h ∈ L2 (R+ ). In particular, it follows from relations (3.2) and (3.3) that .

(R(z)h)(0) = ϕz (0) h, θz¯ 

(3.4)

(R(z)h) (0) = ϕz (0) h, θz¯ 

(3.5)

and .

where .ϕz (0) and .ϕz (0) are defined by equalities (2.1) or (2.2). A proof of the following statement is close to the construction of the resolvent for essentially self-adjoint Schrödinger operators. Theorem 3.3 Let inclusions (2.3) hold true. For all .z ∈ C, we have R(z) : L2 (R+ ) → D(Hmax )

.

(3.6)

Semiclassical Analysis in the Limit Circle Case

691

and (Hmax − zI )R(z) = I.

(3.7)

.

Proof Let .h ∈ L2 (R+ ) and .u(x) = (R(z)h)(x). Boundary condition (1.4) is a direct consequence of relations (3.4) and (3.5). Differentiating (3.3), we see that 



(p(x)u (x)) =

.

(p(x)θz (x))

+ (p(x)ϕz (x))



∞ x



x

ϕz (y)h(y)dy 0

  θz (y)h(y)dy + p(x) θz (x)ϕz (x) − θz (x)ϕz (x) h(x). (3.8)

Since the Wronskian .{ϕz , θz } = 1, the last term in the right-hand side equals .−h(x). Putting now together equalities (3.2) and (3.8) and using Eq. (1.1) for the functions .ϕz (x) and .θz (x), we find that .

− (p(x)u (x)) + q(x)u(x) − zu(x) = h(x)

where .h ∈ L2 (R+ ). Together with boundary condition (1.4) this implies that .Hmax u − zu = h. This yields both (3.6) and (3.7). 

Note that solutions .u(x) of differential equation (1.1) satisfying condition (1.4) are given by the formula .u(x) = ϕz (x) for some . ∈ C. Therefore we can state Corollary 1 All solutions of the equation (Hmax − zI )u = h

.

where

z∈C

and

h ∈ L2 (R+ )

for .u ∈ D(Hmax ) are given by the formula u = ϕz + R(z)h for some

.

 = (z; h) ∈ C.

(3.9)

A following asymptotic relation for .(R(z)h)(x) is a direct consequence of definition (3.2) and condition (2.3): (R(z)h)(x) = θz (x) h, ϕz¯  + o(|ϕz (x)| + |θz (x)|)

.

as

x → ∞.

(3.10)

4 Self-adjoint Extensions and Their Resolvents Here we find an asymptotic behavior as .|x| → ∞ of all functions .u(x) in the domain of the maximal operator .Hmax . This allows us to give an efficient description of all self-adjoint extensions of the operator .Hmin .

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D. R. Yafaev

4.1 Domains of Maximal Operators Recall that boundary condition (1.4) at .x = 0 is included in our definition of the minimal operator .Hmin . Our goal now is to find a similar condition for .x → ∞ distinguishing self-adjoint extensions of .Hmin . The starting point of our construction is asymptotic formula (1.5) for functions .u ∈ Hmax . Recall that the amplitude .a(x) and the phase .(x) were defined by formulas (2.4). The coefficients .σ± (z), .τ± (z) are given by equalities (2.25), and the number .(z; h) is determined by formula (3.9). Theorem 4.4 Let the assumptions of Theorem 2.1 be satisfied. Then an arbitrary function .u ∈ D(Hmax ) has asymptotics (1.5) with some coefficients .s± = s± (u). They can be constructed by relations s+ (u) = (z; (H − zI )u)σ+ (z) + (H − zI )u, ϕz¯ τ+ (z), .

(4.1)

s− (u) = (z; (H − zI )u)σ− (z) + (H − zI )u, ϕz¯ τ− (z)

where the number .z ∈ C is arbitrary. Conversely, for arbitrary .s+ , s− ∈ C, there exists a function .u ∈ D(Hmax ) such that asymptotics (1.5) holds. Proof According to Corollary 1 a function .u ∈ D(Hmax ) admits representation (3.9) where the operator .R(z) is defined by equality (3.2). In view of relation (3.10) and asymptotics (2.27), (2.28) we have   (R(z)h)(x) = a(x) τ+ (z)ei(x) + τ− (z)e−i(x) h, ϕz¯  + o(a(x)),

.

x → ∞, (4.2)

for all functions .h ∈ L2 (R+ ). Therefore it follows from (3.9) that   u(x) =a(x)(z; (H − zI )u) σ+ (z)ei(x) + σ− (z)e−i(x)   + a(x) τ+ (z)ei(x) + τ− (z)e−i(x) (H − zI )u, ϕz¯  + o(a(x))

.

as .x → ∞. This yields relation (1.5) with the coefficients .s± defined by (4.1). Conversely, given .s+ and .s− and fixing some .z ∈ C, we consider a system of equations s+ = σ+ (z) + h, ϕz¯ τ+ (z), .

s− = σ− (z) + h, ϕz¯ τ− (z).

(4.3)

for . and . h, ϕz¯ . According to (2.30) the determinant of this system is not zero so that . and . h, ϕz¯  are uniquely determined by .s+ and .s− . Then we take any h such that its scalar product with .ϕz¯ equals the found value of . h, ϕz¯ . Finally, we

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define u by formula (3.9). Asymptotics as .x → ∞ of .ϕz (x) and .(R(z)h)(x) are given by formulas (2.27) and (4.2), respectively. In view of Eq. (4.3) this leads to asymptotics (1.5). 

Theorem 4.4 yields a mapping .D(Hmax ) → C2 defined by the formula u → (s+ (u), s− (u)).

.

(4.4)

The construction of Theorem 4.4 depends on the choice of .z ∈ C, but this mapping is defined intrinsically. In particular, we can set .z = 0 in all formulas of Theorem 4.4. Note that mapping (4.4) is surjective. Under the assumptions of Theorem 2.1 the right-hand side of (3.1) can be expressed in terms of the coefficients .s+ and .s− . Proposition 3 For all .u, v ∈ D(Hmax ), we have an identity  

Hmax u, v − u, Hmax v = 2i s+ (u)s+ (v) − s− (u)s− (v) .

.

(4.5)

Proof Let us proceed from equality (3.1). It follows from formula (1.5) that .

  − ip(x)u (x)v(x) ¯ = s+ (u)ei(x) − s− (u)e−i(x)   × s+ (v)e−i(x) + s− (v)ei(x) + o(1) = s+ (u)s+ (v) − s− (u)s− (v) + s+ (u)s− (v)e2i(x) − s− (u)s+ (v)e−2i(x) + o(1).

and, similarly,   ip(x)u(x)v¯  (x) = s+ (u)ei(x) + s− (u)e−i(x)   × s+ (v)e−i(x) − s− (v)ei(x) + o(1)

.

= s+ (u)s+ (v) − s− (u)s− (v) − s+ (u)s− (v)e2i(x) + s− (u)s+ (v)e−2i(x) + o(1). Let us take the sum of the last two expressions and observe that the terms containing e2i(x) and .e−2i(x) cancel each other. This yields

.

.

− ip(x)(u (x)v(x) ¯ − u(x)v¯  (x)) = 2s+ (u)s+ (v) − 2s− (u)s− (v) + o(1).

Passing here to the limit .n → ∞ and using equality (3.1), we obtain identity (4.5). 

We can now characterize the set .D(Hmin ).

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Proposition 4 A vector .v ∈ D(Hmax ) belongs to .D(Hmin ) if and only if .v(x) = o(a(x)) as .x → ∞, that is, s+ (v) = s− (v) = 0.

.

(4.6)

∗ ) if and only if Proof A vector v belongs to .D(Hmax

Hmax u, v = u, Hmax v

.

(4.7)

for all .u ∈ D(Hmax ). According to Proposition 3 equality (4.7) is equivalent to s+ (u)s+ (v) − s− (u)s− (v) = 0.

.

(4.8)

This is of course true if (4.6) is satisfied. Conversely, if (4.8) is satisfied for all u ∈ D(Hmax ), we use that according to Theorem 4.4 the numbers .s+ (u) and .s− (u) are arbitrary. This implies (4.6). 

.

This result shows that (4.4) considered as a mapping of the factor space D(Hmax )/D(Hmin ) onto .C2 is injective.

.

4.2 Self-adjoint Extensions All self-adjoint extensions .Hω of the operator .Hmin are parameterized by complex numbers .ω ∈ T ⊂ C. Let the set .D(Hω ) ⊂ D(Hmax ) of vectors u be distinguished by condition (1.6). Theorem 4.5 Let the assumptions of Theorem 2.1 be satisfied. Then for all .ω ∈ T, the operators .Hω are self-adjoint. Conversely, every operator H such that Hmin ⊂ H = H ∗ ⊂ Hmax

.

(4.9)

equals .Hω for some .ω ∈ T. Proof We proceed from Proposition 3. If .u, v ∈ D(Hω ), it follows from condition (1.6) that .s+ (u)s+ (v) = s− (u)s− (v). Therefore according to equality (4.5) ∗ ∗ ∗ . Hω u, v = u, Hω v whence .Hω ⊂ Hω . If .v ∈ D(Hω ), then . Hω u, v = u, Hω v for all .u ∈ D(Hω ) so that in view of (4.5) equality (4.8) is satisfied. Therefore .s− (u)(ωs+ (v) − s− (v)) = 0. Since .s− (u) is arbitrary, we see that .ωs+ (v) − s− (v) = 0, and hence .v ∈ D(Hω ). Suppose that an operator H satisfies conditions (4.9). Since H is symmetric, it follows from Proposition 3 that equality (4.8) is true for all .u, v ∈ D(H ). Setting here .u = v, we see that .|s+ (v)| = |s− (v)|. There exists a vector .v0 ∈ D(H ) such that .s− (v0 ) = 0 because .H = Hmin . Let us set .ω = s+ (v0 )/s− (v0 ). Then .|ω| = 1 and relation (1.6) is a direct consequence of (4.8). 

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We emphasize that our description of self-adjoint extensions of operator (1.3) defined on .C0∞ (R+ ) by boundary conditions at infinity is similar in spirit to the same procedure at the point .x = 0. Note that (4.4) plays the role of the mapping 2 .  .u → (u(0), u (0)) for functions in the local Sobolev class .H loc

4.3 Resolvent Now it easy to construct the resolvents of the operators .Hω defined in the previous subsection. We previously note that, by definition (1.5), s± (ϕz ) = σ± (z),

.

s± (θz ) = τ± (z)

and .|σ+ (z)| = |σ− (z)| for .Im z = 0, by Proposition 2. Theorem 4.6 Let the assumptions of Theorem 2.1 be satisfied. Then for all .z ∈ C with .Im z = 0 and all .h ∈ L2 (R+ ), the resolvent .Rω (z) = (Hω − zI )−1 of the operator .Hω is given by the equality Rω (z)h = γω (z) h, ϕz¯ ϕz + R(z)h

.

(4.10)

where γω (z) = −

.

τ+ (z) − ωτ− (z) . σ+ (z) − ωσ− (z)

(4.11)

Proof According to Corollary 1 a vector .u = Rω (z)h is given by formula (3.9) where the coefficient . is determined by condition (1.6). It follows from Theorem 4.4 than the function .u(x) has asymptotics (1.5) with the coefficients .s± defined by relations (4.3). Thus, .u ∈ D(Hω ) if and only if   σ+ (z) + τ+ (z) h, ϕz¯  = ω σ− (z) + τ− (z) h, ϕz¯ 

.

whence =−

.

τ+ (z) − ωτ− (z)

h, ϕz¯ . σ+ (z) − ωσ− (z)

Substituting this expression into (3.9), we arrive at formulas (4.10), (4.11).



Corollary 2 For .Im z = 0, the resolvents .Rω (z) belong to the Hilbert-Schmidt class, whence the spectra of the operators .Hω are discrete. This result is well known; see, e.g., Theorem 10 in Chapter VII of the book [3] or Theorem 5.8 in the book [9].

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It follows from formula (4.11) that the spectrum of the operator .Hω consist of the points z where σ+ (z) − ωσ− (z) = 0.

.

(4.12)

Since the functions .σ+ (z) and .σ− (z) are analytic, this again implies that the spectra of .Hω are discrete. Of course the roots z of Eq. (4.12) lie on the real axis because .|σ+ (z)| = |σ− (z)| for .Im z = 0. This fact had of course to be expected since z are eigenvalues of the self-adjoint operator .Hω . We finally note that the discreteness of the spectrum of the operators .Hω is quite natural because their domains .D(Hω ) are distinguished by boundary conditions at both ends of .R+ . Therefore .Hω acquire some features of regular operators.

4.4 Spectral Measure In view of the spectral theorem, Theorem 4.6 yields a representation for the CauchyStieltjes transform of the spectral measure .dEω (λ) of the operator .Hω . Theorem 4.7 Let inclusions (2.3) hold. Then for all .z ∈ C with .Im z = 0 and all h ∈ L2 (R+ ), we have an equality

.



∞

.

−∞

(λ − z)−1 d(Eω (λ)h, h) = γω (z)| ϕz , h|2 + (R(z)h, h).

(4.13)

Recall that the operators .R(z) are defined by formula (3.2). Therefore .(R(z)h, h) are entire functions of .z ∈ C, and the singularities of the integral in (4.13) are determined by the function .γω (z). Thus, (4.13) can be considered as a modification of the classical Nevanlinna formula (see his original paper [4] or, for example, formula (7.6) in the book [7]) for the Cauchy-Stieltjes transform of the spectral measure in the theory of Jacobi operators. We mention however that, for Jacobi operators acting in the space .L2 (Z+ ), there is the canonical choice of a generating vector and of a spectral measure. This is not the case for differential operators in 2 .L (R+ ). We finally note an obvious fact: if .λ is an eigenvalue of an operator .Hω , then the corresponding eigenfunctions equal .cϕλ (x) where .c ∈ C.

4.5 Concluding Remarks In the LC case, self-adjoint extensions of the operator .Hmin are traditionally described by the following procedure; see the classical books [1], Chapter IX, [3], §17, 18, and the recent monograph [7], Sect. 14.4, 15.3 and 15.4. Take any real

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functions .j ∈ D(Hmax ), .j = 1, 2, such that .

  lim p(x) 1 (x)2 (x) − 1 (x)2 (x) = 0,

x→∞

and set .s (x) = s1 (x) + 2 (x) where .s ∈ R, .∞ (x) = 1 (x). Let a set .Ds ⊂ D(Hmax ) be distinguished by the condition .

  lim p(x) u (x)s (x) − u(x)s (x) = 0

x→∞

for u ∈ Ds ,

s ). Then the operators .H s be the restriction of .Hmax on the set .Ds =: D(H s and let .H are self-adjoint, and all self-adjoint extensions of the operator .Hmin coincide with s for some .s ∈ R ∪ {∞}. This construction does not look very one of the operators .H efficient, in particular, because it depends on the choice of the functions .1 , 2 . Another possibility is to use von Neumann formulas. They were conveniently adapted to operators commuting with the complex conjugation in the survey [8], Theorem 2.6, and then applied in this paper to Jacobi operators (see also Sect. 16.3 in [7]). Following [12], we briefly describe here this construction for Schrödinger operators (1.3). Recall that .ϕ0 (x) and .θ0 (x) are the solutions of Eq. (1.1) where .z = 0 satisfying conditions (2.1) or (2.2). We set .θ˜0 (x) = ω(x)θ0 (x) where .ω(x) is a smooth function such that .ω(x) = 0 for small x and .ω(x) = 1 for large x. Define operators .H (t) as the restrictions of .Hmax on direct sums D(Hmin )  {tϕ0 + θ˜0 } =: D(H (t) )

.

for

t ∈R

(4.14)

and .D(Hmin )  {ϕ0 } =: D(H (∞) ). Then the operators .H (t) are self-adjoint, and all self-adjoint extensions of the operator .Hmin coincide with one of the operators (t) for some .t ∈ R ∪ {∞}. A drawback of this construction is that, apparently, .H the set .D(Hmin ) cannot be described efficiently without some assumptions on the coefficients .p(x) and .q(x). Finally, we indicate a link between the operators .H (t) and .Hω considered in this paper. Let .u ∈ D(H (t) ) where .t ∈ R. According to Theorem 2.2 and Proposition 4 it follows from definition (4.14) that .u(x) has asymptotics (1.5) where s+ (u) = tσ+ (0) + τ+ (0)

.

and

s− (u) = tσ− (0) + τ− (0).

Therefore relation (1.6) is satisfied with ω=

.

tσ+ (0) + τ+ (0) . tσ− (0) + τ− (0)

(4.15)

If .u ∈ D(H (∞) ), then this equality holds true with .t = ∞, that is, .ω = σ+ (0)σ− (0)−1 . Note that .|ω| = 1 by virtue of identity (2.26). This shows that .u ∈ D(Hω ); see the definition at the beginning of Sect. 4.2. Conversely, if .u ∈ D(Hω ), then .u ∈ D(H (t) ) with t determined by (4.15).

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Acknowledgement Supported by RFBR grant No. 20-01-00451 A.

References 1. E.A. Coddington, N. Levinson, in Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955) 2. N. Dunford, J.T. Schwartz, in Linear Operators. Part 2 (Interscience Publishers, New York, 1963) 3. M.A. Naimark, in Linear Differential Operators (Ungar, New York, 1968) 4. R. Nevanlinna, Asymptotische Entwickelungen beschränkter Funktionen und das Stieltjessche Momentenproblem. Ann. Acad. Sci. Fenn. A 18(5), 52 (1922) 5. F.W.J. Olver, in Introduction to Asymptotics and Special Functions (Academic Press, New York, 1974) 6. M. Reed, B. Simon, in Methods of Modern Mathematical Physics, vol. II (Academic Press, 1975) 7. K. Schmüdgen, in Unbounded Self-Adjoint Operators on Hilbert Space (Springer, Dordrecht, 2012) 8. B. Simon, The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998) 9. E.C. Titchmarsh, in Eigenfunction Expansions Associated with Second-Order Differential Equations, vol. 1 (Oxford, 1946) 10. D.R. Yafaev, Spectral and scattering theory for differential and Hankel operators. Adv. Math. 308, 713–766 (2017) 11. D.R. Yafaev, Self-adjoint Jacobi operators in the limit circle case. J. Oper. Theory 89(1), 101–117 (2022). arXiv 2104.13609 12. D.R. Yafaev, Self-adjoint differential operators in the limit circle case. SIGMA 17, 13 (2021). 077. arXiv 2105.08641