Schrödinger Operators, Spectral Analysis and Number Theory 9783030684891, 9783030684907

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Springer Proceedings in Mathematics & Statistics

Sergio Albeverio Anindita Balslev Ricardo Weder   Editors

Schrödinger Operators, Spectral Analysis and Number Theory In Memory of Erik Balslev

Springer Proceedings in Mathematics & Statistics Volume 348

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

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Sergio Albeverio · Anindita Balslev · Ricardo Weder Editors

Schrödinger Operators, Spectral Analysis and Number Theory In Memory of Erik Balslev

Editors Sergio Albeverio Institute for Applied Mathematics and Hausdorff Center for Mathematics University of Bonn Bonn, Germany

Anindita Balslev Hoejbjerg, Denmark

Ricardo Weder Departamento de Física Matemática Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas Universidad Nacional Autónoma de México Mexico City, Mexico

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-030-68489-1 ISBN 978-3-030-68490-7 (eBook) https://doi.org/10.1007/978-3-030-68490-7 Mathematics Subject Classification: 46N30, 47B93, 81Q10, 81Q15, 81Q35, 81U24, 11F72 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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 Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Erik Balslev belongs to the rank of mathematicians who most influenced the development of mathematical physics since the sixties of the last century, especially through his groundbreaking work on resonances and complex dilations. His work has also had a strong impact upon other areas, including analytic number theory. To those who had the chance to know him personally, the news of his untimely departure came like a shock and left a sense of disconsolateness and grief. Two of the editors, S. A. and R. W., had been in close friendship with him, and in the case of S. A. also with his wife, Anindita. Through contact with Anindita, the third editor, the idea came about to honor the scientific legacy of Erik. For this purpose, we first considered the possibility of organizing a conference in his memory, with the participation of mathematicians who had worked in the scientific areas of interest of Erik. However, we later decided to bring out a volume in his memory, and we invited mathematicians who were collaborators of Erik, or who worked in areas of Erik’s mathematical interests. The book is organized into two parts, called A and B. Part A consists of sections concerned with the personal life and the scientific work of Erik, including a list of his publications. It is complemented by two important contributions, by Anindita Balslev and André Verbeure. The contribution of Anindita is a moving exposition of Erik’s widow, where she gives a vivid account of their life in common, of Erik’s role as a dedicated father, of Erik’s broad interests beyond mathematics, in Indian philosophy, in classical Western music. All this is intertwined with beautiful anecdotes of their encounters with distinguished personalities of science and philosophy. The contribution of André, a close friend of Erik, presents remembrances of a blossoming period of the history of quantum theory related to the time when Erik and André had been together in Marseille and underlines the strong scientific personality of Erik and his influence altogether. Both are living testimonials of a great person who could well combine his outstanding scientific activity with the unfolding of a sensitive and gentle human personality, with an openness of mind going beyond cultural and geographical borders. v

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Part B consists of contributions by specialists of two main areas of activity of Erik, spectral theory as applied to quantum mechanics and as applied to analytic number theory. It also contains a contribution in another area, by a neurobiologist, Nicholas A. Leibovic, who had collaborated with Erik. The contributions are listed in alphabetic order and are summarized in the introduction to part B. They relate to work by Erik, directly or indirectly. In our perspective they indicate ways in which the legacy of Erik is evolving. Many of the authors had collaborated with him or have known him personally, and it is fair to say that all authors have been influenced by his work. The book could never have been published without the very friendly, steady, patient, and supportive help by Marina Reizakis, of Springer Verlag. Our warm and sincere thanks to her. Hearty thanks also to Timo Weiss, a Ph.D. student of philosophy and mathematics, who in parallel to his work on his thesis, not only took care of a competent type-setting of the manuscript, but also helped us on many occasions with useful suggestions. Bonn, Germany Hoejbjerg, Denmark Ciudad de México, México November 2020

Sergio Albeverio Anindita Balslev Ricardo Weder

Short Biography of Erik Balslev

Erik Balslev was born in Haurum, Denmark on September 27, 1935. His father William Balslev was a minister, an erudite theologian, and his mother, Anna Balslev, was a pianist trained at the Royal Danish Conservatory of Music in Copenhagen. Erik received his Master’s degree from the University of Aarhus in 1961 and his Ph.D. from the University of California, Berkeley in 1963. After that he taught for a year at UCLA and returned to Denmark. He, then, left for France for his postDoctoral work. During that period he was associated with Institut de Poincaré, Paris and IHES, Bures-sur-Yvette as well as with the University of Marseille. It was during his stay in Paris that he met Anindita at the Sorbonne, his wife to be, where she was doing her doctoral work in philosophy. Right after their marriage in 1967, he was invited to the Indian Statistical Institute, Kolkata by Prof. P. C. Mahalanobis, the founder of said institute. From 1968–71, he was a visiting professor at the State University of New York, Buffalo. This was followed by a year in Denmark, France, and India, after which he again returned to USA as a visiting professor at the University of California, Los Angeles from 1972–74. His daughter Eva (M.A. econ, UC Berkeley) was born in Buffalo and son Olav (Master’s in Music Performance (Cello) Royal Danish Conservatory, Copenhagen & E.M.C.I. Copenhagen Business School) was born in Los Angeles. vii

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Short Biography of Erik Balslev

Erik Balslev joined the Institute of Mathematics, Aarhus University, Denmark and served as a professor there from 1974 to 2005. Apart from short term invitations to several places such as the Tata Institute, Mumbai, University of Bielefeld, Germany, and Mittag-Leffler Institute in Stockholm, he was a Fellow at the Institute for Advanced Study, Princeton in 1986–87, and was a visiting professor at the University of Virginia, Charlottesville from 1988 to 1990. As Emeritus professor, he was actively pursuing his intellectual interests. He passed away in Goa, India on January 11, 2013.

Scientific Contributions of Erik Balslev Sergio Albeverio and Ricardo Weder

Erik Balslev obtained his Master’s degree at the University of Aarhus, Denmark, in 1961, and after that he went to the University of California at Berkeley where he obtained his Ph.D. in 1963 under the direction of Prof. Tosio Kato. The title of his dissertation was Perturbation Theory of Differential Operators. In this and in various publications in the years 1962–1968, he established important results concerning operators that are relatively compact with respect to certain ordinary differential operators. The first paper in this raw is [1] where sufficient and necessary conditions are given for β(x)y [k] (where β(x) is a function, and where y [k] is an associated quasi-derivative of order k ≤ 2n − 1 acting in a domain in L 2 (a, b), (a, b) being an open interval) to be L−compact, L being a closed extension of the minimal operator determined by an ordinary differential expression of order 2n. In a paper with T. W. Gamelin [2], Erik Balslev determined the essential spectrum and the index of a class of ordinary differential operators, and Euler’s operator. Let us note that these two papers are both quoted in basic reference bodies like T. Kato, Perturbation Theory of Linear Operators, Second Edition, Springer, (1976), and N. Dunford and J. T. Schwartz, Linear Operators Part III, Wiley, (1971). This work was later extended to the case of elliptic operators acting in Banach spaces L p (Rn ) in [3]. In a related paper [4] the singular spectrum of such operators is discussed. Another important paper along such lines is on the discreteness of the spectrum of a perturbed Schrödinger operator [5]. Part of this work was done during the time following his Ph.D. studies, after wich Erik Balslev first took up a position at the University of California, Los Angeles (1963–1964), and then moved back to Europe. After a brief stay in Aarhus, he visited Institut Henri Poincaré, Paris, Institut des Hautes Études Scientifiques (IHES) at Bures-sur-Ivette, and Centre de Physique Théorique (CPT) of the Centre National de la Recherche Scientifique, Marseille (1966–1967). At the time there was at IHES and CPT a strong and dynamical group of research in mathematical physics under the leadership of Daniel Kastler. This environment proved to be very stimulating for Erik Balslev, and during this time, he made major contributions to mathematical physics. Erik Balslev worked in collaboration with André Verbeure on quasi-free states in Clifford algebras over a Hilbert space. These results were published in [6]. Furthermore, in collaboration with J. Manuceau and A. Verbeure ix

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[7], he established fundamental results on quantum field theory, in the representation of the anticommutation relations and Bogoliubov transformations. The discovery of the method of dilation analyticity for the study of many-body Schrödinger operators appeared in an outstanding publication [8] with J. M. Combes, following an earlier paper by J. Aguilar and J. M. Combes in the two-body case.1 Since then, this is referred to as the Balslev–Combes method of complex scaling. This truly seminal work played henceforth a fundamental role in the development of the theory of complex scaling in resonances and scattering in quantum mechanics, quantum chemistry, and quantum field theory. Erik Balslev’s ground breaking contributions to the development of this theory, see [20–22, 24–27, 30], have had a strong impact on all successive work. A complete documentation of Erik Balslev’s contributions can be found in the list of his publications in this volume. Here, we mention only some of them. He returned to work on many-body Schrödinger operators specially in [9] where he studied spectral properties in the case of two-body relatively compact interactions. The quantum particles that constitute the system can also be identical so that considerations of symmetry become important. The lower threshold of the continuum spectrum is studied and, in the case of dilation–analytic interactions, absence of singular continuous spectrum is established. Also, the structure and location of eigenvalues are exhibited, in particular the presence of infinitely many of them below the continuum. A continuation of this study is in [11, 12] where systems with interactions of the spin-orbit type are investigated, leading to the establishment of a virial theorem. Moreover, sufficient conditions for the holding of results on eigenvalues similar to those of [9] are found for all types of symmetries. Other related work is in [16, 19]. For the study of exponential asymptotic decay of eigenfunctions and in excluding positive eigenvalues embedded in the continuous spectrum (including the Balslev–Simon theorem2 ) see [13]. In [17], he considered scattering theory for two-body systems with dilation–analytic potentials. He proved that the resolvent and the scattering matrix have an analytic continuation to the lower half of the complex plane, and that resonances can equivalently be defined as poles of the resolvent and of the scattering matrix as well as complex eigenvalues of a family of operators considered as a complex dilation of the Hamiltonian. These results constitute also an important contribution to scattering theory with nonselfadjoint operators. In [18], he studied many-body systems below the smallest three-body threshold with dilation–analytic two-body potentials. He constructed the scattering matrix below the smallest three-body threshold. He proved that generally, the poles of the scattering matrix are resolvent resonances, but a resolvent resonance may not be a pole of the scattering matrix. The study of resonances in 3−body systems was pursued in [19, 21]. In [28], Erik Balslev considered two-body scattering and he proved that the generalized eigenfunctions have an analytic extension to a region in the lower

1 Aguilar J; Combes J. M., A class of analytic perturbations for one-body Schrödinger Hamiltonians.

Commun. Math. Phys. 22 (1971), 269–279. is quoted as such in M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic, London, (1978) (Theorem XIII:61, p. 237, and p. 354).

2 This

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half of the complex plane, provided that the scattering matrix has an analytic extension to the same region. In [29], he considered three-body systems with dilation– analytic potentials. He constructed local inverse wave operators and obtained results on the boundedness of resonances. In collaboration with B. Helffer [31], he studied the scattering theory for the Dirac operator with short-range and dilation–analytic potentials. In particular, they prove that the resonances defined as poles of the scattering matrix and as poles of the resolvent coincide. In a further publication with A. Grossmann and T. Paul [23], he provides another interesting characterization of dilation–analytic operators in a representation given by functions that are analytic on the complex upper half-plane. Erik Balslev had a short stay at the Indian Statistical Institute at Kolkata at the end of 1967 and then he was a visiting professor at the Department of Mathematics, State University of New York at Buffalo (1968–1971). There he met the applied mathematician and biophysicist K. N. Leibovic with whom he coworked on three papers on binocular perception (see Leibovic contribution in part B). In 1972–1974, Erik Balslev was a visiting professor at the University of California at Los Angeles where he wrote with D. Babbitt three important papers on dilation analyticity [10, 14, 15]. Then, he returned to Aarhus where since 1974 he was a Professor at the Institute of Mathematics (see also the Short Biography of Erik Balslev in part A in the present book). There he continued first his scientific work on Schrödinger operators that we mentioned above, before opening up in 1997 a new line of research. Let us thus now describe this new phase of the scientific production of Erik Balslev, concerned with a challenging program of applying spectral methods, and particularly his work with complex dilation methods to other problems, namely these at the cutting edge of geometry, complex analysis, and number theory. This is an extremely important work that gave birth to an entire new direction of mathematical investigations. Let us say beforehand that in part B of the present volume, in the contribution of Petridis and Risager, many aspects of this work and its legacy are clearly discussed, and we shall refer to it for more details on such aspects. In the paper [32], Erik Balslev considers certain manifolds M with constant negative curvature −1 outside a compact set, with m cusps, and the Laplacian on M. The latter has continuous spectrum [1/4, ∞) with multiplicity m and finitely many eigenvalues embedded in it.3,4 By applying dilation analytic methods, in particular B. Simon’s work,5 and rotating the continuous spectrum around 1/4 by an angle −2Argλ, λ, being a complex parameter, the embedded eigenvalues are unchanged for |Argλ| < π, and so are the isolated eigenvalues for |Argλ| < π. Balslev proves that embedded eigenvalues become discrete resonances after being crossed by the 3 Phillips,

R.; Sarnak, P., Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc., 5 (1992), no. 1, 1–32. 4 Müller, W. Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math. 109 (1992), 265–305. 5 Simon, B., Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. Math. (2) 97 (1973), 247–274.

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continuous spectrum. In this way, he gives a new proof of “Fermi’s golden rule” for this case (the first proof was given by R. S. Phillips and P. Sarnak by other methods). At the same time, Balslev provides the meromorphic extension of the resolvent of the Laplacian and the Eisenstein series for the manifolds he is studying. The work on this paper was continued, in particular, by Y. N. Petridis and M. Risager.6 See also the references in the contribution by the same authors in part B of this volume. The same contribution also describes how an intensive collaboration of Erik Balslev with Alexei B. Venkov was started in the 1990s and lead to a series of papers on which we shall now report. In [35], an important paper written by Erik Balslev in collaboration with A. B. Venkov, a congruence subgroup  of S L(2, Z) is considered and the non-euclidean Laplacian  on L 2 (\H ),H being the Poincaré upper half-plane, is studied. Selberg proved in 1965 that the first eigenvalue of  is larger than or equal to 3/16 and conjectured that the optimal lower bound is 1/4. After that, Luo, Rudlick, and Sarnak (1995) proved λ1 ≥ 171/184 using the twisted Hecke L−series. Balslev and Venkov use in [35] a different technique to prove bounds to the left of the interval, just below 1/4. This technique uses Rankin–Selberg’s type L− functions to show, e.g., that there are no odd eigenfunctions of  for the group  = 0 (N ) with eigenvalues in the interval (1/4 − C/(logN )2 , 1/4). This work has been continued by Risager7 to which we refer for further discussion. In [33, 34], Balslev and Venkov consider again the situation given by a pair (, χ ) where  is a Fuchsian group of the first kind and χ is an associated character. They study the property of the pair being essentially cuspidal, in the sense that the cuspidal spectrum of the corresponding automorphic Laplacian (, χ ) satisfies Weyl’s asymptotic law. They provide very interesting criteria for the essential cuspidity clarifying the importance of a certain multiplicity assumption made by Phillips and Sarnak8 for  = (2). In particular, they obtain a sufficient condition for an important class of arithmetic groups not to safisfy a Weyl law (Theorem 3). This work was continued by D. Mayer, A. Momeni, and A. Venkov,9 M. S. Risager, (see footnote 7) and by E. Balslev and A. Venkov in [38]. In [36, 37], a situation as in [33, 34] is analyzed for the case where  = 0 (N ) (the standard congruence subgroup of S L(2, Z) of level N ∈ N ) and χ being a character of  defined by a real, even, and primitive character modulo N . Very detailed results are provided. In [38], by Balslev and Venkov, methods related to work by L. Faddeev on the Laplace operator on the upper half-plane H are used to prove a theorem on expansion 6 Petridis

Y. N.; Risager M. S., Dissolving of cusps forms: higher order Fermi’s golden rule, Mathematika 59 (2013), no. 2, 269–301. 7 Risager, M. S., On Selberg’s small eigenvalue conjecture and residual eigenvalues, J. Reine Angew. Math. 656 (2011), 179–211. 8 Phillips, R.; Sarnak, P, Cusp forms for character varieties, Geom. Funct. Anal. 4 (1994), no. 1, 93– 118. 9 Mayer, D.; Momeni, A.; Venkov, A., Congruence properties of induced representations and their applications. Algebra i Analiz 26 (2014), no. 4, 129–147; reprinted in St. Petersburg Math. J. 26 (2015), no. 4, 593–606.

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in terms of its automorphic eigenfunctions. For the case of the automorphic Laplacian for any congruence subgroup of the modular group, Selberg had proven a Weyl law for the asymptotics as λ → ∞ of the number N (λ, ) of eigenvalues that are less or equal to λ and are embedded in the continuous spectrum. The question was asked whether the Weyl law is a characteristic of the congruence subgroup, or whether it may be that the same asymptotic law might be valid also for some more general cofinite groups . To investigate this problem, Phillips and Sarnak10 had already introduced and studied the stability theory for the Laplacian (, χ ) under certain variations of the group  in the Teichmüller space of a given congruence subgroup 0 , and varying the representation χ in the Jacobi manifold of a given congruence character χ0 . Based on this work, Balslev and Venkov in [38] consider the Hecke groups 0 (N ) with primitive congruence character χ0 and the automorphic Laplacian (0 (N ), χ0 ). The perturbations are taken to be of the form α M + α 2 N , with α a (small) parameter, M a first-order differential operator, and N a multiplication operator. For certain choices of N regular perturbations in the Jacobi manifold of χ0 are found, and a lower bound on the number of eigenfunctions that become resonance functions for α = 0 is proven. The last published paper in this area by Balslev, together with Venkov, seems to be [39]. It is a very useful survey of the connection between embedded eigenvalues in automorphic Laplacians and certain problems of number theory. In particular, the results of [36, 37] proving a Fermi’s golden rule are reviewed and commented upon, specially the case of the Hecke subgroup 0 (8) of the modular group. Eisenstein series and zeros of classical Dirichlet L−series are mentioned, continuing Selberg’s and Phillips–Sarnak work, that concerned 0 (4), to the case of 0 (8). Alternatives for the behavior under certain perturbations are enumerated. Expansion of eigenvalues in powers of the perturbing parameter α (as in [38]) is exibited and the Fermi’s golden rule is established. Erik Balslev was the adviser for the following Ph.D. students. • • • • •

Steven Schlosser, State University of New York at Buffalo, 1973. Arne Jensen, Aarhus University, 1979. Erik Skibsted, Aarhus University, 1986. Gorm Salomonsen, Aarhus University, 1996. Morten Risager, Aarhus University, 2003.

A conference to celebrate Erik Balslev’s 75th birthday was organized at Aarhus University during 1–2 October, 2010. The invited speakers were • • • •

Sergio Albeverio, Universität Bonn. Nils Elander, Stockholm University. Ira Herbst, University of Virginia. Masau Hirokawa, Okayama University.

R, S; Sarnak, P., On cusp forms for co-finite subgroups of P S L(2, R), Ivent. Math. 80 (1985), no. 2, 339–364. 10 Phillips,

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• • • • • •

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Dieter Mayer, Technische Universität Clausthal. Werner Müller, Universität Bonn. Gheorge Nenciu, Romanian Academy Bucharest. Thierry Paul, École Polytechnique, Palaiseau. Morten Risager, Copenhagen University. Ricardo Weder, Universidad Nacional Autónoma de México. The members of the organizing committee were

• • • •

Arne Jensen, Aalborg University. Jacob Schach Møller, Aarhus University. Erik Skibsted, Aarhus University. Alexei Risager, Aarhus University.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

E. Balslev, Perturbation of ordinary differential operators. Math. Scand. 11, 131–148 (1962) E. Balslev, T.W. Gamelin, The essential spectrum of a class of ordinary differential operators. Pacific. J. Math. 14, 755–776 (1964) E. Balslev, The essential spectrum of elliptic differential operators in L p (Rn ). Trans. Amer. Math. Soc. 116, 193–217 E. Balslev, The singular spectrum of elliptic differential operators in L p (Rn ). Math. Scand. 19, 193–210 E. Balslev, Discreteness of the spectrum of a second order elliptic differential operator in L 2 (Rn ). Math. Scand. 22, 42–50 (1968) E. Balslev, A. Verbeure, States on Clifford algebras. Comm. Math. Phys. 7, 55–76 (1968) E. Balslev, J. Manuceau, A. Verbeure, Representations of anticommutation relations and Bogolioubov transformations. Comm. Math. Phys. 8, 315–326 (1968) E. Balslev, J.M. Combes, Spectral properties of many-body Schrödinger operators with dilatationanalytic interactions. Comm. Math. Phys. 22, 280–294 (1971) E. Balslev, Spectral theory of Schrödinger operators of many-body systems with permutation and rotation symmetries. Ann. Phys. 73, 49–107 (1972) D. Babbitt, E. Balslev, Dilation-analyticity and decay properties of interactions. Comm. Math. Phys. 35, 173–179 (1974) E. Balslev, Schrödinger operators with symmetries. II. Systems with spin. Rep. Math. Phys. 5, 393–413 (1974) E. Balslev, Schrödinger operators with symmetries. Rep. Math. Phys. 5, 219–280 (1974) E. Balslev, Absence of positive eigenvalues of Schrödinger operators. Arch. Ration. Mech. Anal. 59(4), 343–357 (1975) D. Babbitt, E. Balslev, A characterization of dilation-analytic potentials and vectors. J. Funct. Anal. 18, 1–14 (1975) D. Babbitt, E. Balslev, Local distortion techniques and unitarity of the S-matrix for the 2-body problem. J. Math. Anal. Appl. 54(2), 316–347 (1976) E. Balslev, Decomposition of many-body Schrödinger operators. Comm. Math. Phys. 52(2), 127– 146 (1977) E. Balslev, Analytic scattering theory of two-body Schrödinger operators, J. Funct. Anal. 29, 375– 396 (1978) E. Balslev, Analytic scattering theory for many-body systems below the smallest three-body threshold. Comm. Math. Phys. 77, 173–210 (1980) E. Balslev, Analytic scattering theory of quantum mechanical three-body systems. Ann. Inst. H. Poincaré Sect. A (N.S.) 32(2), 125–160 (1980)

Scientific Contributions of Erik Balslev 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

36. 37. 38.

39.

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E. Balslev, Resonances, resonance functions and spectral deformations. Resonances-models and phenomena (Bielefeld, 1984), Lecture Notes in Phys., vol. 211, pp. 27–63. Springer, Berlin (1984) E. Balslev, Resonances in three-body scattering theory. Adv. Appl. Math. 5(3), 260–285 (1984) E. Balslev, E. Skibsted, Boundedness of two- and three-body resonances. Ann. Inst. H. Poincaré Phys. Théor. 43(4), 369–397 (1985) E. Balslev, A. Grossmann, T. Paul, A characterisation of dilation-analytic operators. Ann. Inst. H. Poincaré Phys. Théor. 45(3), 277–292 (1986) E. Balslev, Bounds on many-body resonances. Ann. Inst. H. Poincaré Phys. Théor. 47(2), 185–198 (1987) E. Balslev, Resonance functions for radial Schrödinger operators. J. Math. Anal. Appl. 123(2), 339–365 (1987) E. Balslev, E. Skibsted, Resonances and poles of the S-matrix. Symposium “Partial Differential Equations” (Holzhau, 1988), Teubner-Texte Math., vol. 112, pp. 24–32, Teubner, Leipzig (1989) E. Balslev, E. Skibsted, Resonance theory of two-body Schrödinger operators. Ann. Inst. H. Poincaré Phys. Théor. 51(2), 129–154 (1989) E. Balslev, Analyticity properties of eigenfunctions and scattering matrix. Comm. Math. Phys. 114, 599–612 (1988) E. Balslev, Wave operators for dilation-analytic three-body Hamiltonians. J. Funct. Anal. 81, 345– 384 (1988) E. Balslev, E. Skibsted, Asymptotic and analytic properties of resonance functions. Ann. Inst. H. Poincaré Phys. Théor. 53(1), 123–137 (1990) E. Balslev, B. Helffer, Limiting absorption principle and resonances for the Dirac operator. Adv. Appl. Math. 13, 186–215 (1992) E. Balslev, Spectral deformation of Laplacians on hyperbolic manifolds. Comm. Anal. Geom. 5(2), 213–247 (1997) E. Balslev, A. Venkov, The Weyl law for subgroups of the modular group, University of Aarhus preprint No 2, 1997. http://math.au.dk/publs?publid=803 E. Balslev, A. Venkov, The Weyl law for subgroups of the modular group. Geom. Funct. Anal. 8(3), 437–465 (1998) E. Balslev, A. Venkov, Selberg’s eigenvalue conjecture and the Siegel zeros for Hecke L-series. Analysis on homogeneous spaces and representation theory of Lie groups, Okayama Kyoto (1997), 19–32, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000 E. Balslev, A. Venkov, Spectral theory of Laplacians for Hecke groups with primitive character. Acta Math. 186(2), 155–217 (2001) E. Balslev, A. Venkov, Correction to: “Spectral theory of Laplacians for Hecke groups with primitive character” [Acta Math. 186(2), 155–217 (2001)]. Acta Math. 192(1), 1–3 (2004) E. Balslev, A. Venkov, On the relative distribution of eigenvalues of exceptional Hecke operators and automorphic Laplacians. Algebra i Analiz 17(1), 5–52 (2005); reprinted in St. Petersburg Math. J. 17(1), 1–37 (2006) E. Balslev, A. Venkov, Perturbation of embedded eigenvalues of Laplacians. Traces in number theory, geometry and quantum fields, Aspects Math., E 38, pp. 23–33. Friedr. Vieweg, Wiesbaden, 2008

List of Publications by Erik Balslev

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3.

4.

5. 6.

7. 8. 9. 10. 11.

12.

13. 14.

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Balslev, Erik, Spectral theory of the Laplacian on the modular Jacobi group manifold. University of Aarhus Preprint No. 03, 2011, https://math.au.dk/publs?publid=918. Balslev, Erik; Venkov, Alexei, Perturbation of embedded eigenvalues of Laplacians. Traces in number theory, geometry and quantum fields, 23-33, Aspects Math., E 38, Friedr. Vieweg, Wiesbaden, 2008. Balslev, E.; Venkov, A., On the relative distribution of eigenvalues of exceptional Hecke operators and automorphic Laplacians. Algebra i Analiz 17 (2005), no. 1, 5–52; reprinted in St. Petersburg Math. J. 17 (2006), no. 1, 1–37. Balslev, Erik; Venkov, Alexei, Correction to: “Spectral theory of Laplacians for Hecke groups with primitive character” [Acta Math. 186 (2001), no. 2, 155–217]. Acta Math. 192 (2004), no. 1, 1–3. Balslev, Erik; Venkov, Alexei, Spectral theory of Laplacians for Hecke groups with primitive character. Acta Math. 186 (2001), no. 2, 155–217. Balslev, Erik; Venkov, Alexei, Selberg’s eigenvalue conjecture and the Siegel zeros for Hecke L-series. Analysis on homogeneous spaces and representation theory of Lie groups, OkayamaKyoto (1997), 19–32, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000. Balslev, E.; Venkov, A., The Weyl law for subgroups of the modular group. Geom. Funct. Anal. 8 (1998), no. 3, 437–465. Balslev, E.; Venkov, A, The Weyl law for subgroups of the modular group, University of Aarhus preprint No 2, 1997. http://math.au.dk/publs?publid=803. Balslev, Erik, Spectral deformation of Laplacians on hyperbolic manifolds. Comm. Anal. Geom. 5 (1997), no. 2, 213247. Balslev, E.; Helffer, B., Limiting absorption principle and resonances for the Dirac operator. Adv. in Appl. Math. 13 (1992), no. 2, 186–215. Balslev, Erik, Ed., Schrödinger operators the quantum mechanical many-body problem: proceedings of a workshop held at Aarhus, Denmark 15 May—1 August 1991, Lecture Notes in Physics, 403, Springer-Verlag, Berlin, 1992. Balslev, E.; Helffer, B., Limiting absorption principle and resonances for the Dirac operator. Rigorous results in quantum dynamics (Liblice, 1990), 75–96, World Sci. Publ., River Edge, NJ, 1991. Balslev, Erik; Skibsted, Erik, Asymptotic and analytic properties of resonance functions. Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 1, 123–137. Balslev, Erik; Skibsted, Erik, Resonances and poles of the S-matrix. Symposium “Partial Differential Equations” (Holzhau, 1988), 24–32, Teubner-Texte Math., 112, Teubner, Leipzig, 1989. Balslev, Erik, Asymptotic properties of resonance functions and generalized eigenfunctions. Schrödinger operators (Sønderborg, 1988), 43–64, Lecture Notes in Phys., 345, Springer, Berlin, 1989. xvii

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20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31.

32. 33. 34. 35. 36. 37. 38. 39. 40.

List of Publications by Erik Balslev Balslev, E.; Skibsted, E., Resonance theory of two-body Schrödinger operators. Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), no. 2, 129–154. Balslev, Erik, A note on the cluster model and complex scaling. Resonances (Lertorpet, 1987), 455–457, Lecture Notes in Phys., 325, Springer, Berlin, 1989. Balslev, Erik, Resonances with a background potential. Resonances (Lertorpet, 1987), 35– 46, Lecture Notes in Phys., 325, Springer, Berlin, 1989. Balslev, Erik; Skibsted, Erik, Resonance functions of two-body Schrödinger operators. JournŽes “Équations aux Dérivées Partielles” (Saint Jean de Monts, 1988), Exp. No. XIV, 12 pp., École Polytech., Palaiseau, 1988. Balslev, Erik, Wave operators for dilation-analytic three-body Hamiltonians. J. Funct. Anal. 81 (1988), no. 2, 345–384. Balslev, Erik, Analyticity properties of eigenfunctions and scattering matrix. Comm. Math. Phys. 114 (1988), no. 4, 599–612. Balslev, Erik, Bounds on many-body resonances. Ann. Inst. H. Poincaré Phys. Théor. 47 (1987), no. 2, 185–198. Balslev, Erik, Resonance functions for radial Schrödinger operators. J. Math. Anal. Appl. 123 (1987), no. 2, 339–365. Balslev, Erik, Wave operators for dilation-analytic three-body Hamiltonians. Schrödinger operators, Aarhus 1985, 39–60, Lecture Notes in Math., 1218, Springer, Berlin, 1986. Balslev, Erik, Ed., Schrödinger operators, Aarhus 1985, Lecture Notes in Math., 1218, Springer, Berlin, 1986. Balslev, E.; Grossmann, A.; Paul, T., A characterisation of dilation-analytic operators. Ann. Inst. H. Poincaré Phys. Théor. 45 (1986), no. 3, 277–292. Balslev, Erik; Skibsted, Erik, Boundedness of two- and three-body resonances. Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), no. 4, 369–397. Balslev, Erik, Resonances, resonance functions and spectral deformations. Resonancesmodels and phenomena (Bielefeld, 1984), 27–63, Lecture Notes in Phys., 211, Springer, Berlin, 1984. Balslev, Erik, Local spectral deformation techniques for Schrödinger operators. J. Funct. Anal. 58 (1984), no. 1, 79–105. Balslev, Erik, Resonances in three-body scattering theory. Adv. in Appl. Math. 5 (1984), no. 3, 260–285. Balslev, E., Comment on: “Analytic scattering theory for many-body systems below the smallest three-body threshold” [Comm. Math. Phys. 77 (1980), no. 2, 173–210]. Comm. Math. Phys. 82 (1981/82), no. 2, 257–260. Balslev, Erik, Editor, 18 th Scandinavian Congress of Mathematicians proceedings, 1980, Birkhäuser, Boston, 1981. Balslev, Erik, Analytic scattering theory for many-body systems below the smallest threebody threshold. Comm. Math. Phys. 77 (1980), no. 2, 173–210. Balslev, Erik, Analytic scattering theory of quantum mechanical three-body systems. Ann. Inst. H. Poincaré Sect. A (N.S.) 32 (1980), no. 2, 125–160. Balslev, Erik, Analytic scattering theory of two-body Schrödinger operators. J. Functional Analysis 29 (1978), no. 3, 375–396. Balslev, Erik, Scattering theory of dilated three-body Schrödinger operators. Int. J. Quantum Chemistry XIV (1978), 361–370. Balslev, E., Decomposition of many-body Schrödinger operators. Comm. Math. Phys. 52 (1977), no. 2, 127–146. Babbitt, D.; Balslev, E., Local distortion techniques and unitarity of the S-matrix for the 2-body problem. J. Math. Anal. Appl. 54 (1976), no. 2, 316–347. Balslev, Erik, Absence of positive eigenvalues of Schrödinger operators. Arch. Rational Mech. Anal. 59 (1975), no. 4, 343–357. Babbitt, D.; Balslev, E., A characterization of dilation-analytic potentials and vectors. J. Functional Analysis 18 (1975), 1–14.

List of Publications by Erik Balslev 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

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Balslev, Erik, Schrödinger operators with symmetries. II. Systems with spin. Rep. Mathematical Phys. 5 (1974), 393–413. Balslev, E., Schrödinger operators with symmetries. Rep. Mathematical Phys. 5 (1974), 219– 280. Babbitt, D.; Balslev, E., Dilation-analyticity and decay properties of interactions. Comm. Math. Phys. 35 (1974), 173–179. Balslev, Erik, Spectral theory of Schrödinger operators of many-body systems with permutation and rotation symmetries. Ann. Physics 73 (1972), 49–107. Balslev E.; Leibovic K.N., Theoretical analysis of binocular space perception. J. Theor. Biol., 31 (1971), 77100. Leibovic K.N.; Balslev E.; Mathieson T. A., Binocular vision and pattern recognition. Kybernetik, 8 (1971), 14–23. Leibovic K. N.; Balslev, E.; Mathieson, T. A., Binocular space and its representation by cellular assemblies in the brain. J. Theor. Biology 28 (1971), 513–529. Balslev, E.; Combes, J. M., Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions. Comm. Math. Phys. 22 (1971), 280–294. Balslev, E.; Manuceau, J.; Verbeure, A., Representations of anticommutation relations and Bogolioubov transformations. Comm. Math. Phys. 8 (1968), 315–326. Balslev, Erik, Discreteness of the spectrum of a second order elliptic differential operator in L 2 (Rn ). Math. Scand. 22 (1968), 42–50. Balslev, Erik; Verbeure, André, States on Clifford algebras. Comm. Math. Phys. 7 (1968), 55–76.

52.

Balslev, Erik, The singular spectrum of elliptic differential operators in L p (Rn ). Math. Scand. 19 (1966), 193–210.

53.

Balslev, Erik, The essential spectrum of elliptic differential operators in L p (Rn ). Trans. Amer. Math. Soc. 116 (1965), 193–217. Balslev, E.; Gamelin, T. W., The essential spectrum of a class of ordinary differential operators. Pacific J. Math. 14 (1964), 755–776. Balslev, Erik, Perturbation of Differential Operaotrs. Thesis (Ph.D.)-University of California, Berkeley. 1963. 92 pp, ProQuest LLC. Balslev, Erik, Perturbation of ordinary differential operators. Math. Scand. 11 (1962), 131– 148.

54. 55. 56.

Erik, My Life Companion A Personal Tribute Anindita Balslev

The few lines that I am about to put on record here are not about the work of the young professor of mathematics who arrived at Aarhus University in Denmark in 1974 from the US, after being a visiting professor at SUNY Buffalo, New York and then at UCLA, California. It is important for my story to retrace the phase before that, when he was in France as a post-Doc fellow, associating with the Institut Henri Poincare and Institut des Hautes Etudes Scientific. Paris is, indeed, the place where we met and got married, Buffalo and Los Angeles being the birthplaces of our two children, Eva and Olav. Ever since, France and USA remain as important as Denmark and India in our lives where we have kept on returning over the years. We settled in Denmark, the country of his birth, and it is there that our children grew up. We enjoyed our annual visits to India and the occasional trips to France and USA; our common prayer was—“Let noble thoughts come to us from all directions” (Rigveda). Indeed, as we flew from one hemisphere to the other in these early years, I began to notice that India—the country where I was born—was embracing colossal changes while at the same time seeking to re-vitalize her ancient cultural heritage, whereas the Danish academia was resisting change especially in the fields of philosophy and religion with their view to sustain its virtually monolithic culture. Evidently students were ready to expand the horizon of knowledge by going beyond the rigid set of mono-cultural academic offerings, the system (rather, the few who control the system) was not. Now more than four decades have passed by and despite that I have been with him during all that time, this is not an account about his colleagues or collaborators or students nor about the pros and cons of our work-situation in Denmark, where the opportunities and disappointments that awaited us were to be a part of the web of our lives. Here, I am only seeking to share with the readers a few snapshots from the album of my life by recalling what Erik was like as my life companion, my closest friend, my spouse. It is easier said than done! Indeed, to capture the impact of such a deep companionship is no less subtle a job than to describe the smell of roses or jasmines, especially to those who have not smelt the scent of these flowers. It seems to be an overwhelming task, as I seek to express what Erik’s presence in my life has really meant for me. xxi

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To begin with, I can honestly claim, that my experience has taught me that it is a huge blessing whenever anyone senses in a relationship—as I did—that there prevails a genuine mutual readiness to receive as well as to render unconditional support. Without a doubt, this engenders that rare form of courage that enables one to go through all kinds of adversities fearlessly and sustain the belief that nothing can ever hinder the blossoming of the spiritual core that makes life worth living. Perhaps, it is a matter of common knowledge that true friendship does not emerge simply out of chance acquaintance, even if such an acquaintance happens to last over a long stretch of time. There seems to be a magical component—hard to pin-point—to this sense of closeness. I never tried to fathom this earlier but now I see that there is, indeed, embedded in such an experience a message of disclosure—which in our case happened within a few days after our first meeting at the Sorbonne—when one knows with a sense of certainty in the depth of one’s heart that one has now found one’s life companion. As I recall, this was clearly signaled by the absence of a semblance of any trace of anxiety at the thought of undertaking together a life-journey that lay ahead of us. It is amazing, especially at a time when most of one’s life is yet to be lived and a vast unexplored territory lies before one’s gaze, that one can unmistakably feel a breeze blowing with confidence and promise. It was, probably, a decade and a half later—since my life conjoined with Erik’s— that I seriously began pondering not only the opportunities that living in a technological civilizational setting provides us today but also how keenly our present situation demands from all of us—cross-culturally—that we let emerge a sense of a larger identity. This calls for “softening” those “hard boundaries” that divide people, be these drawn in the name of religion, nationality, ethnicity, profession, or any other form of group-identity. We now seem to badly need a change in mind-set that will allow us, on the one hand, to respect cultural diversity, local identities and, on the other hand, to recognize that we are part of a larger whole and that we share a common habitat—not only with those who are perhaps somewhat unlike “us” but also with all other non-human living beings. I was hardly aware then that this was precisely the process in which we were both involved in our everyday life experiences, and that my academic reflections on such concerns were born as much of those lived experiences as from my actual research that led me to delve deep into a wide range of cognitive discourses on cultural matters. It is astonishing to think now that on the personal front, we did not even have to try to “soften” any boundary that could signify the presence of a barrier, since the artificiality of those divisive constructions were so obvious to us. Today, as I watch various social scenarios afflicted by “hard” boundaries, it feels like looking at a map of Africa, where those artificial straight lines that unduly divide that vast continent into various countries. Those who draw these lines sitting in remote boardrooms, seem to totally disregard the severe troubles that these cause for the people who live in those regions. A perusal of the global scene in multiple contexts does often reveal that these man-made “hard boundaries” are sustained—more often than not—by those who benefit from these divisive tactics. Given my limited ability and despite that the circumstances at that time were hardly favorable, I began designing a forum and

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promoting a program, designating it as CCC—an abridged form for “cross-cultural conversation”. I was almost obsessed with the idea and shared with Erik my thoughts about how to promote such a multi-dimensional program. The idea was to invite conversational partners from a wide range of disciplines who could—in various stages and phases of this conversation—question all forms of “hard boundaries” that are already put in place in societies. My conviction was that such a process could gradually help to promote a public discourse that would foster a sense of human solidarity. I felt encouraged, as I knew that Erik was listening to me with serious attention, occasionally intervening with comments and questions. Let me note here that the Center for Cultural Research, Aarhus University hosted the first CCC international conference in 1994. Since then I have held these CCC international conferences in various places—several in NewDelhi—followed by significant publications. Erik was present in many of these occasions. As I look back over the forty-five years of our life together, I have no hesitation in saying that I benefitted in multifarious ways from my relationship with Erik and how deeply I appreciate the kind of person that he was. His was a personality that combined intelligence with compassion. I can never forget that one time when he went with me to the US—when I was invited as a visiting professor at the University of Kentucky—in order to make sure that I was comfortably settled for that one semester. I know how extremely fortunate I have been to have a spouse so fully supportive. I was always touched by the empathy with which he sought to understand people from various parts of the globe whom we met whenever we traveled or wherever his work took us. I recall the memories of the year when Erik was invited to the Institute for Advanced Study, Princeton, where he was happily engrossed in research work and we also regularly met with scholars who came there from different corners of the world. Indeed, the changing socio-cultural scenarios to which our life exposed us had significant impact on our minds, and the experience of living in different places taught us how it feels like to be a “foreigner” in different cultural settings. I can still keep charging my batteries from all those memories of a shared past, which remain constantly present in my mind. Indeed, we studied and learned so much as we traveled extensively within the US and in Europe, visited Russia, Thailand, Sri Lanka, Nepal and so many places within India and saw parts of Australia and Africa. We saw the architectural splendors of churches, mosques, temples, and stupas. As we read about the lives and teachings of great intellectuals, artists, sages, and prophets from various traditions, and especially when we came in touch with spiritually advanced persons, we aspired to know more and more. This, however, does not mean that as we walked together and appreciated the chain of breath-takingly beautiful scenic views, there did not arise unexpectedly arid, unpleasant scenes. Surely, we had faced bouts of disease, cases of accident and of death of friends and near-ones, and have even experienced the annoying sense of discomfort that arises from being a victim of envy, malice, and hypocrisy in workplaces or outside. All these are perhaps integral to our earthly lives. In any case, both of us were aware that all these are passing episodes, and that there is no good reason to succumb to any of these as scenes of final destination. We both knew that we must

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remain strong and be open to learning how to respond to various challenges with sensitivity and wisdom. Given the fact that much of one’s life circumstances are not of one’s own making, nevertheless, how one responds to this series of challenging situations certainly has much to do with who one is and who one wants to become. We often used to discuss and reflect on certain events that are documented in the biographies and autobiographies of great persons across cultures and longed to obtain a larger frame for understanding the intricacies of human existence. The Sanskrit word “samsara” (world-process), rooted in “samsaran” (wandering), is suggestive of the idea that this world is a place where we are all in transit. We are all wanderers hence must invariably pass through changing landscapes. Erik always carefully noted whenever he encountered glimpses of wisdom and insights from the Indian traditions. I recall his appreciative smile when at the University of Copenhagen I introduced Professor John Searle by citing from the Sanskritic sources that “a king is adored in his own country whereas the learned is admired everywhere”; and again when in my concluding remarks, as I thanked the professor from Berkeley for his lecture, cited that “of all the gifts that a person may possibly receive, the greatest is the gift of knowledge.” Tradition says that a person may be robbed of all other possessions but not of his/her knowledge—once obtained it will always be there. As I look back, it is amusing to recall that a frequently asked question in our early days was “how and where did we meet”-posed even by the distinguished professor Jean Wahl—one of my mentors at the Sorbonne. We were by then already married for six months. The occasion was my “soutenance” when after almost a 3-hour long Q and A session with three veteran professors, they accepted my doctoral thesis with “mention très honorable”. Erik was very pleased when the aged and venerated professor came forward to meet the husband of the doctoral candidate, uttering “mes felicitations”. Monsieur Wahl stretched out his hand and invited us to his apartment in Paris, which—to our delight—we found to be more filled with books than furniture. He then asked that question. Once he heard that we had actually met in a classroom at the Sorbonne where his colleague Prof. O. Lacombe, my principal advisor, was delivering a course on Indian philosophy, Monsieur Wahl quickly embarked on a discussion with Erik. Probably curious to note how the philosophical views tally with the scientific perspectives, he kept focusing on a range of themes such as perception, consciousness, cosmology, and so forth. This may come as a surprise to those who thought of Erik exclusively as a mathematician. Apart from Mathematics and Physics, there are so many books in our bookshelves on various subjects that he gathered as a young student, including old issues of Psychical Research and journals dealing with scientific investigations into various Yogic experiences and practices. Among books on Philosophy, I still see Mind and Its Place in Nature by C. D. Broad, Creative Evolution by Henri Bergson, Problems of Philosophy by Bertrand Russel, and Perception by Henry Price that belong to his student days and he had with him then. Erik told me how interesting it was to listen to Price whom he heard in Berkeley campus while he was a doctoral student there. To my surprise, I had also noted during those early days of our meeting in Paris that there were Danish translations of Kalidas’s Shakuntala and of the Upanishads in his collection. Many dozens more were added as time passed by.

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Erik has been surrounded by books from his childhood. His father was a minister in a long chain of a well-known priestly family and was obviously a voracious reader. During my first visit to Denmark, I was impressed to hear that his father had excellent knowledge of Greek and Hebrew and what was particularly helpful for me was that he could speak quite a bit of English as well. I had several rounds of discussions with him on various topics but that happened to turn out to be the last time—only a few months before his expiry. Many of his books, beautifully bound, are still there with us for his grandchildren to enjoy. It comes to my mind that one day, as we were musing on where lies the highest peak of creativity of Indian and Western cultures, Erik said with deep conviction that for the West, it surely lies in its achievements in the fields of music and mathematics. Indeed, his love for Western classical music was profound. He was exposed to it from his childhood, his mother being a conservatory-trained pianist herself. As the father of our two children, Erik wanted Eva and Olav to be familiar with that great musical tradition, introducing them to violin and cello, respectively, while they were small. He used to accompany them often on the piano, and I am sure that he will always remain in their hearts—among a host of other things—through musical experience. I can hardly listen to a Bach, a Mozart, or a Beethoven piece today without associating and visualizing him with the experience. When I go to Mumbai to listen to the Symphony Orchestra of India, where our son plays, I feel grateful for the continuation of that great love for music from father to son even on a professional level. Erik was always there for them, always. It is not surprising in a way that I met Erik in an Indian philosophy class in Paris. Indeed, Erik was drawn to India, especially because of his own deep spiritual quest and we were fortunate to meet spiritually advanced personalities there. My dear father’s memory is closely associated with those years into which I cannot go here. Among the many interesting conversations that we have had with several distinguished persons over the years, our meeting with Professor and Mrs. Mahalanobis in Kolkata is particularly memorable. When Erik visited the Indian Statistical Institute just after our marriage, we stayed a few months on the top floor of their residence. We heard with a sense of awe when the professor (founder of the Indian Statistical Institute) talked about his experiences with the famous mathematician Srinivas Ramanujan whom he knew during his young days in Cambridge and Mrs. Mahalanobis (Ranidi, as I used to call her) spoke about Rabindranath Tagore, the great poet, with whom she and her husband were both closely associated. Erik also met there the famous physicist Satyen Bose (Boson, named after Bose-Einstein) and the well-known statistician C. R. Rao. Let me refer to just one remarkable conversation here that I had with the Norwegian mathematician Atle Selberg, whose work Erik greatly admired. We met him several times in Princeton as well as in Aarhus. I recall that once during a dinner we were discussing about philosophical debate concerning the idea of truth and I asked him in that connection whether mathematicians discover or invent truth. His prompt reply was: “we invent methods and discover truth”. Selberg adored Ramanujan and told us about how he came to know about Ramanujan’s works and about his own visits to India.

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Apart from his interest in mathematics and other sciences, Erik was particularly fascinated by several ideas from the Upanisadic tradition that he first heard from an old family friend by the name of Aksel Knudsen. I met this gentleman too during my first visit to Denmark and had the clear feeling that even if he had gone to India for the purpose of missioning, he returned to Denmark somewhat transformed. He corresponded with us for a while and this led to much animated discussions on the large themes of life and death with reference to various religious and philosophical traditions. Erik was always discrete and humble, he never showed off. He did not rebuff persons who took unfair advantage of him. I was sometimes surprised to watch his unusual capacity to retain his own integrity even when he was hurt. He was a special person. I remember vividly the occasion, when we went to the US in the summer of 2012 and attended the meeting at the Institute on Religion in an Age of Science, how he was listening with deep concentration and great devotion to the prayer that I recited from the Upanishads: “Lead me from Non-being to Being, from Darkness to Light, from Death to Deathlessness”. Only a few months after that he passed away in Goa, India in January 2013. To me, he was a friend like no one else, a life companion worth treasuring, my soulmate.

Our Friend, Our Colleague Erik Balslev

Unfortunately I was unaware of the fact that Erik Balslev had suddenly passed away, and that too only after three years. But I do remember Erik very well and I will remember him forever. Moreover, I do still feel him to be very close since we have known each other very well. We met each other during that famous academic year 1966–1967 of mathematical physics at the “Institut des Hautes Etudes Scientifiques” in Bures-sur-Yvette. This was an agreeable small but nice place located in a green domain situated in the south of Paris, next to the other well-known place for scientists, namely, the town of Orsay where there was an important science campus. The fine fleur of mathematical physicists of the globe did find each other that year in that quiet French place. There was an exclusive gathering of famous people. I was a young Belgian mathematical or rather theoretical physicist who had printed in his head that he wanted to specialize in mathematical physics. There I came to meet David Ruelle, a Belgian researcher, who was already known to be a kind of a star in his field. He obtained his Ph.D. on the basis of work done in Zurich, Switzerland, as far as I remember. He already had a serious reputation as a well-known researcher, educated in the Zürich school for mathematical physics and was a picture example of somebody whom I strove to be. Also I do remember that our communication suffered a bit from our different Belgian linguistic origins. There are many jokes in our small Belgian country that describe these different language situations. Anyway that was the reason why my scientific attention did move a bit more in the direction of the other star in the field of mathematical physics, namely, in the direction of Daniel Kastler. Unfortunately Daniel passed away very recently. I may take this occasion to express a salute of honor to this great and rich personality. Daniel was a French mathematical physicist and was a typical product of the French school. He was a person who had an important impact on the mathematics education and studies in his country. For centuries mathematical physics was a very popular subject in France, and it was considered at the universities as a very serious field of study as well. Daniel was an exponential example of a French and European mathematical physicist.

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At the end of the academic year at Bures, nearly everybody was leaving the institute in Bures, only the main staff remained, planning their future over there. Many people were just flying home, also several colleagues were looking for alternative interesting places. A large number of the researchers, certainly the most experienced people, like for instance Erik Balslev, were invited by Daniel for a stay at the brand new university campus of Marseille, which was a part of the university campus of Luminiy, nicely situated near seaside at the south side of Marseille. Moreover, I guess that a number of students of Daniel did benefit very much from this external input which they got in Bures. This was the type of setting in which Daniel asked me also to come to Marseille. With pleasure I did accept his offer. This was exactly my occasion to enter into the brand new world of scientific activity which I liked so much. This was the kind of place which I was looking for and which was now offered to me. At the same time I had the occasion of meeting and collaborating with Erik and to become a personal friend of his. First of all, I learned that Erik was a brilliant pure mathematician, specialized in functional analysis. Moreover Erik was a fantastic human being, who was very much influenced by the humanistic ideal that even the work of a mathematician should be helpful in making progress for the welfare of mankind. Looking into what he was doing, being active in pure analysis, Erik was developing some personal doubts. His questions were of the type: what is my contribution to mankind? Sometimes he was hesitant about the whole domain or field of pure mathematical analysis. Does this pure mathematics really satiate my hunger for contributing something towards mankind? I do remember well that at some moments Erik was not sure that his “pure” mathematics could meet or satisfy his vision for mankind. Very quickly, he came to a conclusion that his mathematical activities should be extended by including contents from other areas of investigation, which contained topics directly applicable to our society and were of practical value for mankind. He decided for himself that “pure maths alone” did not offer this necessary contribution to mankind, where on the other hand, the scientific line, which was called physics, did give him the possibility of contributing directly to the benefit of the human being. In many ways, this reading of his situation convinced Erik that by including activities in mathematical physics in his work could propel it to a higher level, thereby lending it an important humanistic aspect. Erik and I used to have hours and hours of discussions about these matters. Indeed this move meant for him an important change in his work compared to what he had been working on earlier. I was happy as well to see this counter-weight, which was necessary to realize a good friend’s philosophy of achieving a humanistic equilibrium. These humanistic aspects were also at the origin of an important human appreciation by Daniel Kastler for Erik’s work. Daniel reminded me several times that I was a lucky boy to have met Erik as a good friend. I am sure that these feelings of Daniel Kastler prompted him to invite Erik as well as me to continue our joint work in Marseille. Together with Erik we became an efficient pair in Marseille working in the domain of mathematical physics, not in philosophy. Erik was a very experienced mathematician. I was trying to learn more analysis from him and to use mostly Erik’s mathematics in order to realize some work, also with a basis in physics. In all ways

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this common history in Paris and particularly also in Marseille will never disappear from my memory. When we were doing work in mathematical physics at Marseille university, we were all living in Bandol together with several other mathematical physicists. This has been a small but important place since the sixties. In particular it is a very well-known place for many mathematical physicists from all over the world. Many researchers stayed over there, when invited by Daniel Kastler. Many excellent papers were written on the beach of Bandol! Erik and I were lucky to have many experiences of this kind during our research in mathematical physics. The progress of our work was extraordinary. Also I remember a certain day, I decided to join an exploration trip for a weekend into the mountains in the north of Marseille. I was back on the next Sunday evening and I expected to meet Erik already the following Monday morning. A number of days passed and nothing happened. No Erik, no news from Erik. A few days later, a nice joyful girl made her appearance in Bandol. Her name was Anindita. We met each other. Erik did not say very much about the girl, except mentioning her Christian name, although we thought that he knew her, and knew her very well. We talked much all three together. This girl spoke French so well. We realized that Erik did not speak French so well. My English was a bit poor but useful, my French was in view of my origins rather acceptable. The conclusion was that Bandol did become a fantastic place. But nevertheless there was a small language inconvenience. Erik was a gentleman, not being able to use his Northern language but speaking English very well. As an outsider I was speaking a bit of French and a bit of English and thus we could all understand each other. This enables all three of us to become good friends. I learned a bit of philosophy from Anindita using both English and French. It was a time never to forget. So with Erik, and a bit later also with Anindita, together in Bandol-Marseille, we were all spending a warm spring and a hot summer at the Mediterranean Sea. We had a fantastic time together for a number of reasons. It was an atmosphere where a young girl doing her Ph.D. in Paris and a phsyically tall Nordic man with a heart of sugar bread fell in love with each other. Erik was a pure mathematician, who was specialized in classical analysis, now wanting to pursue applied mathematics. When we left our collaboration, we returned to our own work. Erik was a bit tired of his dry mathematics, and had taken up the idea of learning some physics. Not too much physics, but at least something less abstract than pure mathematics. Erik had found full equilibrium in his life. Erik remained an excellent, a very good mathematician. He had always his serious personality. Erik was a fantastic person. I will never forget this extraordinary person, called Erik Balslev. André Verbeure Mathematical Physicist Institut for Theoretical Physics KU Leuven, Leuven, Belgium [email protected]

Contents

Introduction to the Scientific Contributions in the Book . . . . . . . . . . . . . . . Sergio Albeverio and Ricardo Weder Asymptotics of Random Resonances Generated by a Point Process of Delta-Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergio Albeverio and Illya M. Karabash Green’s Functions and Euler’s Formula for ζ (2n) . . . . . . . . . . . . . . . . . . . . Mark S. Ashbaugh, Fritz Gesztesy, Lotfi Hermi, Klaus Kirsten, Lance Littlejohn, and Hagop Tossounian

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On Courant’s Nodal Domain Property for Linear Combinations of Eigenfunctions Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pierre Bérard and Bernard Helffer

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Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anne Boutet de Monvel and Lech Zielinski

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Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L 2 (Rd ) . . . . . . . . . . . . . . 117 Jean Michel Combes and Peter D. Hislop Resonances in the One Dimensional Stark Effect in the Limit of Small Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Richard Froese and Ira Herbst On the Spectral Gap for Networks of Beams . . . . . . . . . . . . . . . . . . . . . . . . . 169 Pavel Kurasov and Jacob Muller Some Notes in the Context of Binocular Space Perception . . . . . . . . . . . . . 181 K. N. Leibovic Symbolic Calculus for Singular Curve Operators . . . . . . . . . . . . . . . . . . . . . 195 Thierry Paul

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Higher Order Deformations of Hyperbolic Spectra . . . . . . . . . . . . . . . . . . . 223 Yiannis N. Petridis and Morten S. Risager On the Generalized Li’s Criterion Equivalent to the Riemann Hypothesis and Its First Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 S. K. Sekatskii Regularising Infinite Products by the Asymptotics of Finite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Mauro Spreafico and Alessandro Zaccagnini Trace Maps Under Weak Regularity Assumptions . . . . . . . . . . . . . . . . . . . . 271 Ricardo Weder

Introduction to the Scientific Contributions in the Book Sergio Albeverio and Ricardo Weder

Abstract This is an introduction to the second part of the book that consists of invited contributions written by mathematicians and physicists that have been working in the main areas of interest of Erik Balslev. Some of the authors of the contributions have known him personally or even collaborated with him, others have been influenced by his work. We summarize the contributions and put them into the perspective of the rest of the book. Roughly they are indicated as belonging to one of the areas “Spectral theory and Schrödinger operators” or “Spectral theory and number theory” with one contribution from the area “Mathematics and binocular space perception”. Keywords Spectral theory · Schrödinger operators · Analytic number theory · Random operators · Graphs-networks-visual space · Applications of functional analysis The second part of this book consists of a collection of articles contributed by specialists in areas of research of Erik Balslev. Many of the contributors are mathematicians and mathematical physicists that have been either directly or at least indirectly associated with him, others inspired by work he has published. As we explain in the section on “Scientific contributions by Erik Balslev” (on p. ix) the main areas of research activity of Erik can be characterized as belonging either to the category “spectral theory and Schrödinger operators” (Chaps. 2, 3, 4, 5, 6, 7, 8, 10, 14) or the category “spectral theory and number theory” (Chaps. 11, 12, 13).

S. Albeverio Institute for Applied Mathematics and Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany e-mail: [email protected] R. Weder (B) Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20–126, Ciudad de México 01000, México e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_1

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But the distinctions are not so sharp and a few contributions have aspects situated at the intersections of these categories. This fits very well with the dynamics of research as pursued by Erik Balslev himself who, in the first part of his life, was working rather in the direction of the first category, in the second part rather in that of the second category. In addition our collection of articles has a contribution (Chap. 9) by a neuroscientist and mathematician, K. N. Leibovic, who has been a close friend of Erik’s; this contribution presents joint work with Erik on binocular space perception in the light of new biological studies in that area. Let us now briefly go through the particular contributions. Sergio Albeverio and Illia Karabash: “Asymptotics of random resonances generated by a point process of delta-interactions” The contribution by Sergio Albeverio and Illia Karabash considers the location of resonances of a Schrödinger Hamiltonian for a quantum mechanical particle in a potential generated by randomly placed points (delta-interactions) in the three-dimensional Euclidean space. The resonances are shown to be zeroes of random exponential polynomials, forming a point process in the complex plane. The detailed asymptotic properties of the locations of these resonances (at infinity in the complex plane) are discussed in general and particularly for the special case where the randomness (of the process of delta-interactions) is of the binomial type. In this case it is proven in particular that the counting function for the number of resonances in a sphere of radius R divided by R converges almost surely for R converging to infinity to the random variable π −1 V (Y ), where Y describes the location of the point interactions and V (Y ) is a “measure of size” associated with Y . Thus, a stochastic version of the “Weyl-type asymptotics for resonances” is established (similarly as for the Weyl asymptotics of the number of eigenvalues of Laplace operators in bounded domains). For a general finite point process of deltainteractions, the decomposition of the random set of resonances into sequences with “logarithmic” asymptotics is studied. It it shown that this decomposition can be described by a finite number of real-valued random parameters. The behavior of the distributions of some of these parameters is also considered under the assumption that the intensity of the binomial process of delta-interactions grows to infinity. This paper is related to interests of Erik Balslev, both through the concern with resonances and by the use of exponential polynomials, familiar in the study of location of zeroes of number-theoretical functions. Mark S. Ashbaugh; Fritz Gesztesy; Lotfi Hermi; Klaus Kirsten; Lance Littlejohn; Hagop Tossounian: “Green’s function and Euler’s formula for ζ (2n)” The contribution by Mark S. Ashbaugh, Fritz Gesztesy, Lotfi Hermi, Klaus Kirsten, Lance Littlejohn, and Hagop Tossounian, provides a new derivation of Euler’s formula for the values of the classical ζ -function of Riemann at even natural numbers, as expressed in terms of the corresponding Bernoulli numbers. It also provides a new explicit formula for the resolvent of arbitrary integer positive powers of the negative Dirichlet Laplacian on the interval (0, 1) involving exponential functions of the n-th

Introduction to the Scientific Contributions in the Book

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roots of unity (a formula for elliptic Sturm-Liouville operators, related to investigations on trace formulas). In the introduction of the paper a beautiful historical account of the subject is given, both from the analytic and from the number-theoretic points of view. Pierre Bérard and Bernard Helffer: “On Courant’s nodal domain property for linear combinations of eigenfunctions. Part II” The paper of Pierre Bérard and Bernard Helffer considers the nodal domain of the eigenfunctions of the Schrödinger operator with potential zero in bounded planar domains with piecewise smooth boundary and with Dirichlet boundary condition in one part of the boundary and Neumann boundary condition in the complementary part of the boundary. In particular, the authors consider the generalization of Courant’s nodal domain theorem, namely the Extended Courant property, that is the statement that a linear combination of the first n eigenfunctions has at most n nodal domains. The eigenvalues are listed in nondecreasing order. In this paper the authors give two new counterexamples to the extended Courant property, namely, the equilateral rhombus and the regular exagon. For this purpose they use some input from numerical computations. Anne Boutet de Monvel and Lech Zielinski: “Asymptotic behavior of large eigenvalues of the two-photon Rabi model” The paper of Anne Boutet de Monvel and Lech Zielinski deals with the spectral theory of quantum mechanical Hamiltonians, a subject that was dear to Erik Balslev. It studies the two-photon Rabi Hamiltonian that corresponds to the two-photon JaynesCummings model where the rotating approximation is not assumed. The quantum Rabi model couples a quantized single-mode radiation field and a two-level quantum system, and it plays an important role in cavity quantum electrodynamics. The authors study the asymptotic behavior of the large eigenvalues of this model, and in particular, they prove that the spectrum consists of two sequences of eigenvalues that satisfy the same two-term asymptotic formula with remainder. Jean Michel Combes and Peter Hislop: “Some remarks on spectral averaging and the local density of states for random Schrödinger operators on L 2 (Rd )” The paper of Jean Michel Combes and Peter D. Hislop studies spectral averaging and its application to the local density of states for random Schrödinger operators in Rd , d ≥ 1. For this purpose, the authors use analytic perturbation theory, a topic that was one of the main interests of Erik Balslev, and where he was one of the leading experts. They also prove an upper bound for the trace of spectral projectors for random Schrödinger operators, by means of Poincaré type inequalities for eigenfunctions. They apply these results to the proof of a Wegner estimate and of a Birman-Solomyak theorem. Richard Froese and Ira Herbst: “Resonances in the one dimensional Stark effect in the limit of small field” The paper of Richard Froese and Ira Herbst studies the resonances for the Stark effect. The authors consider Hamiltonians in one dimension with a bounded potential of compact support and a constant electric field. They define resonances as poles

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S. Albeverio and R. Weder

of the analytic continuation of the resolvent to the lower half of the complex plane. They consider all resonances, and in particular the pre-existing ones, that are the resonances of the Hamiltonian with electric field zero. They study the location of the resonances and their asymptotics as the electric field goes to zero. The topic of resonances in quantum mechanics was one of the main interests of Erik Balslev. The characterization of resonances as poles of the analytic continuation of the resolvent to the lower half of the complex plane is one of the central issues of the celebrated method of complex scaling for the study of resonances. Erik Balslev played a fundamental role in the creation of this method. Pavel Kurasov and Jacob Muller: “On the spectral gap for networks of beams” The contribution by Pavel Kurasov and Jacob Muller discusses fourth order homo4 geneous differential operators dxd 4 (also called beam operators or Bi-Laplacians) in certain Hilbert spaces of functions on metric graphs, satisfying certain conditions called “standard vertex conditions”. Such models have applications in the study of elasticity or in certain models used for the design of medical devices. The main aim of this paper is to establish spectral estimates that can also be used to distinguish certain classes of metric graphs. In particular, the authors obtain results on the spectral gap between the first two eigenvalues, in the form of upper and lower estimates, depending on the type of graph or networks that are considered. In the estimates the interesting technique of graph surgery is used. Ambartsumian-type results relating a graph with the beam operator and the type of graph are also established. K. Nicholas Leibovic: “Some notes in the context of binocular space perception” The author of this contribution is a neuroscientist that had first a formation as mathematician. This contribution has personal reminiscences of his interaction with his friend Erik, with whom he cooperated (with three published joint papers) on the problem of binocular space perception. The problem is explained in the introduction of his contribution, followed by notes on the biographical background. The mathematical considerations in cooperation with Erik are then explained as providing a mapping of physical configurations onto the retina and thence to an area (called V1 area) of the visual cortex. One of the main results of this study is the finding of a new family of curves (besides the known Vieht-Muller circles and Hillebrand hyperbolae) described by isokonic transformations projecting onto the same retinal points and hence onto the same cells in the visual cortex. Experiments to check these theoretical findings are described in the second part, together with considerations about the state of the art of the description of perceived visual space in binocular vision. One of the main conclusions is that percepts in visual space are in general not represented by activity in fixed neural assemblies in the brain. Thierry Paul: “Symbolic calculus for singular curve operators” The paper of Thierry Paul generalizes the Töplitz quantization to the case of Töplitz operators with singular symbol. He uses these results to prove that the singular curve operators in Topological Quantum Field Theory are generalized Töplitz operators of

Introduction to the Scientific Contributions in the Book

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this kind. The results of this paper give a non trivial extension of the Töplitz (antiWick) quantization procedure. Another of the main results of this article is to give a semiclassical setting to all curve-operators in Topological Quantum Field Theory in the case of the once punctured torus or the 4-times punctured sphere. Yiannis N. Petridis and Morten S. Risager: “Higher order deformations of hyperbolic spectra” The contribution by Yiannis N. Petridis and Morten S. Risager exposes work done by the authors on spectral problems of the hyperbolic Laplacians, a topic to which Erik Balslev dedicated much attention in the later part of his life. In the first section the problems are clearly presented and Erik’s work on them is briefly described. This includes a representation of the genesis of their work, namely adapting Erik Balslev and Jean Michel Combes’ fundamental work on analytic dilation of operators in the complex plane to the setting of hyperbolic surfaces. In 1997 Erik Balslev managed to cover much of the theory of Philips and Sarnak, including the establishment of Fermi’s golden rule by techniques of analytic dilation. By this approach, eigenvalues of hyperbolic Laplacians that are embedded in the continuous spectrum [ 14 , +∞) turn into discrete eigenvalues (for appropriate angles of deformation). From the early 90’s a fruitful collaboration between Erik and Alexei Venkov started, reinforced after 2001 when Alexei joined the faculty in Aarhus (some results of their brilliant collaboration are summarized in the section on “Scientific contributions by Erik Balslev”). The present paper continues along this line and answers a question asked by Erik Balsev, namely whether a certain condition appearing in work by Philips and Sarnak is sufficient for ensuring that an eigenvalue in the spectrum of a hyperbolic Laplacian dissolves by a deformations of the type of analytic dilation. This investigation has continued previous work by Erik Balsev and Morten Risager (from the time of their collaboration—Risager has been a PhD student of Erik Balsev). The main theorem is stated in Sect. 2.4. It considers a discrete subgroup of PSL2 (R) acting on the complex half plane by linear fractional transformations and a character χ of it. The induced automorphic Laplacian is considered, denoted by L, a square integrable eigenfunction of which transforms as f (γ z) = χ (γ ) f (z), for z ∈ H, γ ∈  (Maas form). The authors extend Philips’ and Sarnak’s result to the case of higher order Maas forms, ensuring also in this case a positive answer to the question raised by Erik Balslev. Sergei K. Sekatskii: “On the generalized Li’s criterion equivalent to the Riemann hypothesis and its first applications” The paper by Sergei K. Sekatskii presents new results and applications of a result by the same author in 2014 proving that the holding of certain inequalities that he called “generalized Li’s criterion” involving numbers cn given in terms of the n first derivatives of the classical Riemann ζ -function at s = 1 + b for some value b > − 21 is equivalent with the verification of the classical Riemann hypothesis. This is a considerable extension of the original Li’s criterion that corresponded to the particular value b = 0. In the present paper the author computes the asymptotic behavior for large n of the Li coefficients cn . He also formulates both sufficient conditions for

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the holding of the Riemann hypothesis and for its failure. He also obtains results on asymptotics for the case where the classical Riemann hypothesis is assumed to be valid. He then discusses how the results can be useful to gather numerical evidence for the holding and failure, respectively, of the Riemann hypothesis. Mauro Spreafico and A. Zaccagnini: “Regularizing infinite products by the asymptotics of finite products” In many considerations both in mathematics and physics it is interesting to try to associate a good notion of infinite products even in absence of absolute convergence. This is e. g. the case when one tries to associate a determinant to selfadjoint operators A (the determinant being the product of eigenvalues in the case of a discrete spectrum). The ζ -function regularization is one way to achieve this, by setting  det ζ A = e−ζ (0,A) , ζ (s, A) being the ζ -function associated with the eigenvalues of A evaluated in the point s of the complex plane. The present paper considers the setting of invariants in differential topology. In a previous paper by M. Spreafico the asymptotics of the determinants of discrete (Hodge) Laplacians on the circle has been studied in terms of the ζ -function regularization of the continuous Laplacian. Motivated by this the present paper considers the determinant associated with an infinite sequence S of numbers, of the form {an 2 + bk 2 }, b > 0, or {an α + b}, α, a, b ∈ R, n ∈ N. E. g. the authors manage to express  the logarithm of det ζ (an 2 + 2 bk ) asymptotically for n → ∞ in terms of log η(i ab ), η being the classical Dedekind function. Further interesting explicit asymptotic expansions of products of a large number of terms are provided. Ricardo Weder: “Trace maps under weak regularity assumptions” The paper by Ricardo Weder considers trace maps for hypersurfaces in Sobolev spaces under weak regularity assumptions on the hypersurfaces. For hypersurfaces on the whole space it is only assumed that the hypersurfaces are Lebesgue measurable, and in the case of bounded domains it is only required that the boundary be continuous. The method to prove these results is fundamentally different from the ones in the classical theory of trace maps and this makes it possible to prove these results under weak regularity assumptions. These trace maps are applied to the Dirichlet problem and to a coarea formula, where the level sets are only required to be Lebesgue measurable. Trace maps for functions in Sobolev spaces play a crucial role in stationary scattering theory, a subject where Erik Balslev worked intensively and where he made fundamental contributions. In stationary scattering theory the trace maps allow to take a sharp spectral parameter in the spectral representation of the Hamiltonian, and to prove the limiting absorption principle. We hope that these contributions can provide the reader with some insights about the vitality and beauty of the areas of research to which Erik Balslev gave important contributions, furthering their development.

Asymptotics of Random Resonances Generated by a Point Process of Delta-Interactions Sergio Albeverio and Illya M. Karabash

This paper is dedicated to the dear memory of Erik Balslev. Erik has done groundbreaking seminal work on the relations between Schrödinger operators, complex scaling, spectral theory, and number theory. The first named author has had the great pleasure to meet him in Princeton in 1970 and received much inspiration from him. Erik has been a great mathematician and a very kind, deep, and open person, a very dear friend.

Abstract We introduce and study the following model for random resonances: we take a collection ϒ of point interactions ϒ j generated by a simple finite point process in R3 and consider  the resonances of associated random Schrödinger Hamiltonians Hϒ = − + “ m(α)δ(x − ϒ j )”. These resonances are zeroes of a random exponential polynomial, and so form a point process (Hϒ ) in the complex plane C. We show that the counting function for the set of random resonances (Hϒ ) in C-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process ϒ, and obtain an explicit formula for the limiting distribution, as m → ∞, of the leading parameter of the asymptotic chain of ‘most narrow’ resonances generated by a sequence of uniform binomial processes ϒ m with m points. We also pose a general question about the limiting behavior of the point process formed

S. Albeverio Institute for Applied Mathematics, Rheinische Friedrich-Wilhelms Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany e-mail: [email protected] Hausdorff Center for Mathematics, Endenicher Allee 60, 53115 Bonn, Germany I. M. Karabash (B) Fakultät für Mathematik, TU Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany e-mail: [email protected] Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Dobrovolskogo st. 1, Slovyansk 84100, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_2

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by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of m samples of a random point in the 3-D space and the related statistics of extreme values. Keywords Anderson-Poisson Hamiltonian · Random Schrödinger operator · Asymptotically narrow resonances · Random scattering · Distribution of scattering poles · Zero-range interactions · Point processes and point interactions · Limits of random asymptotic structures 2010 Mathematics Subject Classification (Primary) 82B44, 35B34, 35P20, 60G70, (Secondary) 60G55, 35P25, 35J10, 60H25, 47B80, 81Q80

1 Introduction This paper is written in Memory of Erik Balslev. Erik’s pioneering works have been very influential in many areas of analysis, mathematical physics, and analytical number theory. Some of his research topics are closely connected with the present paper. In fact, one of Erik’s major contributions into mathematical physics is the clarification of the very concept of resonance [11–14, 16]. Moreover, the technique of complex dilations that he developed in cooperation with Jean-Michel Combes [15] has played an important role in making accessible tools of analytic perturbation theory in problems involving resonances and eigenvalues embedded in continuous spectra. The connection of spectral theory with problems of analytic number theory, that Erik discovered and masterly developed in a series of important papers [17– 20], has inspired us to the study of the interplay between resonances, exponential polynomials, and point interactions. In this paper we introduce a 3-D continuous model for random resonances using Hamiltonians of a generalized Schrödinger type involving point interactions. To make the Hamiltonian random, we assume that the interactions are generated by a point process with suitable properties. The main goal of this paper is to introduce main notions and problems for related random resonances and consider some of their asymptotic properties on relatively simple examples of binomial point processes. While Schrödinger operators with random point interactions have been introduced in mathematical papers and their self-adjoint spectra have been investigated (see [3, 28, 33, 35, 38] and references therein), it seems that the resonances for such models are not yet adequately mathematically studied. For other models of random resonances, the mathematical theory has been attracting an increasing attention during recent years. It is worth to mention the monograph [51] and the paper [39]. One of the problems suggested in the introduction to [51] as a promising direction of future research concerns the connection between Weyl asymptotics and asymptotics of random resonances. We would like to note that Sect. 3 of the present paper addresses a somewhat connected problem in the context

Asymptotics of Random Resonances Generated …

9

of Schrödinger operators with random point interactions. Namely, we prove that the Weyl-type asymptotics, which has been recently introduced for deterministic point interactions in [44], takes place almost surely (a.s.) for our stochastic example. From a more general perspective, random resonance effects were intensively studied in Physics (see, e.g., the literature in [39]). One of the first mentioning in the mathematical context of resonances of random Schrödinger operators known to us is in the paper [32], where the question of estimation of the support of distribution of random resonances served as one of the motivations for a resonance optimization problem (see also [37] for a more recent discussion of this interplay). We shall present now in detail the model of random resonances that will be studied in this paper. Let ϒ be a point process on R3 , let (, F, P) be the underlying complete probability space, and let ηϒ be the random (counting) measure associated with ϒ. Throughout the paper we assume that (A0) the point process ϒ is simple and finite. (A point process  on Rd is said to be finite if it satisfies η (R3 ) < ∞ almost surely; for the definition of simple process see (1.2) below). Any locally-finite point process on Rd is proper. For the finite point process ϒ, this means that there exist random variables ν :  → N0 = {0} ∪ N and ϒ j :  → R3 , j ∈ N, such that ϒ can be considered as a finite collection of random points ϒ(ω) = {ϒ j (ω)}νj=1 for almost all (a.a.) ω ∈  (with respect to the measure P).  In particular, ηϒ = #ϒ j=1 δ(· − ϒ j (ω))dx a.s., where δ(x)dx is Dirac’s delta measure and #S is the number of in a set (or in a multiset) S. When j2 < j1 elements j1 0 or ν = 0, it is assumed that j= j2 = 0 or, resp., {ϒ j } j=1 = ∅. For basic definitions and facts concerning point processes, we refer to [43] (see also [30, 34]). We associate with ϒ a random Hamiltonian Hϒ in the following way. Let α be a complex number, which is fixed throughout the paper. For a deterministic set 3 Y = {y j }#Y j=1 consisting of #Y ∈ N distinct points in R , the linear operator HY in the 2 3 complex Hilbert space L C (R ) is the point interaction Hamiltonian corresponding to the formal differential expression − u(x) + “

#Y 

m(α)δ(x − Y j )u(x)”, x ∈ R3 ,

(1.1)

j=1

with the ‘strength-type’ parameter α. Here the Dirac measure δ(· − y j ) placed at a center y j ∈ R3 of a point interaction is symbolically multiplied by a normalization parameter m(α), see [3, 5, 6, 53] for details and Sect. 2 for the rigorous definition of this deterministic operator. If #Y = 0, we assume that Y = ∅ and HY = −, where   = 3j=1 ∂x2j is the Laplace operator in R3 . The aim of this paper is to study resonances of the random operator Hϒ . For a deterministic Hamiltonian H , (continuation) resonances k are defined as poles of the resolvent (H − k 2 )−1 extended in a generalized sense through the essential spectrum into the lower complex half-plane C− := {z ∈ C : Im z < 0} [5, 48]. The collection of all resonances (H ) ⊂ C associated with H (in short, resonances of

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H ) is actually a multiset, i.e., a set in which an element e can be repeated a finite number of times. We denote this number mult e and call it the multiplicity of e. An element e is called simple (multiple) if mult e = 1 (resp., mult e ≥ 2). A multiset is said to be simple if every of its elements is simple; a point process is called simple if it is a.s. simple.

(1.2)

The (algebraic) multiplicity of a resonance k can be defined as the multiplicity of the corresponding generalized pole of (H − k 2 )−1 (e.g., [25]), as the multiplicity of an eigenvalue obtained by a complex dilation [15, 25, 52], or as the multiplicity of a zero of a certain analytic function built from the resolvent of H and generating resonances as its zeros [5]. In the present paper, we follow the latter approach to the definition of multiplicity (see [3, 5, 6] and Sect. 2). For ω belonging to the event {ω ∈  : ηϒ (R3 ) = ∞} or to the event {ω ∈  : ϒ(ω) is non-simple}, which both have zero probability according to (A0), we do not define the Hamiltonian Hϒ(ω) . So, (Hϒ (·)) is defined only almost surely. Remark 1.1 The reason for this convention is the following. While HY can be defined in some cases when #Y = ∞ and Y is simple, e.g., in the case where inf j = j |Y j − Y j | > 0 [3, 46] (see [35] for a relaxation of this condition in deterministic and stochastic settings), the notion of resonances in such settings is not well understood. It is natural, e.g., to expect that the resonances can be defined in a certain way for the case of a periodic lattice, but the classification of singularities of the resolvent and their physical meaning requires an additional study (cf. [40, 41]). We believe that in some of the cases with Y having multiple points it is also possible to give some meaning to the operator HY (moreover, this question is important for the optimization of resonances of HY by a modification of the positions of centers, cf. [6, 36]), but we are not aware of such studies. The random collection of resonances (Hϒ ) is a proper point process in C (generally, with multiplicities), see Theorem 2.2 below. In this paper, we concentrate mainly on the study of the asymptotic behaviour of this random process (Hϒ ) near ∞ in the complex plane of the spectral parameter k. In Sect. 3, we study this behaviour on the ‘rough level’ of the asymptotics of counting function η(Hϒ ) (D R ) as R → +∞, and the (normalized) asymptotic density of resonances defined by Ad(Hϒ ) := lim

R→∞

η(Hϒ ) (D R ) R

(see [7, 8, 44]),

where D R = {z ∈ C : |z| < R}. The class (m, Br ) of the binomial processes of the sample size m with a uniform sampling distribution in a ball Br := {x ∈ R3 : |x| < r } provides us with a simple example of a point process ϒ with ‘good’ diffusive properties (see, e.g., [43]). We show in Theorem 3.1 that the corresponding asymptotic density Ad(Hϒ ) is equal a.s. to the random variable V (ϒ) , where π

Asymptotics of Random Resonances Generated …

V (Y ) := max σ∈S N

N 

11

|Y j − Yσ( j) |

j=1

is the ‘size’ of a deterministic collection Y of N points (see [44]) and S N is the symmetric group of degree N consisting of permutations σ. (The above notation considers a permutation as a bijective map σ : {1, . . . , N } → {1, . . . , N }). Slightly modifying the terminology of [44], we say that a deterministic operator ) . So, we prove in HY has the Weyl-type asymptotics of resonances if Ad(HY ) = V (Y π the present paper that Weyl-type asymptotics holds a.s. for the case ϒ ∈ (m, Br ), but our proof can be easily extended to a wide class of mixed binomial processes. In Sect. 4, we consider the fine structure of asymptotical behaviour of the resonance point process (Hϒ ) near k = ∞. This structure can be better seen by the use of the logarithmic (asymptotic) density function Adlog (h), i.e., the normalized density corresponding to semi-logarithmic strips {−h ln(| Re k| + 1) ≤ Im k} (the real parameter h characterize, roughly speaking, the “wideness” of this “strip”). This logarithmic density function Adlog (h) is defined in [7] by log

Adlog (h) = Adlog (ϒ, h) := lim

R→∞

N Hϒ(ω) (h, R) R

,

(1.3)

via the associated logarithmic counting function log

N Hϒ(ω) (h, R) := #{k ∈ (Hϒ(ω) ) : −h ln(| Re k| + 1) ≤ Im k and |k| ≤ R}. (1.4) It is easy to see that Adlog (h) is a random variable for each h ∈ R. The main result of Sect. 4 says that the sample paths of Adlog (h), h ∈ R, are almost surely piecewise constant functions with a finite number of jumps and that after multiplication by πh the associated random measure dAdlog (h) becomes N0 valued and defines a finite point process K := {K j }#K j=1 on R+ . Thus, this point process describes the structure of asymptotic sequences of random resonances “going to ∞” in the k-plane. In Sect. 5, we take a sequence of uniform binomial processes ϒ [m] ∈ (m, Br ) in a fixed 3-D ball Br and consider the limiting behavior of the process K when the number of points m grows to ∞. In particular, we obtain an explicit formula for the normalized limiting distribution of min K(ϒ [m] ) =

min

1≤ j≤ #K(ϒ [m] )

K j (ϒ [m] )

as m → ∞. We also obtain probabilistic estimates for the distributions of the asymptotic densities Ad(Hϒ [m] ) for large m using the Weyl-type asymptotics result of Sect. 3.

12

S. Albeverio and I. M. Karabash

Notation. The notation #S stands for the number of elements in a set or multiset S (in the case of a multiset S, the cardinality #S counts each elements according to its multiplicity). For a set Y ∈ Rd , its diameter is defined by diam Y := inf{ ≥ 0 :  ≥ |y0 − y1 | for all y0 , y1 ∈ Y } (we want diam ∅ := 0). The following standard sets are used: the set Z of integers, the set N0 = N ∪ {0} of nonnegative integers, half-lines R± = {x ∈ R : ±x > 0}, discs D R = {z ∈ C : |x| < R} in the complex plane, open half-planes C± = {z ∈ C : ± Im z > 0}, and balls Br := {x ∈ R3 : |x| < r } in the 3-D real space. Sometimes, the complex plane C is considered as the Euclidean space R2 . For a subset S of a normed space U , we denote its closure by S and by ∂ S its boundary. For u 0 ∈ U and z ∈ C, we write zS + u 0 := {zu + u 0 : u ∈ S}. The class of random vectors with the standard normal distribution in R3 is denoted by N (0, IR3 ). The function Ln(·) is the branch of the natural logarithm multi-function ln(·) in C \ (−∞, 0] fixed by Ln 1 = i Arg0 1 = 0. For z ∈ R− , we put Ln z = Ln |z| + iπ. By S N we denote the symmetric group of degree N , and by e the identity permutation.

2 Point Process of Random Resonances Let {an }n∈N0 be a sequence of C-valued random variables on probability the complete n a z is said to be a space (, F, P). Then the random power series F(z) = ∞ n n=0 random entire function (on C) if the radius of convergence of F is a.s. equal to ∞. The random entire function F is said to be a random polynomial if a.s. an = 0 only for a finite number of indices n ∈ N0 (we refer to [9, 10, 23] for basic facts of the theories of random analytic functions and random polynomials). In this section, it will be shown that the multiset (Hϒ ) of resonances of the random operator Hϒ is a.s. the multiset of zeros of a random entire function. Then it is easy to see that (Hϒ ) is a point process in C. Definition 2.1 Consider a finite simple point process β = {β j } j=1 on C. Let { p j }∞ j=1 be a sequence of (complex valued) random polynomials. Then any random function of the form #β  eβ j z p j (z) #β

j=1

is a.s. defined on the whole complex plane C and is said to be a random exponential polynomial. It is easy to check that a random exponential polynomial is a random entire function (concerning the theory of deterministic exponential polynomials we refer to [21, 22]).

Asymptotics of Random Resonances Generated …

13

In the rest of the paper we suppose that the assumption (A0) holds. Then from the definition of resonances for deterministic point interaction Hamiltonians HY [5], one sees that the multiset of random resonances (Hϒ ) is a.s. the multiset of zeros of a specially constructed random exponential polynomial. For convenience of the reader, let us recall the construction of this exponential polynomial and the definition of the operator HY associated with (1.1) in the deter3 ministic settings. Let Y = {Y j }#Y j=1 be a simple finite collection of points in R , i.e., the interaction centers Y j are distinct and their number #Y ∈ N ∩ {0} is finite. We assume that all point interactions are of the same ‘strength’ which is described by a fixed parameter α ∈ C. Various approaches to the definitions of the operator HY associated with (1.1) were given, e.g., in [1–4, 6, 53]. The operator HY is a closed operator in the complex Hilbert space L 2C (R3 ) and it has a nonempty resolvent set which can be obtained from C \ [0, +∞) after possible exclusion of a finite number of points [3, 6]. The resolvent (HY − z 2 )−1 of HY is defined in the classical sense on the set of z ∈ C+ such that z 2 is not in the spectrum. Its integral kernel has the form (HY − z 2 )−1 (x, x ) = G z (x − x ) +

#Y 

G z (x − Y j ) [ Y ]−1 j, j G z (x − Y j ),

j, j =1

(2.1) iz|x−x |

e where x, x ∈ R3 \ Y and x = x , see e.g. [3, 6]. Here G z (x − x ) := 4π|x−x

| is 2 −1 the integral kernel associated with the resolvent (− − z ) of the kinetic energy

Hamiltonian − in L 2C (R3 ); [ Y ]−1 j, j denotes the j, j -element of the inverse to the matrix   #Y G z (x), x = 0 iz  



. Y (z) = α − 4π δ j j − G z (Y j − Y j ) j, j =1 , where G z (x) := 0, x = 0 (2.2)

Remark 2.1 Actually, the Krein-type formula (2.1) for the difference of the perturbed and unperturbed resolvents of operators HY and − can be used as a definition of HY [3, 29]. For other equivalent definitions of HY and for the renormalization procedure giving a meaning to m(α) in (1.1) and to the ‘strength’ parameter α, we refer to [1–3] in the case α ∈ R, and to [4, 6] in the case of complex α ∈ / R. Note that, in the case α ∈ R, the operator HY is self-adjoint in L 2C (R3 ); and in the case α ∈ C− , HY is closed and maximal dissipative (e.g., in the sense of [27], or in the sense that iHY is maximal accretive). In the future, we will consider the finite collection Y of distinct points as a finite simple multiset in R3 . If #Y ≥ 1, the set of resonances (HY ) of the deterministic operator HY is by definition the set of zeroes of the determinant det Y (·) , which we call the characteristic determinant. The multiplicity of a resonance k will be understood as the

14

S. Albeverio and I. M. Karabash

multiplicity of a corresponding zero of det Y , which is an analytic function in the complex variable z [3, 5]. Equipped with the multiplicity of any resonance, the set (HY ) becomes a multiset. Remark 2.2 While it is widely assumed among specialists that the various definitions of (algebraic) multiplicities of resonances coincide (e.g., in [5] and in [25] for HY ; we point out that HY can be easily placed into the black box formalism of [52]), we do not know to what extent this assumption is actually checked. In particular, the answer to this question for the multiplicity of a zero resonance depends on the way how this resonance is handled (in some of the works, 0 is excluded from the set of resonances by definition). The characteristic determinant det Y (·) is obviously an exponential polynomial. Let us consider in more detail the structure of its simple modification DY (z) := (−4π)#Y det Y (z),

(2.3)

which we call the modified characteristic determinant and which is also an exponential polynomial. The multiplication by (−4π)#Y will simplify the appearance of the formulae below. The statement that DY is an exponential polynomial means that it has the form #B 

PB j (z)eiB j z ,

(2.4)

j=1 B where B = {B j }#j=1 is a finite sequence of complex numbers with # B ∈ N, and PB j (·) are polynomials. We will say that a function is an exponential monomial if it has the form eiB0 z p(z) with B0 ∈ C and a nontrivial polynomial p (nontrivial in the sense that p(·) ≡ 0). Expanding by the Leibniz formula the determinant det Y (z), one sees that DY (z) is a sum of terms of the form

eizVσ (Y ) p σ,Y (z)

(2.5)

taken over all permutations σ in the symmetric group S N with N = #Y . Here the numbers Vσ (Y ) and the polynomials p σ,Y (·) have the form Vσ (Y ) :=

#Y j=1

|y j − yσ( j) |,

p σ,Y (ζ) := σ C1 (σ, Y )



(iz − 4πα), (2.6)

j:σ( j)= j

where C1 (σ, Y ) := j:σ( j) = j |y j − yσ( j) |−1 (in the case where σ is equal to the identity permutation e, we put C1 (e, Y ) := 1) and σ is the permutation sign (the LeviCivita symbol).

Asymptotics of Random Resonances Generated …

15

B For the particular case of DY as given by (2.3), the sequence {B j }#j=1 and polynomials PB j in (2.4) can always be chosen such that B • the sequence {B j }#j=1 consists of increasing nonnegative numbers and • each of the polynomials PB j (·) is nontrivial.

In such a case, we say that (2.4) is the canonical form and that B j are the frequencies of the exponential polynomial DY . Similarly, Vσ (Y ) is called the frequency of the exponential monomial (2.5) (in the terminology of [8], Vσ (Y ) is the metric length of the directed graph associated with Y and a permutation σ). Note that we always have B1 = 0 and P0 (z) = PB1 (z) = (iz − 4πα)#Y .

(2.7)

Consider now the random operator Hϒ , where ϒ is assumed to be a finite point process on R3 satisfying (A0). In particular, ϒ is a proper point process (see, e.g., [43]). Consequently, there exist an N0 -valued random variable ν and R3 -valued random variables ϒ j , j ∈ N, such that ϒ can be considered a.s. as a finite collection of random R3 -points, namely, ϒ = {ϒ j }νj=1 a.s. In the case #Y = 0 (recall that # denotes the cardinality), one has HY = − and (HY ) = ∅, and so we put DY (z) := 1. With the above deterministic definitions, the random exponential polynomial Dϒ (z) is now a.s. defined and generates the random multiset of resonances (Hϒ(ω) ) as the multiset of its zeros. Example 2.1 Consider a mixed binomial process ϒ with mixing distribution V given by V({ j}) = 1/3 for j = 0, 1, 2 and the standard multivariate normal distribution N (0, IR3 ) as the sampling distribution Q. That is, ϒ = {ϒ j }νj=1 , where ν, ϒ1 , and ϒ2 are mutually independent random variables, ϒ1 and ϒ2 are normally distributed with the law N (0, IR3 ), and P{ν = j} = 1/3, j = 0, 1, 2. Let us introduce the random variables  = |ϒ1 − ϒ2 | and √ diam ϒ (so diam ϒ is equal to  when ν = 2, and to 0 when ν ≤ 1). Note that / 2 has a χ3 -distribution as its law. We observe that for #ϒ = ν = 0, we have (Hϒ ) = ∅. If #ϒ = 1, then iz and so (Hϒ ) consists of one point (−i)4πα of multiplicdet ϒ (z) = α − 4π ity 1. Each of the two aforementioned events has probability 1/3. In the event {#ϒ = 2,  = 0} having zero probability, the Hamiltonian Hϒ(ω) (and so (Hϒ(ω) )) is formally not defined. Assume now that the event ω ∈ {#ϒ = 2,  > 0} (with probabilty 1/3) takes place. Then the multiset (Hϒ ) consists of zeroes of the exponential polynomial  2 Dϒ(ω) (z) = (iz − 4πα)2 − eiz(ω) /(ω) . So (Hϒ ) is a countable sequence with an accumulation point at ∞. A more detailed description of the set of zeros of this transcendental function of z in terms of α and (ω) can be found in [3, 5, 7] (see also Sect. 4 below). The above considerations easily lead to the following result.

16

S. Albeverio and I. M. Karabash

Theorem 2.2 Assume that the point process ϒ satisfies (A0) (i.e., ϒ is simple and finite). Then: (i) The random multiset of resonances (Hϒ ) is a proper point process on C. (ii) ⎧ ⎨ ∞, for ω such that #ϒ(ω) ≥ 2; #(Hϒ(ω) ) = 1, for ω such that #ϒ(ω) = 1; a.s. (2.8) ⎩ 0, for ω such that #ϒ(ω) = 0; (here #(Hϒ(ω) ) = ∞ means that (Hϒ(ω) ) is countably infinite). Proof (i) Obviously, Dϒ(ω) (·) ≡ 0 and Dϒ(ω) is a random entire function for a.a. ω ∈ . This implies that the multiset of zeros of Dϒ (and so the multiset (Hϒ )) is a locally finite proper point process in C (see [50, pp. 338–340]; the scheme of the proof in the particular case of random polynomials can be found in [9, 23]). (ii) The case #ϒ(ω) = 0 follows from the fact that (H∅ ) = (−) = ∅. Assume that #ϒ(ω) = 1. Then a direct computation gives that (Hϒ(ω) ) consists of one point (−i)4πα of multiplicity 1. Finally, assume that 2 ≤ #ϒ(ω) < ∞ and ϒ(ω) is simple (recall that according to the assumption (A0), ϒ(ω) is simple a.s.). Then it is easy to see that, in the Leibniz expansion of Dϒ(ω) , the exponential monomials (2.5) with positive frequencies Vσ (Y ), σ = e, cannot completely cancel each other [6, 7]. So, additionally to the zero frequency (2.7), the exponential polynomial Dϒ(ω) (·) has at least one positive frequency B1 . The existence of two different frequencies implies the existence of a (countably) infinite number of zeroes of Dϒ(ω) [21, 44]. 

3 Asymptotic Density and Weyl-Type Asymptotics with Probability 1 A substantial part of the mathematical studies of deterministic resonances is devoted to the asymptotics of the their counting function N HY (R) = #{k ∈ (HY ) : |k| ≤ R}.

(3.1)

In [44], the asymptotics N HY (R) = Cπ R + O(1) as R → ∞ with a certain constant C ≥ 0 was established for deterministic Hamiltonians HY with  #Y = N ∈ N point interactions and it was proved that C ≤ V (Y ) := maxσ∈S#Y #Y j=1 |Y j − Yσ( j) |. The number V (Y ) was called in [44] the size of the set Y . In the case C = V (Y ), it was said (slightly changing the wording in [44]) that the Weyl-type asymptotics of N HY (R) takes place. We use in the present paper the terminology of [7] and say that Ad(HY ) := C/π is the total asymptotic density of resonances of HY . This is motivated by the equality Ad(HY ) = lim

R→∞

N HY (R) R

(see (3.1) and the line following it).

(3.2)

Asymptotics of Random Resonances Generated …

17

By Theorem 2.2, the total asymptotic density of random resonances Ad(Hϒ ) is an [0, +∞]-valued random variable for any point process ϒ satisfying (A0). Combining this with the deterministic result of [44] one sees that Ad(Hϒ ) is a [0, +∞)-valued random variable and that Ad(Hϒ ) ≤

V (ϒ) a.s. π

The main result of this section says, roughly speaking, that for point processes ϒ with good enough ‘diffuse’ sampling distributions the Weyl-type asymptotics for random resonances of Hϒ holds with the probability 1. For the sake of simplicity, we prove this result only for the uniform binomial processes (m, Br ) in R3 -balls (see Sect. 1). Theorem 3.1 If ϒ ∈ (m, Br ) with m ∈ N, then a.s. we have Ad(Hϒ ) = V (ϒ)/π. We obtain this theorem in Sect. 3.1 below from the strengthened version of the deterministic result of [8]. Namely, for a deterministic HY with N ∈ N point interactions, it follows from [8] that the Weyl-type asymptotics is generic in the sense described below. We consider Y = {Y j } Nj=1 as an N -tuple of R3 vectors and identify it with a vector in the space (R3 ) N = R3N with the standard 2 -metric. The assumption that the interaction centers Y j are distinct means that Y belongs to the family A of admissible N -tuples that is defined by A := {Y ∈ (R3 ) N : Y j = Y j for j = j } and that is considered as an induced metric space and an induced measurable space with (3N-dimensional) Lebesgue measure. Then [8] implies that the set of Y such that N HY has a Weyl-type asymptotics is nowhere dense. This result is not enough to prove Theorem 3.1 (because there exist nowhere dense subsets of R3N with positive Lebesgue measure). Recall that S is a proper analytic subset of an open set O ⊂ Rd if there exists a real analytic function f on O such that f ≡ 0 on O and S = {x ∈ O : f (x) = 0}. We prove in Sect. 3.1 the following strengthening of the aforementioned result of [8]. Theorem 3.2 Let the set A0 ⊂ (R3 ) N consist of all admissible N -tuples Y ∈ A so that N HY has non-Weyl-type asymptotics (i.e., so that Ad(HY ) < V (Y )/π). Then: (i) A0 is a subset of a certain proper analytic subset of A; (ii) A0 is a set of zero Lebesgue measure. Remark 3.1 Lojasiewicz’s theory about the structure of analytic varieties [45] yields much stronger restrictions (than those of Theorem 3.2 (ii)) on a proper analytic subset containing the non-Weyl-type asymptotics set A0 . We refer to [42] for the discussion of the theory of real analytic sets.

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S. Albeverio and I. M. Karabash

3.1 Proofs of Theorems 3.1 and 3.2 The equivalence classes of edge-equivalent permutations σ ∈ S N for the N -tuple Y were introduced in [8]. This definition was given in terms of directed and undirected graphs associated with permutations. We refer to [8, Sect. 3] for the details and would like to notice here that the following fact was also proved: two permutations σ, σ ∈ S N are edge-equivalent if and only if Vσ (Y ) = Vσ (Y ) for every Y ∈ A. We recall that Vσ is defined in (2.6) (it is the the metric length of the aforementioned directed metric graph associated with σ and Y ). Let us denote by  n ∈ N the number of edge-equivalence classes in S N and let us n , in each of them. We use also the following take one representative  σ j , j = 1, . . . , observation of [8]: if Y belongs to the set A1 := {Y ∈ A : Vσ j (Y ) = Vσm (Y ) if j = m}, then there is no cancellation of the exponential monomials (2.5) with the highest possible frequency V (Y ) after the summation of (2.5) required by the Leibniz formula for det Y . Thus, for every Y ∈ A1 the Weyl-type asymptotics takes place. Proof of Theorem 3.2 For each permutation σ, the function Vσ (·) is a sum of terms of the form  |Y j − Y j | =

3 

1/2 [Y j,m − Y j ,m ]

2

,

m=1

where j = σ( j) and where Y j,m , m = 1, 2, 3, are the R3 -coordinates of Y j , 1 ≤ j ≤ N . Therefore, Vσ (·) is a real analytic function in the variables Y j,m (1 ≤ j ≤ N , m = 1, 2, 3) on A. n , of edge-equivalent classes Let us take now the representatives  σ j , j = 1, . . . , of permutations, which were described above. We see that the function f j,m (Y ) = Vσ j (Y ) − Vσm (Y ) is real analytic on A. Moreover, if j = m this function is not trivial j,m on A, and so the set A0 := {Y ∈ A : f j,m (Y ) = 0} of its zeroes is a proper analytic subset of A. We will use the following well-known fact (see e.g., [42]): a proper analytic subset of an open set in Rd has measure zero. j,m

(3.3)

Thus, each of the sets A0 with j = m is of measure zero and so is their union  j,m  A0 = 1< j 0, m ∈ N. Then: [m] → 2r1 in probability. (i) As m → +∞, we have Kmin [m] (ii) The (rescaled) limit distribution of the random variable Kmin − 2r1 is given by    1 3 3 [m] ≤ t → 1 − e−48r t as m → ∞, t > 0 P m 2/3 Kmin − 2r [m] (recall that Kmin −

1 2r

(5.3)

≥ 0 a.s.).

[m] is equal Proof (i) is obvious from Corollary 5.1 (ii), i.e., from the fact that Kmin 1 . Combining this fact with in distribution to the random variable max |ξ j − ξi | 1≤i, j≤m

the result of [47, Theorem 1.1] on the limiting distribution of max |ξ j − ξi |, one 1≤i, j≤m

obtains statement (ii).



Asymptotics of Random Resonances Generated …

23

5.2 Estimates on the Growth of the Total Asymptotic Density [m] The study of the limit law as m → ∞ for the maximal leading parameter Kmax is a more difficult problem. This parameter is connected with the total asymptotic density of resonances Ad(Hϒ [m] ) and so, due to Theorem 3.1, with the size V (ϒ [m] ) of the random set ϒ [m] . A simple version of this connection is given by the inequality m [m] a.s. (see Corollary 5.1 (iii)). A more precise dependence in the Kmax ≥ V (ϒ [m] ) deterministic case can be seen from [7, formula (3.6)]. The following theorem describes the rate of growth as m → ∞ of the total asymptotic densities Ad(Hϒ [m] ) and of the sizes V (ϒ [m] ), which according to Theorem 3.1 [m] are connected by Ad(Hϒ [m] ) = V (ϒπ ) a.s.

Theorem 5.3 Let r > 0 and ϒ [m] ∈ (m, Br ), m ∈ N. Then, for any t ∈ R,  lim inf P m→∞

 √ V (ϒ [m] ) 36 2 87 √ > m+ t m ≥ 1 − (t), r 35 35

(5.4)

t 2 where (t) = (2π)−1/2 −∞ e−s /2 ds (the standard normal distribution function). In particular, the following estimate is valid for the asymptotic density Ad(Hϒ [m] ) of resonances  mr  → 1 as m → ∞ (5.5) lim P Ad(Hϒ [m] ) > m→∞ π Proof For convenience of the notation, we replace each process ϒ [m] by the process  [m] = {ξ j }mj=1 defined in (5.1). This does not influence the estimates below. ϒ Let m ∗ = 2m/2, i.e., m ∗ = m if m is even, and m ∗ = m − 1 if m is odd. Then, from the definition of V (·), we have  [m] ) ≥ 2Sm ∗ /2 , V (ϒ

where Sm =

m 

|ξ2 j−1 − ξ2 j |.

j=1

The R+ -valued random variables λ j := given by E(λ1 ) = 18/35,

|ξ2 j−1 −ξ2 j | 2r

E(λ21 ) = 3/10

Hence, the variance of λ j is Var λ j = we get

are i.i.d. with the first two moments

(see [31] for the general formula).

 √ 2 87 √ 35 2

. Applying the Central Limit Theorem,

24

S. Albeverio and I. M. Karabash



√  √ Sm 18 87 − m ≤ t m √ P → (t) 2r 35 35 2 as m → ∞. This implies (5.4) and, in turn, (5.5).



Acknowledgements IK is grateful to Richard Froese for inspiring and educative discussions about resonances and random spectral theory, to Jürgen Prestin for the hospitality of the University of Lübeck, to Baris Evren Ugurcan and the Hausdorff Research Institute for Mathematics (HIM) of the University of Bonn for their hospitality during the trimester program “Randomness, PDEs and Nonlinear Fluctuations” at HIM in 2020. IK was supported by the VolkswagenStiftung project “Modeling, Analysis, and Approximation Theory toward applications in tomography and inverse problems” and, during the workshop “Analytical Modeling and Approximation Methods” (Berlin, 04–08.03.2020), by the VolkswagenStiftung project “From Modeling and Analysis to Approximation”.

References 1. S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, Singular perturbations and nonstandard analysis. Trans. Amer. Math. Soc. 252, 275–295 (1979) 2. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, The low energy expansion in nonrelativistic scattering theory. Annales de l’IHP Physique théorique 37(1), 1–28 (1982) 3. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, 1st edn (New York, NY, Springer, 1988), Russian transl. (Moscow, Mir, 2005), 2nd edn, with an appendix by P. Exner (AMS Chelsea Publishing, Providence, RI, 2012) 4. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, L. Streit, Charged particles with short range interactions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38(3), 263–293 (1983) 5. S. Albeverio, R. Høegh-Krohn, Perturbation of resonances in quantum mechanics. J. Math. Anal. Appl. 101, 491–513 (1984) 6. S. Albeverio, I.M. Karabash, Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions. Oper. Matri. 11(4), 1097–1117 (2017) 7. S. Albeverio, I.M. Karabash, On the multilevel internal structure of the asymptotic distribution of resonances. J. Diff. Eq. 267(11), 6171–6197 (2019) 8. S. Albeverio, I.M. Karabash, Generic asymptotics of resonance counting function for Schrödinger point interactions, in Analysis as a Tool in Mathematical Physics: In Memory of Boris Pavlov, ed. P. Kurasov, A. Laptev, S. Naboko, B. Simon, Oper. Theory Adv. Appl., vol. 276 (Birkhäuser, Cham, 2020), pp. 80–93 9. L. Arnold, Random Power Series (Michigan State University, Statistical Laboratory, Department of Statistics, 1966) 10. L. Arnold, Über die Konvergenz einer zufälligen Potenzreihe. J. Reine Angew. Math. 222, 79–112 (1966) 11. E. Balslev, Resonance functions for radial Schrödinger operators. J. Math. Anal. Appl. 123(2), 339–365 (1987) 12. E. Balslev, Wave operators for dilation-analytic three-body Hamiltonians. J. Funct. Anal. 81(2), 345–384 (1988) 13. E. Balslev, Asymptotic properties of resonance functions and generalized eigenfunctions, in Schrödinger Operators (Sønderborg, 1988). Lecture Notes in Phys. 345 (Springer, Berlin, 1989), pp. 43–64 14. E. Balslev, A note on the cluster model and complex scaling, in Resonances (Lertorpert, 1987). Lecture Notes in Phys. 345 (Springer, Berlin, 1989), pp. 455–457

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15. E. Balslev, J.M. Combes, Spectral properties of many body Schrödinger operators with dilation analytic interactions. Comm. Math. Phys. 22(4), 280–294 (1971) 16. E. Balslev, E. Skibsted, Resonances and poles of the S-matrix, in Symposium “Partial Differential Equations” (Holzhau, 1988). Teubner-Texte Math. 112 (Teubner, Leipzig, 1989), pp. 24–32 17. E. Balslev, A. Venkov, The Weyl law for subgroups of the modular group. Geom. Funct. Anal. 8(3), 437–465 (1998) 18. E. Balslev, A. Venkov, Spectral theory of Laplacians for Hecke groups with primitive character. Acta Math. 186(2), 155–217 (2001) 19. E. Balslev, A. Venkov, On the relative distribution of eigenvalues of exceptional Hecke operators and automorphic Laplacians. Algebra i Analiz 17(1), 5–52 (2005); reprinted in St. Petersburg Math. J. 17(1), 1–37 (2006) 20. E. Balslev, A. Venkov, Perturbation of embedded eigenvalues of Laplacians, in Traces in Number Theory, Geometry and Quantum Fields, ed. S. Albeverio, M. Marcolli, J. Plazas, S. Paycha (Vieweg, 2008), pp. 23–34 21. R.E. Bellman, K.L. Cooke, Differential-Difference Equations (Academic Press, New York, London, 1963) 22. C.A. Berenstein, R. Gay, Complex Analysis and Special Topics in Harmonic Analysis (Springer Science & Business Media, 2012) 23. A.T. Bharucha-Reid, M. Sambandham, Random Polynomials (Academic Press, Orlando, 1986) 24. S. Cox, E. Zuazua, The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44(2), 545–573 (1995) 25. S. Dyatlov, M. Zworski, Mathematical Theory of Scattering Resonances (American Mathematical Soc., 2019) 26. V. Enss, Summary of the conference and some open problems, in Resonances—Models and Phenomena, ed. by S. Albeverio, L.S. Ferreira, L. Streit (Springer, Berlin, Heidelberg, 1984) 27. P. Exner, Open Quantum Systems and Feynman Integrals (Springer Science & Business Media, Berlin, 2012) 28. R. Figari, H. Holden, A. Teta, A law of large numbers and a central limit theorem for the Schrödinger operator with zero-range potentials. J. Stat. Phys. 51(1–2), 205–214 (1988) 29. A. Grossmann, R. Høegh-Krohn, M. Mebkhout, The one particle theory of periodic point interactions. Comm. Math. Phys. 77(1), 87–110 (1980) 30. M. Haenggi, Stochastic Geometry for Wireless Networks (Cambridge University Press, 2012) 31. J.M. Hammersley, The distribution of distance in a hypersphere. Ann. Math. Stat. 21(3), 447– 452 (1950) 32. E.M. Harrell, R. Svirsky, Potentials producing maximally sharp resonances. Trans. Amer. Math. Soc. 293, 723–736 (1986) 33. P.D. Hislop, W. Kirsch, M. Krishna, Eigenvalue statistics for Schrödinger operators with random point interactions on Rd , d = 1, 2, 3. J. Math. Phys. 61, 092103 (2020) 34. O. Kallenberg, Foundations of Modern Probability (Springer Science & Business Media, 2006) 35. M. Kaminaga, T. Mine, F. Nakano, A self-adjointness criterion for the Schrödinger operator with infinitely many point interactions and its application to random operators. Ann. Henri Poincaré 21, 405–435 (2020) 36. I.M. Karabash, Pareto optimal structures producing resonances of minimal decay under L 1 -type constraints. J. Diff. Eq. 257, 374–414 (2014) 37. I.M. Karabash, O.M. Logachova, I.V. Verbytskyi, Nonlinear bang-bang eigenproblems and optimization of resonances in layered cavities. Integr. Equ. Oper. Theory 88(1), 15–44 (2017) 38. W. Kirsch, F. Martinelli, On the spectrum of Schrödinger operators with a random potential. Comm. Math. Phys. 85(3), 329–350 (1982) 39. F. Klopp, Resonances for large one-dimensional “ergodic” systems. Anal. & PDE 9(2), 259– 352 (2016) 40. E. Korotyaev, The propagation of the waves in periodic media at large time. Asymptot. Anal. 15(1), 1–24 (1997)

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41. E. Korotyaev, Resonance theory for perturbed Hill operator. Asymptot. Anal. 74(3–4), 199–227 (2011) 42. S.G. Krantz, H.R. Parks, A Primer of Real Analytic Functions (Springer Science & Business Media, 2002) 43. G. Last, M. Penrose, Lectures on the Poisson Process (Cambridge University Press, 2017) 44. J. Lipovský, V. Lotoreichik, Asymptotics of resonances induced by point interactions. Acta Physica Polonica A 132, 1677–1682 (2017) 45. S. Lojasiewicz, Introduction to Complex Analytic Geometry (Birkhäuser, 1991) 46. M.M. Malamud, K. Schmüdgen, Spectral theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions. J. Funct. Anal. 263(10), 3144–3194 (2012) 47. M. Mayer, I. Molchanov, Limit theorems for the diameter of a random sample in the unit ball. Extremes 10(3), 129–150 (2007) 48. R.B. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle. Journées équations aux dérivées partielles 8, article no. 3 (1984) 49. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic Press, New York, London, 1978) 50. T. Shirai, Limit theorems for random analytic functions and their zeros. RIMS Kûkyûroku Bessatsu 34, 335–359 (2012) (Functions in Number Theory and Their Probabilistic Aspects, Kyoto, 2010) 51. J. Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials. Mémoires de la SMF 136, 150 (2014) 52. J. Sjostrand, M. Zworski, Complex scaling and the distribution of scattering poles. J. Amer. Math. Soc. 4(4), 729–769 (1991) 53. Albeverio, S., Kurasov, P.: Singular perturbations of differential operators: solvable Schrödinger-type operators (Cambridge University Press, 2000)

Green’s Functions and Euler’s Formula for ζ (2n) Mark S. Ashbaugh, Fritz Gesztesy, Lotfi Hermi, Klaus Kirsten, Lance Littlejohn, and Hagop Tossounian

Abstract In this note, we calculate the Green’s function for the linear operator (− D )n , where − D is the one-dimensional Dirichlet Laplacian in L 2 ((0, 1); d x) defined by Dedicated to the memory of Erik Balslev (9-27-1935–1-11-2013). K. K. was supported by the Baylor University Summer Sabbatical and Research Leave Program. M. S. Ashbaugh Department of Mathematics, University of Missouri, Columbia, MO 65211, USA e-mail: [email protected] URL: https://www.math.missouri.edu/people/ashbaugh F. Gesztesy (B) · L. Littlejohn · H. Tossounian Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA e-mail: [email protected] URL: http://www.baylor.edu/math/index.php?id=935340 L. Littlejohn e-mail: [email protected] URL: http://www.baylor.edu/math/index.php?id=53980 H. Tossounian e-mail: [email protected] L. Hermi Department of Mathematics and Statistics, Florida International University, 11200 S.W. 8th Street, Miami, FL 33199, USA e-mail: [email protected] URL: http://faculty.fin.edu/~lhermi K. Kirsten GCAP-CASPER, Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA e-mail: [email protected] URL: http://www.baylor.edu/math/index.php?id=54012 H. Tossounian Center for Mathematical Modeling, Universidad de Chile, Beauchef 851, Edificio el Norte, Santiago, Chile © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_3

27

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− D f = − f  with (Dirichlet) boundary conditions f (0) = f (1) = 0. As a consequence of this computation, we obtain Euler’s formula ζ (2n) =

 k∈N

k −2n =

(−1)n−1 22n−1 π 2n B2n , n ∈ N, (2n)!

where ζ (·) denotes the Riemann zeta function and Bn is the nth Bernoulli number. This generalizes the example given by Grieser [29] for n = 1. In addition, we derive  its z-dependent generalization for z ∈ C\ (kπ )2n k∈N , 

(kπ )2n − z

−1

k∈N

=



n−1   1/2 1 1/2 n− ω j z 1/(2n) cot ω j z 1/(2n) , n ∈ N, 2nz j=0

where ω j = e2πi j/n , 0 ≤ j ≤ n − 1, represent the nth roots of unity. In this context −1  we also derive the Green’s function of (− D )n − z I , n ∈ N. Keywords Dirichlet Laplacian · Green’s function · Trace class operators · Trace formulas · Riemann zeta function · Bernoulli numbers · Bernoulli polynomials 2010 Mathematics Subject Classification Primary: 11M06, 47A10, Secondary: 05A15, 47A75

1 Introduction An astute student in a sophomore differential equations course can compute the set of eigenvalues of the Dirichlet boundary value problem −  D f = − f  ,

f (0) = f (1) = 0.

(1.1)

Indeed, the underlying linear operator − D is the positive, self-adjoint operator in the Hilbert space L 2 ((0, 1); d x), (− D f )(x) = − f  (x) for a.e. x ∈ (0, 1),   f ∈ dom(− D ) = g ∈ L 2 ((0, 1); d x)  g, g  ∈ AC([0, 1]); g(0) = g(1) = 0; (1.2)   2 g ∈ L ((0, 1); d x) (AC([0, 1]) denotes the set of absolutely continuous functions on [0, 1]), with purely discrete spectrum,

Green’s Functions and Euler’s Formula for ζ (2n)

σ (− D ) = {λk = (kπ )2

29

 k∈N

.

(1.3)

The eigenspace of each λk is one-dimensional and spanned by the normalized eigenfunctions −  D u k = (kπ )2 u k , u k (x) = 21/2 sin(kπ x), 0 ≤ x ≤ 1, u k  L 2 ((0,1);d x)) = 1, k ∈ N.

(1.4)

  In particular, the collection 21/2 sin(kπ x) k∈N forms a complete orthonormal basis / σ (− D ), (− D )−1 exists and is explicitly given by in L 2 ((0, 1); d x). Since 0 ∈  (− D )−1 f (x) =



1

dy K 1 (x, y) f (y),

f ∈ L 2 ((0, 1); d x),

(1.5)

0

where K 1 ( · , · ) denotes the Green’s function for − D given by

K 1 (x, y) =

x(1 − y), 0 ≤ x ≤ y ≤ 1, y(1 − x), 0 ≤ y < x ≤ 1.

(1.6)

This operator (− D )−1 is a bounded, self-adjoint, compact operator with eigenval∞ ∞ ues {λ−1 k }k=1 and associated eigenfunctions {u k }k=1 . Moreover, as discussed below, −1 (− D ) is a trace class operator and this implies  k∈N

λ−1 k



1

=

d x K 1 (x, x) =

0

1 6

(1.7)

from which the solution to the famous “Basel problem” quickly emerges:  1 π2 . = k2 6

(1.8)

k∈N

This example, and methodology, was studied by Grieser [29] in a paper centered on trace formulas. In this note, we generalize Grieser’s work by explicitly computing the Green’s function K n (·, ·) associated with (− D )n for any integer n ≥ 1 and from this we deduce, as a corollary, Euler’s formula for ζ (2n): ζ (2n) =

 1 (−1)n−1 22n−1 π 2n B2n , n ∈ N; = k 2n (2n)! k∈N

(1.9)

here ζ ( · ) denotes the Riemann zeta function and {Bn }n∈N0 (N0 = N ∪ {0}) denotes the sequence of Bernoulli numbers.  final section we derive a z-dependent  In our generalization of (1.9) for z ∈ C\ (kπ )2n k∈N ,

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n−1   −1  1/2 1 1/2 n− (kπ )2n − z = ω j z 1/(2n) cot ω j z 1/(2n) , n ∈ N, 2nz j=0 k∈N

(1.10) where ω j = e2πi j/n , 0 ≤ j ≤ n − 1, represent the nth roots of unity. To derive (1.10)  −1 we explicitly compute the Green’s function K n (z; · , · ) of (− D )n − z I , n ∈ N. There are several excellent expository articles in the literature concerning the history of the zeta function and, more specifically, Euler’s formula (1.9). The paper by Ayoub [6] on the zeta function includes discussion of some of Euler’s original proofs of the computation of ζ (2) (the Basel problem). Kline [39] gives a thorough account of Euler’s early influence on infinite series, including Euler’s evaluation of ζ (2n) for n ∈ N. Two important sources on the calculation of ζ (2) are the article [53] and book [54, Chap. 21] by Sandifer who details three of Euler’s solutions of the Basel problem beginning in 1735. We remark that the book by Roy [52] notes that Euler gave eight solutions of the Basel problem in his career. As noted in [7], the Basel problem was open for 91 years before Euler supplied the first known proof. Indeed, Pietro Mengoli posed this problem in 1644 (in 1655, John Wallis commented on this problem in his book Arithmetica Infinitorum). Behind the scenes of many known proofs of (1.9) are methods relying on Cauchy’s residue calculus, Weierstrass’ product theorem, Parseval’s theorem, and Fourier expansions. There are at least three ‘most common’ proofs of (1.9) given in various texts. One of them is the original proof given by Euler [24] (see also [40] [Chap. VI, Sect. 24]) in 1741. His proof of (1.9) involved matching coefficients for two different representations of π z cot(π z), one a power series expansion and the other a partial fractions decomposition; see [3] for a short, but concise, explanation of Euler’s method. A second well-known proof of (1.9) is obtained by setting z = 2n in Riemann’s functional equation ζ (1 − z) = 2(2π )−z (z) cos(π z/2)ζ (z)

(1.11)

and using the fact that ζ (1 − 2n) = −B2n /(2n); see [1] [23.2.6 and 23.2.15]. A third standard proof is to evaluate the Fourier cosine series of B2n (x), the Bernoulli polynomial of degree 2n, at x = 0; for example, see [1] [23.1.18] or [4] [Theorem 12.19]. There are several additional contributions in the literature regarding Euler’s formula (1.9). Some of these references deal with new proofs for the special case n = 1, the Basel problem case (which we call group 1) and other proofs discuss the more general n ∈ N case (group 2). We briefly discuss the contents of both groups. For group 1, we note that, in his book [22] [Chap. 3], Dunham illustrates, in detail, one of Euler’s solutions by computing ζ (2) using the infinite product expansion of sin(x)/x, namely,   x2 sin(x)  1− 2 2 . (1.12) = x k π k∈N

Green’s Functions and Euler’s Formula for ζ (2n)

31

Specifically, using the Maclaurin series for sin(x), expanding the right-hand side of (1.12) into a Maclaurin series and then comparing coefficients of x 2 on both sides, the value of ζ (2) is found. Another proof of the computation of ζ (2) was given by Papadimitriou in [48]; his remarkable method stems from the simple inequality sin(x) < x < tan(x) on (0, π/2); see also [34] and [57] which both use ideas similar to those used by Papadimitriou. Choe [12] computes ζ (2) by first cleverly showing that  1 π2 (1.13) = 2 (2k + 1) 8 k∈N 0

and noting, by absolute convergence, that ζ (2) =

 k∈N0

∞  1 1 1 4 + = ; (2k + 1)2 (2k)2 3 (2k + 1)2 k∈N

(1.14)

k∈N0

(it should be noted that Euler also used this ‘trick’ to compute ζ (2) in one of his solutions). Kimble [38] offers a variation of this proof by employing the Maclaurin series of arcsin(x) and using the identity 4 (arcsin(1))2 4 π2 = = 6 3 2 3

0

1

arcsin(x) dx √ . 1 − x2

(1.15)

In [5], Apostol gives another proof of Euler’s formula for n = 1 by evaluating the double integral 1 1 1 . (1.16) d xd y 1 − xy 0 0 In [59, 62], Stark also supplies new proofs of ζ (2). Likewise, the following contributions all deal with interesting techniques, some new and some variations of older methods, of computing ζ (2): Benko [7], Benko and Molokach [8], Chapman [10], Daners [17], Giesey [26], Harper [31], Hirschhorn [32], Hofbauer [33], Ivan [36], Knopp and Schur [41], Kortram [42], Marshall [44], Matsuoka [45], Passare [49], and Vermeeren [67]. We comment briefly on the contents of group two, the set of papers dealing with proofs of the computation of ζ (2n) for each n ∈ N. Apostol [3] generalizes Papadimitriou’s method (n = 1) to compute ζ (2n). Two different proofs of (1.9) were given by Berndt in [9]; one proof uses a special case of the Riemann–Lebesgue lemma and his second proof is, essentially, a further generalization of Apostol’s proof in [3]. More generally, the identity  k∈N

 1 − π z 1/2 cot π z 1/2 1 = , z ∈ C\N. k2 − z 2z

(1.17)

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already appeared in Titchmarsh [65] [p. 113]; it was later employed in Dikii [21] [Eq. (3.9)] in his seminal paper on (regularized) trace formulas for Sturm–Liouville operators and eigenvalue asymptotics. In a recent text, Teschl [64] [p. 85] uses trace methods to derive (1.17), and, by comparing the power series of both sides of (1.17) at z = 0, he deduces (1.9). The recent contribution by Alladi and Defant [2] establishes (1.9) by applying Parseval’s identity to the Fourier coefficients of the periodic function x k and using induction on k; see also [43]. Other interesting, and diverse, proofs of (1.9) are given by Chen [11], Chungang and Yonggao [13], Ciaurri, Navas, Ruiz and Varona [14], de Amo, Díaz Carillo and Fernández-Sánchez [19], Estermann [23], Hovstad [35], Osler [47], Robbins [51], Stark [60, 61], Tsumura [66], Williams [69], and Williams [70]. For zeta functions associated to general Sturm–Liouville operators we refer to the classical work of Dikii [20, 21] (see also the recent article [25] and the extensive literature cited therein). Lastly, we note that there are q version results of the identity in (1.9). Indeed, Goswami [30] recently gave a natural family of identities whose limits as q → 1 with |q| < 1 give Euler’s formula for each n ∈ N; see also the contribution [63] by Sun.

2 Bernoulli Polynomials and Bernoulli Numbers For properties of Bernoulli polynomials and Bernoulli numbers, we refer the reader to the standard sources [1] [Chap. 23] and [27] [Sects. 9.6 and 9.7] as well as the online Digital Library of Mathematical Functions https://dlmf.nist.gov/ and the accompanying book [46]. The Bernoulli polynomials {Bn (x)}n∈N0 can be defined through the generating function  ze x z zn = , |z| < 2π, x ∈ R. B (x) n e z − 1 n∈N n!

(2.1)

0

For the record, the first few Bernoulli polynomials are given by 1 1 B1 (x) = x − , B2 (x) = x 2 − x + , 2 6 1 B4 (x) = x 4 − 2x 3 + x 2 − , etc. 30

B0 (x) = 1,

3 1 B3 (x) = x 3 − x 2 + x, 2 2 (2.2)

Among several properties that these polynomials satisfy, we will make use of the fact that 1 d x B2n (x) = 0, n ≥ 1, (2.3) 0

which follows from the identities (see [1][23.1.8 and 23.1.11])

Green’s Functions and Euler’s Formula for ζ (2n)

33

Bn (1 − x) = (−1)n Bn (x), n ≥ 1,

(2.4)

and

x

du Bn (u) =

0

Bn+1 (x) − Bn+1 (0) , n ≥ 1. n+1

(2.5)

We also recall the Fourier cosine series expansion of B2n (x) (see [1] [23.1.18] or [4] [Theorem 12.19]), which is given by (−1)n−1 (2n)!  cos(2kπ x) = B2n (x), 0 ≤ x ≤ 1, n ∈ N. 22n−1 π 2n k∈N k 2n

(2.6)

The Bernoulli numbers {Bn }n∈N0 are defined as Bn = Bn (0). For example, B0 = 1, B1 = −1/2, B2 = 1/6, B4 = −1/30, B6 = 1/42, B8 = −1/30, B10 = 5/66, etc., B2n+1 = 0 for n ≥ 1.

(2.7)

We will see in Sect. 4, that the Green’s function K n (·, ·), associated with the n th power of the operator − D defined in (1.1), can be given explicitly in terms of Bernoulli polynomials.

3 On Trace Class Operators A classical result in an elementary linear algebra course states that, for an arbitrary n × n matrix B = (B j,k )1≤ j,k≤n of complex numbers with eigenvalues {βk }nk=1 (counting algebraic multiplicities), one has tr(B) =

n  j=1

B j, j =

n 

βk .

(3.1)

k=1

This result generalizes to an important class of compact operators in an infinitedimensional, separable Hilbert space (H, ( · , · )H ), the so-called trace class operators. Specifically, a bounded linear operator T : H → H is trace class if, for some (and hence for all) orthonormal basis {e j } j∈N of H, the sum  (e j , (T ∗ T )1/2 e j )H

(3.2)

j∈N

is finite; see [56] [Sect. 3.6] for an in-depth discussion of this topic. In this case, the trace of T is defined to be

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tr (T ) =



(e j , T e j )H ;

(3.3)

j∈N

this finite-valued sum is absolutely convergent and independent of the choice of orthonormal basis {e j } j∈N . In addition, if {τk }k∈J (with J ⊆ N an appropriate index set) represent the eigenvalues of T, counting algebraic multiplicities, then by Lidskii’s theorem (see, e.g., [55] [Chap. 3], [56] [Sect. 3.12]) tr (T ) =



(e j , T e j )H =

j∈N



τk .

(3.4)

k∈J

For the purpose of this note, we discuss a particular set of trace class integral operators. Let [a, b] ⊂ R be a compact interval and K (· , · ) : [a, b] × [a, b] → C be a continuous function. Define T : L 2 ((a, b); d x) → L 2 ((a, b); d x) by (T f ) (x) =

b

dy K (x, y) f (y),

f ∈ L 2 ((a, b); d x),

(3.5)

a

and assume that T ≥ 0 (implying K (x, x) ≥ 0, x ∈ [a, b]). Then by Mercer’s Theorem (see, e.g.., [18] [Proposition 5.6.9], [56] [Theorem 3.11.9]), T is trace class and b tr (T ) = d x K (x, x) ∈ [0, ∞). (3.6) a

We now see that the formulas given in (3.4) and (3.6) afford us two different means to compute the trace of an integral operator of the type defined in (3.5); this observation is fundamental in our pursuit of establishing Euler’s formula (1.9). We now connect these trace class results to the inverse of the self-adjoint operator (− D )n , where n ∈ N and − D is the second-order differential operator defined in (1.1). It is straightforward to see that the two-point boundary value problem defined by (− D )n is given by  (3.7) (− D )n f (x) = (−1)n f (2n) (x) for a.e. x ∈ (0, 1),  (k)  n 2  f ∈ dom (− D ) = {g ∈ L ((0, 1); d x) g ∈ AC([0, 1]), 0 ≤ k ≤ 2n − 1; g (2 j) (0) = g (2 j) (1) = 0, 0 ≤ j ≤ n − 1; g (2n) ∈ L 2 ((0, 1); d x)}. The spectrum of (− D )n is given by   n  n   = λk = (kπ )2n k∈N , σ (− D )n = σ −  D

(3.8)

 where λk was introduced in (1.3). Furthermore, since 0 ∈ / σ (− D )n for all n ∈ N, −1  −2n one infers that (− D )n = (− D )−n exists with eigenvalues λ−n ,k ∈ k = (kπ )

Green’s Functions and Euler’s Formula for ζ (2n)

35

−n N, and corresponding eigenfunctions {u k }∞ : k=1 defined in (1.4). The inverse (− D ) 2 2 n L ((0, 1); d x) → L ((0, 1); d x) of (− D ) is a bounded, self-adjoint, and compact integral operator given by

 (− D )−n f (x) =



1

dy K n (x, y) f (y),

f ∈ L 2 ((0, 1); d x),

(3.9)

0

where K n (·, ·) is the unique Green’s function for (− D )n in the sense of [15] [Theorem 7.2.2]. In particular, the Green’s function K n (·, ·) has the following properties: (i) K n (·, ·) is symmetric on [0, 1] × [0, 1] and for fixed y ∈ [0, 1], K n (·, y) and its first (2n − 1) derivatives are continuous on [0, y) and (y, 1]. At the point x = y, K n (·, ·) and its first (2n − 2) derivatives have removable singularities (i.e., the left and right limits exist and equal each other), while the (2n − 1)st derivative has a jump discontinuity satisfying ∂ 2n−1 K n (y + , y) ∂ 2n−1 K n (y − , y) − = (−1)n . ∂ x 2n−1 ∂ x 2n−1

(3.10)

(ii) As a function of x ∈ [0, 1]\{y}, K n ( · , y), is a solution of the differential equation  (− D )n f (x) = 0, x ∈ [0, 1]\{y},

(3.11)

and as a function of x ∈ [0, 1], K n ( · , y), satisfies the boundary conditions f (2 j) (0) = f (2 j) (1) = 0, 0 ≤ j ≤ n − 1,

(3.12)

for each y ∈ [0, 1].  (iii) For each f ∈ L 2 ((0, 1); d x), the solution u ∈ dom (− D )n of (− D )n u = f is of the form 1

u(x) =

dy K n (x, y) f (y) for a.e. x ∈ [0, 1].

(3.13)

0

4 The Green’s Function for (− D )n , n ∈ N Following the general method developed, for example in [16] [Sect. 6.7] (see also [58]), some (increasingly) tedious computations yield the following explicit Green’s functions K n (x, y) for n = 2, 3, and 4:

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M. S. Ashbaugh et al.

⎧1 ⎪ ⎨ x(y − 1)(x 2 + y 2 − 2y), 0 ≤ x ≤ y ≤ 1, (4.1) K 2 (x, y) = 6 ⎪ ⎩ 1 y(x − 1)(y 2 + x 2 − 2x), 0 ≤ y < x ≤ 1, ⎧6 ⎪ 1 x(1 − y)(8y − 20x 2 y + 10x 2 y 2 + 8y 2 − 12y 3 +3y 4 + 3x 4 ), ⎪ ⎪ ⎪ ⎪ ⎨ 360 0 ≤ x ≤ y ≤ 1, K 3 (x, y) = 1 ⎪ 2 2 2 2 3 ⎪ y(1 − x)(8x − 20x y + 10x y + 8x − 12x +3x 4 + 3y 4 ), ⎪ ⎪ ⎪ ⎩ 360 0 ≤ y < x ≤ 1, (4.2) and ⎧ 1 ⎪ ⎪ x(y − 1)(−32y + 56x 2 y − 42x 4 y + 56x 2 y 2 −84x 2 y 3 ⎪ ⎪ 15120 ⎪ ⎪ 2 ⎪ +21x y 4 + 21x 4 y 2 + 3x 6 − 32y 2 + 24y 3 + 24y 4 −18y 5 + 3y 6 ), ⎪ ⎪ ⎨ 0 ≤ x ≤ y ≤ 1, K 4 (x, y) = 1 ⎪ ⎪ y(x − 1)(−32x + 56x y 2 − 42x y 4 + 56x 2 y 2 −84x 3 y 2 ⎪ ⎪ ⎪ 151204 2 ⎪ ⎪ +21x y + 21x 2 y 4 + 3y 6 − 32x 2 + 24x 3 + 24x 4 −18x 5 + 3x 6 ), ⎪ ⎪ ⎩ 0 ≤ y < x ≤ 1. (4.3) Calculations show that 0

1

1 , d x K 2 (x, x) = 90



1 0

1 , d x K 3 (x, x) = 945



1

d x K 4 (x, x) =

0

1 , (4.4) 9450

and the values ζ (4), ζ (6), and ζ (8) quickly follow. However, this approach of directly calculating K n (·, ·) for each positive integer n does not seem to lend itself to a general formula. Alternatively, for n ∈ N and y ≥ x, K n (x, y) satisfies the boundary value problem ∂ 2 h(x, y) = −K n−1 (x, y), ∂x2 ∂h (0, 0) = 0; h(0, y) = h(x, 1) = ∂x

(4.5)

consequently, knowing K n−1 ( · , · ) allows us to determine K n (·, ·). This approach, as well, is not helpful in determining K n (·, ·) for all n ∈ N. Fortunately, the following considerations permit us to compute K n ( · , · ) explicitly for any n ∈ N. First, we recall the general (absolutely and uniformly convergent) Green’s function formula (here I = I L 2 ((0,1);d x) )

Green’s Functions and Euler’s Formula for ζ (2n)

37

 sin(kπ x) sin(kπ y)  −1 , (− D )n − z I (x, y) = K n (z; x, y) = 2 [(kπ )2n − z] k∈N

(4.6)

x, y ∈ [0, 1], in terms of the normalized (simple) eigenfunctions u k (x) = 21/2 sin(kπ x), 0 ≤ x ≤ 1, k ∈ N of − D . Indeed, equation (4.6) follows instantly from the spectral theorem −1  of − D , yielding for self-adjoint operators applied to the function (− D )n − z I  −1   −1 (kπ )2n − z (u k , · )H u k . (− D )n − z I =

(4.7)

k∈N

Taking z = 0 in (4.6) yields K n (x, y) = 2

 sin(kπ x) sin(kπ y) k∈N

(kπ )2n

, 0 ≤ x, y ≤ 1.

(4.8)

We are now in position to prove one of the principal results of this note. Theorem 1 For each n ∈ N, ⎧     (−1)n−1 22n−1 y−x x ⎪ ⎪ B − B ⎪ 2n 2n ⎪ ⎪ (2n)! 2 ⎪ ⎨     K n (x, y) = (−1)n−1 22n−1 x−y x ⎪ ⎪ ⎪ B2n − B2n ⎪ ⎪ (2n)! 2 ⎪ ⎩

+y 2 +y 2



, 0 ≤ x ≤ y ≤ 1, 

, 0 ≤ y < x ≤ 1,

(4.9) where B2n (x) is the Bernoulli polynomial of degree 2n. As a corollary of (4.9) one recovers Euler’s formula for ζ (2n) (cf. (1.9)), 1  1  −2n −n = = tr (− D ) d x K n (x, x) = ζ (2n)π (kπ )2n 0 k∈N 1 (−1)n−1 22n−1 = dx [B2n − B2n (x)] (2n)! 0 (−1)n−1 22n−1 B2n . (4.10) = (2n)! Proof Since sin(kπ x) sin(kπ y) = [cos(kπ(x − y)) − cos(kπ(x + y))]/2, one infers from (2.6) and (4.8) that

(4.11)

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2  sin(kπ x) sin(kπ y) π 2n k∈N k 2n 1  cos(kπ(x − y)) − cos(kπ(x + y)) π 2n k 2n k∈N   1  cos(kπ(x − y))  cos(kπ(x + y)) − π 2n k∈N k 2n k 2n k∈N     

x−y x+y (−1)n−1 22n−1 π 2n 1 (−1)n−1 22n−1 π 2n B2n − B2n π 2n (2n)! 2 (2n)! 2   

  x−y x+y (−1)n−1 22n−1 B2n − B2n . (4.12) (2n)! 2 2

K n (x, y) = = = = =

In particular, K n (x, x) =

(−1)n−1 22n−1 [B2n − B2n (x)] . (2n)!

(4.13)

To obtain (4.10) one employs (2.3), (4.13), and the identities in (3.4) and (3.6) applied to T = (− D )−n . 

5 Some Generalizations In our final section we probe some (z-dependent) extensions of Theorem 1. More precisely, for x, y ∈ [0, 1], n ∈ N, we will be considering  sin(kπ x) sin(kπ y)  −1 (− D )n − z I (x, y) = K n (z; x, y) = 2 , [(kπ )2n − z] k∈N   z ∈ C (kπ )2n k∈N ,

(5.1)

and 

− D − z I

−n

n (z; x, y) = 2 (x, y) = K

 sin(kπ x) sin(kπ y) k∈N

=

[(kπ )2 − z]n

  d n−1 1 K 1 (z; x, y), z ∈ C (kπ )2 k∈N , n−1 (n − 1)! dz

noting that,   n (z; x, y) , (− D )−n (x, y) = K n (x, y) = K n (z; x, y)z=0 = K z=0   2  K 1 (z; x, y) = K 1 (z; x, y), z ∈ C (kπ ) k∈N .

(5.2) (5.3)

Green’s Functions and Euler’s Formula for ζ (2n)

39

n (z; · , · ), n ∈ N, and recall the trace formula, We start with the case of K  −μ (kπ )2 − z k∈N

=



∞

  −1 d x eπ x − 1 x μ−(1/2) Iμ−(1/2) z 1/2 x , (2z 1/2 )μ−(1/2) (μ) 0    (μ) > 1/2,  z 1/2  < π, π 1/2

(5.4)

obtained as an elementary consequence of [27] [No. 6.6247] or [68] [Eq. (9) on p. 386] (originally due to Kapteyn [37]), where (cf. [46] [No. 10.27.6]) Iν (ζ ) = e∓i(π/2)ν Jν (±iζ ), (ν) ≥ 0, −π ≤ ±Arg(ζ ) ≤ π/2,

(5.5)

where Arg( · ) represents the single-valued principal value of the argument function on C\(−∞, 0]. Here Jν ( · ) (resp., Iν ( · )) denotes the Bessel function (resp., modified Bessel function) of order ν (cf., e.g.., [1] [Chap. 9], [46] [Chap. 10]). Employing Iν (ζ ) = (ζ /2)ν

 k∈N0

(ζ /2)2k , (ν) ≥ 0, ζ ∈ C, k! (ν + k + 1)

(5.6)

and its asymptotics as ζ → 0, one confirms (cf. [1] [No. 23.2.7], [46] [No. 25.5.1]) that ∞   −1 1 −2μ (kπ ) = ds eπs − 1 s 2μ−1 = π −2μ ζ (2μ), (μ) > 1/2. (2μ) 0 k∈N (5.7) Next, one notes that the explicit formula for K 1 ,

  1 sin z 1/2 x sin z 1/2 (1 − y) , 0 ≤ x ≤ y ≤ 1,    K 1 (z; x, y) = 1/2 z sin z 1/2 sin z 1/2 y sin z 1/2 (1 − x) , 0 ≤ y ≤ x ≤ 1, (5.8) readily leads to the well-known fact (see (1.17) and the subsequent paragraph),  −1  2 −1 (kπ ) − z = = tr (− D − z I L 2 ((0,1);d x) ) k∈N

1

d x K 1 (z; x, x)

0

    1 1 − z 1/2 cot z 1/2 , z ∈ C (kπ )2 k∈N . = 2z

(5.9)

(This also follows from [25] [Theorem 3.4, Example 3.15].) Here we used the elementary fact (see, e.g.., [27] [No. 2.5324]),

40

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1

d x sin(ax) sin(a(1 − x)) = (2a)−1 sin(a)[1 − a cot(a)], a ∈ C.

(5.10)

0

  Lemma 1 Let n ∈ N and z ∈ C (kπ )2 n∈N . Then  −n  (kπ )2 − z = tr (− D − z I L 2 ((0,1);d x) )−n = k∈N

   1/2  d n−1 1  1 1/2 1 − z . = cot z (n − 1)! dz n−1 2z

1

n (z; x, x) dx K

0

(5.11)

In addition, n (z; x, y) = K

  ∞  m n K m (x, y) z m−n , |z| < π 2 , n m=n

(5.12)

and   ∞  −n  m (−1)m+1 22m−1 B2 m m−n z (kπ )2 − z = n , |z| < π 2 . (2 m)! n m=n

(5.13)

k∈N

Proof Differentiation of (5.9) (n − 1) times with respect to z yields (5.11). Employing the resolvent expansion 

(− D ) − z I

−1

=



(− D )−m z m−1 , |z| < π 2 ,

(5.14)

m∈N

yields K 1 (z; x, y) =



K m (x, y) z m−1 , |z| < π 2 ,

(5.15)

m∈N

and hence,

 1 ∂ m K 1 (z; x, y)  K m+1 (x, y) =  . m! ∂z m z=0

Taking the trace of either side in (5.12) (cf. (4.10)) implies (5.13).

(5.16) 

Combining (5.4) (for μ = n ∈ N) and (5.11) then yields the evaluation of the following integral

Green’s Functions and Euler’s Formula for ζ (2n)

41

∞   −1 π 1/2 d x eπ x − 1 x n−(1/2) In−(1/2) z 1/2 x (2z 1/2 )n−(1/2) [(n − 1)!] 0    −n  1/2  d n−1 1  1 2 1/2 1 − z cot z (kπ ) − z , = = (n − 1)! dz n−1 2z k∈N    n ∈ N,  z 1/2  < π.

(5.17)

Although elementary, we were not able to find formula (5.17) in the standard literature on (integrals of) Bessel functions. Next, we turn to a discussion of K n (z; · , · ), n ∈ N, and derive the z-dependent analogs of (4.9) and (4.10).   Theorem 2 Let n ∈ N and z ∈ C\ (kπ )2n n∈N . Then K n (z; x, y) =

n−1 

1 nz 1−(1/n)

 ω j K 1 ω j z 1/n ; x, y , x, y ∈ [0, 1],

(5.18)

j=0

and  −1  −1 2n n (kπ ) − z = = tr (− D ) − z I k∈N

1

d x K n (z; x, x)

0



n−1   1/2 1/(2n) 1 1/2 1/(2n) = n− , ωj z cot ω j z 2nz j=0

where ω j = e2πi j/n , 0 ≤ j ≤ n − 1, represent the nth roots of unity. In addition,  K nm (x, y) z m−1 , |z| < π 2n . K n (z; x, y) =

(5.19)

(5.20)

m∈N

and  −1  (−1)n( +1)+1 22n( +1)−1 B2n( +1) z , |z| < π 2n . (5.21) (kπ )2n − z = [(2n( + 1)]! k∈N

∈N 0

Proof We recall the partial fraction expansion n−1    ωj 1 1 1 = = , z ∈ C\ λn , (5.22)  n−1 1−(1/n) 1/n 1/n λn − z nz λ − ω z ] j j=0 [λ − ω j z j=0

a consequence of the following well-known facts: Introducing

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f n (z) = λn − z =

n−1 

 λ − ω j z 1/n , z ∈ C

(5.23)

j=0

(fixing the branch of z 1/n ), one obtains n−1  n −1 d d   = ln λ − ω j z 1/n ln( f n (z)) = − λ − z dz dz j=0 n−1   −1   1 1 = − 1−(1/n) ω j λ − ω j z 1/n , z ∈ C\ λn . nz j=0

(5.24)

Thus, by the functional calculus implied by the spectral theorem for self-adjoint operators,   −1 n−1 −1 n (− D ) − z I = −  D − ω j z 1/n I j=0

=

1 nz 1−(1/n)

n−1 



ω j −  D − ω j z 1/n I

−1



, z ∈ C\ (kπ )

(5.25)

 2n n∈N

,

j=0

implying (5.18). The fact (5.10) then yields (5.19). Relation (5.20) follows from the resolvent expansion (cf. (5.14))  −1  (− D )n − z I = (− D )−nm z m−1 , |z| < π 2n .

(5.26)

m∈N

Finally, to prove (5.21) we employ Euler’s identity (cf. [28] [No. 6.87]), ζ cot(ζ ) =

 (−1) 22 B2 ζ 2 , |ζ | < π, (2 )!

∈N

(5.27)

0

implying 1 − ζ cot(ζ )  (−1) +1 22 −1 B2 2 = ζ , |ζ | < π. 2 (2 )!

∈N Thus,

(5.28)

Green’s Functions and Euler’s Formula for ζ (2n)

43

n−1   −1   1/2 1  1/2 1 − ω j z 1/(2n) cot ω j z 1/(2n) (kπ )2n − z = 2nz j=0 k∈N

=

 n−1  1 (−1) +1 22 −1 B2 −1+( /n) ω j z n j=0 (2 )!

(5.29)

(5.30)

∈N

=

 (−1)n +1 22n −1 B2n

∈N

utilizing

(2n )!

z −1 , |z| < π 2n ,

n−1 1 1, ≡ 0 (mod n), ω = n j=0 j 0, otherwise.

(5.31)

(5.32) 

One notes that taking the trace on either side of (5.20) (cf. (4.10)) yields an alternative derivation of (5.21). The cases n = 1, 2 in (5.19) are known and can be found in [50] [No. 5.1.25.4], [64] [p. 85], [65] [p. 113] for n = 1, and in [50] [No. 5.1.27.3] for n = 2; we have not found the cases n ≥ 3 in the literature. Taking z → 0 in (5.21) yields another derivation of Euler’s formula (4.10).

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On Courant’s Nodal Domain Property for Linear Combinations of Eigenfunctions Part II Pierre Bérard and Bernard Helffer

Abstract Generalizing Courant’s nodal domain theorem, the “Extended Courant property” is the statement that a linear combination of the first n eigenfunctions has at most n nodal domains. In a previous paper (Documenta Mathematica, 2018, Vol. 23, pp. 1561–1585), we gave simple counterexamples to this property, including convex domains. In the present paper, using some input from numerical computations, we pursue the investigation of the Extended Courant property with two new examples, the equilateral rhombus and the regular hexagon. Keywords Eigenfunction · Nodal domain · Courant nodal domain theorem 2010 Mathematics Subject Classification 35P99 · 35Q99 · 58J50

1 Introduction 1.1 Notation Let  ⊂ R2 be a piecewise smooth bounded open domain (we will actually only work with convex polygonal domains), with boundary ∂ = 1  2 , where 1 , 2 are two disjoint open subsets of ∂. We consider the eigenvalue problem The authors express their hearty thanks to V. Bonnaillie-Noël for providing numerical simulations at an earlier stage of their research. P. Bérard Institut Fourier CS 40700, Université Grenoble Alpes and CNRS, 38058 Grenoble cedex 9, France e-mail: [email protected] B. Helffer (B) Laboratoire Jean Leray, Université de Nantes and CNRS, 44322 Nantes Cedex, France e-mail: [email protected] Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, 91405 Orsay Cedex, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_4

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⎧ ⎨ −u = μ u in , u = 0 on 1 , ⎩ ν · u = 0 on 2 ,

(1.1)

where ν is the outer unit normal along ∂ (defined almost everywhere). Let {μi (, dn), i ≥ 1} (resp. sp(, dn)) denote the eigenvalues (resp. the spectrum) of problem (1.1). We always list the eigenvalues in non-decreasing order, with multiplicities, starting with the index 1. We simply write μi , and skip mentioning the domain , or the boundary condition dn, whenever the context is clear. Examples of eigenvalue problems with mixed boundary conditions appear in Sects. 2 and 3. Let E (μ) denote the eigenspace associated with the eigenvalue μ. Define the min-index κ(μ) of the eigenvalue μ as κ(μ) = min {m | μ = μm } .

(1.2)

1.2 Courant’s Nodal Domain Theorem Let φ be an eigenfunction of (1.1). The nodal set Z(φ) of φ is defined as the closure of the set of (interior) zeros of φ, Z(φ) := {x ∈  | φ(x) = 0}.

(1.3)

A nodal domain of φ is a connected component of the set \Z(φ). Call β0 (φ) the number of nodal domains of φ. We recall the following classical theorem [12, Chap. VI.6]. Theorem 1.1 (Courant 1923) Assume that the eigenvalues of (1.1) are listed in non-decreasing order, with multiplicities, μ1 < μ2 ≤ μ3 ≤ · · · .

(1.4)

Then, for any eigenfunction φ ∈ E(μ) of (1.1), associated with the eigenvalue μ, β0 (φ) ≤ κ(μ).

(1.5)

In particular, any φ ∈ E(μk ) has a most k nodal domains, Courant’s theorem is a partial generalization, to higher dimensions, of a classical theorem of C. Sturm (1836). Indeed, in dimension 1, a kth eigenfunction of the 2 Sturm-Liouville operator − ddx 2 + q(x) in ]a, b[, with Dirichlet, Neumann, or mixed Dirichlet-Neumann boundary condition at {a, b}, has exactly k nodal domains in ]a, b[. In dimension 2 (or higher), Courant’s theorem is not sharp. On the one hand, Stern (1925) proved that for the square with Dirichlet boundary condition, or for the 2-sphere, there exist eigenfunctions of arbitrarily high energy, with exactly two or

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three nodal domains. On the other hand, Pleijel (1956) proved that, for any bounded domain in R2 , there are only finitely many Dirichlet eigenvalues for which Courant’s theorem is sharp. We refer to [7, 24] for more details, and to [20] for Pleijel’s estimate under Neumann boundary condition. Another remarkable theorem of Sturm states that any non trivial linear combi 2 nation u = nk=m a j u j of eigenfunctions of the operator − ddx 2 + q(x) has at most (n − 1) zeros (counted with multiplicities), and at least (m − 1) sign changes in the interval ]a, b[, see [10]. A footnote in [12, p. 454], states that Courant’s theorem may be generalized as follows: Any linear combination of the first n eigenfunctions divides the domain, by means of its nodes, into no more than n subdomains. See the Göttingen dissertation of H. Herrmann, Beiträge zur Theorie der Eigenwerten und Eigenfunktionen, 1932. For later reference, we introduce the following definition. Definition 1.2 We say that the Extended Courant property is true for the eigenvalue problem (, b), or simply  if, for any  m ≥ 1, and for any that the ECP(, b) is true, linear combination v = μ j ≤μm u μ j , with u μ j ∈ E μ j (, b) , β0 (v) ≤ κ(μm ) ≤ m.

(1.6)

The footnote statement in the book of Courant and Hilbert, amounts to saying that ECP() is true for any bounded domain. Already in 1956, Pleijel [24, p. 550] mentioned that he could not find a proof of this statement in the literature. In 1973, Arnold [2, 4] related the statement in Courant-Hilbert to Hilbert’s 16th problem. Indeed, should ECP(RP N , g0 ) be true (where g0 is the usual metric), then the comof any algebraic hypersurface of degree n in RP N would have at most plement N  + 1 connected components. Arnold pointed out that while ECP(RP2 , g0 ) is N +n−2 indeed true, ECP(RP3 , g0 ) is false due to counterexamples produced by Viro [26]. More recently, Gladwell and Zhu [14, p. 276] remarked that Herrmann in his dissertation and subsequent publications had not even stated, let alone proved the ECP. They also produced some numerical evidence that the ECP is false for non-convex domains in R2 with the Dirichlet boundary condition, and conjectured that it is true for convex domains. Our motivations to look into the Extended Courant property came from reading the papers [3, 14, 18]. In [9], we gave simple counterexamples to the ECP for domains with the Dirichlet or the Neumann boundary conditions (equilateral triangle, hypercubes, domains and surfaces with cracks). This was made possible by the fact that the eigenvalues and eigenfunctions of these domains are known explicitly. In [11], we proved that ECP(, n) is false for a continuous family of smooth convex domains in R2 , with the symmetries of, and close to the equilateral triangle. In the present paper, we continue our investigations of the Extended Courant property by studying two examples, the equilateral rhombus Rh e and the regular hexagon H, which are related to the equilateral triangle. The eigenvalues and eigenfunctions of these domains are not known explicitly (except for a small subset of them). Using the symmetries of these domains, and some input from numerical computations, we

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are able to describe the nodal patterns of the first eigenfunctions, and conclude that the equilateral rhombus and the regular hexagon provide counterexamples to the ECP. The paper is organized as follows. In Sect. 2, we analyze the structure of the first eigenvalues and eigenfunctions of the equilateral rhombus Rh e with either the Neumann or the Dirichlet boundary condition. Sections 2.1, 2.2 and 2.3 provide technical ideas which are used in Sect. 3 as well. In Sect. 2.5, we prove that ECP(Rh e , n) is false. In Sect. 2.7, we give numerical evidence that ECP(Rh e , d) is false as well. In Sect. 3, we analyze the structure of the first eigenvalues and eigenfunctions of the regular hexagon H with either the Neumann or the Dirichlet boundary condition. In Sect. 3.4, we give numerical evidence that ECP(H, d) is false. In Sect. 3.6, we give numerical evidence that ECP(H, n) is false. In Sect. 4, we explain our numerical approach, and we make some final remarks and conjectures.

2 The Equilateral Rhombus 2.1 Symmetries and Spectra In this subsection, we analyze how symmetries influence the structure of the eigenvalues and eigenfunctions. The analysis is carried out for the equilateral rhombus, but the basic ideas work for the regular hexagon as well, and will be used in Sect. 3. In the sequel, we denote by the same letter L a line in R2 , and the mirror symmetry with respect to this line. We denote by L ∗ the action of the symmetry L on functions, L ∗ φ = φ ◦ L. A function φ is even (or invariant) with respect to L if L ∗ φ = φ. It is odd (or anti-invariant) with respect to L if L ∗ φ = −φ. In the former case, the line L is an anti-nodal line for φ, i.e., the normal derivative ν L · φ is zero along L, where ν L denotes a unit field normal to L along L. In the latter case, the line L is a nodal line for φ, i.e., φ vanishes along L. Let Rh e √be the interior of√the equilateral rhombus with sides of length 1, and vertices (− 23 , 0), (0, − 21 ), ( 23 ), 0) and (0, 21 ). Call D and M its diagonals (resp. the longer one and the shorter one). The diagonal M divides the rhombus into two equilateral triangles. The diagonals D and M divide the rhombus into four hemiequilateral triangles. In the sequel, we use the generic notation Te (resp. Th ) for any of the equilateral triangles (resp. hemiequilateral triangles) into which the rhombus decomposes, see Fig. 1. For L ∈ {D, M}, define the sets 



S L ,+ = φ ∈ L 2 (Rh e ) | L ∗ φ = + φ , S L ,− = φ ∈ L 2 (Rh e ) | L ∗ φ = − φ .

(2.1)

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Fig. 1 The equilateral rhombus Rh e , and its diagonals

Then, we have the orthogonal decomposition, ⊥

L 2 (Rh e ) = S L ,+ ⊕ S L ,− ,

(2.2)

with respect to the L 2 -inner product. Indeed, any φ ∈ L 2 (Rh e ) can be decomposed as 1 1 (2.3) φ = (I + L ∗ )φ + (I − L ∗ )φ, 2 2 where I denotes the identity map. The symmetries D and M commute M ◦ D = D ◦ M = Rπ ,

(2.4)

where Rθ denotes the rotation with center 0 (the center of the rhombus), and angle θ . It follows that D ∗ leaves the subspaces S M,± globally invariant, and that M ∗ leaves the subspaces S D,± globally invariant. As a consequence, we have the orthogonal decomposition, ⊥

⊥

⊥

L 2 (Rh e ) = S+,+ ⊕ S−,− ⊕ S+,− ⊕ S−,+ ,

(2.5)



Sσ,τ := φ ∈ L 2 (Rh e ) | D ∗ φ = σ φ and M ∗ φ = τ φ ,

(2.6)

where

for σ, τ ∈ {+ , −}. Similar decompositions hold for H 1 (Rh e ) and H01 (Rh e ), the Sobolev spaces which are used in the variational presentation of the Neumann (resp. Dirichlet) eigenvalue problem for the rhombus. In the following figures, anti-nodal lines are indicated by dashed lines, and nodal lines by solid lines. Figure 12 displays the nodal and anti-nodal lines common to all functions in H 1 (Rh e ) ∩ Sσ,τ , where σ, τ ∈ {+, −} (Fig. 2). Because the Laplacian commutes with the isometries D and M, the above orthogonal decompositions descend to each eigenspace of − for Rh e , with the boundary condition b ∈ {d, n} on ∂Rh e . The eigenfunctions in each summand correspond to eigenfunctions of − for the equilateral triangle (decomposition (2.2) with L = M), or for the hemiequilateral triangle (decomposition (2.6)), with the boundary condi-

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Fig. 2 Spaces Sσ,τ for Rh e Fig. 3 Labelling the sides of Te and Th

tion b on the side supported by ∂Rh e , and with mixed boundary conditions, either Dirichlet or Neumann, on the sides supported by the diagonals. To be more explicit, we need naming the eigenvalues as in Sect. 1.1. For this purpose, we partition the boundaries of Te and Th into their three sides. For Th , we number the sides 1, 2, 3, in decreasing order of length, see Fig. 3. For example, μi (Th , ndn) denotes the i-th eigenvalue of − in Th with Neumann boundary condition on the longest (1) and shortest (3) sides, and Dirichlet boundary condition on the other side (2).

2.2 Riemann-Schwarz Reflection Principle In this subsection, we recall the “Riemann-Schwarz reflection principle” which we will use repeatedly in the sequel.  Consider the decomposition Rh e = Te,1 Te,2 , with M(Te,1 ) = Te,2 . Choose a boundary condition a ∈ {d, n} on ∂Rh e . Given an eigenvalue λ of − for (Rh e , a),

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and σ ∈ {+, −}, consider the subspace E(λ) ∩ S M,σ of eigenfunctions φ ∈ E(λ) such that M ∗ φ = σ φ. If 0 = φ ∈ E(λ) ∩ S M,σ , then φ|Te,1 is an eigenfunction of − for (Te,1 , aab), with b = n if σ = +, and b = d if σ = −, associated with the same eigenvalue λ. Conversely, let ψ be an eigenfunction of (Te,1 , aab), with eigenvalue μm (Te,1 , aab), ˇ e,1 = ψ and ψ|T ˇ e,2 = for some m ≥ 1. Define the function ψˇ on Rh e such that ψ|T ˇ σ ψ ◦ M. This means that ψ is obtained by extending ψ across M to Te,2 by symmetry, in such a way that M ∗ ψˇ = σ ψˇ . It is easy to see that the function ψˇ is an eigenfunction of − for (Rh e , a) (in particular it is smooth in a neighborhood of M), with eigenvalue μm (Te,1 , aab), so that ψˇ ∈ E(μm ) ∩ S M,σ . The above considerations prove the first two assertions in the following proposition. The proof of the third and fourth assertions is similar, using the symmetries D and M, and the decomposition of Rh e into hemiequilateral triangles Th, j , 1 ≤ j ≤ 4. Proposition 2.1 (Reflection principle) For any a ∈ {d, n} and any λ ∈ sp(Rh e , a), (i) E(λ, (Rh e , a)) ∩ S M,+ = {0} if and only if λ ∈ sp(Te , aan), and the map φ → φ|Te,1 is a bijection from E(λ, (Rh e , a)) ∩ S M,+ onto E(λ, (Te , aan)); (ii) E(λ, (Rh e , a)) ∩ S M,− = {0} if and only if λ ∈ sp(Te , aad), and the map φ → φ|Te,1 is a bijection from E(λ, (Rh e , a)) ∩ S M,− onto E(λ, (Te , aad)). More generally, define (n) = + and (d) = −. Then, for any λ ∈ sp(Rh e , a), and any b, c ∈ {d, n}, (iii) E(λ, (Rh e , a)) ∩ S(b),(c) = {0} if and only if λ ∈ sp(Th , abc), and the map φ → φ|Th,1 is a bijection from E(λ, (Rh e , a)) ∩ S(b),(c) onto E(λ, (Th , abc)). Furthermore, the multiplicity of the number λ as eigenvalue of (Rh e , a) is the sum, over b, c ∈ {d, n}, of the multiplicities of λ as eigenvalue of (Th , abc) (with the convention that the multiplicity is zero if λ is not an eigenvalue).

2.3 Some Useful Results In this subsection, we recall some known results for the reader’s convenience.

2.3.1

Eigenvalue Inequalities

The following proposition is a particular case of a result of Lotoreichik and Rohleder. Proposition 2.2 [22], (Proposition 2.3) Let  ⊂ R2 be a polygonal bounded domain whose boundary is decomposed as ∂ = 1  2  3 , where the i ’s are nonempty open subsets of ∂. Consider the eigenvalue problems for − in , with

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the boundary condition bi ∈ {d, n} on i , and list the eigenvalues μ j (, b1 b2 b3 ) in non-decreasing order, with multiplicities, starting from the index 1. Then, for any j ≥ 1, the following strict inequalities hold. 

and



μ j (, nnn) < μ j (, ndn) < μ j (, ndd), μ j (, nnn) < μ j (, nnd) < μ j (, ndd),

(2.7)

μi (Th , dnn) < μi (Th , ddn) < μi (Th , ddd), μi (Th , dnn) < μi (Th , dnd) < μi (Th , ddd).

(2.8)

The preceding inequalities can in particular be applied to the triangle Th . In this particular case, when j = 1, we have the following more precise inequalities which are due to Siudeja. Proposition 2.3 [25], (Theorem 1.1) The eigenvalues of Th with mixed boundary conditions are denoted by μi (abc), with the sides listed in decreasing order of length. They satisfy the following inequalities. 0 = μ1 (nnn) < μ1 (nnd) < μ1 (ndn) = μ2 (nnn) < μ1 (dnn) · · · < μ1 (ndd) < μ1 (dnd) < μ1 (ddn) < μ1 (ddd). Remark 2.4 We do not know whether there are any general inequalities between the eigenvalues μi (Th , ndn) and μi (Th , nnd), or between the eigenvalues μi (Th , ddn) and μi (Th , dnd), for i ≥ 2.

2.3.2

Eigenvalues of Some Mixed Boundary Value Problems for Th

For later reference, we describe the eigenvalues of four mixed eigenvalue problems for Th . This description follows easily from [8] or [9, Appendix A]. The eigenvalues of the equilateral triangle Te , with either the Dirichlet or the Neumann boundary condition on ∂Te , are the numbers λˆ (m, n) =

16π 2 2 (m + mn + n 2 ), 9

(2.9)

with (m, n) ∈ N × N for the Neumann boundary condition, and (m, n) ∈ N• × N• for the Dirichlet boundary condition (here N• = N\{0}). The multiplicities are given by,

(2.10) mult(λˆ 0 ) = # (m, n) ∈ L | λˆ (m, n) = λˆ 0 , with L = N × N for the Neumann boundary condition, and L = N• × N• for the Dirichlet boundary condition.

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One can associate one or two real eigenfunctions with such a pair (m, n). When m = n, there is only one associated eigenfunction, and it is D-invariant (here D denotes the bisector of one side of Te , see Fig. 1). When m = n, there are two associated eigenfunctions, one invariant with respect to D, the other one anti-invariant. As a consequence, one can explicitly describe the eigenvalues and eigenfunctions of the four eigenvalue problems (Th , nnn), (Th , ndn) (they arise from the Neumann problem for Te ), and (Th , dnd), (Th , ddd) (they arise from the Dirichlet problem for Te ). The resulting eigenvalues are given in Table 1. Remark 2.5 As far as we know, there are no such explicit formulas for the eigenvalues of the other mixed boundary value problems for Th . Tables 2, 3 display the first few eigenvalues, the corresponding pairs of integers, and the corresponding indexed eigenvalues for the given mixed boundary value problems for Th . Remark 2.6 For later reference, we point out that the eigenvalues which appear in Tables 2 and 3 are simple.

Table 1 Four mixed eigenvalue problems for Th Eigenvalue problem Eigenvalues λˆ (m, n), for 0 ≤ m (Th , nnn) λˆ (m, n), for 0 ≤ m (Th , ndn) (Th , dnd) λˆ (m, n), for 1 ≤ m (Th , ddd) λˆ (m, n), for 1 ≤ m

Table 2 First eigenvalues for (Th , nnn) and (Th , ndn) Eigenvalue Pairs (Th , nnn) 0 16π 2 9

3× 4× 7× 9×

16π 2 9 16π 2 9 16π 2 9 16π 2 9

(0,0)

μ1

(0,1), (1,0)

μ2

≤n

< 3

>




16π 2 9

>

(−, +) (Th , ndn) Prop. 2.2 (−, −) (Th , ndd)

0

μ2 16π 2 9

>

(+, +) (Th , nnn) Prop. 2.2 (+, −) (Th , nnd) Prop. 2.3

μ1

>

(Th , nab)

>

(σ, τ )

≤

{μi (Th , nab) for 1 ≤ i ≤ 6 and a, b ∈ {d, n}} .

(2.14)

Among these numbers, the eigenvalues of (Th , nnn) and (Th , ndn) are known explicitly, and they are simple, see Table 2. Although the eigenvalues and eigenfunctions of (Th , nnd) and (Th , dnn) are, as far as we know, not explicitly known, they satisfy some inequalities: the obvious inequalities μ1 < μ2 ≤ · · · , and the inequalities provided by Proposition 2.2 (see [22]), and Proposition 2.3 (see [25]). Table 4 summarizes what we know about the four first eigenvalues of the problems (Th , nab), for a, b ∈ {d, n}. Known values and inequalities are given by Propositions 2.2 and 2.3. The empty cells correspond to eigenvalues, listed with multiplicities, for which we have no a priori information, except the trivial inequalities. Remark 2.8 Note that we only display the first four eigenvalues in each line, because this turns out to be sufficient for our purposes. Remark 2.9 The reason why there are white empty cells in the 5th row is explained in Remark 2.4. • We know that ν1 = 0, and that this eigenvalue is simple. • From Table 4, we deduce that ν2 ∈ {μ2 (Th , nnn), μ1 (Th , nnd)}, with no other possibility. On the other hand, μ1 (Th , nnd) < μ1 (Th , ndn) = μ2 (Th , nnn). It follows that ν2 = μ1 (Th , nnd), and that this eigenvalue is simple, ν2 < ν3 . • From Table 4 and the knowledge of ν1 and ν2 , we deduce that ν3 ∈ {μ2 (Th , nnn), μ1 (Th , ndn)},

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Table 5 Rh e , Neumann boundary condition

7.16




>

>

>




μ1 0 >

(σ, τ ) (Th , nab) (+, +) (Th , nnn)

(−, −) (Th , ndd) 47.63 < 110.36 ≤ 189.52 ≤ 224.68

with no other possibility. Since μ2 (Th , nnn) = μ1 (Th , ndn), we have ν3 = ν4 <  ν5 . The proposition follows. Note: For the reader’s information, Table 5 displays either approximate eigenvalues computed with matlab, or the numerical values of the explicitly known eigenvalues. Remark 2.10 One can also deduce Proposition 2.7 from the proof of Corollary 1.3 in [25] which establishes that the first four Neumann eigenvalues of a rhombus Rh(α) with smallest angle 2α > π3 are simple, and describes the nodal patterns of the corresponding eigenvalues. When 2α = π3 the eigenvalues ν3 and ν4 become equal, see also Remarks 4.1 and 4.2 in [25].

2.5 ECP(Rh e , n) Is False As a corollary of Proposition 2.7, we obtain, Proposition 2.11 The Extended Courant property is false for the equilateral rhombus with Neumann boundary condition. More precisely, there exists a linear combi nation of eigenfunctions in E(ν1 ) E(ν3 ) with four nodal domains. Proof Proposition 2.7, Assertion (ii) tells us that E(ν3 ) contains an eigenfunction which arises from a second D-invariant Neumann eigenfunction of Te,1 = Te . It suffices to apply the arguments of [9, Sect. 3.1], where we prove that ECP(T0√, n) is false. Here, T0 is the equilateral triangle with vertices (0, 0), (1, 0) and ( 21 , 23 ). A second D-invariant Neumann eigenfunction for T0 is given by  φ(x, y) = 2 cos

2π x 3

     2π x 2π y cos + cos √ − 1. 3 3

(2.15)

The linear combination φ + 1 vanishes on the line segments {x = 34 } ∩ T0 and {x + √ 3 y = 23 } ∩ T0 .

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Fig. 4 Nodal pattern of u 3 + 1, four nodal domains

Fig. 5 (Rh e , n): ECP false in E (ν1 ) ⊕ E (ν3 )

Transplant the function φ to Te,1 by rotation and, using the symmetry with respect to M, extend it to an M-invariant eigenfunction u 3 for (Rh e , n). The linear combination u 3 + 1 vanishes on two line segments which divide Rh e into four nodal domains, see Fig. 4. The proposition is proved.  Figure 5 illustrates the variation of the number of nodal domains (the eigenfunction produced by matlab is proportional to u 3 , not equal, so that the bifurcation value is not 1 as in the proof of Proposition 2.11).

2.6 Numerical Results for the ECP(Rh e , n) In Sect. 2.4, we have identified the first four eigenvalues of (Rh e , n), in particular ν2 = μ1 (Th , nnd). The numerical computations in Table 5 indicate that the next eigenvalues are ν5 = μ2 (Th , nnd) = μ1 (Th , ndd) = ν6 , so that the Neumann eigenvalues of the equilateral rhombus satisfy, 0 = ν1 < ν2 < ν3 = ν4 < ν5 < ν6 < . . . ,

(2.16)

with corresponding nodal patterns shown in Fig. 6. Looking at linear combinations of the form u 5 + au 2 , see Fig. 7, we obtain the following numerical result. Statement 2.12 Numerical computations of the eigenvalues and of the eigenfunctions indicate that the ECP(Rh e , n) is false in E(ν2 ) ⊕ E(ν5 ). More precisely, there exist linear combinations with six nodal domains.

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Fig. 6 (Rh e , n): nodal patterns u 2 – u 5

Fig. 7 (Rh e , n): ECP false in E (ν2 ) ⊕ E (ν5 )

Remark 2.13 This counterexample can also be interpreted as a counterexample to the ECP for the equilateral triangle with mixed boundary conditions, Neumann on two sides, and Dirichlet on the third side. We first look at nodal patterns in E (μ1 (Th , nnd)) ⊕ E (μ2 (Th , nnd)), see Fig. 8. The corresponding nodal patterns in E (μ1 (Te , nnd)) ⊕ E (μ2 (Te , nnd)) are obtained using the symmetry with respect to the horizontal side. Remark 2.14 We refer to Sect. 4 for comments on our numerical approach.

2.7 Numerical Results for the ECP(Rh e , d) Table 6 is the analogue of Table 4 for the Dirichlet problem in Rh e . Although one can identify the first two Dirichlet eigenvalues of Rh e as δ1 (Rh e ) = μ1 (Th , dnn)

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Fig. 8 (Th , nnd): nodal patterns in E (μ1 ) ⊕ E (μ2 )

and δ2 (Rh e ) = μ1 (Th , dnd), it is not possible to rigorously identify the following eigenvalues. We have to rely on numerical computations. Table 7 provides the numerical eigenvalues computed with matlab, and numerical approximations of the explicitly known eigenvalues. From Table 7, we deduce that the Dirichlet eigenvalues of Rh e satisfy 0 < δ1 < δ2 < δ3 < δ4 < δ5 = δ6 < δ7 . . . . Table 6 Rh e , Dirichlet boundary condition μ3

16π 2 9

7

< 12

16π 2 9

< 13

16π 2 9

>




>

≤

>

(Th , ddd) 7

16π 2 9

16π 2 9

< 13

16π 2 9

≤

< 19

16π 2 9

>

≤ >

< >

(+, −) (Th , dnd) 3 Prop. 2.3 (−, +) (Th , ddn) Prop. 2.2 (−, −)

μ2


μ1

>

(σ, τ ) (Th , dab) (+, +) (Th , dnn) Prop. 2.2

< 21

16π 2 9

Table 7 Rh e , Dirichlet boundary condition

(−, +) (Th , ddn)

71.71

μ3 μ4 ≤ 140.50 ≤ 169.20 >

52.64

μ2 83.83

>

(+, −) (Th , dnd)




μ1 24.90

< 122.82 < 210.55 < 228.10

>

>

< 169.80 ≤ 234.10 ≤ 292.70 >

>

>

>

(σ, τ ) (Th , dab) (+, +) (Th , dnn)

(−, −) (Th , ddd) 122.82 < 228.10 < 333.37 < 368.47

(2.17)

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Fig. 9 (Rh e , d): nodal patterns u 2 – u 5

Fig. 10 (Rh e , d): ECP false in E (δ2 ) ⊕ E (δ5 )

More precisely, we find that δ2 (Rh e ) = μ1 (Th , dnd) = δ1 (Te ) (the first Dirichlet eigenvalue of the equilateral triangle Te ). An eigenfunction u 2 associated with δ2 (Rh e ) arises from a first Dirichlet eigenfunction of Te . We also find that δ5 (Rh e ) = μ2 (Th , dnd) = μ1 (Th , ddd) = δ2 (Te ). Eigenfunctions associated with δ5 (Rh e ) arise from second Dirichlet eigenfunctions of Te , one of them u 5 is invariant with respect to D, the other is anti-invariant. The nodal patterns of u 2 and u 5 are given in Fig. 9 (first and last pictures). In [9, Sect. 3], we proved that ECP(Te , d) is false: there exists a linear combination of a first eigenfunction and a second D-invariant eigenfunction of (Te , d), with three nodal domains. The same example transcribed to (Rh e , d) yields a linear combination in E(δ2 ) ⊕ E(δ5 ) with 6 nodal domains: for the Dirichlet problem in Rh e , we have the following (numerical) analogue of Proposition 2.11, see Fig. 10. Statement 2.15 The numerical approximations of the eigenvalues δ j (Rh e ) deduced from Table 7 indicate that the ECP(Rh e , d) is false in E(δ2 ) ⊕ E(δ5 ).

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Remark 2.16 We refer to Sect. 4 for comments on our numerical approach.

3 The Regular Hexagon 3.1 Symmetries and Spectra Let H denote the interior of the regular hexagon with center at the origin, and sides of unit length. The diagonals Di , i = 1, 2, 3, joining opposite vertices, and the medians M j , j = 1, 2, 3, joining the mid-points of opposite sides, are lines of mirror symmetry of the hexagon H, see Fig. 11. We consider the diagonals D1 and M2 , and the associated mirror symmetries of H. They commute, M 2 ◦ D1 = D1 ◦ M 2 = R π ,

(3.1)

and we can therefore apply the methods of Sect. 2.1. It follows that D1∗ leaves the subspaces S M2 ,± globally invariant, and that M2∗ leaves the subspaces S D1 ,± globally invariant. As a consequence, we have the following orthogonal decomposition of L 2 (H), ⊥

where

⊥

⊥

L 2 (H) = S+,+ ⊕ S−,− ⊕ S+,− ⊕ S−,+ ,

(3.2)



Sσ,τ := φ ∈ L 2 (H) | D1∗ φ = σ φ and M2∗ φ = τ φ ,

(3.3)

for σ, τ ∈ {+ , −}. Similar decompositions hold for the Sobolev spaces H 1 (H) and H01 (H), which are used in the variational presentation of the Neumann (resp. Dirichlet) eigenvalue

Fig. 11 The hexagon and its mirror symmetries

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Fig. 12 Spaces Sσ,τ for σ, τ ∈ {+, −}

problem for the hexagon. Since the Laplacian commutes with the isometries D1 and M2 , such decompositions also hold for the eigenspaces of − in H, with the boundary condition b ∈ {d, n} on the boundary ∂H. In the following figures, anti-nodal lines are indicated by dashed lines, and nodal lines by solid lines. Figure 12 displays the nodal and anti-nodal lines common to all functions in H 1 (H) ∩ Sσ,τ , where σ, τ ∈ {+, −}. Denote by R the rotation R 2π3 , 

R = D2 ◦ D1 = M 2 ◦ M 1 = . . . , R −1 = D1 ◦ D2 = M1 ◦ M2 = . . . .

(3.4)

This is an isometry of H, and the action R ∗ of R on functions is an isometry of L 2 (H) with respect to the L 2 -inner-product. Lemma 3.1 Let 

S 0 := ker(R ∗ − I ) , and S 1 := ker(R ∗2 + R ∗ + I ),

as subspaces of L 2 (H). Then

(3.5)

On Courant’s Nodal Domain Property for Linear Combinations …



   ⊥ S 0 = img R ∗2 + R ∗ + I = ker R ∗2 + R ∗ + I , S 1 = img(R ∗ − I ) = ker(R ∗ − I )⊥ , ⊥

L 2 (H) = S 0 ⊕ S 1 .

65

(3.6)

(3.7)

Here, as usual, img( f ) and ker( f ) denote respectively the image and the kernel of the linear map f , and E ⊥ the subspace orthogonal to E. Proof The following polynomial identities hold. x 3 − 1 = (x − 1)(x 2 + x + 1),

(3.8)

3 = (x 2 + x + 1) − (x − 1)(x + 2).

(3.9)

Furthermore, the rotation R satisfies R 3 = I.

(3.10)

From (3.8) and (3.10), we deduce that img(R ∗2 + R ∗ + I ) ⊂ ker(R ∗ − I ),

(3.11)

img(R ∗ − I ) ⊂ ker(R ∗2 + R + I ).

(3.12)

and

From (3.9), we deduce that L 2 (H) = img(R ∗ − I ) + img(R ∗2 + R + I ),

(3.13)

and hence, using (3.11) and (3.12)

Clearly,

L 2 (H) = ker(R ∗ − I ) + ker(R ∗2 + R + I ).

(3.14)

ker(R ∗ − I ) ∩ ker(R ∗2 + R + I ) = {0},

(3.15)

so that, using (3.11) and (3.12), img(R ∗ − I ) ∩ img(R ∗2 + R + I ) = {0}.

(3.16)

Let φ ∈ img(R ∗ − I ) and ψ ∈ img(R ∗2 + R + I ). Using the fact that R ∗ is an isometry and (3.10), we conclude that φ, ψ = 0 (the L 2 inner product). Therefore,

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img(R ∗ − I ) = img(R ∗2 + R + I )⊥ .

(3.17)

From the previous identities, we deduce that ⊥

L 2 (H) = img(R ∗ − I ) ⊕ img(R ∗2 + R + I ), ⊥

(3.18)

L 2 (H) = ker(R ∗ − I ) ⊕ ker(R ∗2 + R + I ),

(3.19)

img(R ∗ − I ) = ker(R ∗2 + R + I ),

(3.20)

img(R ∗2 + R + I ) = ker(R ∗ − I ).

(3.21) 

The lemma is proved. Lemma 3.2 For σ, τ ∈ {+, −}, using the notation (3.5), define the subspaces 

Define the map



0 Sσ,τ := Sσ,τ ∩ S 0 , 1 Sσ,τ := Sσ,τ ∩ S 1 .

(3.22)

T : L 2 (H) → L 2 (H), T (φ) = R ∗ φ − R ∗2 φ.

(3.23)

Then, (1) ker(T ) = S 0 and ker(T )⊥ = S 1 . (2) T 2 = (R ∗2 + R ∗ + I ) − 3I ; T ◦ T |S 1 = −3I ; T (S 1 ) = S 1 ; T is a bijection from S 1 onto S 1 . (3) T ◦  =  ◦ T , so that T leaves the eigenspaces of  globally invariant. 0 satisfies (4) For all σ, τ ∈ {+, −}, the subspace Sσ,τ

0 Sσ,τ = φ ∈ L 2 (H) | Di∗ φ = σ φ, M ∗j φ = τ φ, 1 ≤ i, j ≤ 3 .

(3.24)

  (5) For all σ, τ ∈ {+, −}, T S σ,τ ⊂ S−σ,−τ . 0 1 , and img(T |Sσ,τ ) ⊂ S−σ,−τ . (6) For all σ, τ ∈ {+, −}, ker T |Sσ,τ = Sσ,τ (7) For all σ, τ ∈ {+, −}, ⊥

0 1 Sσ,τ = Sσ,τ ⊕ Sσ,τ ,

(3.25)

1 1 onto S−σ,−τ . and T is a bijection from Sσ,τ

Proof Assertion (1) If φ ∈ ker(T ), then R ∗2 φ = R ∗ φ, so that φ = R ∗3 φ = R ∗2 φ = R ∗ φ, and φ ∈ S 0 . The converse is clear. The second equality follows from Lemma 3.1. Assertion (2) The first two equalities are clear. If φ ∈ S 1 , then (R ∗2 + R ∗ + I )T (φ) = (R ∗2 + R ∗ + I )(R ∗ − I )R ∗ φ = 0, and T (φ) ∈ S 1 . If φ ∈ S 1 , then

On Courant’s Nodal Domain Property for Linear Combinations …

67

T (T (φ)) = −3φ, so that φ = T (ψ) with ψ = − 13 T (φ) ∈ S 1 . This implies that T (S 1 ) = S 1 . On the other hand, if T (φ) = 0 and φ ∈ S 1 , then φ ∈ S 0 ∩ S 1 = {0}. Assertion (3) This assertion is clear because R is an isometry, so that R ∗ commutes with . It follows that T commutes with  as well, and hence that T leaves each eigenspace E(λ) globally invariant. 0 . Then R ∗ φ = φ and D1∗ φ = σ φ. Since R = D1 ◦ D3 , it Assertion (4) Let φ ∈ Sσ,τ ∗ ∗ ∗ follows that φ = R φ = D3 D1 φ = σ D3∗ φ, so that D3∗ φ = σ φ. The other equalities are established in a similar way. On the other hand, if D1∗ φ = D3∗ φ = σ φ, then R ∗ φ = (D1 ◦ D3 )∗ φ = σ 2 φ = φ . Assertion (5) Let φ ∈ Sσ,τ , i.e., D1∗ φ = σ φ and M2∗ φ = τ φ. Then, D1∗ (T (φ)) = D1∗ R ∗ φ − D1∗ R ∗2 φ = D1∗ (D2 ◦ D1 )∗ φ − D1∗ (D3 ◦ D1 )∗ φ = D2∗ φ − D3∗ φ = (D1 ◦ D1 ◦ D2 )∗ φ − (D1 ◦ D1 ◦ D3 )∗ φ = (D1 ◦ D2 )∗ D1∗ φ − (D1 ◦ D3 )∗ D1∗ φ = σ R ∗2 φ − σ R ∗ φ = −σ T (φ). Similarly, one shows that M2∗ (T (φ)) = −τ T (φ). Assertion (6) The first equality follows from Assertion (1). The second equality follows from Assertion (5) and the fact that img(T ) ⊂ S 1 because R ∗3 = I . Assertion (7) Take φ ∈ Sσ,τ . Then T (φ) ∈ S 1 ∩ S−σ,−τ and hence T 2 (φ) ∈ S 1 ∩ Sσ,τ . We also have T 2 (φ) = (R ∗2 + R ∗ + I )(φ) − 3φ, which implies that (R ∗2 + R ∗ + I )(φ) ∈ S 0 ∩ Sσ,τ . The initial equality can be rewritten φ = 13 (R ∗2 + R ∗ + I )(φ) − 13 T 2 (φ) which implies that Sσ,τ = S 0 ∩ Sσ,τ ⊕ S 1 ∩ Sσ,τ . We have T (S 1 ∩ Sσ,τ ) ⊂ S 1 ∩ S−σ,−τ . If φ ∈ S 1 ∩ Sσ,τ and T (φ) = 0, then φ ∈ 0 S ∩ S 1 = {0}. If φ ∈ S 1 ∩ S−σ,−τ , then φ = T (ψ) with ψ = − 13 T (φ) ∈ S 1 ∩ Sσ,τ . This proves that T is bijective.  Figure 13 displays the nodal and anti-nodal lines common to all functions in 0 H 1 (H) ∩ Sσ,τ , with σ, τ ∈ {+, −}. The Laplacian  commutes with isometries. It follows that the eigenspaces of the Laplacian  in H, with either the Neumann or Dirichlet boundary condition on ∂H, decompose orthogonally according to the spaces Sσ,τ , S 0 and S 1 . More precisely, if E(λ) is the eigenspace of − for the eigenvalue λ in the Neumann (resp. Dirichlet) spectrum of , then E(λ) =

⊥  σ,τ ∈{+ ,−}

 ⊥  0 1 E(λ) ∩ Sσ,τ . ⊕ E(λ) ∩ Sσ,τ

(3.26)

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0 for σ, τ ∈ {+, −} Fig. 13 The spaces Sσ,τ

1 1 Remark 3.3 If E(λ) ∩ Sσ,τ has dimension p, then by Lemma 3.2, E(λ) ∩ S−σ,−τ has dimension p. It follows that E(λ) has dimension at least 2 p.

Remark 3.4 Let λ be a simple eigenvalue. Then, any associated eigenfunction φ is either invariant or anti-invariant under any mirror symmetry L which leaves H 0 invariant, and invariant under R ∗ . It follows that φ ∈ Sσ,τ for some pair (σ, τ ). 0 Remark 3.5 Assume that φ ∈ E(λ) ∩ Sσ,τ . Then, by Courant’s theorem, we have 6 ≤ β0 (φ) ≤ κ(λ) if (σ, τ ) = (+, −) or (−, +), and 12 ≤ β0 (φ) ≤ κ(λ) if (σ, τ ) = 0 , then φ arises from an eigenfunction of Th with Neumann bound(−, −). If φ ∈ S+,+ ary condition on the sides 1 and 2.

3.2 Symmetries and Boundary Conditions on Sub-domains Let Q (resp. P) denote the interior of the quadrilateral (resp. the pentagon) which appears in Fig. 14. Let R (resp. Th ) denote the interior of the quadrilateral (resp. of the hemiequilateral triangle) which appears in Fig. 15. Then, Q (resp. P) is a fundamental domain of the action of the mirror symmetry D1 (resp. M2 ), and R is a fundamental domain for the action of the group generated by D1 and M2 .

On Courant’s Nodal Domain Property for Linear Combinations …

69

Fig. 14 The sub-domains Q and P

Fig. 15 The sub-domains R and Th

Using the notation of Sect. 2.1, we consider the following mixed eigenvalue problems in the domains H, P, Q and R. • For the hexagon H, we do not decompose the boundary, ∂H = H,1 ,

(3.27)

and we consider the eigenvalue problem (H, b) with b ∈ {n, d}. • For the quadrilateral Q, we decompose the boundary as ⎧ ⎨ ∂Q = Q,1  Q,2 , with = Q ∩ D1 ,  ⎩ Q,1 Q,2 = Q ∩ ∂H,

(3.28)

and we consider the eigenvalue problems (Q, ab), with a, b ∈ {n, d}. • For the pentagon P, we decompose the boundary as ⎧ ⎨ ∂P = P,1  P,2 , with = P ∩ M2 ,  ⎩ P,1 P,2 = P ∩ ∂H, and we consider the eigenvalue problems (P, ab), with a, b ∈ {n, d}. • For the quadrilateral R, we decompose the boundary as

(3.29)

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⎧ ∂R ⎪ ⎪ ⎨ R,1 ⎪  ⎪ ⎩ R,2 R,3

= R,1  R,2  R,3 , with = R ∩ M2 , = R ∩ D1 , = R ∩ ∂H,

(3.30)

and we consider the eigenvalue problems (R, abc), with a, b, c ∈ {n, d}. • We also consider the hemiequilateral triangle Th , its sides ordered in decreasing order of length, and the eigenvalue problems (Th , abc), with a, b, c ∈ {n, d}. For the equilateral triangle Te , up to isometry, it is not necessary to order the sides, and we consider the eigenvalue problems (Te , abc) with a, b, c ∈ {n, d}. The boundary decompositions for the domains P, Q, R, and for the hemiequilateral triangle Th , are illustrated in Figs. 14 and 15. Consider the eigenvalue problem (H, c) for the hexagon, with c ∈ {n, d}. Let E(μ, c) be an eigenspace of − for (H, c). If φ ∈ E(μ, c) ∩ Sσ,τ , then the restriction φ|R, of the function φ to the domain R, is an eigenfunction of − in (R, ε(σ )ε(τ )c), where ε(+) = n and ε(−) = d,

(3.31)

associated with the same eigenvalue μ. Conversely, let ψ be an eigenfunction of − in (R, abc), associated with the eigenvalue μ, where c is the given boundary condition on ∂H, and a, b ∈ {d, n} are boundary conditions on the sides M2 , D1 . Extend ψ to a function ψˇ defined on H, by symmetry (resp. anti-symmetry) with respect to M2 , if a = n (resp. if a = d), and by symmetry (resp. anti-symmetry) with respect to D1 , if b = n (resp. if b = d). Then, the function ψˇ is an eigenfunction of − for (H, c), associated with the eigenvalue μ, and belongs to Sσ,τ with ε(σ ) = a and ε(τ ) = b. As in Sect. 2.1, we have, Proposition 3.6 The eigenvalues and eigenfunctions of (H, c) in Sσ,τ are in bijection with the eigenvalues and eigenfunctions of (R, abc), with the boundary condition a on M2 , with a = d, if σ = −, and a = n, if σ = +; and, similarly, with the boundary condition b on D1 , with b = d, if τ = −, and b = n, if τ = +. Similar statements hold for P, Q and Th respectively.

3.3 Identification of the First Dirichlet Eigenvalues of the Regular Hexagon Throughout this section, we fix the Dirichlet boundary condition d on ∂H, and we denote the Dirichlet eigenvalues of H by δ1 (H) < δ2 (H) ≤ δ3 (H) ≤ · · · ≤ δ6 (H) ≤ δ7 (H) ≤ · · · ,

(3.32)

On Courant’s Nodal Domain Property for Linear Combinations …

71

Table 8 R-shape, mixed boundary conditions, first four approximate eigenvalues (R, abd) μ1 < μ2 ≤ μ3 ≤

ddd




70.14

≤

94.33 ?

≤

>




>


ndd

≤

>

dnd

32.45

>




nnd

μ4

≤

122.82

and the Dirichlet spectrum of the hexagon by sp(H, d).

3.3.1

Numerical Computations

Numerical approximations for the Dirichlet eigenvalues of the regular hexagon have been obtained by several authors, see for example [5, 13, 16], or the recent paper [17]. The main idea, in order to make the identification of multiple Dirichlet eigenvalues of H easier, is to take the symmetries of H (see Sect. 3.2) into account from the start. For this purpose, one computes the eigenvalues of the domains R and Th , for mixed boundary conditions abd, with a, b ∈ {d, n}. Table 8 displays the first four eigenvalues of (R, abd), as computed with matlab, and contains some useful relations between these eigenvalues. Remark 3.7 The eigenvalues in Table 8 are partially ordered ‘vertically’. Indeed, for i ≥ 1, we have the strict inequalities, 

μi (R, nnd) < μi (R, dnd) < μi (R, ddd), μi (R, nnd) < μi (R, ndd) < μi (R, ddd),

(3.33)

which follow from Proposition 2.2, see [22, Proposition 2.3]. These inequalities are indicated in the table by the (rotated) strict inequality signs. Note that it is in general not possible to compare the eigenvalues μi (R, dnd) and μi (R, ndd). This is indicated in the table by the black question marks. Table 9 displays some eigenvalues of (Th , abd), for a, b ∈ {d, n}. The lower bound in the second line follows from Dirichlet monotonicity (see Sect. 3.3.2). In the third line, we have used the fact due to Pólya (see [19]) that the first Dirichlet eigenvalue of a kite-shape is bounded from below by the first Dirichlet eigenvalue of a square with the same area. In the last two lines, the eigenvalues are known explicitly. Remark 3.8 The figures in Table 8 suggest that the Dirichlet eigenvalues of H come into four well separated sets {δ1 (H)}, {δ2 (H), δ3 (H)}, {δ4 (H), δ5 (H)} and {δ6 (H)}.

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Table 9 Some eigenvalues of the hemiequilateral triangle S (Th , abd) Eigenvalue

Value

0 S+,+ 0 S+,+ 0 S+,− 0 S−,+ 0 S−,−

3.3.2

(Th , nnd) (Th , nnd)

μ1 μ2

μ1 ≈ 7.16 μ2 ≈ 37.49 > 26.37

(Th , ndd)

μ1

μ1 ≥

(Th , dnd)

μ1

μ1 =

(Th , ddd)

μ1

μ1 =

2 4π √ ≈ 3 2 3 16π 9 2 7 16π 3

22.79 ≈ 52.64 ≈ 122.82

Lower and Upper Bounds for the Dirichlet Eigenvalues √

The hexagon H is inscribed in the unit disk D, and contains the disk with radius 23 . By domain monotonicity for the Dirichlet eigenvalues, we have the following lower and upper bounds for the Dirichlet eigenvalues of H, δ j (D) < δ j (H)


dnd

μ3

≤ >

δ 1 (H ) >

nnd

≤

>

Table 12 Possible choices for δ4 (H) (R, abd) μ1 < μ2

≤

no

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3.4 Numerical Results and ECP(H, d) Using the numerical approximations given in Table 8, we infer the (numerical) lower bound δ6 (H) > 35.17. This implies that δ6 (H) is simple. It follows that u 6 arises from the second eigenfunction of Th , with mixed boundary condition nnd (Dirichlet on the smaller side of Th , Neumann on the other sides). This provides the following numerical extension of Proposition 3.9, Statement 3.11 The Dirichlet eigenvalues of H satisfy, δ1 (H) < δ2 (H) = δ3 (H) < δ4 (H) = δ5 (H) < δ6 (H) < δ7 (H),

(3.42)

and δ4 (H) = δ5 (H) = δ1 (R),

(3.43)

The eigenspace E(δ4 ) has dimension 2, and is generated by an eigenfunction u 4 which arises from the first eigenfunction of (R, ddd) and the function u 5 = T (u 4 ). The eigenfunction u 6 associated with δ6 (H) arises from the second eigenfunction of (Th , nnd), and its nodal set is a simple closed curve enclosing the center of the hexagon. Figure 18 displays the nodal patterns of first six Dirichlet eigenfunctions of H. Plotting the nodal set of the linear combination u 6 + a u 1 for several values of a, one finds some values of a for which this function has 7 nodal domains, see Fig. 19. Statement 3.12 Figure 19 provides a numerical evidence that the ECP(H, d) is false. Remark 3.13 For Statement 3.12, we do not really need to separate δ6 (H) from δ5 (H). It suffices to use Proposition 3.9, and more precisely the fact that there

Fig. 18 (H, d): nodal structure for the first six eigenfunctions

On Courant’s Nodal Domain Property for Linear Combinations …

77

Fig. 19 (H, d): the ECP is false in E (δ1 ) ⊕ E (δ6 )

exists an eigenfunction in E(δ4 ) ⊕ E(δ5 ) ⊕ E(δ6 ), which arises from a second eigenfunction of (Th , nnd). As in Sect. 2.6, we then need to know the nodal patterns in E (μ1 (Th , nnd)) ⊕ E (μ2 (Th , nnd)), or equivalently the nodal patterns in E (μ1 (Te , nnd)) ⊕ E (μ2 (Te , nnd)), see Remark 2.13 and Fig. 8.

3.5 Identification of the First Neumann Eigenvalues of the Regular Hexagon 3.5.1

Numerical Computations and Preliminary Remarks

We did not find numerical computations of the Neumann eigenvalues of the hexagon in the literature. We use the same method as in Sect. 3.3. Given an eigenspace E(λ) of − for (H, n), we apply Lemma 3.2, and write E(λ) =

⊥ 

 σ,τ ∈{+,−}

⊥  0 1 E(λ) ∩ Sσ,τ . ⊕ E(λ) ∩ Sσ,τ

(3.44)

This means that to determine the eigenvalues of (H, n), it suffices to list the eigenvalues of (R, abn), with a, b ∈ {d, n}, and to re-order them in non-decreasing order. Table 13 displays the approximate values of the first four eigenvalues of (R, abn), as calculated by matlab.

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ddn







>


ndn

≤

>

dnn

10.87

≤

>




nnn

μ3

>

Table 13 First four eigenvalues for (R, abn), a, b ∈ {d, n} (R, abd) μ1 < μ2 ≤

≤

71.71

Value

(Th , nnn)

μ2

μ2 =

(Th , ndn)

μ1

μ1 =

(Th , dnn)

μ1

μ1 >

(Th , ddn)

μ1

μ1 >

16π 2 9 ≈ 17.55 16π 2 9 ≈ 17.55 2 4π √ > 22.79 3 16π 2 3 > 52.64

Remark 3.14 The following inequalities follow from Proposition 2.2, see [22], 

μi (R, nnn) < μi (R, dnn) < μi (R, ddn), μi (R, nnn) < μi (R, ndn) < μi (R, ddn).

(3.45)

These inequalities are indicated in Table 13 by the (rotated) strict inequality signs. The question marks indicate that one cannot compare the other values. 0 correspond to eigenfunctions of − for (Th , abn) Eigenfunctions in E(λ) ∩ Sσ,τ with a = d (resp. a = n) if τ = − (resp. τ = +), and similarly for b, with σ . Table 14 displays the first non trivial eigenvalue of (Th , abn). Remark 3.15 (1) The first eigenvalue μ1 (Th , nnn) is 0. The second eigenvalue μ2 (Th , nnn) is also the second eigenvalue of an equilateral triangle with Neumann boundary condition. The corresponding eigenfunction has a nodal line which is a curve from side 1 to side 2 of Th . (2) In the third line of Table 14, we use the fact that μ1 (Th , dnn) is the first Dirichlet eigenvalue of an equilateral rhombus. It is bounded from below by the first Dirichlet eigenvalue of a square with the same area (Pólya, see [19]). (3) In the fourth line of Table 14, we use the fact that μ √1 (Th , ddn) is the first Dirichlet eigenvalue of an isosceles triangle with sides (1, 1, 3). It is bounded from below by the first Dirichlet eigenvalue of the equilateral triangle with the same area (Pólya, see [19]). Note that μ1 (Th , ddn) > μ1 (Th , dnn) according to Proposition 2.2. One can also compute the eigenvalues of (H, n) directly, without taking the symmetries into account. The first Neumann eigenvalues of the hexagon are given in Table 15.

On Courant’s Nodal Domain Property for Linear Combinations … Table 15 First non-trivial Neumann eigenvalues of H Eigenvalue of H Approximation ν2 (H) ν3 (H) ν4 (H) ν5 (H) ν6 (H) ν7 (H) ν8 (H)

≈ 4.04 ≈ 4.04 ≈ 10.87 ≈ 10.87 ≈ 17.55 ≈ 17.55 ≈ 24.90

79

Eigenvalue of R μ1 (R, dnn) μ1 (R, ndn) μ1 (R, ddn) μ2 (R, nnn) μ2 (R, dnn) μ3 (R, nnn) μ2 (R, ndn)

Fig. 20 (H, n): nodal patterns u 2 – u 7

Figure 20 displays the nodal patterns of eigenfunctions associated with the eigenvalues νi (H), 2 ≤ i ≤ 7. Remark 3.16 The figures in Table 13 suggest that the Neumann eigenvalues of the hexagon come into well separated sets: ⎧ ν1 (H) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ {ν2 (H), ν3 (H)} ⊂ ]3, 5[, {ν4 (H), ν5 (H)} ⊂ ]6, 14[, ⎪ ⎪ {ν6 (H), ν7 (H)} ⊂ ]15, 20[, ⎪ ⎪ ⎩ ν8 (H) > 21.

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In the following subsections, we analyze the possible eigenspaces and, more precisely, the double eigenvalues. Note that for Neumann eigenvalues we do not have monotonicity inequalities as the ones we used for Dirichlet eigenvalues in Sect. 3.3.2, so that we have to rely on the numerical evidence provided by Remark 3.16.

3.5.2

Analysis of the Possible Eigenspaces of (H, n)

We divide the analysis into several steps. Step 1: eigenvalue ν1 (H). The first Neumann eigenvalue is zero, and simple, with a 0 , and ν1 (H) = corresponding eigenfunction u 1 which is constant. We have u 1 ∈ S+,+ μ1 (Th , nnn). Step 2: eigenvalue ν2 (H). Let E2 = E (ν2 (H)) be the corresponding eigenspace. • We claim that E2 ∩ S 0 = {0}.

(3.46)

0 = Indeed, Courant’s nodal domain theorem and Lemma 3.2 imply that E2 ∩ Sσ,τ 0 {0} unless (σ, τ ) = (+, +). Assume that there exists some 0 = φ ∈ E2 ∩ S+,+ . The restriction of φ to Th would be an eigenfunction of − for (Th , nnn). Because ν2 (H) is the least non zero eigenvalue, we would have ν2 (H) = μ2 (Th , nnn), whose eigenfunction is known, with nodal set an arc from the side 1 to the side 2. The function φ would have a closed nodal line bounding a nodal domain strictly contained in the interior of H, and we would have ν2 (H) > δ1 (H), contradicting the fact that ν3 (H) ≤ δ1 (H) according to [21, Theorem 4.2].

As a by-product of (3.46), Lemma 3.2 tells us that the map T , defined by (3.22), is a bijection from E2 to E2 . • We claim that 1 1 = {0} and E2 ∩ S+,+ = {0}. E2 ∩ S−,−

(3.47)

The first assertion is clear by Courant’s theorem. The second assertion follows 1 1 onto S−,− which commutes from the fact that the map T is a bijection from S+,+ with . • We claim that     1 1 = dim E2 ∩ S−,+ = 1, dim E2 ∩ S+,− and hence that mult (ν2 (H)) = 2 .

(3.48)

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1 1 Indeed, using the map T again, we see that the spaces E2 ∩ S+,− and E2 ∩ S−,+ have the same dimension. According to [15], see the statement p. 1170, line (-8), the multiplicity of ν2 (H) is less than or equal to 3, and we can conclude that this dimension must be 1. Here is an alternative argument for the case at hand. It suffices to prove that the dimension of E2 cannot be larger than or equal to 4. Indeed, assume that dim E2 ≥ 4. One could then find a point x0 ∈ H, and an eigenfunction u 4 ∈ E2 such that u 4 (x0 ) = 0. The subspace E2,x0 = {u ∈ E2 | u(x0 ) = 0} would have dimension 3, with a basis u 1 , u 2 , u 3 . The three vectors ∇u 1 (x0 ), ∇u 2 (x0 ), ∇u 3 (x0 ) ∈ R2 would be linearly dependent, and we would then find a nontrivial u ∈ E2 such that u(x0 ) = ∇u(x0 ) = 0. The nodal set of u would contain at least four semi-arcs emanating from x0 , and we would reach a contradiction with the fact that u has only two nodal domains by Courant’s theorem. Because μ2 (R, nnn) is an eigenvalue of (H, n), we have proved the following lemma.

Lemma 3.17 The eigenvalue ν2 (H) has multiplicity 2, ν2 (H) = μ1 (R, dnn) = μ1 (R, ndn),

(3.49)

and corresponding eigenfunctions u 2 , u 3 arise from the first eigenfunctions of − for (R, dnn) and (R, ndn). Furthermore, ν2 (H) = ν3 (H) < μ2 (R, nnn). Step 3: eigenvalue ν4 (H). Let E4 = E (ν4 (H)) be the eigenspace associated with the eigenvalue ν2 (H). • We claim that E4 ∩ S 0 = {0}.

(3.50)

Indeed, by Courant’s theorem and Lemma 3.2, 0 = {0}, E4 ∩ Sσ,τ

(3.51)

0 . Then, we unless (σ, τ ) = (+, +). Assume that there exists some 0 = φ ∈ S+,+ 2 16π would have ν4 (H) = μ2 (Th , nnn) = 9 . Observe that ν2 (Te ) = μ2 (Th , nnn) = 0 having 6 μ1 (Th , ndn). This means that E4 would also contain a function in S+,− nodal domains which would contradict Courant’s theorem. From (3.50) and Proposition 2.3 ([25, Theorem 1.1]), we deduce that

ν4 < μ2 (Th , nnn) = μ1 (Th , ndn) < μ1 (Th , dnn) < μ1 (Th , ddn).

(3.52)

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• We claim that 1 1 = {0} and E4 ∩ S−,+ = {0}. E4 ∩ S+,−

(3.53)

1

= {0} or, equivalently using the map T , that E4 ∩ Indeed, assume that E4 ∩ S+,−

= {0}. Then, we would have

1 S−,+

ν4 = μ2 (R, ndn) = μ2 (R, dnn).

(3.54)

These eigenvalues are strictly larger than μ2 (R, nnn) by (3.45), and this would contradict the fact that ν3 (H) < μ2 (R, nnn), see Step 2, because μ2 (R, nnn) is an eigenvalue for (H, n). As a by-product, we have the inequalities 

ν4 < μ2 (R, dnn) < μ2 (R, ddn), ν4 < μ2 (R, ndn) < μ2 (R, ddn).

(3.55)

• It follows from the above arguments that we must have,     1 1 = dim E 4 ∩ S+,+ > 0, dim E 4 ∩ S−,−

(3.56)

and hence that dim E4 ≥ 2, so that ν4 (H) = ν5 (H) = μ1 (R, ddn) = μ2 (R, nnn), with corresponding eigenfunction u 4 , u 5 for (H, n). Remark 3.18 According to Table 13 and Remark 3.16, dim E4 = 2. Step 4: eigenvalue ν6 (H). So far, we have established the following facts ν1 (H) < ν2 (H) = ν3 (H) < ν4 (H) = ν5 (H) ≤ · · · or μ1 (R, nnn) < μ1 (R, dnn) = μ1 (R, ndn) < μ2 (R, nnn) = μ1 (R, ddn). The next eigenvalue ν6 (H) should belong to the set, {μ2 (R, dnn), μ2 (R, ndn), μ2 (R, ddn), μ3 (R, nnn)}

(3.57)

We can exclude μ2 (R, ddn) because it is larger than both μ2 (R, dnn) and μ2 (R, ndn) according to (3.45) ([22, Proposition 2.3]). The eigenvalues of (Th , abn), a, b ∈ {d, n}, are eigenvalues of (H, n). Using Table 14 and Remark 3.16, we can conclude that ν6 (H) = ν7 (H) = μ1 (Th , ndn) = μ2 (Th , nnn),

(3.58)

and that associated eigenfunctions u 6 , u 7 arise from the first and second eigenfunctions of (Th , ndn).

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Statement 3.19 From the numerical evidence in Remark 3.16, we conclude that νi (H) have multiplicity 2 for i ∈ {2, 4, 6}, and that ν6 (H) and ν7 (H) arise from eigenvalues of (Th , abn).

3.6 Numerical Computations and ECP(H, n) The first Neumann eigenvalue of the hexagon, ν1 (H), is 0, with associated eigenfunction u 1 ≡ 1. As sixth Neumann eigenfunction u 6 of the hexagon, we can choose the function which arises from an eigenfunction for μ2 (Th , nnn), or equivalently from a D-invariant second eigenfunction ψ of (Te , n). It follows from [9, Sect. 3] that ECP(Te , n) is false, i.e., that there exists some real value a such that ψ + a has three nodal domains in Te . It follows that u 6 + a has seven nodal domains, so that ECP(H, n) is false. Alternatively, we can look at μ3 (R, nnn) = μ2 (Th , nnn). Figure 21 displays the nodal pattern and the level lines of an eigenfunction for μ3 (R, nnn). By reflection with respect to the lines D1 and M2 , one obtains a Neumann eigenfunction u H of H, associated with ν6 (H) = ν7 (H), whose nodal set is a closed simple curve around O, and whose level lines are displayed in Fig. 22; some level lines of u H have six connected components, one component near each vertex of the hexagon, so that ECP(H, n) is false. Statement 3.20 The ECP(H, n) is false in E(ν1 ) ⊕ E(ν6 ). Fig. 21 (R, nnn): nodal set and level lines for u 3

Fig. 22 Level lines of u H

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4 Final Comments 4.1 Numerical Computations In Sect. 3.6, we used numerical approximations of the first eigenvalues of the problems (R, nab) and (Th , abn) in order to identify the first eight eigenvalues of (H, n), and to conclude that ECP(H, n) is false (we also used the fact that some eigenfunctions are known explicitly), see Table 16. We did not find tables providing the first eigenvalues of (H, n) in the literature. We used the symmetries, and computed the eigenvalues of the problems (R, abn) and (Th , abn) with matlab. We checked the accuracy of our computations in two ways. (1) First, using the symmetries, we computed the eigenvalues of (R, abd) and (Th , abd) in order to obtain the Dirichlet eigenvalues of the hexagon. We then compared the results with the tables in [13], see Table 17. (2) Second, we computed the eigenvalues of (Th , abc), and compared the results both with explicitly known eigenvalues, and with the tables in [16], see Tables 18 and 19.

Table 16 Neumann eigenvalues of H R-nnn R-dnn μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8

0.00000 10.87154 17.54612 33.44804 52.63932 54.77368 70.18642 95.730376

4.04324 17.54612 32.91475 49.90025 70.18634 80.45171 96.57468 122.82951

Table 17 Dirichlet eigenvalues of H R-nnd R-dnd μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8

7.15537 32.45240 37.49207 70.14288 87.53590 90.06261 120.87511 145.49213

18.13185 47.63049 60.10698 94.33006 110.36078 125.05361 152.67570 183.45321

R-ndn

R-ddn

4.04324 24.89890 32.91476 49.90023 80.45178 83.83234 96.57456 130.09479

10.87153 33.44806 54.77365 71.71384 95.73076 111.78681 128.35718 159.56644

H-Neumann 0.00000 4.04327 4.04327 10.87171 10.87172 17.54658 17.54660 24.89989

R-ndd

R–ddd

H-Dirichlet CurKut1999

18.13186 52.63929 60.10704 94.32994 122.82935 125.05369 152.67575 183.45364

32.45236 70.14278 87.53571 122. 82902 145.49250 165.52862 187.35944 228.12299

7.15545 18.13234 18.13236 32.45398 32.45400 37.49405 47.63410 52.64351

7.15534 18.13168 18.13168 32.45186 32.45186 37.49136 47.62937 52.63789

On Courant’s Nodal Domain Property for Linear Combinations … Table 18 Eigenvalues of Th (Dirichlet on shortest side) Dirichlet N N ×16π 2 /9 nnd-oee nnd-oee nnd-oee nnd-oee ndd-ooe ndd-ooe ndd-ooe ndd-ooe dnd-oeo dnd-oeo dnd-oeo dnd-oeo ddd-ooo ddd-ooo ddd-ooo ddd-ooo

μ1 μ2 μ3 μ4 μ1 μ2 μ3 μ4 μ1 μ2 μ3 μ4 μ1 μ2 μ3 μ4

3 7 12 13 7 13 19 21

52.63789 122.82174 210.55156 228.09752 122.82174 228.09752 333.37330 368.46523

Table 19 Eigenvalues of Th (Neumann on shortest side) Neumann N N ×16π 2 /9 nnn-eee nnn-eee nnn-eee nnn-eee ndn-eoe ndn-eoe ndn-eoe ndn-eoe dnn-eeo dnn-eeo dnn-eeo dnn-eeo ddn-eoo ddn-eoo ddn-eoo ddn-eoo

μ1 μ2 μ3 μ4 μ1 μ2 μ3 μ4 μ1 μ2 μ3 μ4 μ1 μ2 μ3 μ4

0 1 3 4 1 4 7 9

0.00000 17.54596 52.63789 70.18385 17.54596 70.18385 122.82174 157.91367

85

ML-μ

Jones93-μ-LR

7.15536 37.49188 90.06160 120.87283 47.63020 110.35921 189.52747 224.69274 52.63891 122.82735 210.56758 228.11669 122.82707 228.11639 333.41196 368.51464

7.15568 37.49395 90.07301 120.88563 47.63622 110.38024 189.60464 224.68511 52.63793 122.82181 210.55171 228.09759 122.82181 228.09759 333.37282 368.46338

ML-μ

Jones93-μ-LR

0.00000 17.54608 52.63891 70.18573 17.54608 70.18561 122.82718 157.92305 24.89880 83.83145 140.45719 169.23939 71.71299 169.78103 234.07541 292.73528

0.00000 17.54596 52.63793 70.18385 17.54596 70.18385 122.82181 157.91441 24.90489 83.83966 140.61691 169.22888 71.74751 169.84084 234.37467 292.71446

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Remark 4.1 The tables in [13, 16] are organized according to the symmetries, and they provide the square roots of the eigenvalues. In [16], the labelling of the sides of Th is different from ours: we use d, n to indicate the boundary condition on each side, while Jones uses the notation e, o (for even and odd). For the reader’s convenience, we indicate both labellings in the first column of Tables 18 and 19. The fourth column of each table contains the eigenvalues which are known explicitly; the fifth column contains our computations. The sixth column of each table contains the values deduced from [16], Tables 7, 8, 9, 10, 11, 12, 13, 14. Remark 4.2 Our purpose in this paper is to identify eigenvalues, and their relations with the symmetries, not to find high precision approximations as in [16, 17]. The approximated values which appear in the tables indicate that the approximations are indeed sufficient to identify the eigenvalues (because we took the symmetries into consideration from the start, and identified multiple eigenvalues). In Sects. 2.5 and 3.4, we also used numerical approximations of the first and second eigenfunctions of (Th , nnd) in order to show that the ECP(Rh e , n) is false in E(ν2 ) ⊕ E(ν5 ), and that ECP(H, d) is false in E(δ1 ) ⊕ E(δ6 ).

4.2 Final Remarks The estimates in Table 10 are valid for the regular polygon Pn with n sides, inscribed in the circle of radius 1. The upper bounds get better when n increases, and for n ≥ 9, they are sufficient to separate δ6 (Pn ) from δ5 (Pn ). This shows that δ6 (Pn ) is a simple eigenvalue for n ≥ 6, and that an associated eigenfunction u 6 arises from the first eigenfunction of a right triangle with smallest angle πn , hypotenuse of length 1, with Dirichlet condition on the smallest side and Neumann condition on the other sides. Equivalently, the eigenfunction u 6 arises from a first eigenfunction of an isosceles , with equal sides of length 1, Dirichlet condition triangle whose apex angle is 2π n on the smallest side and Neumann condition on the equal sides. Note that δ6 (D) corresponds to the second radial eigenfunction of the disc. Based on our computations, we conjecture that the ECP(Pn , a) is false for any regular polygon Pn ⊂ R2 with n ≥ 6 sides, and a ∈ {d, n}, with some linear combination u 6 + au 1 of a sixth and a first eigenfunctions providing a counterexample with (n + 1) nodal domains. Using [23][Theorem B], one can show that ECP(Pn , n) is false for n sufficiently large, see [6]. The simulations show that the first six Dirichlet eigenfunctions of Pn look very much like the first six Dirichlet eigenfunctions of the disk D. The above considerations do not provide any counter-example to the ECP when the number of sides is 4 or 5. It is not clear whether the ECP is false for the square and for the regular pentagon. It is not clear either whether the ECP is false for the disk.

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In the Neumann case, the present paper is also relevant to the investigation of the level lines of Neumann eigenfunctions. Such investigations arise when studying the hot spots conjecture.

References 1. G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Comment. Math. Helvetici 69, 142–154 (1994) 2. V. Arnold, The topology of real algebraic curves (the works of Petrovskii and their development). Uspekhi Math. Nauk. 28(5), 260–262 (1973). English translation in [4] 3 3. V. Arnold, Topological properties of eigenoscillations in mathematical physics. Proc. Steklov Inst. Math. 273, 25–34 (2011) 4. V. Arnold, Topology of real algebraic curves (Works of I.G. Petrovskii and their development). Translated by Oleg Viro. in A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassilev, O.Y. Viro (Eds.) Collected Works, Volume II. Hydrodynamics, Bifurcation Theory and Algebraic Geometry, 1965–1972 (Springer, 2014) 5. L. Bauer, E.L. Reiss, Cutoff wavenumbers and modes of hexagonal waveguides. SIAM J. Appl. Math. 35(3), 508–514 (1978) 6. P.Bérard, P. Charron, B. Helffer, Non-boundedness of the number of nodal domains of a sum of eigenfunctions. To appear in J. d’analyse mathématique. arXiv:1906.03668 7. P.Bérard, B. Helffer, Nodal sets of eigenfunctions, Antonie Stern’s results revisited. Séminaire de théorie spectrale et géométrie (Grenoble) 32, 1–37 (2014–2015). http://tsg.cedram.org/item? id=TSG_2014-2015__32__1_0 8. P. Bérard, B. Helffer, Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle. Lett. Math. Phys. 106, 1729–1789 (2016) 9. P. Bérard, B. Helffer, On Courant’s nodal domain property for linear combinations of eigenfunctions, Part I. Documenta Mathematica 23, 1561–1585 (2018). arXiv:1705.03731 10. P. Bérard, B. Helffer, Sturm’s theorem on zeros of linear combinations of eigenfunctions. Expositiones Mathematicae (in press). https://doi.org/10.1016/j.exmath.2018.10.002. arXiv:1805.01335(expanded version) 11. P. Bérard, B. Helffer, Level sets of certain Neumann eigenfunctions under deformation of Lipschitz domains. Application to the Extended Courant Property. To appear in Annales de la Faculté des Sciences de Toulouse. http://afst.cedram.org/. arXiv:1805.01335 12. R. Courant, D. Hilbert, Methods of Mathematical physics. First English edition, vol. 1. Interscience, New York (1953) 13. L.M. Cureton, J.R. Kuttler, Eigenvalues of the Laplacian on regular polygons and polygons resulting from their disection. J. Sound Vib. 220(1), 83–98 (1999) 14. G. Gladwell, H. Zhu, The Courant-Herrmann conjecture. ZAMM-Z. Angew. Math. Mech. 83(4), 275–281 (2003) 15. T. Hoffmann-Ostenhof, P. Michor, N. Nadirashvili, Bounds on the multiplicities of eigenvalues for fixed membranes. GAFA Geom. Func. Anal. 9, 1169–1188 (1999) 16. R.S. Jones, The one-dimensional three-body problem and selected wave-guide problems: solutions of the two-dimensional Helmholtz equation. PhD Thesis, The Ohio State University (1993). Retyped 2004, available at http://www.hbelabs.com/phd/ 17. R.S. Jones, Computing ultra-precise eigenvalues of the Laplacian with polygons. Adv. Comput. Math. 43, 1325–1354 (2017). arXiv:1602.08636v1 18. N. Kuznetsov, On delusive nodal sets of free oscillations. Newsletter Eur. Math. Soc. 96, 34–40 (2015) 19. R. Laugesen, B. Siudeja, Triangles and other special domains, in A. Henrot (ed.) Chapter 1 in Shape optimization and spectral theory (De Gruyter, Berlin 2017)

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20. C. Léna, Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions. Annales de l’institut Fourier 69(1), 283–301 (2019). arXiv:1609.02331 21. H. Levine, H. Weinberger, Inequalities between Dirichlet and Neumann eigenvalues. Arch. Rational Mech. Anal. 94, 193–208 (1986) 22. V. Lotoreichik, J. Rohledder, Eigenvalue inequalities for the Laplacian with mixed boundary conditions. J. Diff. Equ. 263, 491–508 (2017) 23. Y. Miyamoto, A planar convex domain with many isolated “hot spots” on the boundary. Japan J. Indust. Appl. Math. 30, 145–164 (2013) 24. Å. Pleijel, Remarks on Courant’s nodal theorem. Comm. Pure. Appl. Math. 9, 543–550 (1956) 25. B. Siudeja, On mixed Dirichlet-Neumann eigenvalues of triangles. Proc. Am. Math. Soc. 144, 2479–2493 (2016) 26. O. Viro, Construction of multi-component real algebraic surfaces. Soviet Math. dokl. 20(5), 991–995 (1979)

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model Anne Boutet de Monvel and Lech Zielinski

Dedicated to the memory of Erik Balslev

Abstract We investigate the asymptotic behavior of large eigenvalue for the twophoton Rabi Hamiltonian, i.e., for the two-photon Jaynes–Cummings model without the rotating wave approximation. We prove that the spectrum of this Hamiltonian − ∞ consists of two eigenvalue sequences (E n+ )∞ n=0 , (E n )n=0 , satisfying the same two−1/3 ) when n tends to infinity. We also term asymptotic formula with remainder O(n propose a conjecture on a three-term asymptotic formula modeled on the GRWA for the one-photon Rabi model. Keywords Quantum Rabi model · Large eigenvalue asymptotics · Unbounded Jacobi matrices 2010 Mathematics Subject Classification Primary 81Q80 · Secondary 81Q10 · 47B36 · 47A75

A. Boutet de Monvel (B) Institut de Mathématiques de Jussieu-PRG, Université de Paris, bâtiment Sophie Germain case 7012, 75205 Paris Cedex 13, France e-mail: [email protected] L. Zielinski Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville EA 2597, Université du Littoral Côte d’Opale, 62228 Calais, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_5

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1 Introduction The simplest interaction between a two-level atom and a classical light field is described by the Rabi model [20, 21]. The quantum Rabi model couples a quantized single-mode radiation and a two-level quantum system according to the idea that each photon creation accompanies an atomic de-excitation, and each photon annihilation accompanies atomic excitation (see [23] for the microscopic derivation of the quantum Rabi model in Cavity Quantum Electrodynamics). In the pioneer work [17], E. T. Jaynes and F. W. Cummings introduced the rotating wave approximation (RWA) of the quantum Rabi model. The model of [17] (with the RWA) is explicitly diagonalizable and explains a range of experimental phenomena, but further developments in engineering of quantum systems, shows the necessity of understanding the full Rabi Hamiltonian given by the formula (1.3). We refer to [26] for an exhaustive overview of theoretical and experimental works in relation with the quantum Rabi model and its various generalizations (see also [6]). In this paper we investigate the two-photon quantum Rabi model. The corresponding Hamiltonian is given by the formula (1.2) which differs from (1.3) by the fact that the atomic excitation/de-excitation appears via annihilation/creation of two photons. This model was used in [13] to describe a two-level atom interacting with squeezed light and we refer to [8] for the overview of works about the two-photon quantum Rabi model (see also [11]). The purpose of this paper is to investigate the asymptotic behavior of large eigenvalues of the two-photon quantum Rabi Hamiltonian. It appears that this problem can be reduced to the analog problem for some infinite Jacobi matrices. The asymptotic behavior of large eigenvalues of classes of infinite Jacobi matrices with discrete spectrum was initiated by Janas and Naboko [16] and continued in [1, 15, 18]. However, in the case of a non-degenerated two-level system, the corresponding Jacobi matrices do not belong to the classes of operators considered in [1, 15, 16, 18] and the asymptotic behavior of large eigenvalues is quite different. It appears that one deals with Jacobi matrices that have diagonal entries perturbed by periodic oscillations, but the contribution of these oscillations to the behavior of the n-th eigenvalue tends to 0 when n → ∞. This phenomenon for the quantum Rabi Hamiltonian (1.3), was first investigated by Tur [24, 25] (see also [2–5, 27]). In this paper we prove that a similar phenomenon still holds for the two-photon Rabi Hamiltonian (1.2). The main result of this paper is the two-term asymptotic formula (1.4) given in Theorem 1.1. However, our approach allows us to conjecture the three-term asymptotic formula (1.6) which is the two-photon version of the three-term asymptotic formula proved in [4].

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1.1 The Two-Photon Rabi Model The two-photon Rabi model is given by a Hamiltonian Hˆ Rabi acting in a complex Hilbert space HRabi = Hatom ⊗ Hfield . This Hamiltonian is of the form Hˆ Rabi = Hˆ atom ⊗ IHfield + IHatom ⊗ Hˆ field + Hˆ int ,

(1.1)

where Hˆ atom is the Hamiltonian of the two-level atomic system, Hˆ field is the Hamiltonian of the light field, and Hˆ int is the interaction term. The two-level atomic system is defined by a self-adjoint operator Hˆ atom acting in the two-dimensional complex Hilbert space Hatom = C2 . We denote by E g ≤ E e its eigenvalues, that is, the energies of the ground state and of the excited state. To simplify the model we assume the eigenvalues are ± 21  where  := E e − E g is the level separation energy. Identifying a basis of eigenvectors {ee , eg } with the canonical basis of Hatom = C2 we have  Hˆ atom = σ3 , 2  0  is the usual Pauli matrix. Henceforth we denote e1 := ee where σ3 = σz := 01 −1 and e−1 := eg . Hfield is a complex Hilbert space equipped with an orthonormal basis {en }∞ 0 . The Hamiltonian Hˆ field is the self-adjoint operator in Hfield characterized by Hˆ field en = nen , n = 0, 1, 2, . . . The photon annihilation and creation operators are defined in Hfield by aˆ en =

√

n en−1 , aˆ † en =

√ n + 1 en+1 , n = 0, 1, 2, . . .

(with aˆ e0 = 0). In particular, aˆ † aˆ = Nˆ , where Nˆ is the photon number operator characterized by Nˆ en = n en . Therefore Hˆ field can be written as ˆ Hˆ field = Nˆ = aˆ † a. Finally, the interaction term Hˆ int is given by   Hˆ int = gσ1 ⊗ aˆ 2 + (aˆ † )2 , where g > 0 is the coupling constant and σ1 = σx := To summarize, (1.1) takes the explicit form

0 1 10

is the Pauli matrix.

   Hˆ Rabi = σ3 ⊗ IHfield + IHatom ⊗ Nˆ + gσ1 ⊗ aˆ 2 + (aˆ † )2 . 2

(1.2)

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Remark The usual (one-photon) quantum Rabi Hamiltonian is given by  σ3 ⊗ IHfield + IHatom ⊗ Nˆ + gσ1 ⊗ (aˆ + aˆ † ). 2

(1.3)

1.2 Main Result Our aim is to prove Theorem 1.1 (large eigenvalues of Hˆ Rabi ) Let Hˆ Rabi be the two-photon Rabi Hamiltonian given by (1.2), with parameters  ≥ 0 and g > 0. We make the additional assumption g < 21 . Then Hˆ Rabi is self-adjoint, has discrete spectrum, its eigenvalues − ∞ can be enumerated in two nondecreasing sequences {E n+ }∞ 0 and {E n }0 : − ∞ σ ( Hˆ Rabi ) = {E n+ }∞ 0 ∪ {E n }0 ,

and the large n behavior of these eigenvalues is given by  1  1 E n± = n + 1 − 4g 2 − + O(n −1/3 ). 2 2

(1.4)

Remark In the degenerated case  = 0 we have an exact formula for E n+ = E n− :   1  1 E n± = n + 1 − 4g 2 − . 2 2

(1.5)

The proof of (1.5), based on a paper by Emary and Bishop [10], is given in Sect. 6.2. By analogy with the Generalized Rotating-Wave Approximation (GRWA) proposed by Irish [14] for the one-photon Rabi model (see also Feranchuk et al. [12]), and proved in [4], we propose the following conjecture for the two-photon Rabi model. Conjecture For μ = 0, 1 and

1 2

< ρ < 1, we have the three-term asymptotics

ωμ ± 0 E 2k+μ = E 2k+μ ±  √ cos(αμ k + θμ ) + O(k −ρ ) as k → ∞, k where

(1.6)

 1  1 0 1 − 4g 2 − , := 2k + μ + E 2k+μ 2 2

and ωμ , αμ , and θμ are real constants, depending only on g (see Sect. 5.3 for explicit expressions of these constants).

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Comment As we will show in Sect. 5.3 below, the validity of this three-term asymptotics can be derived from the following improvement of the estimate (5.8) from Lemma 5.6: λ j (L δγ ) = λ j (L 0γ ) + δ Rγ ( j, j) + O(| j|−ρ ) as | j| → ∞.

(1.7)

1.3 Plan of the Paper In Sect. 2, by splitting the canonical basis B of HRabi into four parts Bν,μ , ν = ±1, μ = 0, 1 we introduce a natural orthogonal splitting HRabi = ⊕ν,μ Hν,μ = H1,0 ⊕ H−1,0 ⊕ H1,1 ⊕ H−1,1

(1.8)

which is invariant under the two-photon Rabi Hamiltonian. Moreover, using Bν,μ we show that the restriction of the two-photon Rabi Hamiltonian to Hν,μ is unitary equivalent to some operator defined in 2 (N) by a Jacobi matrix Jν,μ . Thus the initial problem is reduced to the asymptotic analysis of large eigenvalues of these Jacobi matrices Jν,μ . In Sect. 3 we reduce this analysis to that of a simpler class of Jacobi matrices ⎞ δ gγ 0 0 ... ⎜gγ 1 − δ g(1 + γ ) 0 . . .⎟ ⎟ ⎜ ⎟ ⎜ Jγδ := ⎜ 0 g(1 + γ ) 2 + δ g(2 + γ ) . . .⎟ ⎜0 0 g(2 + γ ) 3 − δ . . .⎟ ⎠ ⎝ .. .. .. .. .. . . . . . ⎛

(1.9)

Lemma 3.2 shows that suitable estimates for large eigenvalues of the associated operator Jˆγδ imply the assertion of Theorem 1.1. In Sect. 4 we replace the operator Jˆγδ acting in 2 (N) with its natural extension J˜γδ acting in 2 (Z). It appears that the spectrum of J˜γ0 was explicitly described in [9, 19]. Following [9] we use the Fourier transform F0 : L2 (−π, π ) → 2 (Z) defined by (4.1) and consider the operator L δγ := F0−1 J˜γδ F0

(1.10)

acting in L2 (−π, π ). In the case δ = 0 this operator, L 0γ , is a first order differential operator in L2 (−π, π ) with periodic boundary conditions (or a differential operator with smooth coefficients on the circle). Our analysis of L 0γ given in Sect. 4 is a modification of the approach used in [9]. The main difference is that we use a unitary equivalence in order to obtain an explicit expression for an orthonormal basis of eigenvectors of L 0γ .

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In Sect. 5 we use the form of eigenvectors of L 0γ to estimate the difference between the eigenvalues of L δγ and L 0γ thanks to an idea coming from [27]. Since the operator L δγ is unitary equivalent to J˜γδ , we obtain the asymptotic behavior of the eigenvalues of J˜γδ . In Sect. 6 we complete the proof of Theorem 1.1, using that Jˆγδ is the restriction of J˜γδ to 2 (N).

1.4 Notations and Conventions 2 (Z) is the Hilbert space of square-summable complex sequences x˜ : Z → C indexed by the set of integers Z and equipped with the scalar product

x, ˜ y˜  :=



x(k) ˜ y˜ (k), x, ˜ y˜ ∈ 2 (Z)

k∈Z

 and the associated norm x ˜ := x, ˜ x. ˜ We denote by {˜em }m∈Z the canonical basis of 2 (Z), that is e˜ m (m) = 1 and e˜ m (k) = 0 when k = m. The space 2 (N) of square-summable complex sequences indexed by the set of nonnegative integers N is identified with the closed subspace of 2 (Z) generated by {˜em }m∈N . Thus 2 (N) is a Hilbert space with the scalar product

x, y =



x(k)y(k), x, y ∈ 2 (N)

k∈N

and the canonical basis {˜em }m∈N which we also denote by {em }m∈N . If L is an operator acting in the Hilbert space H, let σ (L) denote its spectrum. We say that L has discrete spectrum if and only if every point λ ∈ σ (L) is an isolated eigenvalue of finite multiplicity. Assume that L has discrete spectrum. We say that {λ j (L)} j∈J is a complete eigenvalue sequence of L if and only if any λ ∈ σ (L) is some λ j (L) and has multiplicity m λ := dim ker(L − λI ) = card{ j ∈ J : λ j (L) = λ}. Assume that a self-adjoint operator L has discrete spectrum such that inf σ (L) = −∞ and sup σ (L) = +∞. Then we can choose a complete eigenvalue sequence {λ j (L)} j∈Z such that {λ j (L)} j∈Z is a nondecreasing sequence such that λ j (L) → ±∞ as j → ±∞. Moreover, there exists an orthogonal basis { f j } j∈Z such that L f j = λ j (L) f j holds for all j ∈ Z and any other nondecreasing complete eigenvalue sequence is of the form {λ j+κ (L)} j∈Z with a shift κ ∈ Z. Assume now that L is a self-adjoint operator with discrete spectrum such that inf σ (L) > −∞ and sup σ (L) = +∞. Then there exists a unique nondecreasing complete eigenvalue sequence {λn (L)}n∈N which we will call the eigenvalue sequence of L. Moreover, λn (L) → +∞ as n → ∞ and there exists an orthogonal basis { f n }n∈N such that L f n = λn (L) f n holds for all n ∈ N.

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

95

2 Formulation in Terms of Jacobi Matrices 2.1 Jacobi Matrices Let J be a real symmetric tridiagonal matrix ⎛ d(0) ⎜b(0) ⎜ ⎜ J =⎜ 0 ⎜ 0 ⎝ .. .

b(0) d(1) b(1) 0 .. .

0 0 b(1) 0 d(2) b(2) b(2) d(3) .. .. . .

⎞ ... . . .⎟ ⎟ . . .⎟ ⎟ . . .⎟ ⎠ .. .

(2.1)

∞ whose entries {d(k)}∞ k=0 and {b(k)}k=0 satisfy the two conditions

d(k) −−−→ +∞,

(2.2a)

k→∞

lim sup k→∞

|b(k)| + |b(k − 1)| < 1. d(k)

(2.2b)

Let D = diag{d(k)}∞ k=0 and B be the diagonal and off-diagonal parts of J : J = D + B.

(2.3)

Then it is known (see [7]) that under conditions (2.2) there exist constants C0 > 0 and 0 < c0 < 1 such that, in 2 (N), ± x, Bx ≤ c0 x, Dx + C0 x 2

(2.4)

and the Jacobi matrix (2.1) defines a self-adjoint operator Jˆ characterized by Jˆek = d(k)ek + b(k)ek+1 + b(k − 1)ek−1 , k = 0, 1, . . .

(2.5)

where Jˆe0 = d(0)e0 + b(0)e1 for k = 0.

2.2 Jacobi Matrices and the Two-Photon Hamiltonian Henceforth, ν = ±1 and μ = 0, 1. We reindex the orthogonal basis B = {eν ⊗ en } where ν = ±1 and n = 0, 1, 2, . . . as follows: emν,μ := eν(−1)m ⊗ e2m+μ . ν,μ

Let Hν,μ denote the closed subspace of HRabi generated by Bν,μ := {em }m≥0 . An easy calculation shows that

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   eν(−1)m ⊗ e2m+μ Hˆ Rabi emν,μ = 2m + μ + ν(−1)m 2  + g (2m + μ)(2m + μ − 1)e−ν(−1)m ⊗ e2m+μ−2  + g (2m + μ + 1)(2m + μ + 2)e−ν(−1)m ⊗ e2m+μ+2 , ν,μ

(2.6)

ν,μ

where eν(−1)m ⊗ e2m+μ = em and e−ν(−1)m ⊗ e2m+μ±2 = em±1 . Therefore, each sub space Hν,μ is Hˆ Rabi -invariant and the partition B = ν,μ Bν,μ defines the Hˆ Rabi invariant splitting (1.8): HRabi = H1,0 ⊕ H1,1 ⊕ H−1,0 ⊕ H−1,1 . Let Hˆ ν,μ denote the restriction of Hˆ Rabi to Hν,μ . Thus the spectrum of Hˆ Rabi splits into four parts: σ ( Hˆ Rabi ) = σ ( Hˆ 1,0 ) ∪ σ ( Hˆ 1,1 ) ∪ σ ( Hˆ −1,0 ) ∪ σ ( Hˆ −1,1 ). Moreover, each Hν,μ is isometric to 2 (N) through the surjective isometry ν,μ Uν,μ : Hν,μ → 2 (N) defined by Uν,μ em = em . This isometry transforms Hˆ ν,μ into −1 the operator Jˆν,μ := Uν,μ Hˆ ν,μ Uν,μ in 2 (N), so that both operators have the same spectrum: σ ( Hˆ ν,μ ) = σ ( Jˆν,μ ).

−1 is associated with a Jacobi matrix Lemma 2.1 The operator Jˆν,μ := Uν,μ Hˆ ν,μ Uν,μ whose diagonal and off-diagonal entries are given by d(m) = dν,μ (m) and b(m) = bμ (m), m ∈ N, with

 dν,μ (m) := 2m + μ + ν(−1)m , 2  bμ (m) := g (2m + μ + 1)(2m + μ + 2). Proof Follows from (2.6) given (2.5).

(2.7a) (2.7b) 

Lemma 2.2 Assume that 0 < g < 1/2. Let Jˆν,μ , ν = ±1, μ = 0, 1 be the operator in 2 (N) defined in Lemma 2.1. Then, (i) Jˆν,μ is self-adjoint, bounded from below, and has discrete spectrum. (ii) Moreover, its eigenvalue sequence {λn ( Jˆν,μ )}∞ n=0 satisfies cn − C ≤ λn ( Jˆν,μ ) ≤ C(n + 1), where C and c are some positive constants.

(2.8)

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

97

Proof (i) By the previous lemma it suffices to show that both conditions (2.2) are fulfilled when the diagonal and off-diagonal entries are given by (2.7). By (2.7a) we clearly have d(m) → +∞ if m → ∞. Moreover, for d(m) and b(m) given by (2.7): lim

m→∞

|b(m)| + |b(m − 1)| = 2g < 1. d(m)

(ii) The estimate (2.8) follows from (2.3), (2.4), and the min-max principle.



Proposition 2.3 Assume that 0 < g < 21 . Let ν = ±1 and μ = 0, 1 be given, and let Jˆν,μ be the self-adjoint operator defined in Lemma 2.1. If {λk ( Jˆν,μ )}∞ k=0 is its eigenvalue sequence we have the large n asymptotic estimate  1  1 λn ( Jˆν,μ ) = 2n + μ + 1 − 4g 2 − + O(n −1/3 ). 2 2

(2.9)

Proof Lemma 3.2 will reduce the proof to that of Proposition 3.1, and the proof of Proposition will be completed in Sect. 6.1.  The next lemma shows that Proposition 2.3 implies Theorem 1.1. Lemma 2.4 Estimates (2.9) for ν = ±1 and μ = 0, 1 imply estimate (1.4), that is, Theorem 1.1. ± Proof We deduce (1.4) from (2.9) setting E 2k+μ := λk ( Jˆ±1,μ ) for μ = 0, 1.



3 A New Family of Jacobi Matrices In this section we introduce a new family of Jacobi matrices and we will prove Proposition 2.3, hence Theorem 1.1 in the degenerated case  = 0.

3.1 Jacobi Matrices Jγδ For γ , δ ∈ R, let Jˆγδ be the self-adjoint operator in 2 (N) defined by the Jacobi matrix (1.9). Its diagonal and off-diagonal entries d(m) = dδ0 (m) and b(m) = bγ0 (m) are defined by (m ∈ N)

Thus, Jˆγδ is characterized by

dδ0 (m) := m + (−1)m δ,

(3.1a)

bγ0 (m)

(3.1b)

:= g(m + γ ).

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Jˆγδ em = dδ0 (m)em + bγ0 (m)em+1 + bγ0 (m − 1)em−1 .

Proposition 3.1 Assume 0 < g < 1/2. The eigenvalue sequence {λn ( Jˆγδ )}∞ n=0 satisfies the large n estimate   1  ( 1 − 4g 2 − 1) + O(n −1/3 ). λn ( Jˆγδ ) = n 1 − 4g 2 + γ − 2 Proof The proof is completed in Sect. 6.1.

(3.2) 

Lemma 3.2 Let ν = ±1, μ = 0, 1 be given. Estimate (3.2) for λn ( Jˆγδ ) with γ = μ + 43 and δ = ν/4 implies estimate (2.9) for λn ( Jˆν,μ ). 2 The proof of Lemma 3.2 is given in the next section and uses the following consequence of [22, Theorem 1.1]: Theorem 3.3 (Rozenbljum [22]) Let L 0 be a self-adjoint operator which is bounded from below and has discrete spectrum. Let {λn (L 0 )}∞ n=0 be its eigenvalue sequence. Assume that 0 ∈ / σ (L 0 ). Let also R be a bounded symmetric operator such that R|L 0 |ρ is bounded for a certain ρ > 0. Then L := L 0 + R is a self-adjoint operator which is bounded from below and has discrete spectrum. Moreover, its eigenvalue sequence {λn (L)}∞ n=0 satisfies λn (L) = λn (L 0 ) + O(|λn (L 0 )|−ρ ) as n → ∞.

3.2 Proof of Lemma 3.2 We assume we know estimate (3.2) for λn ( Jˆγδ ). The diagonal entries dν,μ (m) and dδ0 (m) of Jν,μ and Jγδ are given by (2.7a) and (3.1a), respectively. They are related by 1 0 (dν,μ (m) − μ) = dν/4 (m). (3.3) 2 The off-diagonal entries bμ (m) and bγ0 (m) are given by (2.7b) and (3.1b), respectively, and they are related by 1 bμ (m) = bγ0 (μ) (m) + rμ (m), 2

(3.4)

where rμ (m) = O( m1 ) as m → ∞ and γ (μ) :=

μ 3 + . 2 4

(3.5)

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

99

To prove (3.4) it suffices to observe that  1 1 μ 1 μ  m+ + = g(m + γ (μ)) + O bμ (m) = g m + 1 + 2 2 2 2 m holds with γ (μ) given by (3.5). Therefore, using (3.3) and (3.4) we can write  1ˆ ν/4 Jν,μ − μ = Jˆγ (μ) + Rˆ μ , 2 where Rˆ μ em = rμ (m)em+1 + rμ (m − 1)em−1 . The operator Rˆ μ Jˆν,μ is clearly bounded due to rμ (m) = O( m1 ). Therefore, using the estimate (2.8) and Theorem 3.3 with ρ = 1, we find  1 1 ν/4 λn ( Jˆν,μ ) − μ = λn ( Jˆγ (μ) ) + O . 2 n

(3.6)

Combining (3.6) with (3.2) we obtain   λn ( Jˆν,μ ) − μ = 2n 1 − 4g 2 + (2γ (μ) − 1)( 1 − 4g 2 − 1) + O(n −1/3 ). (3.7) However the equality 2γ (μ) − 1 = μ +

1 2

allows us to write (3.7) in the form

 1  1 1 − 4g 2 − μ − + O(n −1/3 ), λn ( Jˆν,μ ) − μ = 2n + μ + 2 2 which is exactly the estimate (2.9).

3.3 Proof of Proposition 2.3 in the Case  = 0 It’s about proving the estimate (2.9) in that case. Due to Lemma 3.2 for δ = ν/4 = 0 it suffices to prove the large n asymptotics (3.2) for δ = 0. We will actually prove that   1  ( 1 − 4g 2 − 1) + O(n −N ) λn ( Jˆγ0 ) = n 1 − 4g 2 + γ − 2

(3.8)

holds for any integer N ≥ 1. Let Jˆγδ,1 be the self-adjoint operator in 2 (N) defined by Jˆγδ,1 em = dδ1 (m)em + bγ1 (m)em+1 + bγ1 (m − 1)em−1 , where dδ1 (m) = dδ0 (m) for m ≥ 1 and bγ1 (m) = bγ0 (m) for m ≥ 1. Next we set bγ1 (0) := 0, which implies Jˆγδ,1 e0 = dδ1 (0)e0 . Then we take dδ1 (0)  0 to ensure that

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λ0 ( Jˆγ0,1 ) = dδ1 (0). Since Rˆ := Jˆγδ − Jˆγδ,1 is defined by a 2 × 2 block, Theorem 3.3 ensures that λn ( Jˆγ0 ) = λn ( Jˆγ0,1 ) + O(n −N ) holds for any integer N ≥ 1. Let 2 (N∗ ) be identified with the closed subspace of ˆ0 2 (N) generated by {em }∞ m=1 . This subspace is invariant under Jγ ,1 and we denote by 0  −1 2 ∗ Jˆγ the restriction of g Jˆγ ,1 to (N ). It turns out that this operator Jˆγ was investigated by Janas and Malejki in [15]. The result of [15, Theorem 3.4], as completed in [15, Sect. 4], and applied for δ = g −1 and c = γ , says that the eigenvalue sequence {λn ( Jˆγ )}∞ n=0 satisfies    1  −2 λn ( Jˆγ ) = n g −2 − 4 + γ + g − 4 − g −1 + g −1 + O(n −N ) 2 for any fixed integer N ≥ 1. Thus for n ≥ 1 one has     1  λn ( Jˆγ0,1 ) = gλn−1 ( Jˆγ ) = (n − 1) 1 − 4g 2 + 1 + γ + 1 − 4g 2 − 1 + O(n −N ), 2

completing the proof of (3.8).

4 Extension to 2 (Z) For γ , δ ∈ R, we consider the self-adjoint operator J˜γδ in 2 (Z) characterized by J˜γδ e˜ m = (m + (−1)m δ)˜em + g(m + γ )˜em+1 + g(m − 1 + γ )˜em−1 , m ∈ Z, where {˜em }m∈Z denotes the canonical basis in 2 (Z). The following theorem is a special case of the result of [9] (see also [19]). Theorem 4.1 (see [9, 19]) The complete eigenvalue sequence {λ j ( J˜γ0 )} j∈Z of the operator J˜γ0 can be ordered so that  1  1 1 − 4g 2 − γ + . λ j ( J˜γ0 ) = j + γ − 2 2 In this section we present a modification of the proof given in [9]. Our purpose is to give an explicit expression of a basis of eigenvectors of J˜γ0 .

4.1 Preliminaries Notations (a) We denote by L2 (−π, π ) the Hilbert space of Lebesgue square integrable functions [−π, π ] → C equipped with the scalar product

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

f, g :=

1 2π



π

101

f (ξ ) g(ξ ) dξ

−π

√ and with the norm f := f, f . (b) For f ∈ L2 (−π, π ) we denote by f  its mean value

f  :=

1 2π



π

−π

f (η)dη.

(c) We denote by F0 : L2 (−π, π ) → 2 (Z) the Fourier transform defined by (F0 f )( j) =

1 2π



π −π

f (ξ )e−i jξ dξ.

(4.1)

2 (d) We denote by C∞ per ([−π, π ]) the set of functions u ∈ L (−π, π ) whose 2π periodic extension is a smooth function R → C. (e) We denote by { f j0 } j∈Z the orthonormal basis given by f j0 (η) := ei jη . (f) We denote by L 0 the self-adjoint operator in L2 (−π, π ) defined by

L 0 f j0 = j f j0 for j ∈ Z. We denote by D(L 0 ) the domain of L 0 and observe that L 0 is the closure of the d defined on C∞ operator Dη := 1i dη per ([−π, π ]). Proposition 4.2 Let q ∈ C∞ per ([−π, π ]) be a real valued function and let L q denote the self-adjoint operator defined by L q f := L 0 f + q f,

f ∈ D(L 0 ).

(4.2)

Then L q is unitary equivalent to L 0 + q. Proof We define a unitary operator Uq in L2 (−π, π ) by ˜ f (η), (Uq f )(η) = eiq(η)−i qη

where q˜ is a primitive of q. Since q˜ − q belongs to C∞ per ([−π, π ]), the subspace ±1 C∞ ([−π, π ]) is invariant under U . Moreover, for f ∈ C∞ per q per ([−π, π ]), Uq (Dη + q)Uq−1 f = (Dη + q) f. To complete the proof it remains to observe that L q is the closure of the operator  f → Dη f + q f defined on C∞ per ([−π, π ]).

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q

Corollary 4.3 Let { f j } j∈Z be the orthonormal basis of L2 (−π, π ) given by f j := Uq−1 f j0 . Then, for every j ∈ Z we have ˜ f j (η) = ei( j+ q)η−iq(η) q

and

q

q

L q f j = ( j + q) f j .

4.2 A Unitary Change of Variable Assume that p ∈ C∞ per ([−π, π ]). Further on p(ξ )Dξ + hc denotes a symmetric operator in L2 (−π, π ) defined on C∞ per ([−π, π ]) by ( p(ξ )Dξ + hc) f := p Dξ f + Dξ ( p¯ f ).

(4.3)

The right-hand side of (4.3) can be expressed in the form 2(Re p)Dξ f − (Im p  + i Re p  ) f.

(4.4)

Assumptions Further on we assume that  : [−π, π ] → [−π, π ] is a bijection  which has a derivative  ∈ C∞ per ([−π, π ]) with  ≥ c0 > 0 for some positive constant c0 . These assumptions ensure the property ±1 ∈ C∞ f ∈ C∞ per ([−π, π ]) =⇒ f ◦  per ([−π, π ]).

Moreover, we define a unitary operator U˜  in L2 (−π, π ) by (U˜  f )(ξ ) =  (ξ )1/2 f ((ξ )), ˜ ±1 and observe that the subspace C∞ per ([−π, π ]) is invariant under U . Proposition 4.4 Assume that q ∈ C∞ per ([−π, π ]) and L q is given by (4.2). Then for ([−π, π ]) one has the relation any h ∈ C∞ per 1   (ξ )−1 Dξ + hc h + (q ◦ )h. U˜  L q U˜ −1 h = 2

(4.5)

Proof We assume that η → f (η) belongs to C∞ per ([−π, π ]) and write η = (ξ ). Then U˜  (q f )(ξ ) =  (ξ )1/2 q((ξ )) f ((ξ )) = (q ◦ )(ξ )(U˜  f )(ξ )

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

103

and U˜  (Dη f )(ξ ) =  (ξ )1/2 (Dη f )((ξ )) =  (ξ )−1/2 Dξ ( f ((ξ ))).

(4.6)

Let us consider the quantity  (ξ )−1 Dξ (U˜  f )(ξ ) :=  (ξ )−1 Dξ ( (ξ )1/2 f ((ξ ))).

(4.7)

However, the right hand side of (4.7) can be written in the form  (ξ )−1/2 Dξ ( f ((ξ ))) +  (ξ )−1

(−i)  −1/2   (ξ )  (ξ ) f ((ξ )) 2

(4.8)

and the first term of (4.8) equals U˜  (Dη f )(ξ ) due to (4.6). Thus the quantity (4.7) can be written in the form i  (ξ )−1 Dξ (U˜  f )(ξ ) = U˜  (Dη f )(ξ ) −  (ξ )−2  (ξ )(U˜  f )(ξ ). 2

(4.9)

Then using  (ξ )−2  (ξ ) = −(1/ ) (ξ ) we find that (4.9) gives i U˜  (Dη f ) = (1/ )Dξ (U˜  f ) − (1/ ) (U˜  f ). 2 Using f = U˜ −1 h and taking into account (4.4) with p = 1/(2 ) we obtain 1  (1/ ) Dξ + hc h U˜  Dη U˜ −1 h = 2 for any h ∈ C∞ per ([−π, π ]).



Corollary 4.5 The operator L q := U˜  L q U˜ −1 is essentially self-adjoint on ∞  C∞ per ([−π, π ]). If h ∈ Cper ([−π, π ]) then L q h is given by (4.5) and one has L q f q,j = ( j + q) f q,j , where { f q,j } j∈Z is the orthonormal basis of L2 (−π, π ) given by f q,j := U˜  f j , i.e. q

˜ )) . f q,j (ξ ) =  (ξ )1/2 ei( j+ q)(ξ )−iq((ξ

(4.10)

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4.3 Proof of Theorem 4.1 We consider the operator L 0γ := F0−1 J˜γ0 F0 introduced in (1.10). Following [9] we find that L 0γ is given by the formula L 0γ =

 1 2

  + geiξ Dξ + hc + γ g(eiξ + e−iξ ).

Using (4.4) we find L 0γ =

1 2

 (1 + 2g cos ξ )Dξ + hc + (2γ − 1)g cos ξ.

Now we define (ξ ) :=



 1−

ξ

4g 2 0

dξ  1 + 2g cos ξ 

(4.11)

and observe that an easy computation (see [9]) gives (±π ) = ±π . Thus  satisfies the assumptions of Sect. 4.2 and one has 1/ (ξ ) = (1 − 4g 2 )−1/2 (1 + 2g cos ξ ). Introducing

 1 2g cos(−1 (η)) qγ (η) := (1 − 4g 2 )−1/2 γ − 2

(4.12)

we find that L 0γ =



1 − 4g 2 U˜  L qγ U˜ −1 .

Combining the last formula with Corollary 4.5 we obtain Proposition 4.6 Let  and qγ be given by (4.11) and (4.12), respectively. For each j ∈ Z we denote f γ , j := f qγ , j , where f q,j is given by (4.10). Then { f γ , j } j∈Z is an orthonormal basis of L2 (−π, π ) and, for every j ∈ Z, L 0γ f γ , j =



1 − 4g 2 ( j + qγ ) f γ , j .

Due to Proposition 4.6, the proof of Theorem 4.1 will be complete if we show that   1 1  1−  .

qγ  = γ − 2 1 − 4g 2

(4.13)

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

105

However using the change of variable η = (ξ ) in

qγ  =

γ−1  2 2π 1 − 4g 2



π −π

2g cos(−1 (η)) dη

(4.14)

we can express the quantity (4.14) in the form γ−1  2 2π 1 − 4g 2



π

γ − 21 2g (ξ ) cos ξ dξ = 2π −π 



π −π

2g cos ξ dξ 1 + 2g cos ξ

(4.15)

and the right hand side of (4.15) equals      π   1 1 1 1 1 . 1− dξ = γ − 1−  γ− 2 2π −π 1 + 2g cos ξ 2 1 − 4g 2

5 Eigenvalue Asymptotics for L δγ In this section we investigate the operator L δγ introduced in (1.10). We observe that we can express L δγ = F0−1 J˜γδ F0 = L 0γ + δTπ ,

(5.1)

where Tπ is defined in L2 (−π, π ) by the formula  (Tπ f )(ξ ) =

f (ξ − π ) if 0 ≤ ξ ≤ π, f (ξ + π ) if − π ≤ ξ ≤ 0.

The operator Tπ is unitary and self-adjoint because it satisfies Tπ2 = I . We will prove: Proposition 5.1 The operator L δγ has discrete spectrum and its complete eigenvalue sequence {λ j (L δγ )} j∈Z can be ordered as a nondecreasing sequence such that  λ j (L δγ ) = ( j + qγ ) 1 − 4g 2 + O( j −1/3 ) as | j| → ∞,

(5.2)

where qγ  is given by (4.13). By (5.1) the operators J˜γδ and L δγ are unitary equivalent, hence Proposition 5.1 gives the following corollary: Corollary 5.2 The operator J˜γδ has discrete spectrum and its complete eigenvalue sequence {λ j ( J˜γδ )} j∈Z can be ordered as a nondecreasing sequence such that

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 λ j ( J˜γδ ) = ( j + qγ ) 1 − 4g 2 + O( j −1/3 ) as | j| → ∞,

(5.3)

where qγ  is given by (4.13).

5.1 Auxiliary Results The following result was proved by Janas and Naboko (see [16, Lemma 2.1]): Theorem 5.3 (Janas and Naboko [16]) We fix 0 > 0 and assume that (μn )∞ n=0 is a real sequence such that μn → +∞, |μn | ≤ |μn+1 | and |μn − μm | ≥ 0 if μn = μm (n, m ∈ N). Let {en0 }∞ n=0 be an orthonormal basis of the Hilbert space H and denote by L 0 the self-adjoint operator satisfying L 0 en0 = μn en0 for any n ∈ N. If R is compact in H, then the operator L := L 0 + R has discrete spectrum and its eigenvalue sequence {λn (L)}∞ n=0 can be ordered so that one has the estimate |λn (L) − μn | ≤ C R ∗ en0 , where C > 0 is a large enough constant and R ∗ denotes the adjoint of R. Theorem 5.3 was used by E. A. Yanovich to prove the following result ([27, Theorem 2.2]): Theorem 5.4 (Yanovich [27]) Let {en0 }∞ n=0 be an orthonormal basis of the Hilbert space H and let L 0 be the self-adjoint operator satisfying L 0 en0 = nen0 for n ∈ N. If R is a symmetric bounded operator in H satisfying 0 , Ren0  = 0 lim ek+n

n→∞

for every k ∈ Z, then the operator L := L 0 + R has discrete spectrum and the eigenvalue sequence {λn (L)}∞ n=0 satisfies the estimate |λn (L) − n − en0 , Ren0 | ≤ Csn1/2 for all n ∈ N, where C > 0 is some constant and sn :=

 | e0 , Re0 |2 k

n

(k − n)2

k=n

.

We will also use the following version of van der Corput Lemma. Lemma 5.5 For h,  ∈ C∞ per ([−π, π ]) and τ ∈ R denote Jh (τ ) :=

1 2



π −π

eiτ (ξ ) h(ξ ) dξ.

(5.4)

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

107

Assume that  is real-valued and satisfies the condition   (ξ ) = 0 =⇒   (ξ ) = 0.

(5.5)

Then there is a constant C0 such that for all τ ∈ R∗ one has    J (τ ) ≤ C0 |τ |−1/2



h

 |h(0)| +

π

−π





|h (t)| dt .

5.2 Proof of Proposition 5.1 We begin by Lemma 5.6 Let { f γ , j } j∈Z be the orthonormal basis of L2 (−π, π ) defined in Proposition 4.6 and denote ( j, k ∈ Z) Rγ ( j, k) := f γ , j , Tπ f γ ,k .

(5.6)

lim Rγ (k + j, j) = 0,

(5.7)

If for every k ∈ Z one has | j|→∞

then L δγ has discrete spectrum and its complete eigenvalue sequence {λ j (L δγ )} j∈Z can be ordered in nondecreasing order so that we have the estimate |λ j (L δγ ) − λ j (L 0γ ) − δ Rγ ( j, j)| ≤ Cs j

1/2

for all j ∈ Z,

(5.8)

where C > 0 is some constant and s j :=

 |Rγ ( j, k)|2 k= j

(k − j)2

.

Proof Define the operators L˜ 0 := (1 − 4g 2 )−1/2 L 0γ − qγ , R˜ := (1 − 4g 2 )−1/2 δTπ + qγ . Then L˜ 0 f γ , j = j f γ , j for all j ∈ Z and (5.8) becomes 1/2 |λ j ( L˜ 0 ) − j − f γ , j , R˜ f γ , j | ≤ C1 s j .

(5.9)

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This estimate is similar to the estimate (5.4), except that indices belong to Z instead of N. However we obtain the result in the case of Z by reasoning as in [27] and replacing Theorem 5.3 with its generalized version proved by Malejki [18]. In fact the assumption μn → +∞ from Theorem 5.3 can be replaced with |μn | → ∞. Therefore, choosing a bijection Z → N, j → n j , one can use the sequence {μn j } j∈Z and the basis {en j } j∈Z .  Since Proposition 5.1 follows from Lemma 5.6, it remains to show (5.7) and s j = O(| j|−2/3 ) as j → ∞,

(5.10)

where s j is given by (5.9). However, (5.7) follows immediately from the following Lemma 5.7 Assume that Rγ ( j, k) is given by (5.6) and s j by (5.9). Then there exists a positive constant C such that |Rγ ( j, j + k)| ≤ C(1 + |k|)(1 + | j|)−1/2 holds for all j, k ∈ Z. Proof In order to simplify the expression of 1 2π

Rγ ( j, k) =



π −π

f γ ,k (ξ ) (Tπ f¯γ , j )(ξ ) dξ

we denote pγ (ξ ) := ei qγ (ξ )−iqγ ((ξ )) (1 + 2g cos ξ )−1/2 (1 − 4g 2 )1/4 where qγ is a primitive of qγ , and qγ (0) = 0. Then, f γ , j (ξ ) = ei j(ξ ) pγ (ξ ) and we can express Rγ ( j, k) =

1 2π



π

−π

ei(k− j Tπ )(ξ ) pγ (ξ ) (Tπ p¯ γ )(ξ ) dξ.

In order to apply van der Corput Lemma 5.5 we observe that R( j, j + k) = Jhγ ,k ( j) holds with (ξ ) = (ξ ) − (Tπ )(ξ ), h γ ,k (ξ ) = eik(ξ ) pγ (ξ ) (Tπ p¯ γ )(ξ ). It remains to check the condition (5.5). For this purpose we observe that

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109

 1 − 4g 2 1 − 4g 2 − ,  (ξ ) = 1 + 2g cos ξ 1 − 2g cos ξ 



hence   (ξ ) = 0 ⇐⇒ cos ξ = 0, and  

cos ξ = 0 =⇒  (ξ ) =

    2g 1 − 4g 2 2g 1 − 4g 2 + sin ξ = ±4g 1 − 4g 2 = 0. 2 2 (1 + 2g cos ξ ) (1 − 2g cos ξ )

 In order to prove (5.10) we first observe that 

|Rγ ( j, j + k)|2 =

k∈Z



| Tπ f γ , j , f γ ,m |2 = ||Tπ f γ , j ||2 = 1

(5.11)

m∈Z

holds due to Parseval’s equality. Next we observe that using (5.11) we can estimate 

|k|−2 |Rγ ( j, j + k)|2 ≤ | j|−2/3

|k|>| j|1/3



|Rγ ( j, j + k)|2 = O(| j|−2/3 )

k∈Z

and using Lemma 5.7 we get the estimate 

|k|−2 |Rγ ( j, j + k)|2 ≤

1≤|k|≤| j|1/3



|k|−2 C(1 + |k|)2 (1 + | j|)−1 = O(| j|−2/3 ).

1≤|k|≤| j|1/2

5.3 About the Asymptotics Conjecture We will show that the improved estimate (1.7) implies the validity of the three-term asymptotics (1.6). For this purpose we write λ j (L δγ ) = λ j ( jˆγδ ) and λ j (L 0γ ) = λ j ( jˆγ0 ) in (1.7). However, in the case of large positive indices, reasoning as in Sect. 6.1, we can replace ( jˆγδ ) and ( jˆγ0 ) with ( jˆγδ ) and ( jˆγ0 ), respectively, and so we get λk ( Jˆγδ ) = λk ( Jˆγ0 ) + δ Rγ (k, k) + O(k −ρ ) as k → ∞.

(5.12)

± = λk ( Jˆ±1,μ ) for μ = 0, 1; 1. In Sect. 3.2 (see (3.6)) we proved that Recall that E 2k+μ ±/4

± = μ + 2λk ( Jˆγ (μ) ) + O(k −1 ) E 2k+μ

holds with γ (μ) =

μ 2

+ 43 . Therefore, combining (5.12) and (5.13) we get

± 0 = E 2k+μ ± E 2k+μ

 Rγ (μ) (k, k) + O(k −ρ ) 2

(5.13)

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and observe that the stationary phase formula allows us to express ω Rγ (k, k) = √ cos(αk + θ ) + O(k −1 ), k where ω, α, and θ are constants depending on γ and g. Indeed, we observe that ξ = ± π2 are non-degenerated critical points of ,  (νπ/2) = ν

π/2 −π/2

   1 − 4g 2 1 + 2g  dξ = 4ν arctan , 1 + 2g cos ξ  1 − 2g

and    (νπ/2) = 4gν 1 − 4g 2 . Moreover q˜γ ((ξ )) = (2γ − 1)g(1 − 4g 2 )−1/2 sin ξ gives θ˜ := q˜γ ((π/2)) − q˜γ ((−π/2)) = 2g(2γ − 1)(1 − 4g 2 )−1/2 and the stationary phase formula gives  ei(k+ qγ )(νπ/2)+iνβ+iνπ/4 + O(k −1 ) √  (νπ/2)| 2kπ | ν=±1   1 2 cos (k + qγ )(π/2) + θ˜ + π/4 + O(k −1 ) =√ 2(2πg)1/2 (1 − 4g 2 )1/4 k

(1 − 4g 2 )−1/2 Rγ (k, k) =

  with qγ  = γ − 21 (1 − (1 − 4g 2 )−1/2 ) according to (4.13).

6 End of the Proof of Theorem 1.1 6.1 Proof of Proposition 3.1 Due to Lemma 3.2, to prove Theorem 1.1 it suffices to prove Proposition 3.1, that is, estimate (3.2) for λn ( Jˆγδ ). We will use the following consequence of [22, Theorem 1.2]: Theorem 6.1 (Rozenbljum [22]) Let L 0 be a self-adjoint operator with discrete spectrum. We assume that 0 ∈ / σ (L 0 ), inf σ (L 0 ) = −∞, and sup σ (L 0 ) = ∞. We denote by {λ j (L 0 )} j∈Z a nondecreasing complete eigenvalue sequence of L 0 . Let R be a bounded symmetric operator and denote L := L 0 + R.

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

111

If R|L 0 |ρ is bounded for a certain ρ > 0, then L has discrete spectrum and one can order a complete eigenvalue sequence {λ j (L)} j∈Z so that λ j (L) = λ j (L 0 ) + O(| j|−ρ ) as | j| → ∞. Let δ0 > 0 be an arbitrary fixed positive real number. Further on we assume that δ ∈ [−δ0 , δ0 ]. By the min-max principle, inf σ ( Jˆγδ ) ≥ −|δ| + inf(Jγ0 ). We choose ρ0 > 0 large enough so that −ρ0 < inf σ ( Jˆγδ ) for δ ∈ [−δ0 , δ0 ]. Further on 2 (Z \ N) denotes the orthogonal complement of 2 (N) in 2 (Z). Then for n 0 ∈ N we denote by Jˆγδ,n 0 the self-adjoint operator in 2 (Z \ N) satisfying Jˆγδ,n 0 e˜ m = dδn 0 (m)˜em + bγn 0 (m)˜em+1 + bγn 0 (m − 1)˜em−1 for m ∈ Z, m < 0, with  dδn 0 (m)

:= 

bγn 0 (m)

:=

m + (−1)m δ if m < −n 0 , if − n 0 ≤ m < 0, −ρ0 (m + γ )g if m < −n 0 , 0 if − n 0 ≤ m < 0.

Then it is clear that we can fix n 0 large enough to ensure sup σ ( Jˆγδ,n 0 ) = −ρ0 for δ ∈ [−δ0 , δ0 ]. Let J˜γδ,n 0 be the self-adjoint operator in 2 (Z) defined as the direct sum J˜γδ,n 0 := Jˆγδ,n 0 ⊕ Jˆγδ and let {λ j ( J˜γδ,n 0 )} j∈Z be a nondecreasing complete eigenvalue sequence of J˜γδ,n 0 . Since all eigenvalues of Jˆγδ,n 0 are smaller than inf σ ( Jˆγδ ) we can order the sequence {λ j ( J˜γδ,n 0 )} j∈Z so that λn ( J˜γδ,n 0 ) = λn ( Jˆγδ ) for n ∈ N. (6.1) Next we observe that R := J˜γδ − J˜γδ,n 0 is given by a finite block matrix. Thus, Theorem 6.1 can be applied for any ρ > 0. Therefore there exists κγ (δ) ∈ Z such that λ j ( J˜γδ,n 0 ) = λ j+κγ (δ) ( J˜γδ ) + O(| j|−ρ ) as | j| → ∞.

(6.2)

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However, using (6.1) and (5.3) in (6.2) we obtain  λn ( Jˆγδ ) = (n + κγ (δ) + qγ ) 1 − 4g 2 + O(n −1/3 ) as n → ∞.

(6.3)

Therefore we can express   κγ (δ) = lim (1 − 4g 2 )−1/2 λn ( Jˆγδ ) − n − qγ  n→∞

and we claim that the function δ → κγ (δ) is continuous. Indeed, |λn ( Jˆγδ1 ) − λn ( Jˆγδ2 )| ≤ |δ1 − δ2 | holds due to the min-max principle. Using the fact that the function κγ : [−δ0 , δ0 ] → Z is continuous, we deduce that the function κγ is constant. Since we have proved that in the case δ = 0 the formula (6.3) holds with κγ (0) = 0, we conclude that (6.3) holds with κγ (δ) = 0 for every δ ∈ R.

6.2 Proof of (1.5) In the case  = 0 the two-photon Rabi Hamiltonian Hˆ Rabi is unitary equivalent to    = aˆ † aˆ + gσz ⊗ aˆ 2 + (aˆ † )2 . Hˆ Rabi ˆ The decomposition BRabi = B1 ∪ B−1 with Bν := {eν ⊗ em }∞ m=0 shows that HRabi   is unitary equivalent to the direct sum Hˆ g ⊕ Hˆ −g , where    := aˆ † aˆ ± g aˆ 2 + (aˆ † )2 . Hˆ ±g  is self-adjoint in Hfield = 2 (N) and its spectrum is discrete If 0 < g < 21 , then Hˆ ±g  ˆ )}∞ and bounded from below. Let {λn ( Hˆ ±g n=0 be the eigenvalue sequence of H±g . It remains to prove

 1  1  )= n+ 1 − 4g 2 − . λn ( Hˆ ±g 2 2 The proof is based on the following result.

(6.4)

 Lemma 6.2 (Emary and Bishop [10]) Denote α := (1 − 1 − 4g 2 )/(±2g) and define aˆ † + α aˆ aˆ + α aˆ † zˆ † := √ , zˆ := √ . (6.5) 1 − α2 1 − α2

Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model

113

Then [ˆz , zˆ † ] = 1 and  1  1  . + = 1 − 4g 2 zˆ † zˆ + Hˆ ±g 2 2 To complete the proof of (6.4) it suffices to check that 1 λn ( Hˆ ±g ) = n + 2 where

(6.6)

 1  . Hˆ ±g := (1 − 4g 2 )−1/2 Hˆ ±g + 2

The idea is well known, but we present the details below. For simplicity let H := Hˆ ±g . We first observe that H x, x ≥ 21 ||x||2 implies σ (H ) ⊂ [ 21 , ∞). We denote by D(H ) the domain of H equipped with the graph norm. Let z † and z denote bounded operators D(H ) → Hfield acting on {em }m∈N according to (6.5). Since H zˆ † − zˆ † H = zˆ † holds on the subspace generated by † −1 − H −1 z † = H −1 z † H −1 . Assume now that λ ∈ σ (H ). {em }∞ m=0 , we deduce z H −1 Then H λx = x holds for a certain x ∈ D(H ) \ {0} and the equality z † H −1 λx − H −1 z † λx = H −1 z † H −1 λx implies H −1 z † λx = z † x − H −1 z † x. We conclude that z † x ∈ D(H ) and (λ + 1)H −1 z † x = z † x, hence λ + 1 ∈ σ (H ) if z † x = 0. However, if z † x = 0, then for every y in the subspace generated by {em }∞ m=0 one has

z † x, zˆ † y = x, zˆ zˆ † y = x, ( 12 + H )y = ( 21 + H )x, y = 0, and ( 21 + H )x = 0 is in contradiction with x = 0. Thus λ ∈ σ (H ) implies λ + 1 ∈ σ (H ). Assume now that λ > 21 . Using H zˆ − zˆ H = −ˆz in similar computations we find that H x = λx implies (λ − 1)H −1 zx = zx, hence λ − 1 ∈ σ (H ) or zx = 0. However, if zx = 0, then for every y in the subspace generated by {em }∞ m=0 one has zx, zˆ y = x, zˆ † zˆ y = x, (H − 21 )y = (H − 21 )x, y = 0 and (H − 21 )x = 0 implies λ = 21 . Thus λ − 1 ∈ σ (H ) holds if λ > 21 . Assume now that there is k ∈ N such that k + 21 < λ < k + 23 and λ ∈ σ (H ). Then λ − m ∈ σ (H ) holds for m = 1, . . . , k + 1. In particular we obtain λ − k − 1 ∈ σ (H ) and λ − k − 1 < 21 is in contradiction with σ (H ) ⊂ [ 21 , ∞). To complete the proof of (6.6) we take λ ∈ σ (H ) and observe that there exists k ∈ N, such that λ = k + 21 . Then we have λ − m ∈ σ (H ) for m = 1, . . . , k, hence 1 ∈ σ (H ) and consequently 21 + m ∈ σ (H ) for every m ∈ N. It remains to show that 2 each eigenvalue of H has multiplicity 1. However the family g → Hg is analytic in the sense of Kato, hence the multiplicity of each λ ∈ Z + 21 is constant. Thus the multiplicity is 1 as in the case g = 0.

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References 1. A. Boutet de Monvel, S. Naboko, L.O. Silva, The asymptotic behavior of eigenvalues of a modified Jaynes–Cummings model. Asymptot. Anal. 47(3–4), 291–315 (2006) 2. A. Boutet de Monvel, L. Zielinski, Asymptotic behavior of large eigenvalues of a modified Jaynes–Cummings model, in Spectral Theory and Differential Equations, American Mathematical Society Translations: Series 2, vol. 233 (Amer. Math. Soc., Providence, RI, 2014), pp. 77–93 3. A. Boutet de Monvel, L. Zielinski, Asymptotic behavior of large eigenvalues for Jaynes– Cummings type models. J. Spectr. Theory 7(2), 559–631 (2017) 4. A. Boutet de Monvel, L. Zielinski, Oscillatory behavior of large eigenvalues in quantum Rabi models. Int. Math. Res. Not. IMRN 59 pp. (2021) (to appear). https://arXiv.org/abs/1711. 03366v2 5. A. Boutet de Monvel, L. Zielinski, On the spectrum of the quantum Rabi model, in Analysis as a Tool in Mathematical Physics: In Memory of Boris Pavlov. Operator Theory: Advances and Applications, vol. 276 (Birkhäuser/Springer, Cham), pp. 183–193 6. D. Braak, Q.-H. Chen, M.T. Batchelor, E. Solano, Semi-classical and quantum Rabi models: in celebration of 80 years [Preface]. J. Phys. A 49(30), 300301, 4 (2016) 7. P.A. Cojuhari, J. Janas, Discreteness of the spectrum for some unbounded Jacobi matrices. Acta Sci. Math. (Szeged) 73(3–4), 649–667 (2007) 8. L. Duan, Y.-F. Xie, D. Braak, Q.-H. Chen, Two-photon Rabi model: analytic solutions and spectral collapse. J. Phys. A 49(46), 464002, 13 (2016) 9. J. Edward, Spectra of Jacobi matrices, differential equations on the circle, and the su(1, 1) Lie algebra. SIAM J. Math. Anal. 24(3), 824–831 (1993) 10. C. Emary, R.F. Bishop, Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamiltonians. J. Math. Phys. 43(8), 3916–3926 (2002) 11. C. Emary, R.F. Bishop, Exact isolated solutions for the two-photon Rabi Hamiltonian. J. Phys. A 35(39), 8231–8241 (2002) 12. I.D. Feranchuk, L.I. Komarov, A.P. Ulyanenkov, Two-level system in a one-mode quantum field: numerical solution on the basis of the operator method. J. Phys. A: Math. Gen. 29(14), 4035–4047 (1996) 13. C.C. Gerry, Two-photon Jaynes–Cummings model interacting with the squeezed vacuum. Phys. Rev. A (3) 37(7), 2683–2686 (1988) 14. E.K. Irish, Generalized rotating-wave approximation for arbitrarily large coupling. Phys. Rev. Lett. 99(17), 173601 (2007) 15. J. Janas, M. Malejki, Alternative approaches to asymptotic behavior of eigenvalues of some unbounded Jacobi matrices. J. Comput. Appl. Math. 200(1), 342–356 (2007) 16. J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach. SIAM J. Math. Anal. 36(2), 643–658 (electronic) (2004) 17. E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51(1), 89–109 (1963) 18. M. Malejki, Asymptotics of the discrete spectrum for complex Jacobi matrices. Opuscula Math. 34(1), 139–160 (2014) 19. D.R. Masson, J. Repka, Spectral theory of Jacobi matrices in l 2 (Z) and the su(1, 1) Lie algebra. SIAM J. Math. Anal. 22(4), 1131–1146 (1991) 20. I.I. Rabi, On the process of space quantization. Phys. Rev. 49(4), 324 (1936) 21. I.I. Rabi, Space quantization in a gyrating magnetic field. Phys. Rev. 51(8), 652 (1937) 22. Rozenbljum, G.V.: Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle (Russian). Trudy Maskov. Mat. Obshch 36, 59–84 (1978) 23. M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997) 24. È.A. Tur, Jaynes–Cummings model: solution without rotating wave approximation. Opt. Spectrosc. 89(4), 574–588 (2000)

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25. E.A. Tur, Jaynes–Cummings model without rotating wave approximation. Asymptotics of eigenvalues, 12 pp. (2002). https://arXiv.org/abs/math-ph/0211055 26. Q. Xie, H. Zhong, M.T. Batchelor, C. Lee, The quantum Rabi model: solution and dynamics. J. Phys. A 50(11), 113001, 40 (2017) 27. E.A. Yanovich, Asymptotics of eigenvalues of an energy operator in a problem of quantum physics, in Operator Methods in Mathematical Physics. Operator Theory: Advances and Applications, vol. 227 (Birkhäuser/Springer Basel AG, Basel, 2013), pp. 165–177

Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L 2 (Rd ) Jean Michel Combes and Peter D. Hislop

Dedicated to the memory of Erik Baslev

Abstract We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes  ⊂ Rd . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes  ⊂ Rd , for d  1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit. Keywords Spectral averaging · Density of states · Random schrödinger operators

PDH thanks the Centre de Physique Théorique, CNRS, Marseille, France, for support during the time this work was begun. J. M. Combes Centre de Physique Théorique, CNRS Luminy Case 907, Marseille Cedex 9, 13009 Marseille, France e-mail: [email protected] P. D. Hislop (B) Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_6

117

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J. M. Combes and P. D. Hislop

1 Statement of the Problem and Result This note is another presentation of spectral averaging with applications to the study of the local density of states (DOS) for random Schrödiner operators on cubes  ⊂ Rd , for d  1. Spectral averaging is revisited using tools from analytic perturbation theory, an area in which Erik Balslev was an expert. We also prove an upper bound on the trace of spectral projectors using a Poincaré-type inequality for eigenfunctions. We present three applications: (1) we prove the Wegner estimate with an explicit form of the constant, (2) we prove that the local density of states function is Lipschitz continuous in the energy, independent of localization, and (3) we give a simple proof of the Birman-Solomyak Theorem. The spectral averaging result applies to selfadjoint operators of the form Hω = H0 + ωu 2 on a separable Hilbert space where H0 has discrete spectrum. A version of the Birman-Solomyak formula for the spectral shift function is proved in this setting. The random Schrödinger operators that we study in the applications have the form Hω := H0 + Vω ,

(1.1)

on L 2 (Rd ), where H0 is a self-adjoint operator such as the Laplacian H0 = − or a magnetic Schrödinger operator, and the potential Vω is a random, ergodic process described as follows. d Hypothesis 1 [H1]. Single-site potential: Let u 0 (x) ∈ L ∞ 0 (R ; R) be a compactlysupported function satisfying

0  κχ0  u 20  1, for some κ > 0, and where χ0 is the characteristic function on the unit cube C0 := [0, 1]d . Hypothesis 2 [H2]. Random variables: Let ω := {ωk }k∈Zd denote a family of independent, identically distributed (iid) random variables with ω0  0 with common probability density ρ having compact support. For k ∈ Zd , we denote by u k the translate of u 0 by k, that is, u k (x) := u 0 (x − k). Similarly, Ck denotes the translation of C0 by k ∈ Zd and we write χk for the characteristic function on the unit cube Ck . The random potential Vω is defined to be  ωk u k (x). (1.2) Vω (x) := k∈Zd

We work with a restricted version of the random potential in Sect. 4: Hypothesis 3 [H3]. The single-site potential u 0 = κχ0 , for some κ > 0, where χ0 is the characteristic function of the unit cube C0 := [0, 1]d . The single-site probability measure is the uniform measure on the interval [0.1]. We need local operators Hω obtained from Hω by restricting to cubes  L := [−L , L]d , for L ∈ N, and imposing self-adjoint boundary conditions, such as Dirich-

Spectral Averaging and the Local Density of States

119

let, Neumann, or periodic boundary conditions. The unperturbed operator H0 is associated with the nonnegative quadratic form:  f ∈ Q() →



|∇ f |2 ,

(1.3)

for f in the appropriate form domain Q() determined by the boundary conditions. From the construction of Vω , this potential is relatively H0 -bounded with relative H0 bound less than one, so Hω is self-adjoint on the same domain as H0 . Furthermore, Hω has a compact resolvent so the spectrum of Hω is discrete. We write Pω (I ) for the spectral projector for Hω and the interval I ⊂ R. The DOS measure μ is defined as the number of eigenvalues of Hω in the interval I = [I− , I+ ] ⊂ R per unit volume: μ (I ) :=

1 E {Tr Pω (I )}. ||

(1.4)

The density of states measure for the infinite-volume operator Hω is obtained by taking || → ∞. It exists almost surely, see, for example, [7, 8]. The Wegner estimate [4] in this setting is the bound E {Tr Pω (I )} = ||μ (I )  C W (I+ )|||I |. This bound shows that the measure μ is absolutely continuous with respect to Lebesgue measure. The locally bounded density of the DOS measure is denoted n  (E).

1.1 Contents In Sect. 2, we prove an upper bound on the trace of a spectral projector of a local Schrödinger operator. The upper bound is expressed in terms of the matrix elements of the spectral projector with respect to the eigenfunctions of the Neumann Laplacian of the unit cube. The spectral averaging result is derived in Sect. 3 using analytic perturbation theory for one-parameter families of self-adjoint operators. An application is given relating the spectral shift function to the local DOS proving a form of the Birman-Solomyak formula. Finally, in Sect. 4, we prove the local Lipschitz continuity of the DOS for random Schrödinger operators restricted to finite domains.

2 Trace Estimates from the Poincaré Inequality Let h 0,k denote the Neumann Laplacian on the unit cube Ck ⊂ Rd that is the translate of the unit cube C0 := [0, 1]d by k ∈ Zd . The L 2 -eigenfunctions of the self-adjoint

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operator h 0,k are ϕ j,k with eigenvalues E j,k , listed including multiplicity. The set of  eigenvalues is {( dm=1 n 2m )π 2 | (n 1 , . . . , n d ) ∈ {0, 1, 2, . . .}d }. The set of eigenfunctions {ϕ j,k } forms an orthonormal basis of L 2 (Ck ). The spectral representation of h 0,k is ∞  E j,k ϕ j,k , h 0,k = j=0

where ϕ j,k is the projection onto the vector ϕ j,k ∈ L 2 (Ck ). In general, we let ψ denote the projection onto ψ in the appropriate Hilbert space. In the following, we denote by  the cube  := [−L , L]d , with L ∈ N, and we  the integer lattice points in  so that   :=  ∩ Zd . denote by  Theorem 2.1 We assume [H1] and [H2]. Let I = [a, b] with b  (n + 1)2 π 2 , for some n ∈ N ∪ {0}. Then, we have  Tr Pω (I )  1 −

b (n + 1)2 π 2

−1   n ˜ k∈

 ϕ j,k , Pω (I )ϕ j,k 1 −

j=0

j2 (n + 1)2

 . (2.1)

In particular, if n = 0 so b < π , we have 2

  b −1  ϕ0,k , Pω (I )ϕ0,k . Tr Pω (I )  1 − 2 π

(2.2)

˜ k∈

Proof Let {ψ j } be an orthonormal basis of eigenfunction of Hω with corresponding eigenvalues E j . Although the eigenvalues are random variables, the randomness does not play a role in Theorem 2.1. We begin by expanding the trace with respect  to the orthonormal basis of eigenfunctions {ψ j } and use the decomposition χ = k∈  χk of the identity on  giving Tr Pω (I ) =

   { j:E j ∈I } k∈

Trψ j χk =

    { j:E j ∈I } k∈

|ψ j (x)|2 d x.

(2.3)

Ck

Assuming Lemma 2.1, the proof now easily follows by summation over eigenvalues  ˜ and over lattice points  k ∈ . The self-adjoint boundary conditions of Hω guarantee that the sum k∈  Bk (ψ E ) = 0, where the boundary term associated with C k , Bk (ψ E ), is defined in (2.4).  We now turn to Lemma 2.1 and its proof that is based on a Poincaré-type inequality (2.8). Lemma 2.1 We assume [H1] and [H2]. Let ψ E be a normalized eigenfunction of ˜ we Hω with eigenvalue E ∈ [0, (n + 1)2 π 2 ], for some n ∈ N. Then, for all k ∈ , have

Spectral Averaging and the Local Density of States

121

  κωk − E −1 |ψ E (x)| d x  1 + (n + 1)2 π 2 Ck ⎡  n  ×⎣ | ψ E , ϕ j,k |2 1 −



2

j=0

2

j (n + 1)2

⎤

 +

Bk (ψ E ) ⎦ , (2.4) (n + 1)2 π 2

where κ > 0 is the constant in [H1], and the boundary terms Bk (ψ E ) given by  Bk (ψ E ) := satisfy

∂Ck



ψ E (x)νˆ · ∇ψ E (x),

(2.5)

Bk (ψ E ) = 0.

(2.6)

 k∈

Proof 1. Working with k = 0 for simplicity, we define the projector Pn by Pn ψ E := χ0 ψ E −

n  ϕ j,0 , ψ E ϕ j,0 .

(2.7)

j=0

The vector Pn ψ E is the projection of χ0 ψ E onto the spectral subspace of h 0,0 spanned by eigenstates of h 0,0 with energy at least (n + 1)2 π 2 . As a consequence, we have the Poincaré-type inequality for Pn ψ E : 

1 |Pn ψ E | d x  (n + 1)2 π 2 C0

 |∇ Pn ψ E |2 d x.

2

(2.8)

C0

This inequality follows from the expansion of Pn ψ E in the orthonormal basis {ϕ j,0 } of eigenfunctions of h 0,0 and noting that C0 |∇ϕ j,0 |2 = E j,0 . Consequently, we obtain  |∇ Pn ψ E | d x = 2

C0

∞ 

 E j,0 | ψ E , ϕ j,0 |  (n + 1) π 2

|Pn ψ E |2 ,

2 2 C0

j=n+1

from which (2.8) follows. 2. Decomposing χ0 ψ E with respect to the basis ϕ j,0 , and using the Poincaré-type inequality (2.8), we have 

 |ψ E (x)|2 d x = C0

|Pn ψ E (x)|2 d x + C0



1 (n + 1)2 π 2



n 

| ψ E , ϕ j,0 |2

j=0

|∇ Pn ψ E (x)|2 d x + C0

n  j=0

| ψ E , ϕ j,0 |2 , (2.9)

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and 

 |∇ Pn ψ E (x)| d x =

|∇ψ E (x)| d x −

2

C0

2

C0

n 

( jπ)2 | ψ E , ϕ j,0 |2 .

(2.10)

j=0

3. Finally, from the assumptions on H0 , integration by parts results in 

 |∇ψ E (x)| d x = −

(ψ E (x))ψ E (x) d x + B0 (ψ E )   (E − κω0 ) |ψ E (x)|2 d x + B0 (ψ E ),

2

C0

C0

(2.11)

C0

where the boundary term B0 is 



B0 (ψ E ) :=

∇ · (ψ E (x)∇ψ E (x) d x = C0

∂C0

ψ E (x) νˆ · ∇ψ E (x) d S(x), (2.12)

and d S denotes the surface measure. Using expression (2.11) in (2.10) we obtain, 

 C0

|∇ Pn ψ E (x)|2 d x  (E − κω0 )

C0

|ψ E (x)|2 d x + B0 (ψ E ) −

n 

( jπ)2 | ψ E , ϕ j,0 |2 .

j=0

(2.13) Substituting the right side of (2.13) into the right side of (2.9) yields the result (2.4) for k = 0. 4. To verify the second result (2.6), we note that the equality on the first line of (2.11)  replacing 0: holds for any k ∈  

 |∇ψ E (x)| d x = −

(ψ E (x))ψ E (x) d x + Bk (ψ E ).

2

Ck

(2.14)

Ck

 replacing 0. We note that  = with Bk (ψ E ) defined as in (2.12) with k ∈  Int ∪k∈  C k . Because of the self-adjoint boundary conditions, the quadratic form associated with H0 in (1.3) satisfies:  Q  (ψ E ) =

|∇ψ E (x)|2 d x   = |∇ψ E (x)|2 d x =



 k∈

Ck

 k∈

Ck

 

(−ψ E )(x)ψ E (x) d x,

(2.15)

Spectral Averaging and the Local Density of States

123

so comparing (2.15) with the sum of (2.14), we have 

Bk (ψ E ) = 0,

(2.16)

 k∈



verifying (2.6).

3 An Alternate Approach to Spectral Averaging In this section, we present an alternate approach to spectral averaging based on analytic perturbation theory, and use it to prove a version of the Birman-Solomyak Theorem connecting the DOS with the spectral shift function. We consider a oneparameter family of self-adjoint operators Hω := H0 + ωu 2 on a separable Hilbert space H. We assume that the self-adjoint operator H0 has discrete spectrum, at least locally in a bounded interval I ⊂ R. The perturbation u 2 is a bounded, nonnegative, self-adjoint operator with u 2   1, and the variable ω ∈ R. Theorem 3.1 Let I ⊂ R be a bounded interval and Pω (I ) be the spectral projector for I and Hω . Let ϕ ∈ H be a normalized vector so ϕ = 1. For any τ1 < τ2 , we have  τ2 ϕ, u Pω (I )uϕ dω  |I |ϕ2L 2 (supp u) . (3.1) τ1

Proof 1. The family Hω is a type A analytic family of operators. From the standard results on analytic perturbation theory (see, for example, [6, Chap. VII, Sect. 2]), there are analytic eigenvalues E j (ω) ∈ I , corresponding eigenfunctions ψ j (ω), with ψ j (ω) = 1, and rank-one eigenprojections P j (ω) = ψ j (ω) , such that 

Pω (I ) =

P j (ω),

{ j : E j (ω)∈I }

where the sum over the eigenvalues includes multiplicities. Substituting this into the left side of (3.1), we obtain 

τ2 τ1

 ϕ, u Pω (I )uϕ dω =

τ2 τ1

⎡ ⎣



⎤ ϕ, u P j (ω)uϕ ⎦ dω.

(3.2)

{ j : E j (ω)∈I }

2. Concerning the projectors P j (ω), an application of the Feynman-Hellman Theorem implies that (3.3) P j (ω)u 2 P j (ω) = E j (ω)P j (ω).

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If we let A j := P j (ω)u, we form two self-adjoint, rank-one operators: A j Aj = P j (ω)u 2 P j (ω), and Aj A j = u P j (ω)u. The operator Aj A j projects onto uψ j (ω), whereas the operator A j Aj projects onto ψ j (ω). We assume that uψ j (ω) = 0. This follows for local Schrödinger operators, for example, by the unique continuation principle. Since A j Aj and Aj A j are self-adjoint and have the same eigenvalues (except possibly 0), the spectral theorem gives j (ω), Aj A j = u P j (ω)u = E j (ω) P

(3.4)

j (ω) projects onto uψ j (ω). where P 3. The positivity of the left side of (3.3) implies that E j (ω) is monotone increasing. As a consequence, given E ∈ I = [a, b], let ω j (E) ∈ [τ1 , τ2 ] be such that E j (ω j (E)) = E, whenever such an ω j (E) exists. We perform a change of variables from ω ∈ [τ1 , τ2 ] → E ∈ I . With this change of variables and (3.4), an arbitary term of the sum on the right side of (3.2) becomes 

τ2 τ1



ϕ, u P j (ω)uϕ dω = =

τ2

j (ω)ϕ2 P τ1  inf{b,E j (τ2 )} sup{a,E j (τ1 )}

E j (ω) dω

j (ω j (E))ϕ2 d E. P (3.5)

j (ω), it is easy to check that if ω j (E) = ω j (E), 4. With respect to the projectors P then j (ω j (E)) = δ j j P j (ω j (E)) j (ω j (E)) P P

(3.6)

j (ω) by the reduction This also holds if ω j (E) = ω j (E) by construction of the P process as described in [6, Chap. II, Sect. 2.3]. Let us define f j (E) by j (ω j (E))ϕ2 . f j (E) :=  P

(3.7)

From (3.4) and (3.5), it follows that 

τ2

τ1

 ϕ, u Pω (I )uϕ dω  a

⎡ b

⎣



⎤ f j (E)⎦ d E.

(3.8)

{ j | ω j (E)∈[τ1 ,τ2 ]}

 According to the orthogonality condition (3.6), we have j f j (E)  ϕ2L 2 (supp u) , for all E ∈ I = [a, b]. This bound, together with (3.8), proves the result.  There is a situation where we can have equality in Theorem 3. This is when the interval [τ1 , τ2 ] is equal to the real line R. The proof of this requires some basic tools from Birman-Schwinger theory developed, for example, in [1, Appendix B].

Spectral Averaging and the Local Density of States

125

These operators require that for all ω the operators Hω are local in the sense that if Hω ϕ = 0 on any open set in Rd , then ϕ = 0 on that set. The Schrödinger operators considered here are local in this sense. Corollary 3.1 Assume that Hω = H0 + ωu 2 is a local operator, in the sense above, for all ω ∈ R. Assume that I ⊂ R is an interval for which σ(H0 ) ∩ I has zero Lebesgue measure (for example, σ(H0 ) ∩ I is discrete). We then have  R

ϕ, u Pω (I )uϕ dω = |I |ϕ L 2 (supp u) .

(3.9)

We assume that E ∈ / σ(Hω ) and define the Birman-Schwinger kernel by K 0 (E) := u(H0 − E)−1 u. According to Lemma B.2 of [1] the set of ω j (E) in (3.7) are the repeated eigenvalues of −K 0 (E)−1 considered as a self-adjoint operator on j (ω j (E)) in (3.4) are a complete set of eigenL 2 (supp u). Moreover, the projectors P projectors for K 0 (E)−1 . It follows that ϕ2L 2 (supp u) =



j (ω j (E))ϕ2 . P

j

Since this holds for almost every E ∈ I , the result follows from (3.5). Turning to the spectral shift function, from (3.8), we recover some known results about the connection between the spectral shift function (SSF) for the pair (H0 , Hω ) and the local density of states as first proven in [3, 9]. For any ϕ ∈ H, we define ηϕ (E) to be ηϕ (E) := lim+ →0

1 



τ2

τ1

ϕ, u Pω ([E, E + ])uϕ dω

 1  inf{E+,E j (τ2 )} = lim+ f j (s) ds →0  sup{E,E j (τ1 )} j  = f j (E),

(3.10)

j∈(E)

where f j (E) is defined in (3.7) and (E) is the set of indices defined as follows: j ∈ (E) ⇔ ∃ω j ∈ [τ1 , τ2 ] s. t. E j (ω j ) = E ⇔ ω j (E) ∈ [τ1 , τ2 ],

(3.11)

so that card (E) = ξ(E; Hτ2 , Hτ1 ).

(3.12)

That is, the integer card (E) is the number of eigenvalues of Hω crossing E as ω runs from τ1 to τ2 . Let {ϕn }n be an orthonormal basis of H and take ϕ = ϕk to be any element. Then, summing the right side of (3.7) over this basis and, using the fact that

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j (ω j (E))) = 1, the following form of the Birman-Solomyak formula now Tr( P follows from (3.10) and (3.12): ξ(E; Hτ2 , Hτ1 ) = lim+ →0

1 



τ2

τ1

Tr(u Pω ([E, E + ])u) dω.

(3.13)

We note that (3.13) is a version of the Birman-Solomyak formula established solely by analytic perturbation theory. A similar formula was derived by Simon using the Krein trace formula for resolvents [9, Eq. (1)]. A more common version of this formula is   τ2 Tr(u Pω (I )u) dω = ξ(E; Hτ2 , Hτ1 ) d E, τ1

I

as found, for example, in [3]. Formula (3.13) applies to the spectral shift function for local Schrödinger operators with discrete spectrum discussed here. We consider a one-parameter family of Schrödinger operators Hω := H0 + ωu 2 on L 2 (), with u  0 satisfying d u ∈ L∞ 0 (R ) and for a parameter ω ∈ R. The self-adjoint operator H0 is given by H0 = − + j∈Zd \{0} ω j u j . Let Hω denote a self-adjoint restriction of Hω to  ⊂ Rd , similarly for H0 . Then the operators H0 and Hω have discrete spectrum for all ω ∈ R. The Birman-Solomyak formula applies to the pair (H0 , Hω ). We conclude this section with a bound on the SS F that will be used in the proof of Theorem 4.1. Lemma 3.1 Under the hypotheses of Corollary 3.1, the SSF ξ(E; Hτ2 , Hτ1 ) for τ2 > τ1 satisfies the bound, ξ(E; Hτ2 , Hτ1 )  Tr Pτ1 ([E − u2 (τ2 − τ1 ), E]).

(3.14)

Proof Let E(ω) be an eigenvalue of Hω crossing E for some value ω(E) ∈ [τ1 , τ2 ]. If Hω ψ(ω) = E(ω)ψ(ω), with ψ(ω) = 1, then by the Feynman-Hellman Theorem we have E (ω)uψ(ω)2  u2 . It follows that E − E(τ1 )  u2 (ω(E) − τ1 )  u2 (τ2 − τ1 ), which implies the bound (3.14) since ξ(E; Hτ2 , Hτ1 ) = Tr Pτ1 ([0, E]) − Pτ2 ([0, E]), as follows from the definition of the SSF.



Spectral Averaging and the Local Density of States

127

4 Lipschitz Continuity of the Local DOS In this section, we establish local regularity of the finite-volume DOS function n  (E) at low energy without a localization assumption for a restricted family of random potentials. We first mention that under the hypothesis of Theorem 2.1, we can prove the Wegner estimate with an explicit form of the constant. The Wegner estimate for random Schrödinger operators with an absolutely continuous single-site probability measure with density 0  ρ ∈ L ∞ 0 (R) has the form E  {Tr Pω (I )}  C W ρ∞ |||I |,

(4.1)

for I ⊂ R and a finite constant C W > 0 that depends upon I+ = max I . In the next proposition, we give an explicit form of the constant. Proposition 4.1 Assume hypotheses [H1] and [H2]. Let I = [a, b] with b < (n + 1)2 π 2 . We then have

E



{Tr Pω (I )}



b  |I ||| κ ρ∞ (n + 1) 1 − (n + 1)2 π 2 −1

−1 

,

(4.2)

where κ > 0 is the lower bound in [H1]. The proof of the proposition follows from the bound on the trace of the spectral projector in (2.1) and the spectral averaging result (3.1). In order to apply (3.1), we 1 use the bound χk  u k κ− 2 in the inner products on the right side of (2.1). After taking the expectation and spectral averaging, the result follows by summing over . j ∈ {1, . . . , n} and k ∈  We define the local density of states (DOS) function n  (E) by n  (E) := lim+ →0

1 E  {Tr Pω ((E, E + ])}. ||

(4.3)

By the Wegner estimate, (4.2), we have the bound

n  (E)  C W



E := κ ρ∞ (n + 1) 1 − (n + 1)2 π 2 −1

−1 

,

(4.4)

for E < (n + 1)2 π 2 . The local density of states function n  is related to the DOS measure defined in (1.4) by  μ (I ) =

n  (E) d E. I

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J. M. Combes and P. D. Hislop

We next show that n  (E) is Lipschitz continuous in E for energies in the interval [0, E 0 (d)], where E 0 (d) is defined in (A.10), near the bottom of the deterministic spectrum. Theorem 4.1 We assume [H3]: The single-site potential u 0 = κχ0 and the singlesite probability measure is the uniform distribution on [0, 1] so that ρ(s) = χ[0,1] (s). Let n  (E) be the DOS function for the local Hamiltonian Hω , where  = [0, L]d , with L ∈ N. For any 0  E 1 < E 2 < E 0 (d), with E 0 (d) defined in (A.10), there exist a finite constant K 1 > 0, depending only on E 2 and d, so that |n  (E 2 ) − n  (E 1 )|  min {2C W , K 1 ||(E 2 − E 1 )} , where C W > 0 is given in (4.4). Proof 1. Hypothesis [H3] provides the covering condition some κ > 0. For E 2 > E 1 , definition (4.3) implies that



 k∈

(4.5)

u 2k = κ2 χ , for

n  (E 2 ) − n  (E 1 )   1 = lim E Tr Pω ([E 2 , E 2 + ) − Tr Pω ([E 1 , E 1 + ]) →0+ ||  1   = lim 2 E Tr(u 2k Pω ([E 2 , E 2 + )) − Tr(u 2k Pω ([E 1 , E 1 + ])) →0+ κ ||  k∈    1 1  = 2 E ω⊥ lim E ωk Tr(u 2k Pω ([E 2 , E 2 + )) − Tr(u 2k Pω ([E 1 , E 1 + ])) , k κ || →0+   k∈

(4.6) where the interchange of the expectation and the limit may be justified by using the uniform bounds on the ωk -integrals following from (3.13) and Lemma 3.1 so that the Dominated Convergence Theorem applies. 2. By the Birman-Solomyak formula presented in (3.13), we write the limit of the expectation with respect to ωk in (4.6) as   1 E ωk Tr(u 2k Pω ([E 2 , E 2 + )) − Tr(u 2k Pω ([E 1 , E 1 + ]))      , H(ω ) − ξ(E 1 ; H(ω , H(ω ) = ξ(E 2 ; H(ω ⊥ ⊥ ⊥ ⊥ ,ωk =0) ,ωk =1) ,ωk =0) ,ωk =1)

lim

→0+

k

k

k

  = Tr P(ω ([E 2 , E 1 ]) − Tr P(ω ([E 2 , E 1 ]). ⊥ ⊥ ,ωk =0) ,ωk =1) k

k

(4.7)

k

In order to bound the expectation with respect to ωk⊥ of each trace on the last line of (4.7), we use Lemma A.2 and obtain   c(E , d) 2   E ωk⊥ Tr P(ω ([E , E ]) − Tr P ([E , E ])  |||E 2 − E 1 |, ⊥ ⊥ 2 1 2 1 ,ω =0) ,ω =1) (ω k k k k κ2 (4.8) where c(b, d) is defined in (A.12). Combining (4.7)–(4.8), we obtain the bound

Spectral Averaging and the Local Density of States

|n  (E 2 ) − n  (E 1 )|  which is the bound (4.5) with K 1 :=

129

c(E 2 , d) |||E 2 − E 1 |, κ2

c(E 2 ,d) . κ2

(4.9) 

Remark 4.2 We note that the estimate (4.5) is not adequate for controlling the infinite-volume limit. One expects that an additional hypothesis, such as localization, would allow the removal of the volume factor on the right side of (4.5). Indeed, during the completion of this note, a preprint of Dolai et al. [5] was posted in which they use localization and obtain Lipschitz continuity of n  (E), with a volume independent constant, for E in the region of localization and for a smooth probability density ρ. More generally, these authors prove regularity of n  (E), depending on the regularity of the single-site probability measure ρ, for energies in the region of localization, and obtain regularity results for the infinite-volume limit.

Appendix: Some Technical Results We begin with an estimate on the L 2 -norm of an eigenfunction of Hω restricted to a unit cube C0 . Lemma A.1 Assume [H1] and [H2] and that Hω has Dirichlet boundary conditions on  = [−L , L]d . Let ψ E be an eigenfunction of Hω with eigenvalue E: Hω ψ E = Eψ E , with ψ E  = 1 and E > 0. Then, for C0 the unit cube, we have ψ L 2 (C0 ) 

d + 4E . 2d

(A.1)

Proof This result follows by an integration by parts. We write x = (X k , xk ), where X k ∈ [−L , L]d−1 , and xk ∈ [−L , L], and we denote a smaller domain by Tk := {(X k , xk ) | xk ∈ [0, 1], X k ∈ [−L , L]d−1 }. Because of the Dirichlet boundary conditions, we have

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 ∂  ψ E (X k , τ )2 dτ −L ∂τ  xk ∂ψ E (X k , τ ) dτ ψ E (X k , τ ) =2 ∂τ −L 1  xk  1  x k  2 2 2 ∂ψ E 2 2 ψ E (X k , τ ) dτ dτ (X k , τ ) ∂τ −L −L

ψ E (X k , xk )2 =

xk



  21 ∂2ψE 2 ψ E (X k , τ ) dτ (X k , τ ) ψ E (X k , τ ) dτ − ∂τ 2 −L −L  L    L  2 ∂ ψE 1 2 −  ψ E (X k , τ ) dτ + 4 (X k , τ ) ψ E (X k , τ ) dτ . 2 ∂τ 2 −L −L L

2

 21 

L



(A.2) Integrating each term in the last line of (A.2) over Tk (recalling that xk ∈ [0, 1]), we obtain 

  2   1 ∂ ψE 2 , ψE ψ E (X k , xk ) d X k d xk  . (A.3) 1+4 − 2 ∂xk2 Tk L 2 ( L ) Finally, since C0 ⊂ Tk for any k ∈ {1, . . . , d}, it follows from (A.3) and the positivity of the potential Vω , that d     1 1  d + 4 Hω ψ E , ψ E L 2 (L ) . |ψ E (x)| d x  |ψ E (x)|2 d x  d k=0 Tk 2d C0 (A.4) The result follows directly from this and the eigenvalue equation. 



2

We apply Lemma A.1 in order to derive a version of the Wegner estimate for a random Hamiltonian with one random variable fixed. A similar result was obtained in [2, Lemma 4.2] for more general situations but with a less explicit constant. Lemma A.2 Assume [H1] and [H2] and that Hω has Dirichlet boundary conditions on  = [−L , L]d . Let τ  0 and I = [a, b]. Then, there exists an energy E 0 = E 0 (d) < π 2 , and a constant c(b, d) > 0, depending only on b and d, so that for all 0 < b < E 0 , one has E ω0⊥ {Tr Pω0⊥ ,τ (I )}  c(b, d)κ−2 |||I |.

(A.5)

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131

Proof Let ψ E be a normalized eigenfunction of H(ω0⊥ ,τ ) with eigenvalue E ∈ [a, b]. From Lemma A.1, it follows that d + 4E , 2d

| ϕ0,0 , ψ E | 

(A.6)

so that with I = [a, b],  (I )ϕ0,0  ϕ0,0 , P(ω ⊥ ,τ ) 0

d + 4b  Tr P(ω (I ). ⊥ 0 ,τ ) 2d

(A.7)

We now bound the trace according to Theorem 2.1,  Tr P(ω (I ) ⊥ 0 ,τ )



b  1− 2 π

−1  ˜ k∈\{0}



b + 1− 2 π

−1 

 ϕ0,k , P(ω (I )ϕ0,k

⊥ ,τ ) 0

 d + 4b  Tr P(ω (I ), ⊥ 0 ,τ ) 2d

(A.8)

where we used (A.1) for the ϕ0,0 term. We take b < π 2 sufficiently small so that the coefficient of the last trace term on the right in (A.8) is bounded above as  1−

b π2

−1 

d + 4b 2d

 < 1,

(A.9)

so this term can be moved to the left side. Condition (A.9) requires that b satisfy: 0 < b  E 0 (d) :=

1 2



π2 d 2π 2 + d


0. Here V is a real bounded potential of compact support. The potential f x represents an electric field of strength f in the negative x direction acting on a particle of charge 1 and mass 1/2. There have been several simple models considered which contribute to the understanding of the existence and non-existence of resonances with an emphasis on the fate of pre-existing resonances of H = −d 2 /d x 2 + V (x) (see [2–4]) as f → 0. None of these references treat the model given by H f above although in [3] a specific model is treated where V is replaced by the sum of two delta functions, and [4] contains an instability result for the model where V is replaced by a suitable finite rank operator. All these results rely on the analytic structure of the resolvent kernel of the Stark operator which follow from a representation using Airy functions. In this paper we are not only interested in pre-existing resonances which mostly disappear in the limit f → 0, rather we attempt to calculate the limiting behavior of all the resonances of R. Froese (B) Department of Mathematics, University of British Columbia, Vancouver, Canada e-mail: [email protected] I. Herbst Department of Mathematics, University of Virginia, Charlottesville, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_7

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H f for small f > 0 (at least those in a compact set of the plane). We would like to mention papers [18, 19] which treat the resonances of H f for f = 1 at high energy with a compact support V satisfying some assumptions. There is a relation between high energy with f = 1 and small f with fixed energy but with a much changed potential V . We define a resonance as a pole in the meromorphic continuation of the resolvent √ (H f − z)−1 in the open lower half plane. Sometimes we will refer to k = z as a resonance when z is a resonance. Our results show that as f → 0 resonances are of four different types in any compact set of the closed lower half plane: (1) Resonances crowd densely along the positive real axis converging to this axis. We compute these in Sect. 6. (2) Resonances crowd densely along the line arg z = −2π/3. These are computed in Sect. 7. (3) Resonances converge to points z where an analytic continuation of a certain reflection coefficient for scattering with the potential V vanishes. This occurs only when in addition −2π/3 < arg z < 0 and these points are the only limits of resonances of H f in this sector. This is discussed in Sect. 3. (4) Resonances move continuously into the lower half plane from the negative eigenvalues of H as f increases from 0. The imaginary parts are exponentially small as f → 0. They are computed in Sect. 5. For small f there are no resonances in the region −π + δ < arg z < −2π/3 − δ for δ > 0 (see Theorem 5 for a finer description of this region). We also have some results for higher dimensions but they represent only a start on the problem. See Sect. 8.

2 Resonance—Free Regions We write V = V1 V2 with V j real and bounded and H f,0 = H f − V . An easy computation verifies that for Imz = 0, I − V1 (H f − z)−1 V2 = (I + V1 (H f,0 − z)−1 V2 )−1 . It is easy to see that K 0, f (z) := V1 (H f,0 − z)−1 V2 has an entire analytic continuation from the upper half plane to C (see (3) and the discussion below) and thus, since the analytic continuation of K 0, f (z) is compact, I − V1 (H f − z)−1 V2 has a meromorphic continuation to C. We define a resonance as a pole of this meromorphic continuation. Note the resolvent equation (H f − z)−1 = (H f,0 − z)−1 − (H f,0 − z)−1 V2 (1 + K 0, f (z))−1 V1 (H f,0 − z)−1 (1)

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from which we can see, in particular, that if there is a pole in the meromorphic continuation of V1 (H f − z)−1 V2 then there is a pole in the meromorphic continuation of the integral kernel of (H f − z)−1 and vice versa. R 3 Using the Airy function defined as Ai(x) = lim R→∞ (2π )−1 −R ei(k /3+kx) dk, it 3 follows that the unitary operator, e±i p /3 ( p = −id/d x), has an integral kernel:  3 exp(±i p /3) f (x) = Ai(±(x − y)) f (y)dy. (2) We obtain the kernel K 0, f (x, y; z) of the operator K 0, f (z) for Imz = 0 using the 3 3 identity ei p /3 f f xe−i p /3 f = p 2 + f x, K 0, f (x, y; z) = f −1/3 V1 (x)



 Ai( f 1/3 (x − t))(t − z/ f )−1 Ai( f 1/3 (y − t))dt V2 (y).

(3)

Using the fact that the Airy function is entire we can analytically continue the operator K 0, f (z) = V1 (H f,0 − z)−1 V2 into the lower half plane. This can be accomplished by distorting the contour in (3) locally into the lower half plane in a neighborhood of Re(z/ f ) and picking up a pole term so that the analytic continuation of V1 (H f,0 − z)−1 V2 into the lower half plane has kernel K 0, f,c (x, y; z) given by K 0, f,c (x, y; z) = 2πi f −1/3 g1 (x; z)g2 (y; z) + K 0, f (x, y; z)

(4)

where in (4) Im z < 0 and g j (x; z) = V j (x) Ai( f 1/3 (x − z/ f )).

(5)

With the notation (V1 (H f,0 − z)−1 V2 )c for the analytic continuation of V1 (H f,0 − z) V2 into the lower half plane we have, −1

I + (V1 (H f,0 − z)−1 V2 )c = I + Q + V1 (H f,0 − z)−1 V2 Q = (e2 , ·)e1 e1 = 2πi f −1/3 g1 e2 = g 2 . The operator with integral kernel K 0, f,c has eigenvalue −1 when z = z 0 in the lower half plane if and only if z 0 is a resonance. This follows from the Fredholm alternative. Thus the resonances are points z in the lower half plane for which there is a non-trivial solution to (I + Q + K 0, f (z))ψ = 0.

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This equation has a solution only if (I + K 0, f (z))ψ = −Qψ = ce1 . Substituting back into   (I + Q(I + K 0, f (z))−1 (I + K 0, f (z))ψ = 0 shows that we can have c = 0 if and only if 1 + (e2 , (I + K 0, f (z))−1 e1 ) = 0 which gives Theorem 1 Define A2 (x) = Ai( f 1/3 (x − z/ f )), e1 (x) = 2πi f −1/3 V1 (x)A2 (x), e2 (x) = V2 (x)A2 (x) and G(z, f ) = 1 + (e2 , (I − V1 (H f − z)−1 V2 )e1 ).

(7)

The equation G(z, f ) = 0 determines the resonances of H f in the lower half plane. It is easy to find a formula for the continued resolvent of H f using a simple formula for (I − S)−1 where S is a rank one operator satisfying S 2 = αS for some α = 1. We have (I − S)−1 = I + (1 − α)−1 S. With the notations K f = V1 (H f − z)−1 V2 and K f,c = (V1 (H f − z)−1 V2 )c we obtain K f,c = K f +

(I − K f )|e1 e2 |(I − K f ) , G(z, f )

where a subscript c indicates meromorphic continuation across the real axis from the upper half plane to the lower half plane. √ Let k = z and arg k ∈ [−π/2, 0]. We need to obtain the asymptotics of G(z, f ) for small f . Using expansions in Abramowitz and Stegun [1] (10.4.59, 10.4.60), for x in a compact set and |k| > δ1 > 0 we have f 1/6 (8) Ai( f 1/3 (x − z/ f )) = √ e(iζx −iπ/4) (1 + O( f )); arg k ∈ [−π/2, −δ) 2 πk f 1/6 = √ ei(ζx −π/4) (1 + O( f ) + ie−2iζx ); arg k ∈ (−π/3 + δ, 0] 2 πk (9) where ζx =

2k 3 3f

− kx. We can use (9) to conclude that if arg k ∈ [−π/6, 0] and Imk
0 so that if both f > 0 and |e−4ik /3 f | are small enough ||V1 (H f − z)−1 V2 − V1 (H − z)−1 V2 || ≤ C|e−4ik

3

/3 f

| + O( f ).

Here z = k 2 . Proof The integral kernel of the resolvent (H0, f − z)−1 is for x < y given by A˜1 (x)A2 (y)/W where A˜ 1 (x) = Ai(e−2πi/3 f 1/3 (x − z/ f )) ∈ L 2 (−∞, 0), A2 (x) = Ai( f 1/3 (x − z/ f )) ∈ L 2 (0, ∞) and the Wronskian W = f 1/3 eiπ/6 /2π . We calculate for x < y, and for x and y in a compact set 3 A˜1 (x)A2 (y)/W = (e−ik|x−y| /2k)[1 + O( f ) + ie−i(4k /3 f −2yk) ](1 + O( f )).

It follows that ||V1 (H0, f − z)−1 V2 − V1 ( p 2 − z)−1 V2 || ≤ C1 |e−4ik

3

/3 f

| + O( f ).

(10)

We have (1 − V1 (H f − z)−1 V2 ) − (1 − V1 (H − z)−1 V2 ) = (1 + V1 (H0, f − z)−1 V2 )−1 − (1 + V1 ( p 2 − z)−1 V2 )−1 . We use the perturbation formula (1 + A)−1 − (1 + B)−1 = −(1 + B)−1 (A − B)(1 + (1 + B)−1 (A − B))−1 (1 + B)−1 .

The result now follows from Eq. (10) and a bound on (1 + B)−1 = 1 − V1 (H −  z) V2 near the real axis which follows from limiting absorption estimates. −1

Let

F(k) = (V2 , e−ikx (I − V1 (H − z)−1 V2 )e−ikx V1 ), z = k 2 .

In the next proposition we get a handle on the behavior of G(z, f ) not too far from the positive real axis when |Imk|/ f is large enough. The next proposition will be effective only when F(k) does not vanish.

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Proposition 3 Suppose 0 < δ < |k| < δ −1 and arg k ∈ (−π/6, 0). Then G(z, f ) = 1 + (e2 , (1 − V1 (H f − z)−1 V2 )e1 ) = 1 + e4ik |E| ≤ C + O( f )|e4ik

3

/3 f

3

/3 f

(2k)−1 F(k) + E

|

for some constant C > 0. Proof The proposition follows by inserting the estimates in (9) and in Lemma 2 into the definition of G(z, f ) given in Theorem 1.  Let d(z) = dist(z, σ (H )) Proposition 4 G(z, f ) = 1 + (2k)−1 e

4ik 3 3f

[F(k) + O( f )]

(11)

−1

if Imz < −δ, |k| < δ . G(z, f ) = 1 + (2k)−1 e

(12)

4ik 3 3f

[F(k) + O( f )] 3 f log 1/ f if arg k ∈ [−π/6, 0), Imk ≤ − , and δ < |k| < δ −1 . 8|k|2  4ik 3  G(z, f ) = 1 + (2k)−1 e 3 f F(k) + O( f /d(z)2 )

(13) (14)

−1

if arg k ∈ [−π/2, −5π/12), δ < |k| < δ , and d(z) > C1 f where C1 is large enough.

(15)

Proof The first estimate follows directly from (8) and the fact that if Imz < −δ, ||V1 ((H f − z)−1 − (H − z)−1 )V2 || ≤ C f follows easily. The second estimate again follows from the equation in (8) which is valid when arg k ∈ [−π/6, 0] as long as 1/ f and |k| > δ and from Lemma 2. The latter gives Imk ≤ − f log 8|k|2 ||(I − V1 (H f − z)−1 V2 ) − (I − V1 (H − z)−1 V2 )|| ≤ C f for these k. The third estimate follows from the fact that ||V1 (H0, f − z)−1 V2 − V1 ( p 2 − z)−1 V2 || ≤ C f for Rez < −δ, arg k ∈ [−π/2, −5π/12] (one can use the explicit asymptotic behavior of Airy functions or see Sect. 4). Writing K f = V1 (H0, f − z)−1 V2 and K 1 = V1 ( p 2 − z)−1 V2 , We have I + K f = (I + K 1 )(I − (I + K 1 )−1 (K 1 − K f )). So if we let B f = (I + K 1 )−1 (K 1 − K f ) = (I − V1 (H − z)−1 V2 )(K 0 − K f ), we have ||B f || ≤ C f d(z)−1 (where we have assumed for simplicity that |k| < δ −1 ).

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It follows that V1 (H − z)−1 V2 − V1 (H f − z)−1 V2 = B f (I − B f )−1 (I − V1 (H − z)−1 V2 ). It follows that if f d(z)−1 is small enough (so that ||B f || < 1/2 for example),  we have ||V1 (H − z)−1 V2 − V1 (H f − z)−1 V2 || ≤ C f /d(z)2 . The next theorem is a corollary of Proposition 4, Proposition 3, and the behavior of 3 |e4ik /3 f | as f ↓ 0. Note the definition d(z) = dist(z, σ (H )). Theorem 5 (i) Given δ > 0 and C1 > 0, if C0 is large enough then for small f > 0, H f has no resonances in the set {z : δ −1 > |z| > δ > 0, arg k ∈ [−π/6, 0), Imk ≤ −C0 f, |F(k)| > C1 }. (ii) Given δ > 0, if C0 is large enough then for small f > 0, H f has no resonances log 1/ f , in the set {z : δ −1 > |z| > δ > 0, arg k ∈ [−π/6, 0), Imk ≤ − 3 f 8|k| 2 |F(k)| > C0 f }. (iii) Given δ > 0, if C0 is large enough then for small f > 0, H f has no resonances in the set {z : δ −1 > |z| > δ > 0, arg z ∈ [−π, −5π/6], d(z) > C0 f }. (iv) Given δ > 0, if C0 is large enough then for small f > 0, H f has no resonances in the set {k : |k| < δ −1 , arg k ∈ [−π/2 + δ, −π/3], k = e−iπ/3 (k0 − iκ), k0 > δ, κ > C0 f }. (v) Given δ > 0, if C0 is large enough then for small f > 0, H f has no resonances in the set {k : |k| < δ −1 , arg k ∈ [−π/3, −δ], k = e−iπ/3 (k0 + iκ), k0 > δ > 0, κ > C0 f, |F(k)| > δ}. (vi) Given δ > 0, if C0 is large enough then for small f > 0, H f has no resonances in the set {k : |k| < δ −1 , arg k ∈ [−π/3, −δ], k = e−iπ/3 (k0 + iκ), k0 > δ > f) , |F(k)| > C0 f }. 0, κ > f log(1/ 8|k|2 Proof (i) and (ii) follow from Proposition 3 using |e4ik

3

/3 f

| ≥ e8|k|

2

|Imk|/3 f

.

In the proof of (iii), we use (14) and the fact that for k in the stated set |e √ −2 2|k|3 /3 f so if C0 is large enough we have |G(z, f )| > 1/2. e 4ik 3

4ik 3 3f

|
1/2 for small f > 0. The proofs of (v) and (vi) are very similar to those of (i) and (ii).  2

3 Vanishing of the Reflection Coefficient In Theorem 5 in the region {k : arg k ∈ (−π/3, 0)} we demanded the non-vanishing of F(k) to conclude the absence of resonances. According to [16], p. 139, the S-matrix for potential scattering from momentum k ∈ R to momentum k ∈ R is given as k |S|k = δ(k − k) − 2πiδ(k 2 − k 2 )k |T (k 2 + i0)|k where

T (z) = V − V (H − z)−1 V

(16)

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√ is the “off-shell” T-matrix. Noting that |k = eikx / 2π we see that F(k) = (V2 , e−ikx (I − V1 (H − k 2 )−1 V2 e−ikx ) is the analytic continuation of 2π −k|T (k 2 + √ i0)|k from k = z around the branch point z = 0 to the region we are interested in, −2π/3 < arg z < 0. Note that as z makes a 2π revolution, k changes sign. Thus the vanishing of F(k) is the vanishing of the analytic continuation of the amplitude for reflection for an incoming particle with momentum k. √ We have learned that if k0 = z 0 with arg k ∈ (−π/3, 0) and F(k0 ) = 0, then z 0 is not a limit of resonances of H f as f → 0. On the other hand, if F(k0 ) = 0 and arg k ∈ (−π/3, 0) then generically k0 will be the limit of resonances of H f . We state this as a theorem: Theorem 6 If k0 = 0 with −π/3 < arg k0 < 0, and  F(k) := (V2 , e−ikx I − V1 (H − k 2 )−1 V2 e−ikx V1 ) is 0 at k0 while the derivative F (k0 ) = 0 then as f ↓ 0, k0 is a limit of resonances of H f . If k0 = 0 with −π/3 < arg k0 < 0 and F(k0 ) = 0 or if k0 = 0 with −π/2 < arg k0 < −π/3 then there are no resonances of H f near k0 for f > 0 small. Note that there is more detailed information about the absence of resonances in Theorem 5. Proof After the discussion above what is left to prove is that if k0 = 0, −π/3 < arg k0 < 0, F(k0 ) = 0, F (k0 ) = 0, then k0 is a limit of resonances of H f . Let H (k, f ) = 2ke−4ik = 2ke−4ik

3

/3 f

3

/3 f

+ 2ke−4ik

3

/3 f

(e2 , (I − V1 (H f − z)−1 V2 )e1 )

+ (V2 , a(x, k, f )(I − V1 (H f − z)−1 V2 )a(x, k, f )V1 ).

(17)

√ 3 4π keiπ/4 f −1/6 e−2ik /3 f Ai( f 1/3 (x − z/ f )) → e−ikx

(18)

where a(x, k, f ) =

as f ↓ 0. We want to solve H (k, f ) = 0 for k near k0 given f > 0 and small. We can write

V1 ( p 2 + f x − k 2 )−1 V2 = −i



∞

V1 eit f x/2 ei( p

2

−k 2 )t it f x/2

e

V2 eit

3

f 2 /12

dt

(19)

0

if −π/2 < arg k < 0. We thus see that this quantity is C ∞ in the variable f for f ∈ R. Since I − V1 (H f − z)−1 V2 = (I + V1 (H f,0 − z)−1 V2 )−1 , it follows that I − V1 (H f − z)−1 V2 is C ∞ for small f . The function a(x, k, f ) is at least C ∞ in (k, f ) in a region of the form Br (k0 ) × [0, f 0 ) where Br (k0 ) is a small ball centered at k0 where H (k0 , 0) = 0 (see the Appendix). Here and in the following we define 3 e−4ik /3 f and all its derivatives to be zero at f = 0. This makes the latter function

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C ∞ for f ∈ [0, ∞) and k near k0 . To make reference to the standard implicit function theorem directly we can consider H (k, f 2 ) instead which, as is required, is C 1 in (k, f ) for (k, f ) in an open set containing (k0 , 0). Our assumption is that H (k0 , 0) = F(k0 ) = 0, but its k derivative F (k0 ) is non-zero at this point. Thus by the implicit  function theorem, for small f , H f has a resonance near k0 . For reference we have F (k0 ) = − 2i



V (x)e−2ik0 x xd x

+ i(V, e−ik0 x R0 e−ik0 x x V ) + i(x V, e−ik0 x R0 e−ik0 x V ) − 2k0 (V, e−ik0 x R02 e−ik0 x V )

(20)

where R0 = (H − k02 )−1 .

4 Resolvent Convergence The purpose of this section is to prove that in a complex neighborhood of a point on the negative real axis we have convergence of the analytically continued resolvent ( p 2 + f x − z)−1 to ( p 2 − z)−1 with exponential weights. We use the method of E. Mourre [6] as expounded in Perry-Sigal-Simon [7] in proving bounds on ( p 2 + f x − z)−1 with weights for Rez in some compact set of the negative reals, uniformly for Im z = 0. Lemma 7 Let D = (| p|2 + 1)−1/2 . Then for f > 0 and Imz = 0, ||D( p 2 + f x − z)−1 D|| ≤ C f −1 where C is a universal constant. Proof We take Imz ≤ 0. Let G (z) = ( p 2 + f x + i f − z)−1 and F (z) = DG (z)D. Clearly ||G (z)|| ≤ ( f )−1 . Note that f = [i p, p 2 + f x]. We have ||G (z)φ||2 = (φ, G (z)∗ 2 f G (z)φ)/2 f ≤ (φ, G (z)∗ (2 f − 2Imz)G (z)φ)/2 f = (−i/2 f )(φ, (G (z)∗ − G (z))φ) ≤ |(φ, G (z)φ)|/ f

Thus ||G D|| ≤ ( f )−1/2 ||DG D||1/2 = ( f )−1/2 ||F ||1/2 . Similarly

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||DG || ≤ ( f )−1/2 ||F ||1/2 . Then d F /d = D( p 2 + f x + i f − z)−1 [ p, p2 + f x + i f − z]( p 2 + f x + i f − z)−1 D.

Thus

||d F /d || ≤ ||DG || + ||G D|| ≤ 2( f )−1/2 ||F ||1/2

Starting from ||F || ≤ ( f )−1 and iterating, we get the result for some universal constant C.  We can now show the convergence of x−1 ( p 2 + f x − z)−1 x−1 to x−1 ( p 2 − z) x−1 uniformly for Imz = 0 and Rez ≤ −δ for δ > 0. −1

Lemma 8 Suppose Imz = 0 and Rez ≤ −δ with δ > 0.Then ||x−1 ( p 2 + f x − z)−1 x−1 − x−1 ( p 2 − z)−1 x−1 || ≤ C f where C depends only on δ. Proof We use the resolvent equation twice to obtain  x−1 ( p 2 + f x − z)−1 − ( p 2 − z)−1 x−1 = −x−1 ( p 2 − z)−1 f x( p 2 − z)−1 x−1 + x−1 ( p 2 − z)−1 f x( p 2 + f x − z)−1 f x( p 2 − z)−1 x−1 .

Using the easily established bound ||( p + i)x( p 2 − z)−1 x−1 || ≤ C , with C dependent only on δ, and the previous lemma we obtain the result.



By iterating the equation above we can get an asymptotic expansion for ( p + f x − z)−1 with appropriate weights. To shorten our equations we use the abbreviations W = −x, R0 = ( p 2 − z)−1 , R0, f = ( p 2 + f x − z)−1 . 2

Lemma 9 Suppose the functions ζ1 and ζ2 are bounded and measurable with |x n ζ j (x)| ≤ an < ∞ for all n. Suppose Rez < −δ with δ > 0 and Imz = 0. Then for any N > 0 ζ1 R0, f ζ2 =

2N −1

(ζ1 R0 (W R0 )n ζ2 ) f n + (ζ1 (R0 W ) N f R0, f (W R0 ) N ζ2 ) f 2N −1

n=0

with the remainder ||ζ1 (R0 W ) N f R0, f (W R0 ) N ζ2 || ≤ b N where b N depends only on N , δ, ζ1 , and ζ2 .

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Lemma √ 10 Suppose z = −α − iβ with α ≥ δ > 0 and α ≥ 8|β|. Choose > 0 so that α ≥ 8|β|. Let g ,z (x) = e− |x| Ai( f 1/3 (x − z/ f )). Then ||g ,z ||2 ≤ C e− α/4 f for small f > 0. Proof We first write x + α/ f = a, β/ f = b. We obtain 1/2   √    2a − a 2 + b2 / 2, |Im(a + ib)3/2 | Re(a + ib)3/2 = a + a 2 + b2  1/2   √ = a 2 + b2 − a a 2 + b2 + 2a / 2 If the argument w = f 1/3 (a + ib) of Ai is bounded, then for example we must have |x| ≥ α/ f − |a| ≥ f −1/3 ( f 1/3 α/ f − f 1/3 |a|) ≥ f −1/3 ( f 1/3 α/ f − C) = α/ f − f −1/3 C ≥ α/2 f for small f . Thus ||g ,z 1{|w|≤C} ||2 ≤ k e− α/4 f . Therefore in the following we assume |w| is large so we can use asymptotic expansions of the Airy function. First assume that | arg(a + ib)| < π − δ1 for some small δ1 > 0. We write 1δ1 for the indicator function of this set. Then we can use the asymptotic behavior of Ai(w), | Ai(w)| ≤ C|w|−1/4 e−Reζ , where ζ = 2w 3/2 /3. If a ≥ 0  1/2   √ −Rew 3/2 ≤ f 1/2 a 2 + b2 + a a 2 + b2 − a − a / 2 ≤ f 1/2 

 1/2 √ 2 a 2 + b2 − a 2 a 2 + b2 − a / 2 = f 1/2 |b|

 1/2 √ √ √ a 2 + b2 − a / 2 ≤ f 1/2 |b|3/2 / 2 = |β|3/2 / 2 f.

Thus |x|/2 + (2/3)Rew3/2 ≥ (a + α/ f )/2 −

√

2|β|3/2 /3 f.

√ If we demand that α ≥ 4|β| and α ≥ |β| then |x|/2 + (2/3)Rew3/2 ≥ α/4 f + α/4 f − Thus we have ||g ,z 1{|w|≥C,a≥0} 1δ1 ||2 ≤ k e− α/4 f . Suppose now that a ≤ 0. Then

√

2/6 f α 1/2 |β| ≥ α/4 f.

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−Re(a + ib)3/2 =

 1/2   √  1/2 √ a 2 + b2 − |a| a 2 + b2 + 2|a| / 2 ≤ 2|b| a 2 + b2 + |a|

≤ 2|b|(a 2 + b2 )1/4 ≤ 2|b|3/2 + 2|b||a|1/2 and thus since |x| = |a| + α/ f |x| + 2Rew 3/2 /3 ≥ |x|/2 + α/4 f + α/4 f − 4|b|3/2 f 1/2 /3 + |a|/2 − 4|b||a|1/2 f 1/2 /3.

Minimizing the last two terms over |a| we find |x| + 2Rew3/2 /3 ≥ ( |x|/2 + α/4 f ) + α/4 f − 4|β|3/2 /3 f − 8β 2 /9 f √ We now require α ≥ 8|β| and α ≥ 8|β|. Then one can check that the last three terms add to something positive. Thus ||g ,z 1{|w|≥C,a≤0} 1δ1 ||2 ≤ k e− α/4 f . Finally, consider the region where π ≥ | arg(a + ib)| ≥ π − δ1 where δ1 is small. 3/2 Then we can use the asymptotics | Ai(−u)| ≤ C|u|−1/4 e|Im2u /3| where√−u = w = 1/3 1/3 3/2 2 2 f (a + √ ib). Thus u = f√ (|a| − ib) and from above |Imu | = ( a + b − 1/2 2 2 |a|) ( a + b + 2|a|)/ 2. But this has just been estimated and we thus find  ||g ,z 1{|w|≥C} (1 − 1δ1 )||2 ≤ k e− α/4 f . This proves the lemma. Lemma 11 For f > 0 the weighted resolvent ζ1 (x)(H f,0 − z)−1 ζ2 (x) has an analytic continuation from the upper half plane to C as a bounded operator as long as 1/2 |ζ j (x)| ≤ Cγ e−γ |x| for all γ > 0. If Im f > 0, (H f,0 − z)−1 is analytic in f and entire in z and if f 1 > 0 then lim

f → f 1 ,Im f ↓0

  ζ2 (x)|| = 0 ||ζ1 (x) (H f,0 − z)−1 − (H f1 ,0 − z)−1 c

where the subscript c indicates the analytic continuation of ζ1 (x)(H f1 ,0 − z)−1 ζ2 (x) from the upper half z plane. Proof Since the Airy functions are entire functions of their arguments, the lemma just follows from their asymptotic behavior (see [1]).  Note that (1) then shows that if f > 0, ζ1 (x)(H f − z)−1 ζ2 (x) has a meromorphic continuation to C. Proposition 12 Suppose√λ0 < 0. Choose r with 0 < r < dist(λ0 , σ (H ) \ {λ0 }) and r < |λ0 |/9. Then if > 8|λ0 |/9, −1 − |x| || → 0 ||e− |x| ((H f − z)−1 c − (H − z) )e − |x| is the meromoras f ↓ 0 uniformly for |z − λ0 | = r . Here e− |x| (H f − z)−1 c e phic continuation of the resolvent (with weights) from the upper half plane to a neighborhood of the point λ0 .

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Proof We use Lemma 8 to show the convergence of the resolvent for Imz = 0, and even for the limits onto the real axis from above and below. To see the convergence below the real axis in the region stated we use a version of (4) without the weights V1 and V2 , along with Lemma 10. This shows that 2 −1 − |x| || → 0 ||e− |x| ((H0, f − z)−1 c − ( p − z) )e

as f ↓ 0 uniformly for |z − λ0 | = r . In particular it follows that (I + V1 (H f,0 − −1 → (I + V1 ( p 2 − z)−1 V2 )−1 as f ↓ 0. Then we use (1) and Lemma 11 z)−1 c V2 ) to show that (H f − z)−1 (with the weights given in that lemma) has a meromorphic continuation from the upper half plane to C. Finally with the weights e− |x| , we can see using (1) for (H f − z)−1 c , we have the convergence stated in this proposition. 

5 Resonances Near the Negative Real Axis Theorem 13 Suppose λ0 is an eigenvalue of H . Then for small enough f > 0 there is a resonance λ( f ) of H f near λ0 . λ( f ) has an asymptotic expansion in (nonnegative) powers of f . The terms of this expansion can be calculated as if λ( f ) were a real eigenvalue of H f using the Rayleigh-Schrödinger perturbation expansion. Thus the expansion coefficients are real although λ( f ) is not. Proof Let ψ0 be the normalized of H corresponding to λ0 . Since V √ eigenfunction √ has compact support ψ0 = √ae −λ0 x + be− −λ0 x and thus it is clear that λ0 < 0. We see that in fact ψ0 = a± e− −λ0 |x| for large |x| with ±x > 0. Because of the decay rate of ψ0 we learn that for r as in Proposition 12, lim −(2πi)−1 f ↓0

 |z−λ0 |=r

−1 (ψ0 , (H f − z)−1 c ψ0 )dz = −(2πi)

 |z−λ0 |=r

(ψ0 , (H − z)−1 ψ0 )dz = 1.

As we can make r > 0 as small as we like we see that H f has a resonance λ( f ) → λ0 as f ↓ 0. The pole in (H − z)−1 at λ0 corresponds to a simple zero of the Wronskian and similarly from the convergence of projections λ( f ) is a simple pole in the resolvent (H f − z)−1 for Im f > 0. According to the convergence result of Lemma 11 for small f > 0 the pole of (ψ0 , (H f − z)−1 c ψ0 ) is simple. Thus for small enough r and f > 0, |z−λ0 |=r (ψ0 , (z − λ( f ))(H f − z)−1 c ψ0 )dz = 0. It follows (again for small enough r and f ) that 

|z−λ0 |=r (ψ0 , z(H f

λ( f ) = 

|z−λ0 |=r (ψ0 , (H f

− z)−1 c ψ0 )dz − z)−1 c ψ0 )dz

.

(21)

We now substitute the asymptotic expansion of (H0, f − z)−1 c from Lemma 9 into 1 (note that the rank one term as estimated in Lemma 10 has a zero asymptotic

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expansion and can thus be ignored). Thus λ( f ) has an asymptotic expansion in powers of f . If we instead did the same thing with f x replaced by a real bounded function f W with lim|x|→∞ W (x) = 0 we would again get an asymptotic expansion, this time for the eigenvalue of H + f W , which is actually convergent for small f and whose terms are real. The terms of this expansion match up exactly with those of our expansion of λ( f ) except of course with x replaced by W . Thus the asymptotic expansion of λ( f ) is exactly given by Rayleigh - Schrödinger perturbation theory for the non-existent eigenvalue of H f . It is non-existent because no linear combination of the two linearly independent Airy functions is square integrable near −∞. Thus  H f has no eigenvalues. This result has been known for many years [8, 9, 12, 13] in the dilation analytic framework. In the latter framework λ( f ) is an actual eigenvalue of the dilated Hamiltonian unlike in our framework. But of course the dilation analytic framework requires that the potential, V , be dilation analytic in some angle. The result was also proved by Graffi and Grecchi ([11]) for Hydrogen in an electric field using the separability in squared parabolic coordinates. As is well known, for Im f = 0 the resolvent (H0, f − z)−1 is analytic in f and entire in z. For such f the operator H0, f has domain equal to D( p 2 ) ∩ D(x). Let N f = { f t + s 2 : t and s ∈ R}. For all non-real f the spectrum is empty so the resolvent is of course bounded. But in addition we have the explicit bound ||(H0, f − z)−1 || ≤ 1/dist(z, N f ) when z is outside the closure of the numerical range, N f , of H0, f . Of course this bound also holds if f = 0. Proposition 14 Consider the Hamiltonian H f for f non-real. If f is not real the resolvent of H f is meromorphic in C. Suppose C = {z : |z − z 0 | = r } is a circle in C which is disjoint from the spectrum of H and from N f . Then for small enough f , C is disjoint from the spectrum of H f and allowing f → 0 in such a way that the distance between the circle C and N f is bounded away from 0 we have the convergence of projections lim ||(2πi)−1 f →0

 |z−z 0 |=r

(H f − z)−1 dz − (2πi)−1

 |z−z 0 |=r

(H − z)−1 dz|| = 0.

Proof We use the formula (1): (H f − z)−1 = (H f,0 − z)−1 − (H f,0 − z)−1 V2 (1 + V1 (H f,0 − z)−1 V2 )−1 V1 (H f,0 − z)−1 .

The convergence of V1 (H f,0 − z)−1 V2 to V1 ( p 2 − z)−1 V2 in norm is easy to show. Equation (1) then implies that for small f ,(H f − z)−1 has no spectrum on the circle and we can integrate (H f − z)−1 − (H − z)−1 around C. We use (1) for both resolvents and note that the term (H f,0 − z)−1 − ( p 2 − z)−1 occurring in the differ−1 2 −1 ence of resolvents integrates  to zero. Thus consider (H f,0− z) V2 − ( p − z) V2

for example. We have || ( p 2 + f x − z)−1 − ( p 2 − z)−1 V2 || ≤ c| f | where we are

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using the fact that the distance between C and N f is bounded away from zero. The  term V1 (H f,0 − z)−1 is treated in the same way. This completes the proof. Corollary 15 From proposition 14 and the formula (21) we see that for f > 0 and small, λ( f ) is the boundary value from Im f > 0 of a function analytic in f . The analytic continuation of λ( f ) to Im f > 0 with | f | > 0 but small, is a simple eigenvalue of H f . Theorem 16 Suppose λ0 is a negative eigenvalue of H = p 2 + V where V is a real bounded measurable function of compact support. Suppose  is the normalized eigenfunction corresponding to λ0 . Then for small positive f the Hamiltonian H f = H + f x has a resonance λ( f ) near λ0 with real part given as in Theorem 13 and imaginary part √ 4 3/2 1 −Imλ( f ) = √ e− 3 f (−λ0 − f λ1 ) (V e− −λ0 x , )2 (1 + O( f )) 4 −λ0 − where √ λ1 = (, x). We have (V e −λ0 x for x near −∞. κ− e

√

−λ0 x

√ , )= − 2κ− −λ0 where (x) =

Proof We follow Howland in [15] where he computes an exponentially small imaginary part of a resonance caused by a barrier which is becoming infinite in extent. Our situation is similar but a bit more complicated. Nevertheless much of the analysis below is lifted from Howland’s paper. Let Q + ( f, z) = V1 (H f,0 − z)−1 c V2 where the subscript c and the superscript + indicate analytic continuation from the upper half plane. Similarly a − superscript will indicate analytic continuation from the lower half plane. We let L( f, z) = (Q + ( f, z) + Q − ( f, z))/2 and D( f, z) = (Q + ( f, z) − Q − ( f, z))/2. It follows from (4) that D( f, z) is a rank one operator with kernel D( f, z)(x, y) = πi f −1/3 g1 (x; f, z)g2 (y; f, z) with g1 (x; f, z) = V1 (x) Ai( f 1/3 (x − z/ f )) and g2 (x; f, z) = V2 (x) Ai( f 1/3 (x − z/ f )).

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We saw that I + Q + ( f, z) has a one dimensional kernel at the resonance z = λ( f ) converging to λ0 as f ↓ 0. Similarly if we consider I + L( f, z) the same proof shows that this operator has a simple eigenvalue −1 at z = μ( f ) with μ( f ) converging to λ0 as f ↓ 0. We assume that |V1 | + |V2 | = 0 exactly where V = 0 and V1 /V2 is bounded above and below on the set where |V | > 0. We set g = V1 /V2 on the set {|V | > 0} and = 1 on the complement. It follows that g −1 L( f, μ( f ))g = L( f, μ( f ))∗ . Since σ (L( f, μ( f ))) = σ (L( f, μ( f ))∗ ), L( f, z) has eigenvalue −1 for both z = μ( f ) and z = μ( f ). But there is only  one such z near λ0 for small f > 0 and thus μ( f ) is real. Let J ( f ) = (2πi)−1 |z+1|= (z − L( f, μ( f ))∗ )−1 dz for small > 0. Since in the limit f ↓ 0, J ( f ) → J (0) which projects onto V2 , where (H − λ0 ) = 0, φ f = J ( f )V2  satisfies (L( f, μ( f ))∗ + I )φ f = 0 for small f > 0. Similarly

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 define J˜( f ) = (2πi)−1 |z+1|= (z − Q + ( f, λ( f )))−1 dz for small > 0. Then ψ f = J˜( f )V1  is in the kernel of I + Q + ( f, λ( f ) for small f > 0. 0 = (φ f , (I + Q + ( f, λ( f ))ψ f ) = (φ f , (I + L( f, μ( f )) + L ( f, μ( f ))(λ( f ) − μ( f )) + D( f, λ( f )))ψ f ) + O(|λ( f ) − μ( f )|2 ) = (φ f , L ( f, μ( f ))ψ f )(λ( f ) − μ( f )) + (φ f , D( f, λ( f ))ψ f ) + O(|λ( f ) − μ( f )|2 )

Here we have used the fact that L( f, z) is analytic in z with derivatives bounded for z in a neighborhood of λ0 as f ↓ 0. We have (φ f , L ( f, μ( f ))ψ f ) → (V2 , V1 ( p 2 − λ0 )−2 V2 V1 ) = ||||2 and thus λ( f ) − μ( f ) = −(φ f , D( f, λ( f ))ψ f )/(φ f , L ( f, μ( f ))ψ f ) + O(|λ( f ) − μ( f )|2 )

Consider D( f, λ( f )). We have 3/2 √ 1 − 2(−λ(3 ff )) −x −λ( f ) f −1/6 g j (x) = V j (x) √ e e (1 + O( f )) 2 π (−λ( f ))1/4

so (φ f , D( f, λ( f ))ψ f ) = 3/2  √ √  − 4(−λ(3 ff )) (i/4 −λ( f ))e φ f (x)V1 (x)e− −λ( f )x (1 + O( f ))e− −λ( f )y V2 (y)ψ f (y)dxdy

From above we have Imλ( f ) = O( f n ) for all n and λ( f ) has an asymptotic expansion in f given by the Rayleigh-Schrödinger series. Thus λ( f ) = λ0 + f (, x) + O( f 2 ) (here we normalize |||| = 1.) Thus   √ 4(−λ( f ))3/2 −i λ( f ) − μ( f ) = √ (V e− −λ0 x , )2 + O( f ) e− 3 f 4 −λ0 √ √  , ) = − [( p 2 − λ0 )(x)]e− −λ0 x d x using√inteWe can compute (V e− −λ0 x √ √ gration by parts. We find (V e− −λ0 x , ) = −2κ− −λ0 ,where (x) = κ− e −λ0 x for x near −∞. Then

 4(−λ0 − f (,x))3/2 3f λ( f ) − μ( f ) = −i −λ0 κ−2 e− (1 + O( f )). Since μ( f ) is real we learn that  4(−λ0 − f (,x))3/2 3f Imλ( f ) = − −λ0 κ−2 e− (1 + O( f )).

(23)

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149

 Note that in the above proof μ( f ) = Reλ( f ) up to a function with zero asymptotic expansion.

6 Resonances Near the Positive Real Axis Let A1 (x) = Ai(e2πi/3 f 1/3 (x − z/ f )) A2 (x) = Ai( f 1/3 (x − z/ f )) / For fixed f > 0 and Imz > 0, A1 (x) ∈ L 2 (−∞, 0) while if Imz < 0 A1 (x) ∈ L (−∞, 0). On the other hand, for fixed f > 0, A2 (x) ∈ L 2 (0, ∞) for all z. We know that if z is a resonance then there is a solution of Schrödinger’s equation −ψ + (V (x) + f x)ψ = zψ which has the form c A2 (x) for large positive x and c A1 (x) for large negative x. We note the following obvious but important fact: If we are given (ψ(a), ψ (a)) = (α, β) , then for any b ∈ R, both ψ(b) and ψ (b) are entire functions of the variables (z, f, α, β). Suppose z is near the positive real axis and the support of the potential V is contained in (−L , L). Choosing ψ(−L) = 1 and ψ (−L) = A 1 (−L)/A1 (−L) we note that ψ(L) and ψ (L) are analytic functions of f, z, and ψ (−L). It is easily √ seen from [1] that for arg z ∈ (−2π/3 + δ, 0] and k = z with arg(k) ∈ (−π/2, 0]. 2

A 1 (−L)/A1 (−L) =: −iλ(k, f ) = −ik(1 + O( f )) uniformly for |k| > δ > 0. On the other hand from [1]   √ A 2 (L) = −π −1/2 f 1/6 k(1 + O( f )) (1 + O( f 2 )) cos(ζ + π/4) + O( f ) sin(ζ + π/4) 

(24)

A2 (L) = π −1/2 f 1/6 k −1/2 (1 + O( f )) (1 + O( f 2 )) sin(ζ + π/4) + O( f ) cos(ζ + π/4)



(25) with ζ =

2k 3 (1 − f L/k 2 )3/2 3f

To get an idea what we are dealing with we look at the leading order as f ↓ 0 of (ψ(L), ψ (L)) which arises by propagating from −L to L with (ψ(−L), ψ (−L)) = (1, −ik) and f = 0. Thus we are solving the Schrödinger equation −ψ + V (x)ψ = k 2 ψ with ψ(x) = e−ikx for x to the left of the support of V . Then to this leading order ψ (L)/ψ(L) = g(k) is analytic in k unless ψ(L) = 0. To leading order in the sense that we neglect O( f ) terms in (24) and (25) we obtain

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A 2 (L)/A2 (L) = −k/ tan(ζ + π/4), and the equation A 2 (L)/A2 (L) = ψ (L)/ψ(L) which holds when k is a resonance takes the form e−2iζ = i

ψ (L) + ikψ(L) . ψ (L) − ikψ(L)

Let us use unitarity of the S-matrix to get a relation between ψ (L) and ψ(L) for real k given the initial (ψ(−L), ψ (−L)) = (1, −ik). Here we assume f = 0. We have ψ(x) = e−ikx for x to the left of V and c1 eikx + c2 e−ikx to the right of V . Unitarity gives |c2 |2 = 1 + |c1 |2 . (Notice that c1 = r (−k)/t (−k), c2 = 1/t (−k) where t (k) is the transmission amplitude for a particle of momentum k and r (k) is the reflection amplitude for this momentum.) We compute −

ψ (L) + ikψ(L) c1 eik L = = −r (−k)e2ik L . c2 e−ik L ψ (L) − ikψ(L)

Here r (−k) is the analytic continuation of the reflection amplitude from k > 0 to Imk < 0. It follows that    ψ (L) + ikψ(L) 2 |c1 |2    ψ (L) − ikψ(L)  = 1 + |c |2 < 1 1

= 0 exactly when the (right) for k real. And we see that for real k, ψψ (L)+ikψ(L) (L)−ikψ(L) reflection coefficient, r (−k), is zero. Define

t (k) r (−k) S(k) = r (k) t (−k) where the transmission and reflection amplitudes are given by t (k) = 1 − 2πi|2k|−1 k|T (k 2 + i0)|k

(26)

r (k) = −2πi|2k|−1 −k|T (k 2 + i0)|k.

(27)

Then for real k these quantities satisfy the usual unitarity relations |r (k)|2 + |t (k)|2 = 1 r (−k)t (k) + r (k)t (−k) = 0 or in matrix form S(k)∗ S(k) = I

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This is a consequence of the unitarity of the S − matrix as given in (16). Note that it follows from the symmetry of the resolvent kernel (H − z)−1 (x, y) that t (k) = t (−k) and thus from the unitarity relations that |r (k)| = |r (−k)|. Going back to f > 0, we see that the functions ψ (L) ± ikψ(L) are of the form h ± (k, f, λ(k, f )) where the functions h ± are analytic in their arguments. If arg k ∈ (−π/3 + δ, 0) we have ψ (L) ± ikψ(L) = h ± (k, 0, k) + O( f ). Consider a real point k0 > 0 for which the (right) reflection coefficient for scattering in potential V is non-zero. In a neighborhood of k0 , say |k − k0 | ≤ , ψ (L) + ikψ(L) = G(k)e2ik L (1 + b(k, f )) ψ (L) − ikψ(L) where G is analytic in this neighborhood and non-zero while for real k, |G(k)| < 1. The function b(k, f ) is C 1 in k with ∂b(k, f )/∂k bounded for δ −1 > |k| > δ (δ > 0), k near the positive real axis, and f > 0 small. (See Appendix 2). It follows that a resonance k of H f in this neighborhood obeys G(k) = −ie−4ik

3

/3 f

(1 + a(k, f )). 3

(28) 3

(1 − f L/k 2 )3/2 = 4k − where we have cancelled out the factor e2ik L using 2ζ = 4k 3f 3f 2Lk + O( f ). It follows that for some integer j and some branch of the logarithm k 3 = (3i f /4) log(i G(k)) + 3π f j/2 − (3 f i/4) log(1 + a(k, f )) = (3i f /4) log(i G(k)) + 3π f j/2 + f c(k, f ) where c(k, f ) = O( f ). Without loss of generality we choose a continuous branch of the logarithm with arg(log i G(k0 )) ∈ (−π, π ]. For f small we will choose j very large so that η0 ( j) := (3π f j/2)1/3

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is not far from k0 . We require |η0 ( j) − k0 | < < k0 /2. We thus obtain k 3 − η03 = (3i f /4) log(i G(k)) + f c(k, f ) or

 1/3 k = η0 1 + (3i f /4η03 ) log(i G(k) + η0−3 f c(k, f )

A simple contraction mapping argument shows that for |k − k0 | < with sufficiently small, there is a unique solution to this equation which we call k( j). This string of resonances satisfies k( j) = η0 ( j) + i( f /4η0 ( j)2 ) log G(iη0 ( j)) + O( f 2 ). We remark on the nature of the solution: Notice that there are order of magnitude f −1 integers j with |η0 ( j) − k0 | < . (More exactly since η0 = , to order 2 , η0 ( j)3 = 3η02 , so there are j = (2η02 /π f ) = (2k02 /π f ) such integers j to order 2 .) Write log(i G(k)) = −l(k) + iθ (k) where for small enough, l(k) > 0. Then we get Imk( j) = − f l(η0 ( j))/4η0 ( j)2 + O( f 2 ) and Rek( j) = η0 ( j) − θ (η0 ( j))/4η0 ( j)2 ) f + O( f 2 ). We have thus shown Theorem 17 Suppose the reflection coefficient is non- zero at k0 > 0. Then if > 0 is small enough, H f has resonances in the disk |k − k0 | < . If k is a such a resonance there exists an integer j such that k is one of the resonances k( j) found above which in particular satisfy   k( j) = η0 ( j) − 4−1 θ (η0 ( j))/η0 ( j)2 + il(η0 ( j))/η0 ( j)2 f + O( f 2 ). Here l(k) > 0 for k near k0 and j is allowed to vary in an interval so that |k − k0 | < . The linear density of resonances along the positive real axis near a point k0 where the reflection coefficient is non-zero is to leading order 2k02 /π f . We now consider how to calculate resonances in the neighborhood of a point k0 > 0 where the reflection coefficient vanishes. The functions h ± = h ± (k, f, λ) are analytic functions of their three arguments in a neighborhood of (k0 , 0, k0 ) but λ(k, f ) is not analytic in f , rather analytic in a product set of the form {k ∈ C : |k − k0 | < } × { f : 0 < | f | < , | arg f | < } and C ∞ in {k ∈ C : |k − k0 | < } × { f : | f | < , | arg f | < } in the sense that the derivatives are continuous in f up to f = 0 (see Appendix 2). Let μ(k, f ) = λ(k, f ) − k (thus μ(k, 0) = 0). Let us write A 2 (L) 1 + a(k, f ))e2i(ζ +π/4) + 1 + b(k, f ) ψ (L) = = −ikψ(L) −ik A(L) (1 + c(k, f )e2i(ζ +π/4) − (1 + d(k, f ))

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where according to Appendix 2 the functions a, b, c, d are C ∞ in f for f ≥ 0 and small and analytic in k for k near a point on the positive real axis. In addition they are all 0 when f = 0. Inverting the linear fractional transformation we have ie2iζ =

ψ (L) − ikψ(L) + dψ (L) − ikbψ(L) ψ (L) + ikψ(L) + cψ (L) + ikaψ(L)

We are interested in resonances near a point k0 > 0 where the reflection coefficient vanishes. This means ψ (L) + ikψ(L) = 0 when f = 0. We have ψ (L) + ikψ(L) = h + (k, f, λ(k, f )) with h + analytic in its three arguments and 0 at (k0 , 0, λ(k0 , 0)). The quantity a˜ := cψ (L) + ikaψ(L) is C ∞ in small f ≥ 0 and analytic in k near k0 . Of course a(k, 0) = 0. Thus the denominator is an analytic function of k, f, μ, and a˜ with a zero at (k0 , 0, 0, 0). We can thus use the Weierstrass preparation theorem to write e2iζ = g(k, f )/prep(k, f ) where ˜ − k0 ) p−1 + · · · + b0 ( f, μ, a) ˜ prep(k, f ) = (k − k0 ) p + b p−1 ( f, μ, a)(k Here p is the order of the zero of h + (k, 0, k). The function g is analytic in k ˜ = bs ( f, k). bs ( f, k) is analytic near k0 and C ∞ in small f ≥ 0. We write bs ( f, μ, a) in k near k0 and C ∞ in small f ≥ 0. We have bs (0, k) = 0. Defining m(k, f ) = 3 2 3/2 g(k, f )e2ik /3 f [(1− f L/k ) −1] , we need to solve e4ik

3

/3 f

=

m(k, f ) . prep(k, f )

(29)

We proceed by iteration. Define k1 ( j) and λ f ( j) by the equations k1 ( j) = λ f ( j) + k0 = (3π f j/2)1//3 .

(30)

Here j is chosen very large for f small so that k0 /2 > δ0 > |λ f | ≥ 2 f (1− )/ p where ∈ (0, 1) and δ0 is small. With this lower bound we see that for small f , |prep(k1 , f )| ≥ 2 f 1− + O( f ). The equation we want to solve is k 3 = 3π f j/2 − (3i f /4)(log m(k, f ) − log prep(k, f )). We set kn3 = k13 − (3i f /4)(log m(kn−1 , f ) − log prep(kn−1 , f )); n ≥ 2 where we take the cube root closest to the positive real axis. Let us assume |k − k0 | ≥ f (1− )/ p and estimate prep(k, f ) and its derivative with respect to k. We have

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prep(k, f ) = (k − k0 ) p +

p 

f

(∂b p− j (s, k)/∂s)(k − k0 ) p− j ds.

0

j=1

Thus |prep(k, f )| ≥ |(k − k0 ) p |(1 −

p

c j f |k − k0 |− j ) ≥ |k − k0 | p (1 − c f ).

j=1

∂prep(k, f )/∂k = p(k − k0 ) +

p−1  f

j=1

p−1

+

p 

f

(∂ 2 b p− j (s, k)/∂s∂k)(k − k0 ) p− j ds

0

j=1

(∂b p− j (s, k)/∂s)( p − j)(k − k0 ) p− j−1 ds

0

p

|∂prep(k, f )/∂k| ≤ p|k − k0 | p−1 [1 + c

f −( j (1− )/ p) + c

j=1

≤ p|k − k0 |

p−1

[1 + c f

p−1

f f ((1− )/ p)(− j+1)

j=1

+(1− )/ p

(1 + O( f

(1− )/ p

)].

It follows that | f ∂ log prep(k, f )/∂k| ≤ p f |k − k0 |−1 (1 + c f ). We easily find |k23 − k13 | ≤ (3 f /4)(C + log 1/ f ) − k13 )/k13 )1/3 . Thus

so

that

(k23

k2 = k1 (1 +

|k2 − k1 | ≤ c ( f log 1/ f )(k0 /2)−2 = C f log 1/ f. Let G(k, f ) = log m(k, f ) − log prep(k, f ). We have  3 = (3i f /4)(kn−1 − kn−2 ) kn3 − kn−1

1

0

∂G (kn−2 + t (kn−1 − kn−2 , f )dt ∂k

Thus  |kn3

−

3 kn−1 |

≤ (3 f /4)|kn−1 − kn−2 |

1

(c + 2 p|kn−2 + t (kn−1 − kn−2 )|−1 )dt

0

−1 )). ≤ (3 f /4)(c|kn−1 − kn−2 | + 2 p log(1/(1 − |kn−1 − kn−2 |kn−2

(31)

Let us assume the Weierstrass preparation theorem holds for |k − k0 | ≤ δ < k0 /2 and that m(k, f ) is analytic (in k and non-zero in this ball for small f . Assume |λ f ( j)| ≤ δ/3. Let us make the inductive hypotheses that 3k0 /2 ≥ |kl | ≥ k0 /2

Resonances in the One Dimensional Stark Effect in the Limit of Small Field

155

and |kl+1 − kl | ≤ (C0 f )l−1 |k2 − k1 | for 1 ≤ l ≤ n − 2 where we take C0 = 2(c + 2 p)/k02 . Then k1 −

n−2

|kl+1 − kl | ≤ |kn−1 | ≤ k1 +

l=1

n−2

|kl+1 − kl |.

l=1

It follows that k1 − (1 − C0 f )−1 |k2 − k1 | ≤ |kn−1 | ≤ k1 + (1 − C0 f )−1 |k2 − k1 | and thus for small f , 2k0 /3 ≥ |kn−1 | ≥ k0 /2. Using (31), the lower bound on |kn−1 |, and the induction hypothesis we obtain for 0 < f < f 0 with f 0 independent of n |kn − kn−1 | ≤ ( f /4|kn−1 |2 )(2c + 4 p)|kn−1 − kn−2 | ≤ C0 f |kn−1 − kn−2 |. The induction is complete. This estimate shows that for f sufficiently small, k = k( j) = limn→∞ kn exists and satisfies (29). Actually k2 ( j) is close enough to the limit to get a good idea of what the string of resonances looks like near k0 (but not too near). Thus   k( j)=k1 ( j) − i( f /4k12 ) log m(k1 ( j), 0)− log prep(k1 ( j), f ) + O(( f log 1/ f )2 ) k1 ( j) = k0 + λ f ( j) = (3π f j/2)1/3 |λ f ( j)| ≥ 2 f (1− )/ p  Since |kn −k0 | ≥ |λ f |− nj=2 |k j − k j−1 |≥2 f (1− )/ p − (1 − C0 f )−1 |k2 − k1 | ≥ f (1− )/ p for small enough f , we have |prep(k( j), f )| ≥ f 1− . We are ready to state and prove. Theorem 18 Suppose the reflection coefficient for scattering vanishes at k0 > 0 of order p. Then the quantities k( j) given above define a string of resonances of H f . Suppose k is a resonance of H f . Then given ∈ (0, 1) and δ > 0 small enough with k0 /2 > δ > |k − k0 | > f (1− )/ p there exists a positive integer j such that k = k( j). Proof Suppose that k is a solution to (29) satisfying |k − k0 | < δ and |k − k0 | > f (1− )/ p . Then there is an integer j such that k 3 = 3π f j/2 − (3i f /4)(log m(k, f ) − log prep(k, f )). Without loss of generality we take the same branches of the logarithms we took in defining k( j). We obtain   k 3 − k( j)3 = −(3i f /4) (log m(k, f ) − log m(k( j), f ) − log prep(k, f )) + log prep(k( j), f )) .

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At this point we are not able to get a good estimate for the difference of the final two terms involving prep so we just estimate the difference by the absolute value of the sum. Thus we obtain |k 3 − k( j)3 | ≤ C f (|k − k( j)| + log 1/ f )). Factoring k 3 − k( j)3 and using the fact that both k and k( j) are close to k0 we obtain |k − k( j)| ≤ C f (|k − k( j)| + log 1/ f ) so that |k − k( j)| ≤ C f log 1/ f . Using this rough estimate we can now estimate the difference of the log prep terms using 

1

log prep(k, f )) − log prep(k( j), f )) = (k − k( j))

∂ log prep(k( j) + t (k − k( j), f )/∂kdt.

0

Here we use |k( j) + t (k − k( j)) − k0 | ≥ (1/2) f (1− )/ p so that |k − k( j)| ≤ C|k − k( j)|( f + f ( p−1+ )/ p ) which implies k = k( j).



7 Resonances Near the Line arg k = −π/3 Recalling the definitions of A1 and A2 at the beginning of the previous section, we know that k is a resonance if there is a non-zero solution ψ(x) to −ψ + (V + f x − k 2 )ψ = 0 satisfying ψ (−L) A (−L) = 1 ψ(−L) A1 (−L) A (L) ψ (L) = 2 . ψ(L) A2 (L)

(32) (33)

In the last section where k is close to a point k0 on the positive real axis, A1 and A 1 had single sum asymptotic expansions for f ↓ 0 while A2 and A 2 have double sum expansions. There we could define ψ to be the solution satisfying (32) and use (33) as the equation that determined the resonances. In this section we consider k close to k0 with arg k0 = −π/3. Then the situation is reversed: Now A2 and A 2 have single sum asymptotic expansions while A1 and A 1 have double sum expansions. We will now take ψ to be the solution satisfying (33) and use (32) as our resonance defining equation. Since we need more information than is provided in the expansions in [1],

Resonances in the One Dimensional Stark Effect in the Limit of Small Field

157

we will use the approximation in Appendix 2 which proves properties of the error term. So let us define ψ(x; k, f, α) to be the solution of −ψ + (V + f x − k 2 )ψ = 0 satisfying ψ(L) = 1 and ψ (L) = α. Let λ(k, f ) = −i A 2 (L)/A2 (L). Then ψ(x, k, f, iλ(k, f )) is the solution satisfying (33). Proposition 19 Let k be close to k0 where arg k0 = −π/3. Then λ(k, f ) = k + a( f, k) where a( f, k) is analytic in k near k0 , C ∞ in f for f ≥ 0 and O( f ) as f ↓ 0. Proof Let w2 = f 1/3 (−L − k 2 / f ). When f ↓ 0, w2 avoids a sector about the negative real axis. We can therefore use the asymptotic formulas e−ζ Ai(w2 ) = √ 1/4 (1 + a1 (k, f )) 2 π w2 Ai (w2 ) = −

w2 e−ζ √ (1 + a2 (k, f )) , 2 π 1/4

where 3/2

ζ = (2/3)w2

(34)

= −i

2k 3 3f

f L 3/2 1+ 2 , k

(35)

(36)

and by Appendix 1 the error terms a1 and a2 are analytic in k near k0 , C ∞ in f and O( f ), and The proposition follows easily from this.  Now we analyze A 1 (−L)/A1 (−L). We have A1 (−L) = Ai(w1 ),

A 1 (−L) = e2πi/3 f 1/3 Ai (w1 )

where w1 = e2πi/3 f 1/3 (−L − k 2 / f ). When arg k = −πi/3, then as f ↓ 0, w1 moves to infinity along the negative real axis. We may use the asymptotic formulas for Ai and Ai in Appendix 2: with k1 = eiπ/3 k and η = k12 (1 + f L/k 2 ) A 1 (−L) e2iη /(3 f ) (−i a˜ 1 ) + e−2iη /(3 f ) (−a˜ 2 ) = η1/2 2iη3/2 /(3 f ) , A1 (−L) e (a1 ) + e−2iη3/2 /(3 f ) (ia2 ) 3/2

3/2

where each ai (k, f ) and a˜ i (k, f ) is smooth in f , analytic in k and equal to 1 when f = 0. We know that k near k0 is a resonance exactly when (32) holds, that is, if the left side of this equation is equal to ψ (−L)/ψ(−L). If this condition holds we

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R. Froese and I. Herbst

can solve the linear fractional transformation for e−4iη resonance when e−4iη

3

/3 f

3

/3 f

, and we find that k is a

= i G(η)(1 + b(η, f ))

(37)



when f = 0 and b(η, f ) is where the analytic function G is a multiple of − ψψ −ikψ +ikψ smooth in f , analytic in k and O( f ) as f ↓ 0. This quantity cannot have a zero at 3 k = k0 since according to (11), near k0 e−4iη /3 f = −2k/(F(k) + O( f )) and F(k) is analytic near k0 . Note that (37) is exactly the same equation as (28) which arose when we found the resonances near the positive real axis in a neighborhood of a point where the reflection coefficient did not vanish. As before it has a solution given by the fixed point of a contraction. Thus setting log i G(η) = −l(η) + iθ (η) we have Theorem 20 Suppose k0 = e−iπ/3 η0 with η0 > 0 a point where G does not have a pole. Define η0 ( j) = (3π f j)1/3 . Then the resonances near k0 are given by k( j) = e−iπ/3 η( j) with   η( j) = η0 ( j) − 4−1 θ (η0 ( j))/η0 ( j)2 + il(η0 ( j))/η0 ( j)2 f + O( f 2 ).

The situation where ψψ −ikψ has a pole at k0 can be treated in essentially the same +ikψ way as was done in Theorem 18.

8 Higher Dimensions 2 We generalize equation (4). Writing H0, f = p12 + f x1 + p⊥ , we write L 2 (Rd ) as a direct integral of functions of the perpendicular momentum with values in L 2 (R). We assume the potential V is factorized as V = V1 V2 where the V j are real, bounded, measurable with compact support in B R = {x ∈ R : |x| < R}. Thus the weighted resolvent of H0, f has an integral kernel

K 0, f (x1 , y1 ; p⊥ , z) = f −1/3 V1 (x1 , ·)



2 Ai( f 1/3 (x1 − t))(t − (z − p⊥ )/ f )−1  Ai( f 1/3 (y1 − t))dt V2 (y1 , ·).

We now analytically continue from the upper to the lower half plane and find the kernel of (V1 (H0, f − z)−1 V2 )c K 0, f,c = Q˜ + K 0, f

Resonances in the One Dimensional Stark Effect in the Limit of Small Field

159

where Q˜ is the integral kernel in the variables (x1 , y1 ) of an operator Q: 2 2 ˜ 1 , y1 ) = 2πi f −1/3 V1 (x1 , ·) Ai( f 1/3 (x1 − (z − p⊥ Q(x )/ f ) Ai( f 1/3 (y1 − (z − p⊥ )/ f )V (y1 , ·).

(38)

As f ↓ 0, f 1/6 3 2 )/ f ) = √ e2iη /3 f e−iηx1 −iπ/4 (1 + O( f )) Ai( f 1/3 (x1 − (z − p⊥ 2 πη  2 where η = z − p⊥ . Here Imz < 0. In the following we think of using the Fourier transform to diagonalize p⊥ , so p⊥ becomes ξ⊥ . First assume that arg z ∈ [−π, −2π/3). If arg(z − ξ⊥2 ) = −π + φ, with φ ∈ [0, π/3 − δ], Re iη3 = −|η|3 cos(3φ/2) = −|η|3 sin(3δ/2). Taking into account the compact support of V j which we assume to be in the ball B R = {x : |x| < R} and assuming Imz < −δ < 0 we have for |x1 | < R | Ai( f 1/3 (x1 − (z − ξ⊥2 )/ f )| ≤ C f 1/6 |η|−1 e−

2|η|3 sin(3δ/2) +|η|R 3f

≤ c1 f 1/6 e−c2 / f for some positive constants c j . We thus have Theorem 21 Given δ > 0, if f is small enough there are no resonances of H f in the region {z : arg z ∈ [−π + δ, −2π/3 − δ], Imz ≤ −δ < 0}. Proof Resonances are points z so that ker(I + Q(I − V1 (H f − z)−1 V2 )) is not {0}. From the above estimates  |(ψ, Qφ)| = | (V1 ψ(x1 ), F(x1 , y1 )V2 φ(y1 ))d x1 dy1 |  ≤

||V1 ψ(x1 )||c12 e−2c2 / f ||V2 φ(y1 )||d x1 dy1

where F(x1 , y1 ) is Q˜ with the V j ’s removed and we have used ||F(x1 , y1 )|| ≤ c12 e−2c2 / f for |x1 |, |y1 | < R. Thus by the Schwarz inequality |(ψ, Qφ)| ≤ R||V1 ψ||||V2 φ||c12 e−2c2 / f which implies (I + Q(I − V1 (H f − z)−1 V2 ) is invertible for small f .



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Now consider the region arg z ∈ (−2π/3 + δ, −δ) for small δ > 0. This is a region where progress should be reasonably simple compared to the remaining regions along the positive real axis and the line arg z = −2π/3. Unfortunately we have nothing to report about the existence of resonances in this region. But we give some information about the operator which needs to be examined to make further progress. We modify Q˜ slightly to define an operator M: Mψ(x1 , ·) = 2πi f −1/3 1[−R,R] (x1 )

 2 − z)/ f )) Ai( f 1/3 (x1 + ( p⊥ 2 − z)/ f ))1 Ai( f 1/3 (y1 +( p⊥ [−R,R] (y1 )ψ(y1 , ·)dy1 .

M has the virtue that it is a multiple of a projection: M 2 ψ(x1 , ξ⊥ ) =

 (2πi f −1/3 )2 1[−R,R] (x1 ) Ai( f 1/3 (x1 + (ξ⊥2 − z)/ f )) Ai( f 1/3 (y1 + (ξ⊥2 − z)/ f ))  1[−R,R] (y1 ) Ai( f 1/3 (y1 + (ξ⊥2 − z)/ f )) Ai( f 1/3 (w1 + (ξ⊥2 − z)/ f )) 1[−R,R] (y1 )ψ(w1 , ξ⊥ )dw1 dy1 = F(ξ⊥2 )Mψ(x1 , ξ⊥ ) where with η =



F(ξ⊥2 ) = 2πi f −1/3

z − ξ⊥2 ,



R −R

e4iη /3 f (sin(2η R)/η)(1 + O( f )). 2η 3

Ai( f 1/3 (x1 + (ξ⊥2 − z)/ f ))2 d x1 =

2 2 Thus M = F( p⊥ )P f where P 2f = P f and P f commutes with F( p⊥ ). Even though 2 we have not indicated it, of course F( p⊥ ) also depends on f and z. We note that Q = V1 M V2 so using the fact that σ (AB) \ {0} = σ (B A) \ {0} we have

Proposition 22 The resonances of H f in the lower half plane are the points z such that the operator 2 )P f (V − V (H f − z)−1 V )P f F( p⊥ has eigenvalue −1. We give some of the asymptotics of P f and F(ξ⊥2 ): Lemma 23 For Imz < 0, ||P f − P0 || = O( f ) where P0 has kernel as an operator on L 2 functions of the perpendicular momentum and the x1 variable

Resonances in the One Dimensional Stark Effect in the Limit of Small Field

161

1[−R,R] (x1 )e−iηx1 (sin 2η R/2η)−1 e−iηy1 1[−R,R] (y1 ). Here η =



z − ξ⊥2 is always in the 4th quandrant.

√ estimates when Rez + |Imz|/ 3 > 0 The function F(ξ⊥2 ) satisfies the following √ and Imz < 0. Set α 2 = Rez + |Imz|/ 3. Then √

|F| ≥ c1 e(2/

3)1/2 |Imz|1/2 (α 2 −ξ⊥2 )/ f

if ξ⊥2 < α 2 . √

|F| ≤ c2 e−(2/

3)1/2 |Imz|1/2 (ξ⊥2 −α 2 )/ f

if β > ξ⊥2 > α 2 . √

|F| ≤ c3 e−(1/

3)1/2 |Imz|1/2 (ξ⊥2 −α 2 )/ f

if ξ⊥2 > β. Note that arg(z − ξ⊥2 ) = −2π/3 exactly when α 2 = 0. When z − ξ⊥2 crosses the line with argument −2π/3 we cross from the region where F blows up as f ↓ 0 to the region where it decays exponentially.

Appendix 1: The Airy Function Ai(z) in the Sector arg z ∈ (−2π/3 + δ, −δ) We believe the results in this appendix are known but could not find a suitable reference. √ √ 3 Theorem 24 With η = w, arg η ∈ (−π/2, π/6), ηe2η /3 Ai(w) extends to an analytic function, J , of ζ −1 , ζ = 2η3 /3 with arg ζ −1 ∈ (−π/2, 3π/2). The function J and all its derivatives extend continuously to the origin in this sector. Proof We start with the integral representation Ai(w) = lim R→∞ (2π )−1  R i(sw+s 3 /3) √ ds where w ≥ 0 and deform the contour to s = i w + u with u ∈ R −R e to obtain e−2η /3 Ai(w) = 2π 3



∞

e−ηu eiu 2

3

/3

du

−∞

√ with η = w. We can analytically continue to arg η ∈ (−π/2, π/2). Thus using polar coordinates with ζ −1 = 3w −3/2 /2 = r eiθ , we have √ 3 2π ηe2η /3 Ai(w) = e−iθ/6



∞

−∞

exp(−e−iθ/3 t 2 )g(t 6r )dt

162

R. Froese and I. Herbst

√ where g(ψ) = cos( 2ψ/27) is a C ∞ function of its argument. With an integration by parts it is easy to verify the polar form of the Cauchy-Riemann equations ((∂/∂r + ir −1 ∂/∂θ )J = 0). We take θ = arg ζ −1 ∈ (−π/2, 3π/2). Because J satisfies the Cauchy-Riemann equations J (n) is given by e−inθ ∂ n J/∂r n which is easily shown to have a limit as r → 0 which is independent of θ .  Thus in particular J has an asymptotic expansion J (ζ −1 ) ∼

∞

cn ζ −n

n=0

which can be differentiated term by term.

Appendix 2: The Airy Function Ai(z) in the Sector arg z ∈ [−π, −π + δ] We follow [2] but the latter reference does not go far enough for our purposes. We write the Airy function Ai(w) for w real as Ai(w) = (2π )−1 lim



R

R→∞ −R

e−i(t

3

/3+wt)

dt.

We set w = f 1/3 (L − z/ f ) = −(z/ f 2/3 )(1 − f L/z) where we first keep z real and positive. We set η = z(1 − f L/z) = z − f L. With a change of variable we have Ai(w) = (2π f 1/3 )−1



e(−i/ f )(s

3

/3−ηs)

ds.

(39)

C

Here we have used the fact the η is real to move the contour into the lower half plane so that C is just s(t) = t − iα with t ∈ R and α > 0. We then see that the Airy function is analytic in η for η in a neighborhood of a real point. We distort the contour further. We use steepest descents near the critical points of the exponential where (s 3 /3 − √ √ ηs) = s 2 − η = 0, namely the points s = ± η. Thus near + η we write s = ζ + √ 3 η. Note that at this critical point we have s /3 − ηs = −2η3/2 /3 and −i(s 3 /3 − √ ηs + 2η3/2 /3) = −i(ζ 3 /3 + ηζ 2 ). The steepest descents curve near ζ = 0 will √ 3 come from setting Re(ζ /3 + ηζ 2 ) = 0. We will use part of the following contour √ √ for s near η: We solve the equation Re(ζ 3 /3 + ηζ 2 ) = 0 and find with ζ = x + i y, y= We have

ν+



−x(γ + x/3) ν2

+ (γ + x)(γ + x/3)

Resonances in the One Dimensional Stark Effect in the Limit of Small Field

163

∞

√ −i( ηζ 2 + ζ 3 /3) = −b2 x 2 + Bn x n n=3

√ where b2 = 2γ −2 |η|( |η| − ν). The series converges for |x| < γ (see [2]). We have the following estimates in the indicated regions: −i(ζ 3 /3 +

 √ 2 ηζ ) ≤ −(γ /3)2 x 2 / ν 2 + (γ + x)(γ + x/3); 0 ≥ x ≥ −γ , ν ≥ 0 ≤ −(γ 2 /18|η|)x 2 ; 0 ≤ x ≤ 4γ , ν ≥ 0 ≤ −γ x 2 ; 0 ≤ x ≤ 10γ , 0 ≥ ν ≥ −γ /10 ≤ −γ x 2 ; −γ ≤ x ≤ 0, 0 ≥ ν.

√ √ For s near − η we write s = ζ − η which gives −i(s 3 /3 − ηs − 2η3/2 /3) = √ √ −i(ζ 3 /3 − ηζ 2 ). Setting Re(ζ 3 /3 − ηζ 2 ) = 0 and ζ = x + i y we find  y = x(ν + ν 2 + (γ − x)(γ − x/3))(γ − x)−1 . We find −i(ζ 3 /3 −

√

∞

ηζ 2 ) = −b˜ 2 x 2 +

B˜ n x n

n=3

where b˜ 2 = 2γ −2 |η|(|η|1/2 + ν). The series converges for |x| < γ . We have the following estimates in the indicated regions: −i(ζ 3 /3 −

√

 ηζ 2 ) ≤ −γ x 2

γ − x/3 ; 0 ≤ x < γ, ν ≥ 0 γ −x

γ − x/3 ; 0 ≤ x < γ, ν < 0 ≤ −γ x 2 (y/x) ≤ −γ x 2  2 2 ν + (γ − x)(γ − x/3) ≤ −γ x 2 /5; −10γ ≤ x ≤ 0, 0 ≤ ν < γ ≤ −(2γ /5)x 2 ; −8γ ≤ x ≤ 0, −γ /3 ≤ ν ≤ 0.

We write Ai(w) = (2π f 1/3 )−1 (



e(−i/ f )(s

3

/3−ηs)

C−

 ds +

e(−i/ f )(s

3

/3−ηs)

We first consider the integral over the part of the contour C+ near  C+

e

(−i/ f )(s 3 /3−ηs)

ds = (2π f

1/3 −1 (2iη3/2 /3 f )

) e

ds).

C+

 B+

e(−i/ f )(ζ

3

√

η:

√ /3+ ηζ 2 )

dζ.

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R. Froese and I. Herbst

√ √ To obtain the part of the contour in the s variable near η we take s = ζ + η. This gives us the contour B+ with ζ = x + i y where x ∈ (β, ∞), y = −α , where √ as mentioned above α > ν. We thus see that this integral is analytic in η as long as −1/2 f > 0. We will see that after multiplying by f we can take the limit as f → 0 uniformly for η near a point k0 > 0. We deform the contour to obtain a new contour + where ζ = x + i y, x ∈ (−γ /2, γ /2), y =

ν+



−x(γ + x/3) ν2

+ (γ + x)(γ + x/3)

.

Thus we have I+ = (2π f 1/3 )−1

J+ = f

−1/2



γ /2



e−i(ζ

3

√ /3+ ηζ 2 )/ f

+

ef

−1

 n (−b2 x 2 + ∞ n=3 Bn x )

−γ /2

dζ =

f 1/6 J+ 2π

(1 + idy/d x)d x

We expand the integrand in a convergent power series keeping the e−b  J+ =

γ f −1/2

e−b

−γ f −1/2

2 2

x

(1 +

2 2

x

to obtain

∞

∞ ∞



( Bn f −1+n/2 x n )k /k!)(1 + i am f m/2 x m )d x k=1 n=3

m=0

The odd powers of x do not contribute and thus we have  J+ = 2(1 + ia0 )

γ f −1/2

e−b

2x2



dx +

0

cn,m f (|n|+m−2l)/2



γ f −1/2

e−b

2x2

x |n|+m d x

0

m+|n|/3≥1,l≥1

Here n is a multi-index n = (n 1 , n 2 , . . . , n l ) and |n| = n 1 + · · · + n l . Each n j ≥ 3 and thus the power of f , (|n| + m)/2 − l is positive and since |n| + m is even this  γ f −1/2 −b2 x 2 |n|+m e x d x is C ∞ in power is a positive integer ≥ (l + m)/2 . Note that 0 2 2 f near 0 as long as we define f −n e−b γ / f to be 0 at f = 0. Thus J+ is C ∞ for f ≥ 0 in the sense that the derivatives have limits as f ↓ 0. In the limit  f ↓ 0 the first term can be calculated (with some effort - see also [2]) to be e−iπ/4 π/η1/2 . We now connect up this curve to infinity as follows: We take x + i y = γ /2 + t − i y(γ /2) with t ∈ [0, ∞). For simplicity take |ν| < γ /10. It is then easy to see that √ the real part of −i(ζ 3 /3 + ηζ 2 ) ≤ −C < 0 independent of t ≥ 0 and independent of η for η near a real point k0 > 0. If the extended curve is called ˜ + This is easily seen to imply that I˜+ = (2π f 1/3 )−1

 ˜ +

e−i(ζ

3

√ /3+ ηζ 2 )/ f

dζ =

f 1/6 ˜ J+ 2π

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165

√ √ with J˜+ a C ∞ function in f for f 0 > f ≥ 0 and analytic in η for η near a point k0 > 0. √ The curve near the critical point − η can be handled similarly. We set s = √ √ 3 3 ζ − η giving −i(s /3 − ηs) = −i(ζ /3 − ηζ 2 ) − 2iη3/2 /3 = and ζ = x + i y. √ We demand that for x ∈ (−γ /2, γ /2) we have Re[−i(ζ 3 /3 − ηζ 2 )] = 0. This results in y=

x(γ − x/3)  . 2 −ν + ν + (γ − x)(γ − x/3)

We connect this curve to infinity and to the imaginary axis just as in the case where we extended the curve + . The result is the same. It remains to connect the two curves on  the imaginary axis. The curve on the  left ends on the imaginary axis at s = i(2ν + ν 2 + 5γ 2 /12) or ζ = γ + i(ν + ν 2 + 5γ 2 /12). The curve on the 2 2 right ends on the imaginary axis at s = i(−2ν  + ν + 5γ /12). Thus we extend 2 2 the curve on the left as ζ = γ + i(u + ν + ν + 5γ /12) or s = γ + i(u + 2ν +  ν 2 + 5γ 2 /12) with u going from 0 to -4ν. It is not hard to see that this piece is C ∞ in f for f ≥ 0 with the function and all derivatives equal to zero at f = 0. (Of course the derivative at zero is the right hand derivative.) It is convenient in the estimates to √ restrict |ν| < γ /10. Even though the contours involve η in a non-analytic way, it √ is not hard to see that they can be distorted in such a way that given η0 the contour √ can be chosen to depend on this quantity but then η can be varied in a small disk around this point and the result is analytic in the disk with estimates similar to what we have derived. Thus what we have shown is that 1 3/2 3/2 (e2iη /3 f a1 (k, f ) + ie−2iη /3 f a2 (k, f )) Ai(w) = f 1/6 e−iπ/4  2 π η1/2 where a j (k, f ) is C ∞ in f for f 0 > f ≥ 0 and analytic in k for k near a point on the positive real axis. Since the limits involved in taking derivatives in f converge uniformly in k, these derivatives are also analytic in k in a neighborhood of a positive real point. We have a j (k, 0) = 1. We also need the derivative of Ai(w) with respect to the spatial variable which we have called L in this appendix. Note that in (39) L occurs only in η = z − f L so that differentiation with respect to L brings down −is in this integral. In the integral √ √ near + η the change of variable is s = ζ + η. −iζ contributes something of at √ most order f while the term −i η gives the main contribution. A similar analysis √ √ near the critical point − η shows that the main contribution is a factor of +i η. Thus we obtain  1/2 3/2 3/2 1/6 −iπ/4 η (−ie2iη /3 f a˜ 1 (k, f ) − e−2iη /3 f a˜ 2 (k, f )) (d/d L) Ai(w) = f e 4π

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where a˜ j (k, f ) is C ∞ in f for f 0 > f ≥ 0 and analytic in k for k near a point on the positive real axis. Since the limits involved in taking derivatives in f converge uniformly in k, these derivatives are also analytic in k in a neighborhood of a positive real point. We have a˜ j (k, 0) = 1. We now must do similar estimates with A1 (x) for z near the line arg = −2π/3. Here the argument of the Airy function Ai is w = f 1/3 (x − z/ f ). We set z 1 = e2π/3 z which is near a positive real point and η = z 1 (1 − f x/z) which is also near a point on the positive real axis when f is small. Thus after a change of variable we can write  3 1/3 −1 e(−i/ f )(s /3−ηs) ds. (40) Ai(w) = (2π f ) C

where the contour C is the same as above. Thus the same analysis as above works in this situation. Note that when we differentiate with respect to the spatial variable the η dependence on x gives a contribution to the exponential from −iηs/ f of i xe2πi/3 . Thus differentiation with respect to x brings down a factor of ie2πi/3 s, and thus a √ √ main contribution of ηie2πi/3 for the integral near the critical point η. Similarly √ 2πi/3 for the integral near the critical point there is a main contribution of − ηie √ − η.

References 1. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Department of Commerce, National Bureau of Standards (1964) 2. I. Herbst, J. Rama, Instability of pre-existing resonances under a small constant electric field. Ann. Henri Poincaré 16, 2783–2835 (2015) 3. I. Herbst, R. Mavi, Can we trust the relationship between resonance poles and lifetimes? J. Phys. A: Math. Theor. 49(19) (2016) 4. A. Jensen, K. Yajima, Instability of resonances under Stark perturbations. arXiv:1804.05620 5. R. Froese, I. Herbst, Resonances - lost and found. J. Phys. A: Math. Theor. 50, 405201 (2017) 6. E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators. Comm. Math. Phys. 78 (1981) 7. P. Perry, I.M. Sigal, B. Simon, Spectral analysis of N-body Schrödinger operators. Ann. Math. 114 (1981) 8. S. Graffi, V. Grecchi, S. Levoni, M. Maioli, Resonances in the one-dimensional Stark effect and continued fractions. J. Math. Phys. 20 (1979) 9. I. Herbst, Dilation analyticity in an electric field. Comm. Math. Phys. 69 (1979) 10. I. Herbst, B. Simon, Dilation analyticity in an electric field, II: the N-body problem, Borel summability. Comm. Math. Phys. 80 (1981) 11. S. Graffi, V. Grecchi, Resonances in Stark effect and perturbation theory. Comm. Math. Phys. 62 (1978) 12. S. Graffi, V. Grecchi, Resonances in Stark effect of atomic systems. Comm. Math. Phys. 79 (1981) 13. E. Harrell, B. Simon, The mathematical theory of resonances whose widths are exponentially small. Duke Math. J. 47, 845–902 (1980) 14. E. Harrell, N. Corngold, B. Simon, The mathematical theory of resonances whose widths are exponentially small, II. J. Math. Anal. Appl. 99, 447–457 (1984)

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15. J. Howland, Imaginary part of a resonance in barrier penetration. J. Math. Anal. Appl. 86, 507–517 (1982) 16. J.R. Taylor, Scattering Theory (John Wiley and Sons, N.Y., 1972), pp. 134–139 17. E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw Hill, N.Y., 1955), p. 150 18. E. Korotyaev, Resonances for 1D stark operators. J. Spec. Th. 7, 669–732 (2017) 19. E. Korotyaev, Asymptotics for resonances for 1d Stark operators. arXiv:1705.08072

On the Spectral Gap for Networks of Beams Pavel Kurasov and Jacob Muller

To the memory of Erik Balslev

Abstract A notion of standard vertex conditions for beam operarators (the fourth derivative) on metric graphs is presented, and the spectral gap (the difference between the first two eigenvalues) for the operator with these conditions is studied. Upper and lower estimates for the spectral gap are obtained, and it is shown that stronger estimates can be obtained for certain classes of graphs. Graph surgery is used as a technique for estimation. A geometric version of the Ambartsumian theorem for networks of beams is proved. Keywords Metric graphs · Spectral gap · Beams 2010 Mathematics Subject Classification Primary 34L15 · 81U40 · Secondary 34L40 · 74K10 · 81V99

1 Introduction In this paper we discuss the problem of estimating the spectral gap for beam oper4 ators. These are operators on metric graphs with differential expression dxd 4 on a certain domain of functions satisfying in addition vertex conditions. We focus on a particular set of vertex conditions, which we call standard, which can be seen as an analogue of the standard vertex conditions for quantum graphs. There are a number P. Kurasov (B) · J. Muller Institute of Mathematics, Stockholm University, Stockholm, Sweden e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_8

169

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of possible candidates for the standard beam operator conditions (see for instance [5, 6]), but we opt for the choice which corresponds to the positive operator with the largest spectral gap. Some estimates for the spectral gap of the standard beam operator are found by minimisation of the Rayleigh quotient. We also extend the notion of graph surgery for quantum graphs [1, 4, 7] to beam operators. In this approach, one uses topological alterations of graphs affecting the eigenvalues in a predictable manner, and this leads to explicit spectral estimates. With these techniques, certain classes of metric graphs permit stronger upper estimates (bipartite graphs), whilst others permit stronger lower estimates (Eulerian graphs). Studies of properties of beam operators with other specific vertex conditions include [3], and spectral asymptotics for general vertex conditions are investigated in [8]. Beam operators correspond to physical networks of beams connected together in some way. Eigenstates of the beam operator represent stationary states of the systems, with the functions themselves describing the transverse displacement of points along the beam (see [9] for more details). The problem of maximising the spectral gap for beam operators also has the significant practical interpretation of increasing the gap between resonant frequencies in a physical system. This may be useful in the context of networks of beams where occurrence of resonant frequencies may have a detrimental impact (for instance Harrington rods for scoliosis).

2 The Operator Let  be a compact finite connected metric graph with edge set E (consisting of N edges E 1 , . . . , E N , identified as intervals from different copies of R) and vertex set V (where the vertices Vm ∈ V are identified with disjoint subsets of the set {x1 , . . . , x2N } of endpoints of the edges). We do not require that the metric graph  can be embedded in Rn . A beam operator on  is an operator A() in L 2 () with differential expres4 sion dxd 4 acting on functions from W24 (\V) satisfying certain vertex conditions. These vertex conditions should not mix between vertices, i.e. the boundary terms at endpoints in one vertex should not depend on those of endpoints in another vertex. Many different vertex conditions can be considered (see [5]), but we are looking for an analogue of standard vertex conditions1 introduced for the Laplacian (second derivative operator) at every vertex Vm ∈ V: 

1 Standard

u is continuous at Vm ,  x j ∈Vm ∂u(x j ) = 0,

conditions are also called Kirchhoff, Neumann or natural by some authors.

(1)

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where the sum is taken over all endpoints joined at the vertex and ∂u denote the normal derivatives.The best way to justify the standard conditions (1) is to consider the Dirichlet form  u  2 dx defined on the functions from W21 ( \ V) which are assumed in addition to be continuous at the vertices. This form is closed and bounded 2 from below. The corresponding operator is L st () = − dxd 2 defined on the functions 2 from W2 ( \ V) satisfying the standard conditions (1) at every vertex. The formal quadratic form of the (positive) beam operator is given by a(u, u) = u  2 =

 

|u  |2 dx.

(2)

with u ∈ W22 ( \ V). One may impose vertex conditions on functions in the domain of this form which then corresponds to a certain positive beam operator. The spectrum of this operator is purely discrete and non-negative, since each operator is a finite rank perturbation (in the resolvent sense) of the operators on single edges with any self-adjoint boundary conditions (see [2, 8]). If no vertex conditions are assumed, 4 then the corresponding operator A() is just dxd 4 defined on the functions satisfying the following conditions at all endpoints x j of the edges: 

∂ 2 u(x j ) = 0, ∂ 3 u(x j ) = 0.

(3)

It is clear that in this case A() is just an orthogonal sum of similar operators on the edges E n : N A(E n ). A() = ⊕n=1 This operator does not correspond to the graph , but rather to a collection of nonconnected edges. In order to produce a model where the edges are connected, one has to introduce vertex conditions on functions in the domain of the quadratic form. It is natural to enforce the continuity condition u(xi ) = u(x j ), provided xi , x j ∈ Vm .

(4)

This condition means that the beams are touching at the vertices. If no further conditions are imposed on the values of the functions at the endpoints (i.e. any further conditions concern only the derivatives) then certainly the constant function ψ1 (x) ≡ 1 lies in the domain of the quadratic form and hence also in the domain of the corresponding positive operator A(). Consequently the first eigenvalue of A() is zero, and the spectral gap—the difference between the first two eigenvalues—is simply equal to the second eigenvalue: λ2 (A()) − λ1 (A()) = λ2 (A()).

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In applications, one may be interested in maximising the distance between eigenvalues. A good starting point would be to maximise the spectral gap. In what follows we are going to consider the quadratic form (2) defined on the functions from W22 ( \ V) satisfying the following conditions: 

u(xi ) = u(x j ), provided xi , x j ∈ Vm ; ∂u(x j ) = 0.

(5)

4

The corresponding operator Ast () = dxd 4 is defined on the domain of functions from W24 ( \ V) satisfying the vertex conditions ⎧ ⎨ u(xi ) = u(x j ), provided xi , x j ∈ Vm ; ∂u(x j ) = 0; ⎩ 3 x j ∈Vm ∂ u(x j ) = 0

(6)

at every vertex Vm . 4

Definition The operator Ast () = dxd 4 with domain equal to the set of functions from W24 ( \ V) satisfying the vertex conditions (6) is called the standard beam operator. We refer to these vertex conditions as standard. This operator is uniquely determined by the graph . One may refer to its spectrum as the spectrum of the graph.

3 On Spectral Properties of Ast (). The standard beam operator Ast () has the smallest quadratic form domain among those obtained by imposing vertex conditions on functions in W22 (); the eigenvalues 4 (and in particular the spectral gap) of the corresponding positive operator dxd 4 are the largest possible as is clear from the following well-known result: Proposition 1 Let A be a self-adjoint operator with quadratic form a, semi-bounded from below, with discrete spectrum. The eigenvalues λ j (A) of A satisfy λ j (A) = =

min

max

V j ⊂dom(a), u∈V j dim(V j )= j

max

V j−1 ⊂dom(a), dim(V j−1 )= j−1

a(u, u) u2

min

u∈V ⊥ j−1

a(u, u) . u2

(7)

(8)

In particular, given a compact finite connected graph , the spectral gap of Ast () can be calculated as

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 |u  |2 dx , λ2 (A ()) = min  2 u∈dom(a),   |u| dx st

(9)

 u dx=0

where now a is the quadratic form for Ast (). The operator Ast () is in fact the Friedrichs extension of the symmetric beam operator with conditions (3) and (5) imposed simultaneously, as discussed in [5]. It has already been mentioned that ψ1 (x) ≡ 1 is a ground state; if the graph is connected then the ground state is unique. To see this directly, assume that u is an eigenfunction corresponding to the eigenvalue λ = 0, then u (4) (x) = 0 ⇒ u(x) = an + bn x + cn x 2 + dn x 3 , x ∈ E n , for each edge E n . This  function also minimises the Rayleigh quotient (7), which means that it satisfies  u  2 dx = 0, and thus u  (x) = 0 ⇒ cn = dn = 0. The function u(x) = an + bn x satisfies the Neumann condition ∂u(x j ) = 0 at each endpoint x j if and only if bn = 0. The continuity condition at the vertices implies that all an are equal, i.e. that the function ψ is constant on the whole graph since graph is connected. One can get some simple upper bounds for the spectral gap by applying Proposition 1. Lemma 2 Let  be a compact finite connected metric graph with total length L() and let n be the length of the edge E n for each n. Then

π λ2 (A ()) ≤ 16 L() st

4  N n=1

L() n

3 .

(10)

If in particular  is a bipartite graph, then λ2 (Ast ()) ≤

π L()

4  N n=1

L() n

3 .

(11)

Proof Let u be the function, in the domain of the quadratic form of Ast (), defined by u(x) = cos( 2πnx ) on each edge E n = [0, n ]. Then  8π 4 

4  N |u  |2 dx L() 3 π n 3n   = = 16 .  n 2 L() n=1 n  |u| dx n 2 Since clearly 1, u =

 

u(x) dx = 0, the estimate (10) follows from Proposition 1.

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If  is bipartite, then applying the same approach to the function u defined by u(x) = cos( πnx ) on E n (parameterised in the appropriate direction such that u is continuous on ) gives the second estimate (11).  The estimates from Lemma 2 are quite bad if  contains some edges which are much shorter than the others. In that case better estimates may be found, for instance by comparison with eigenvalues of other beam operators. Let AF () be the beam operator defined on functions from W24 (\V) satisfying the vertex conditions  u(x j ) = 0, (12) ∂u(x j ) = 0, at every endpoint x j . This is the Friedrichs extension of the minimal symmetric beam operator whose domain consists only of functions from W24 (\V) satisfying (12) and (3) simultaneously. Since Ast () is also a self-adjoint extension of this minimal operator, it follows that λk (Ast ()) ≤ λk (AF ()) for all k. Observe that the smallest eigenvalue of AF () is strictly positive: the eigenvalues of AF () are in fact the eigenvalues of AF (E n ) for each edge E n . Thus λ1 (AF ()) ≤ λ1 (AF (E 1 )) with 4 equality if E 1 is the longest edge. Solutions of dxd 4 ψ = λψ for λ = k 4 > 0 are of the form ψ(x) = aeikx + be−ikx + ce−kx + dekx . By imposing the vertex conditions of AF (E 1 ), one sees that λ = k 4 is an eigenvalue if and only if x = k1 is a positive root of

sin x − cos x + e−x − sin x − cos x + ex ≡ 4(1 − cos x cosh x). det sin x + cos x − e−x sin x − cos x + ex Let μ > 0 be such that x = π μ is the smallest positive root. By numerical approximation, one finds that μ  1.506. Thus λ1 (AF (E 1 )) ≤ μ4

π 1

4 .

Let ψ1 be the corresponding eigenfunction of AF (E 1 ). It can be extended to a function φ in the domain of the quadratic form of the standard beam operator Ast () such that 1, φ = 0 as follows: ⎧ ⎪ if x ∈ E 1 , ⎨ψ1 (x) φ(x) = − 21 ψ1 ( 12x ) if x ∈ E 2 , ⎪ ⎩ 0 otherwise. Here we have assumed that E 1 = [0, 1 ] and E 2 = [0, 2 ] without loss of generality. Then   1 + ( 21 )5 E1 |ψ1 |2 dx 1 + ( 21 )5 |φ  |2 dx    = = λ1 (AF (E 1 )), 2 2 1 + 21 1 + 21  |φ| dx E 1 |ψ1 | dx

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(with the last equality by (7)), and so by Proposition 1, λ2 (A ()) ≤ μ π F

4

4

1 1 1 1 1 − 3 + 2 2− + 4 41 1 2 1 2 1 32 2

.

(13)

For 1 + 2 fixed, this estimate is optimal when 1 = 2 . Assume without loss of generality that 1 > 2 (indeed we may as well assume that 1 is the longest edge).2 This estimate depends only on the lengths of two edges, whilst the estimate (10) tends to +∞ if one shrinks any of the other edges to 0.

4 Surgery for Standard Beam Operators Graph surgery is an umbrella term used to describe a number of operations that one can perform on a graph—like cutting through or gluing together vertices, lengthening edges, or adding edges—whose effects on the spectrum of the operator on the graph (for instance increasing or decreasing eigenvalues) are predictable. A detailed overview of such results can be found in [1, 7]. These techniques can of course be employed for beam operators, and analogous results to those for quantum graphs can be obtained. We present just one of those here, and we shall soon see that it it is convenient for obtaining lower estimates for the spectral gap. Theorem 3 If the graph ˜ is obtained from  by cutting through one or several vertices, then the following inequality holds for the eigenvalues of the standard beam operators: ˜ k = 1, 2, . . . (14) λk (Ast ()) ≥ λk (Ast ()), Proof Every function on the graph  can naturally be considered as a function on ˜ Moreover, if the function satisfies standard conditions on , then its image on ˜ . ˜ Hence the set of n-dimensional subspaces of the satisfies standard conditions on . domain of the quadratic form increases when vertices are cut. Inequality (14) then follows from (7) from Proposition 1.  One may repeat the proofs of several further theorems contained in [1] to obtain additional surgery-related eigenvalue inequalities.

that (13) is certainly better than the upper estimate μ4 ( π2 )4 which can be directly seen as the ground state eigenvalue of AF (E 2 ) and thus an eigenvalue of AF () (second or higher).

2 Observe

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5 Estimates for the Spectral Gap For standard Laplacians on metric graphs we have the following estimate: λ2 (L st ()) ≥

π L()

2 .

(15)

This estimate can be improved in the case of Eulerian graphs (having even degrees of all vertices):

2 π , (16) λ2 (L st ()) ≥ 4 L() or further for two-connected graphs. These are obtained in [7] using graph surgery principles for quantum graphs. A similar result can be achieved by this method for standard beam operators: Theorem 4 Let  be a connected finite compact metric graph with total length L() and let Ast () be the corresponding standard beam operator described by vertex conditions (6). The spectral gap for this operator can be estimated as: λ2 (Ast ()) ≥

π L()

4 .

(17)

If the graph  is Eulerian (i.e. the degrees of all vertices are even), then the above estimate can be improved:

π λ2 (A ()) ≥ 16 L()

4

st

.

(18)

Proof We follow closely the proof presented for the Laplacian in [7]. Let ψ2 be an eigenfunction corresponding to λ2 (Ast ()). Consider the graph  2 obtained from  by adding to each edge E n a parallel edge E N +n of the same length. Let Ast ( 2 ) be the standard beam operator on  2 . The λ2 (Ast ())-eigenfunction ψ2 can be extended to  2 by assigning the same values on the new parallel edges. Obviously the new function is an eigenfunction of Ast ( 2 ) with the same eigenvalue (as it satisfies the vertex conditions on  2 and the differential equation on the edges). It follows that λ2 (Ast ()) ≥ λ2 (Ast ( 2 )). The degree of every vertex of graph  2 is even, and so the graph is Eulerian and there exists a closed path P2L coming along every edge in  2 precisely once. This path can be seen as a graph formed of 2N edges connected pairwise at 2N vertices, where standard conditions for the beam operator are assumed. One can obtain P2L by cutting through vertices in  2 , hence we have

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λ2 (Ast ( 2 )) ≥ λ2 (Ast (P2L )). The eigenvalues of Ast (P2L ) are relatively difficult to calculate: since the corresponding eigenfunction does not satisfy the differential equation at the (degree two) vertices, vertex conditions are assumed instead. These conditions on functions u ∈ W24 (P2L ) are as follows: • continuity of u; • Neumann condition: u  (x j ) = 0 (a pair of conditions at each vertex); • continuity condition for the third derivative u  . The functions from the domain of the quadratic form satisfy just the first two conditions at every vertex on the path. Consider the beam operator A(C2L ) on the loop C2L of length 2L. The loop can be seen as one edge [0, 2L] with the endpoints identified. The corresponding vertex conditions for functions u ∈ W24 (C2L ) are • • • •

continuity of u, continuity of the first derivative u  , continuity of the second derivative u  , continuity of the third derivative u  .

The functions from the domain of the quadratic form satisfy just the first two conditions. Every function satisfying Neumann conditions at a degree two vertex automatically has continuous first derivative there. Hence the domain of the quadratic form for A(C2L ) is larger than the domain of the quadratic form of Ast (P2L ). The following inequality follows: λ2 (Ast (P2L )) ≥ λ2 (A(C2L )). The chain of inequalities together with the fact that λ2 (A(C2L )) =

π L()

4

implies the lower estimate in (17). It is enough to notice that if the graph  is already Eulerian, then there is no necessity to double it building  2 . By cutting through vertices the graph  can be turned into a path PL of the length L. Hence we have;

π λ2 (A ()) ≥ λ2 (PL ) ≥ λ2 (CL ) = 16 L() st

4 . 

Remark In this instance, the estimates in Theorem 4 can be obtained more easily by recalling that Ast () is the Friedrichs extension of the symmetric beam operator with

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continuity of functions and all higher derivatives zero at vertices. The quadratic form domain of all other extensions of this symmetric operator contain the quadratic form domain of Ast (). The beam operator (L st ())2 is one such extension its spectral gap is λ2 ((L st ())2 ) = (λ2 (L st ()))2 . Since both of these beam operators have zero as their first eigenvalue, it follows that the spectral gap of Ast () is necessarily greater than (or equal to) that of (L st ())2 , whence λ2 (Ast ()) ≥ (λ2 (L st ()))2 . The estimates (17) and (18) are then just a direct corollary of the standard Laplacian estimates (15) and (16). Theorem 4 implies that Eulerian graphs have a tendency to have larger spectral gap, which means that their eigenfrequencies are larger. The factor is 16 compared to 4 for Laplacians. Lemma 5 Let C2L be the loop of length 2L. Then λ2 (Ast ([0, L])) = λ2 (Ast (C2L )) =

π 4 L

.

(19)

π 4 ) . The Proof Observe that Ast ([0, L]) = (L st ([0, L]))2 , so λ2 (Ast ([0, L])) = ( L st same relation with the Laplacian does not hold for A (C2L ). However, since C2L is Eulerian, by estimate (18) of Theorem 4, we have the lower bound λ2 (Ast (C2L )) ≥ π 4 π 4 ) . But also cos( πLx ) is a nontrivial eigenfunction, so indeed λ2 (Ast (C2L )) = ( L ) (L as claimed. 

Finally we prove that the estimate (17) is attained if and only if  is an interval. We argue in a similar manner to the proof of Theorem 3 in [7] for Laplacians. Theorem 6 Let  be a connected finite compact metric graph with total length L() and let Ast () be the corresponding standard beam operator described by vertex conditions (6). If

4 π st , (20) λ2 (A ()) = L() then  is an interval. Proof Let ψ2 be an eigenfunction of Ast () with eigenvalue λ2 (Ast ()). It can be extended to a function ψ˜ 2 on  2 by identifying it with ψ2 on the duplicated edges— ˜ E  = ψ2 | En where E n is the duplicate of E n in  2 . Then ψ˜ 2 is an ˜ En = ψ| that is ψ| n eigenfunction of Ast ( 2 ). Now ψ˜ 2 can be identified as a function from W24 (P2L ) satisfying conditions (5) at the vertices. This means in fact that ψ˜ 2 ∈ W22 (C2L ). By Lemma 5,   2 ˜  |ψ  (x)|2 dx C2L |ψ2 (x)| dx = λ2 (Ast (C2L )) =  2  2 dx ˜ 2 (x)|2 dx |ψ (x)| | ψ 2  C 2L

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and so ψ˜ 2 must be the first nontrivial eigenfunction of Ast (C2L ), which by Lemma 5 can, without loss of generality, be identified with cos( πLx ). The original graph  can be recovered by gluing together pairs of points on C2L where the values of ψ˜ 2 are equal. But since the preimage of every value in [−1, 1] under ψ˜ 2 consists of exactly two points on C2L , the gluing can only possibly done in one way, and  is an interval of length L: there could not possibly be any internal  vertices since cos( πLx ) only satisfies ψ  (x) = 0 at x = 0 and x = L. Remark The final gluing stage of this proof is in contrast with the equivalent proof for quantum graphs where there may be (removable) internal vertices of degree 2. This result is a geometric Ambartsumian-type theorem: given only the spectral gap (20) and the total length of the graph, one can recover the beam operator (and hence also the graph). Acknowledgements The work was supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld in the framework of the cooperation group on ‘Discrete and continuous models in the theory of networks’ and by the Swedish Research Council grant D0497301. We thank I. Popov for attracting our attention to this problem and explaining possible implications for the design of medical tools.

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). https://doi.org/10.1090/ tran/7864 2. M.S. Birman, M.Z. Solomjak, Spectral theory of self adjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series), note=Translated from the 1980 Russian original by S. Khrushchëv and V. Peller, D. Reidel Publishing Co., Dordrecht, (1987), xv+301, ISBN: 90-277-2179-3 3. B. Dekoninck, S. Nicaise, The eigenvalue problem for networks of beams. Linear Algebra Appl. 314(1–3), 165–189 (2000). https://doi.org/10.1016/S0024-3795(00)00118-X 4. L. Friedlander, Extremal properties of eigenvalues for a metric graph. Annales de l’institut Fourier 55, (2005). https://doi.org/10.5802/aif.2095 5. F. Gregorio, D. Mugnolo, Bi-Laplacians on graphs and networks. D. J. Evol. Equ. 11, (2019). https://doi.org/8-019-00523-7 6. J.-C. Kiik, P. Kurasov, M. Usman, On vertex conditions for elastic systems. Phys. Lett. A 379(34–35), 1871–1876 (2015). https://doi.org/10.1016/j.physleta.2015.05.017 7. P. Kurasov, S. Naboko, Rayleigh estimates for differential operators on graphs. J. Spectr. Theory 4(2), 211–219 (2014). https://doi.org/10.4171/JST/67 8. P. Kurasov, J. Muller, Higher order differential operators on metric graphs and almost periodic functions. Submitted 9. L.D. Landau, E.M. Lifshitz, Theory of elasticity, Course of Theoretical Physics, Vol. 7. Translated by J. B. Sykes and W. H. Reid, Pergamon Press, London-Paris-Frankfurt (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959), viii+134

Some Notes in the Context of Binocular Space Perception K. N. Leibovic

Abstract Binocular space perception in reduced cue conditions has been a favourite model for investigating visual perception. This paper considers questions such as metrics for representing visual space, iseikonic transformations and insights from the analysis of back projections of image points from retina and visual cortex V1. Keywords Binocular space perception · Metrics in vision · Iseikonic transformations · Frontoparallel lines · Back projections from retina · Visual information acquisition

1 Introduction Erik Balslev was my friend and colleague. He was an excellent mathematician and his wide ranging interests extended to science and philosophy. His work in mathematics was marked by rigor and insight. But above all he was a good human being. We got to know each other during the time Erik and his remarkable wife Anindita spent in Buffalo where Erik was in the Department of Mathematics and I in Biophysics. Although I had also started off as a mathematician my interests had turned to neuroscience, initially to theoretical problems but later also to experimental work. We know much more about brain function now than we did in the mid-twentieth century. Theory was poorly developed in spite of mountains of observational data. It was my aim to try to formalize some of these data while remaining rooted in experiment [9]. It seemed that a good strategy to gain insight on how the brain works was to consider a sensory pathway, such as the visual pathway, and to work one’s way inward from the periphery where one could identify and control the stimulus. This was our thinking when Erik and I decided to take a closer look at binocular space perception. Here we have an example where an external stimulus forms an K. N. Leibovic (B) State University of NewYork, Buffalo, NY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_9

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image on the retina which is mapped topologically point by point onto area V1 of the visual cortex. Psychophysical experiments on binocular vision go back at least to the time of Helmholtz [6]. As regards theoretical work, Luneburg [13] and later Blank [5] developed a theory according to which perceived space obeyed a hyperbolic geometry. So there was a background of experimental data as well as theory to explore.

2 Biophysical Background (a)

Notes on optics

In the human eye (Fig. 1) most of the refraction occurs at the fixed cornea. The eyeball is filled with the aqueous and vitreous humors which have a different index of refraction from the cornea. The lens also has a different index of refraction. It accommodates so as to focus on the object in view. This affects the positions of the nodal and principal points. For many calculations, however, it is sufficient to use a schematic or reduced eye model - such as the Gullstrand eye ([4], vol. 4)—and to have the rays of light pass through the center of rotation of the eye instead of the nodal points. Where we have done so we have also calculated the error thus introduced. (b)

Cortical representation of space

The retinal image is transmitted via the optic nerve to the lateral geniculate nucleus of the thalamus and from there to area V1 of the visual cortex. We shall not deal here with subsidiary connections from the retina to other parts of the brain, such as to nuclei which play a role in eye movements or circadian rhythms. On their way, the nerve fibers originating in the nasal half of each retina cross over to the opposite side at the optic chiasma. As a result the right visual field is represented in the left cortex and the left visual field in the right cortex. This is shown in Fig. 2. The different subdivisions of the visual field and the details of their projections from retina to V1 are illustrated in Fig. 3 (based on Fig. 3 of [1]. Panum’s area is the part of the visual field where the images from the two retinae are seen as one, or are fused. Outside of this area they are seen as double. (c)

Visual cues and binocular disparity

There are various cues to our perception of depth or distance in vision, many of them monocular as well as binocular, as indicated in Fig. 4. In binocular vision, however, which gives us the full power of our space perception, the dominant cue is horizontal disparity, the difference of the image positions on the retinae due to the different positions of the eyes in the head. (Note: Longuet-Higgins [12], showed computationally that vertical disparity could provide information about distance and direction of the fixation point. This may be relevant because many of the experiments in reduced cue conditions were carried

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Fig. 1 Diagram of the eye. N1 , N2 are, the nodal points

Fig. 2 View of optic pathways

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Fig. 3 The Visual field projections onto the two eyes and beyond (not to scale)

out with the stimulus lights on a table slightly below eye level of the subject. On the other hand, the general evidence from random dot stereograms [8] and the Wheatstone stereoscope [14] show that vertical disparity is not a binocular cue.) To isolate the role of disparity experimentally one can reduce other cues to a minimum by putting a subject in a dark room and using only one or more dim lights as a stimulus. For example, to test the perception of egocentric distance one may ask the subject to set up the lights at a fixed distance from him, with himself at the center. It turns out that he does so approximately on the arc of a circle passing through the fixation point—in Panum’s area of binocular fusion—which theoretically can be

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extrapolated to a full circle passing through his eyes. This is the so called ViethMuller circle. Similarly, if the subject is instructed to set up the test lights on straight lines radiating from him, he places them on hyperbolae passing through his eyes, the so called Hillebrand hyperbolae. If the instruction is to set up the lights on a straight line perpendicular to his line of sight, a so called fronto-parallel line (FPL), he sets them up on a curve concave towards him at near distances and convex towards him at far distances. This is illustrated in Fig. 5. It was based on data such as these which led Luneburg [13] to develop his theory of hyperbolic geometry for perceptual space.

Fig. 4 Cues for visual perception

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3 Exploring Binocular Vision in Reduced Cue Conditions (a)

Mathematical considerations

Erik Balslev developed his own formalism for mapping a physical configuration onto the retina and thence to area V1 of visual cortex. As in the theory of Luneburg and especially Blank and coworkers an important part in our analysis is played by iseikonic transformations. Using bipolar coordinates in two dimensions (γ ,ϕ), as shown in Fig. 6, this can be written as γ →γ +ξ φ →ϕ+η where ξ, η are constants. According to Blank et al. [5] an iseikonic transformation of the total configuration will yield the same perceptions to the observer as the original configuration. Treating the eyes as spherical and approximating light rays as passing through the centers of rotation of the eyes, one can determine the retinal coordinates of a point image and hence, because of the one to one mapping of the retina onto V1, one can make statements about the cortical coordinates (compare Fig. 3). Within that framework iseikonic transformations assume new meaning: any two configurations which are connected through an iseikonic transformation give rise to the activity of the same

Fig. 5 Illustrations of the Vieth-Muller circles Hillebrand Hyperbola and Fronto-parallel lines

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set of neurons from retina to V1 as the point of fixation moves over the configuration. To make this more precise mathematically we have the following proposition: Let Tξ,η be the iseikonic transformation defined by P(γ , ϕ) → Tξ,η (P)(γ + ξ, ϕ + η) where P is a point in physical space and ξ, η are constants. Then for every point set {Pi } i ∈ I where i is indexed by I, the configuration {Tξ,η Pi }i ∈ I by fixation on Tξ,η Pi0 , for every i0 ∈ I, gives rise to the activity of the same set of neurons from retina to V1 as the configuration {Pi }i ∈ I by fixation on Pi0 [1, 11]. The Vieth-Muller circles and Hillebrand hyperbolae obey this proposition. But as Erik showed there is, in addition, a whole family of curves which obey this proposition. They are given by the equation aϕ + bγ = c where a, b and c are constants. This could be interesting because such a set of curves could appear to be very stable perceptually. In particular the Vieth-Muller circles and Hillebrand hyperbolae could be the basis for a perceptual polar coordinate system. In our daily experience we see objects in a given direction and at a certain distance and there is some neurophysiological evidence to support the view that polar coordinates play a special role in vision. It has long been known from recordings in the visual cortex that there are cells which are tuned to distance and to direction [2, 3, 7]. (b)

The role of a metric in space perception

There is ample evidence from visual illusions and even experiments in reduced cue conditions using a minimum of stimulus points that perceptual judgements of distance depend on configuration and context. It follows that a global geometry of perceived space cannot be described by a Riemannian geometry which is characterized by ds2 = gij dxi dxj where gij are functions of the coordinates xi , xj . This applies a fortiori to the hyperbolic geometry of Luneburg which at best would be valid as an approximation over a restricted range. Since context and configuration can change in time and from one area of visual space to another, it is difficult to see how one metric can be applicable to all of perceived space and independent of time. However, since the most accurate judgements are made in the relatively restricted area around the fixation point in Panum’s area, it can be possible to devise special local geometries. To gain further insights on the boundary conditions for binocular vision it may be useful to consider some psychophysiological aspects in the context of adaptive advantage. For example, what are the neural and visual counterparts of notions of “intimate”

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versus “personal” versus “social space” and how does that relate to behavior? As a modest step in this direction let us consider the FPL (fronto-parallel line) in more detail.

4 A Closer Look at the FPL As stated before, when a subject is asked to set up a series of lights perpendicular to his line of sight, while fixating a point on his line of sight, he sets them up on a curve concave towards him at near distances and convex at far distances. The subject repeats the procedure with the fixation point at different distances from him. The image points on the retinae are then known and they can be back projected into the physical space from one distance of fixation to another and compared with the points obtained experimentally at that distance. The details of the calculations are given in our paper [11]. When this is done it is found that the experimental points differ substantially from the back projected points as shown in Fig. 7. This is contrary to what one would expect if disparity was the only cue in binocular vision. Put in other words, if disparity was the single defining parameter for FPLs, then the same retinal point on FPLs at different distances should have the same disparity. It follows that there must be other parameters that influence the shape of the percept of a FPL. An obvious parameter that comes to mind is convergence, although it has usually been considered to be of minor importance. Convergence is a function of the distance from the observer, a contextual cue. (Note: It should be noted, as mentioned earlier, that an error is introduced by assuming the light rays pass through the center of rotation of the eye. We have calculated this error if instead the rays pass through the nodal points and found it quite minor in comparison to the difference between the experimental and back projected points.) It should be stated that the FPL’s are not transformable into each other by an iseikonic transformation. But this does not imply that they form different percepts. The instructions for setting them up are the same at all distances from the observer (see also [10], Chap. 6). It is instead the contextual cue of distance which is different. There is another feature which may be of interest. It can be seen that the FPL curvature changes from convex to concave towards the observer at a distance which is within the range of what is considered “personal distance (or space)” in psychology. So at closer range the FPL points in perceived space are closer than the points in Euclidean space and vice versa at a farther range. This could be a strategy for a system designed to use distance cues to enhance attention to objects close by and relative neglect at a safer distance.

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Fig. 6 Illustration of bipolar coordinates (γ ), (ϕ)

5 Acquisition of Visual Information In the experiments on binocular vision involving Vieth-Muller circles, Hillebrand hyperbolae and FPL’s the subject was not limited in time and the results were the same whether he was instructed to keep his eyes fixed or was allowed to move them freely. It raises the question how visual information is acquired by the nervous system.

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Involuntary eye movements

It is a common belief that our eye movements are under voluntary control. But on closer examination it is found that they are never still even when we fixate a point in space. In pioneering studies Yarbus [15] showed that our eyes are in constant motion which appears to be quasi random and consists of saccades and jitters superimposed on high frequency oscillations. Thus, when examining a face our eyes move in saccades from point to point, dwelling longer in a jittering mode on points of high interest, such as the eyes and mouth. Similarly, when reading a page of text our eyes do not move smoothly over each line but instead move in saccades from point to point, pausing briefly at the end of each saccade. This is illustrated in Fig. 8 for viewing a face. (b)

Temporal factors

As regards temporal factors, the photoreceptors have a graded response to light unlike the millisecond pulse response of most neurons. This temporal persistence acts as a kind of memory and limits their reaction time as illustrated in following a flickering

Fig. 7 Fronto-parallel planes

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light: as the flicker rate increases, successive light pulses appear to merge and can no longer be distinguished as separate (Fig. 8). Based on observations such as the above (see also [10], Chap. 6) there emerges a model for the acquisition of visual information. (c)

A model for information acquisition

Bloch’s law in psychophysics states that at the threshold of detection, the product of intensity and duration of a flash of light is constant. This can be paraphrased that a minimum number of photons is required for a flash of light to be detectable.

Fig. 8 Illustration of the eye movements when examining a face. Note the concentration on salient features like eyes and mouth and saccadic motions in between

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The physiological counterpart is that the photoreceptor layer in the retina requires a minimum number of photons to produce a detectable response. However brief the flash, it will be detected if it is bright enough. But when we have a stimulus of more structure than a simple flash, the brain requires a number of samples over time to make sense of it. Observations such as the above suggest the following model. Briefly, the involuntary eye movements function to collect packets of information via a quasi-random sampling at salient points, then darting to another point until sufficient information is acquired to determine what we are looking at. It is only then that an elementary percept emerges. With this picture in mind, it makes sense that it is immaterial how much time the subject takes and whether he is allowed to keep the eyes moving or “fixed” in the experiments described. It also helps to better understand the process by which visual information and binocular information in particular is acquired.

6 Conclusion The starting point of our observations was the problem of how to describe the perceived visual space. Could we begin by investigating binocular vision which supposedly depends primarily on horizontal disparity as a cue? Luneburg [13] had carried out pioneering studies and came to the conclusion that the perceived binocular space obeyed a hyperbolic geometry. On further consideration of experimental data this cannot be the case except, perhaps, as an approximation over a restricted range. It is clear that perceived space is context dependent and does not obey a Riemannian geometry. It seems that we have to “go back to the drawing board” in our quest to develop a mathematical or quantitative framework for binocular vision. We need to ask what are the basic characteristics of binocular vision, what are its evolutionary and environmental advantages and what boundary conditions are set up by experimental observations? One such boundary condition is that perceived space cannot be described by a Riemannian geometry. Another boundary condition is that disparity alone is insufficient for creating a binocular percept. This is demonstrated by our results on the back projections of FPL’s which should be identical if the percept depended only on disparity. On the other hand, these results suggest that distance from the observer, a contextual cue, must be included. There are also some implications for the nature of representation in our work. For example there is the question whether percepts can be represented in fixed neural assemblies in the brain and if so, where? If this was the case in V1 then the neural assembly activated by a geometric figure should remain invariant with change of fixation on the figure. But as we have shown this is only true for iseikonic transformations in the set of configurations which obey the equation aϕ + bγ = c.

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It follows that, at least at the level of V1, percepts are in general not represented by activity in fixed neural assemblies. These remarks demonstrate that binocular space perception touches on many topics as well as being a fascinating study in itself. I was fortunate in having Erik Balslev as my colleague in this endeavor.

References 1. E. Balslev, K.N. Leibovic, Theoretical analysis of binocular space perception. J. Theor. Biol. 31, 77–100 (1971) 2. H.B. Barlow, C. Blakemore, J.D. Pettigrew, The neural mechanism of binocular depth discrimination. J. Physiol. 193, 327–342 (1967) 3. C. Blakemore, The representation of three dimensional visual space in the cat’s striate cortex. J. Physiol. 209, 155–178 (1970) 4. H. Davson (ed.), The Eye, vol. 4 (Academic Press, N.Y, 1962) 5. L.H. Hardy, G. Rand, M.C. Rittler, A.A. Blank, P. Boeder, The biology of Binocular Space Perception. Report to USONR project NR 143-638 (Columbia Univ., NY, 1953) 6. V. Helmholtz H., Handbuch der physiologischen Optik (Voss, Leipzig, 1867) 7. D.H. Hubel, T.N. Wiesel, Stereoscopic vision in macaque monkey. Nature 255(5227), 41–42 (1970) 8. B. Julesz, Foundations of Cyclopean Perception (Univ. of Chicago Press, Chicago, 1971) 9. K.N. Leibovic, Nervous System Theory (Academic Press, N.Y., 1972) 10. K.N. Leibovic, Science of Vision (ed.) Ch. 6 (Springer Verlag, N.Y, 1990) 11. K.N. Leibovic, E. Balslev, T. Mathieson, Binocular vision and pattern recognition. Kybernetik 8(3), 18–21 (1971) 12. H.C. Longuet-Higgins, The role of the vertical dimension in stereoscopic vision. Perception 11(4), 377–386 (1982) 13. R.K. Luneburg, Mathematical analysis of binocular vision (Princeton univ Press, Princeton, N.J., 1947) 14. C. Wheatstone, Contributions to the physiology of vision. Philos. Trans. R. Soc. London. 138 (371), (1838) 15. Yarbus, Eye Movements and Vision (Plenum Press, N.Y., 1967)

Symbolic Calculus for Singular Curve Operators Thierry Paul

In memory of Erik Balslev from whom I learned so much in mathematics and in physics

Abstract We define a generalization of the Töplitz quantization, suitable for operators whose Töplitz symbols are singular. We then show that singular curve operators in Topological Quantum Fields Theory (TQFT) are precisely generalized Töplitz operators of this kind and we compute for some of them, and conjecture for the others, their main symbol, determined by the associated classical trace function. Keywords Topological quantum fields theory · Quantization · Töplitz operators · Semiclassical approximation · Moduli spaces · Noncommutative geometry Mathematics Subject Classification (2010) 81T45 · 81S10 · 53D30 · 81S30 · 14D21

1 Introduction In 1925, Heisenberg invented quantum mechanics as a change of paradigm from (classical) functions to (quantum) matrices. He founded the new mechanics on the well known identity 1 [Q, P] = 1 i that, a few months later, Dirac recognized as the quantization of the Poisson bracket T. Paul (B) CNRS and Laboratoire Jacques-Louis Lions, Sorbonne Université, 4 place Jussieu, 75005 Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Albeverio et al. (eds.), Schrödinger Operators, Spectral Analysis and Number Theory, Springer Proceedings in Mathematics & Statistics 348, https://doi.org/10.1007/978-3-030-68490-7_10

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{q, p} = 1. Again a few years later, Weyl stated the first general quantization formula by associating to any function f (q, p) the operator  F(Q, P) =

f˜(ξ, x)ei

x P−ξ Q 

dξ d x

where f˜ is the symplectic Fourier transform defined analogously by  f (q, p) =

f˜(ξ, x))ei

x p−ξ q 

dξ d x.

Many years after was born the pseudodifferential calculus first establish by Calderon and Zygmund, and then formalized by Hörmander through the formula giving the integral kernel ρ F for the quantization F of a symbol f in d dimensions as  ρ F (x, y) =

f (q, p)ei

p(x−y) 

dp (2π )d

A bit earlier had appeared, both in quantum field theory and in optics (Wick quantization) the (positive preserving) Töplitz quantization of a symbol f  Op [ f ] = T

f (q, p)|q, pq, p|dqdq

where |q, p are the famous (suitably normalized) coherent states. As we see, quantization is not unique. But all the different symbolic calculi presented above share, after inversion of the quantization formulæ written above, the same two first asymptotic features: • the symbol of a product is, modulo , the product of the symbols • the symbol of the commutator divided by i is, modulo  again, the Poison bracket of the symbols. In other words, they all define a classical underlying space (an algebra of functions) endowed with a Poisson (of more generally symplectic) structure. But it is very easy to show that this nice quantum/classical picture has its limits. And one can easily construct quantum operators whose classical limit will not follow the two items exprressed above. Consider for example the well known creation and annihilation operators a + = Q + i P, a − = Q − i P. They act of the eigenvectors h j of the harmonic oscillator by   a + h j = ( j + 21 )h j+1 , a − h j = ( j − 21 )h j−1 .

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Consider now the matrices ⎛

⎞ 0 ... ... 0 . . . . . .⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1 0 . . .⎠ ... ... ...

⎛

⎞ 0 ... ... 0 . . . . . .⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ 0 0 . . .⎠ ... ... ...

0 0 0 ⎜1 0 0 ⎜ ⎜ ... ⎜ + ⎜ ... M1 = ⎜ ⎜ ... ⎜ ⎝ 0 ... 0 ... ... ... and its adjoint

0 1 0 ⎜0 0 1 ⎜ ⎜ ... ⎜ − ... M1 = ⎜ ⎜ ⎜ ... ⎜ ⎝ 0 ... 0 ... ... ... An elementary computation shows that

+ 2 2 −1/2 2 2 −1/2 − M+ , M− a . 1 = a (P + Q ) 1 = (P + Q )

therefore, their (naively)expected leading symbols are f + (q, p) =  p f − (q, p) = q−i or, in polar coordinates q + i p = ρeiθ , f ± = e±iθ . q+i p



q+i p q−i p

and

− If symbolic calculus would work the leading symbol of M+ 1 M1 should be equal + − to 1 and M1 M1 should be therefore close to the identity I as  → 0. But ⎞ ⎛ 0 0 0 0 ... ... ⎜ 0 1 0 0 . . . . . .⎟ ⎟ ⎜ ⎜ ... . . .⎟ ⎟ ⎜ ... . . .⎟ M1+ M1− = ⎜ ⎟ = I − |h 0 h 0 |  I, ⎜ ⎟ ⎜ . . . . . . ⎟ ⎜ ⎝ 0 . . . 0 0 1 . . .⎠ ... ... ... ... ... ...

 The reason for this defect comes from the fact that the function eiθ = zz is not a smooth function on the plane. In fact it is not even continuous at the origin: F(z) can tend to any value in {eiθ , θ ∈ R} when z tends to zero. Note finally that the commutator

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−1 0 0 ⎜ 0 0 0 ⎜ ⎜ ⎜ ... ⎜ − ⎜ [M+ ... 1 , M1 ] = ⎜ ⎜ ... ⎜ ⎜ ⎝ 0 ... 0 ... ... ...

⎞

0 ... ... 0 . . . . . .⎟ ⎟ ⎟ . . .⎟ ⎟ . . .⎟ ⎟ = −|h 0 h 0 | = O(), . . .⎟ ⎟ ⎟ 0 0 . . .⎠ ... ... ...

(1.1)

so that its symbol doesn’t vanish at leading order, as expected by standard symbolic asymptotism. One of the main goal of this paper is to define a quantization procedure which assigns a symbolic calculus to matrices presenting the pathologies analogues to the − ones of M+ 1 , M1 . We will state the results in the framework of quantum mechanics 2 on the sphere S as phase space. The reason of this is the fact that it is this quantum setting which correspond to the asymptotism in Topological Quantum Fields Theory (TQFT) studied, among others, in [12]. This procedure will be a non trivial extension to the Töplitz (anti-Wick) quantization already mentioned, and we will derive a suitable notion of symbol. Indeed, another main goal of this article is to give a semiclassical settings to all curve-operators in TQFT in the case of the once punctured torus or the 4-times punctured sphere. In [12] was established that these curve-operators happen, for almost all colors associated to the marked points, to be Töplitz operators associated to the quantization of the two-sphere, in some asymptotics of large number of colors. It happens that this result applies for every curve whose classical trace function is a smooth function on the sphere. Since this trace function is shown to be the principal Töplitz-symbol of the curve operator, the lack of smoothness ruins the possibility of semiclassical properties for the curve operator in the paradigm of Töplitz quantization (see Sect. 4 below for a very short presentation of TQFT and the main results of [12]). In fact these singular trace functions are not even continuous at the two poles of the sphere, which suggests a kind of blow-up on the two singularities of the classical phase-space. This is not surprising that such a regularization should be done in a more simple way at the quantum level. In the present paper we will show how to “enlarge” the formalism of Töplitz operators to “a-Töplitz operators”, in order to catch the asymptotics of the singular cases by semiclassical methods and compute the leading order symbols of some of them and conjecture them for the general singular curve operators. This principal symbol will be completely determined by the corresponding classical trace function, but will not be equal to it, for the reason that this enlarged a-Töplitz quantization procedure involves operator valued symbols. This construction will also be valid in the regular cases, where in this case the operator valued symbol is just a potential, hence it is defined by a function on the sphere whose leading behaviour is given by the trace function, as expected. Therefore we are able in this paradigm to handle the large coloring asymptotism of all curve operators n the case of the once punctured torus or the 4-times punctured sphere (note that the method we use is able to give some partial results in higher genus cases).

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We also study the natural underlying phase-space of our enlarged paradigm, the corresponding moduli space for TQFT, as a non-commutative space by identification with the non-commutative algebra of operator valued functions appearing at the classical limit for the symbol of the a-Töplitz operators, in the spirit of noncommutative geometry. We will built the construction of the a-Töplitz quantization by showing its necessity on some toy matrices situations in Sect. 3 after having defined in Sect. 2 the new Hilbert space on which these matrices will act, and before to show in Sect. 4 how general curve operators in TQFT enter this formalism. The main results are Theorems 15 and 18, out of Definition 12 and Theorem 20 together with Sect. 4.4 below. The a-Töplitz operators are introduced in Definition 17. Quantization of the sphere is briefly reviewed in Sect. 2.1, we won’t repeat it here. Let us just say that it consists in considering the sphere S 2 as the compactification of the plane C. Hence one expect that the singular phenomenon which appeared above at the origin should now appear twice at the poles of the sphere. The quantum Hilbert space can be represented as the space of entire functions, square-integrable with respect to a measure dμ N given in (3.6). Instead of trying to blow-up these two singularities at a “classical” (namely manifold) level, we will see that there is an easiest way of solving the problem by working directly at the “quantum” level. Namely, instead of considering the quantization process related to the so-called coherent state family ρz defined in (2.3) and which are (micro)localized at the points z ∈ S 2 , we will consider families of

τ (z)t |z|2 states ψza := R a(t)ei  ρeit z √dt2π where τ (z) = 1+|z| 2 and a ∈ S(R) (see Sect. 2.2 a for details). For z not at the poles, ψz is a Lagrangian (semiclassical) distribution (WKB state) localized on the parallel passing through z [15], but for z close to the pole the states ψza catches a different information. The equality (2.13):  C

|ψza ψza |dμ N (z) =

N −1 

|ψnN ψnN |,

(1.2)

n=0

where each ψnN is proportional to the elements of the canonical basis {ϕnN , n = 1, . . . , N − 1}, provides a decomposition of the identity on H N endowed with a different Hilbert structure for which the ψnN s are normalized (see Sect. 2.3). the advantage of working with the left hand side of (1.2) instead of the usual decomposition of the identity using coherent states and leading to standard Töplitz quantization, is the fact that ψza possess an extra parameter: the density a. Therefore one can “act” on ψza not only by multiplication by a function f (z) but by letting an operator valued function of z acting on a. This leads to what is called in this paper a-Töplitz operators, namely operators of the form  C

|ψz (z)a ψza |dμ N (z),

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where now (z) is, for each z, an operator acting on S(R). The precise definition is given in Sect. 3.6 Definition 17, and Theorem 18 shows that matrices like M± 1 are a-Töplitz operators, together with their products whose symbols are, at leading order, the (noncommutative) product of their symbols. Let us remark finally that, even at the limit  = Nπ = 0, the symbol of M± 1 is ± 21 ±iθ = (z/z) . Traces of the noncummutative part of the symbol persist at NOT e the classical limit, as in [14]. Therefore the “classical underlying phase-space” is not the 2-sphere anymore, but rather a noncommutative space identified with a non commutative algebra of such symbols playing the role of the commutative algebra of continuous functions on a standard manifold. A quick description of this space, inspired of course by noncommutative geometry [7], is given in Sect. 3.7. The construction dealing with M1± can be in particular generalized to matrices of the form ⎛

γ1 (0) γ2 (0) γ0 (0) ⎜ γ−1 (0) γ0 (1/N ) γ1 (1/N ) ⎜ ... ⎜ ⎜ ... MγN = ⎜ ⎜ ... ⎜ ⎝γ ... ... −(N −2) (0) ... ... γ−(N −1) (0)

⎞ γ N −1 (0) γ N −2 (1/N ) ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ γ−1 ((N − 2)/N ) γ0 ((N − 2)/N ) γ1 ((N − 2)/N )⎠ γ−2 ((N − 3)/N ) γ−1 ((N − 2)/N ) γ0 ((N − 1)/N ) ... ...

... ...

(1.3)

It has been proven in [5, 17] that such a family of matrices MγN is a Töplitz operator N −1 γk (τ )eikθ if and only if of symbol γ (τ, θ ) = k=1−N |k|

(τ (1 − τ )) 2 γk (τ ) ∈ C ∞ ([0, 1]), k = 1 N , . . . , N − 1.

(1.4)

Condition (1.4) expresses explicitly that γ ∈ C ∞ (S2 ). In Sect. 3.6 Theorem 18 we prove that (more general matrices than) the family MγN are a-Töplitz operators, and we compute their symbols, when (1.4) is replaced by the condition1 γk (τ ) ∈ C ∞ ([0, 1]), k = 1 − N , . . . , N − 1.

(1.5)

Under (1.5) γ ∈ / C ∞ (S2 ) and one has to pass form the Töplitz to the a-Töplitz paradigme (note that M1± indeed satisfy (1.5) and not (1.4)). Let us finish this long introduction by giving the key ideas leading to the setting of our main result, Theorem 20. The reader can found in Sect. 4 a very short introduction to TQFT. Larger basics on TQFT can be found in [12] using the same vocabulary as the present paper together with a substantial bibliography. 1 The

construction works certainly also for conditions of the type, e.g.., (τ (1 − τ ))

α|k| 2

γk (τ ) ∈ C ∞ ([0, 1]), k = 1, . . . , N − 1, 0 ≤ α ≤ 1

(or even more general ones), but since we don’t see any applications of these situation, we concentrate in this paper to the condition (1.5).

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Combinatorial curve operators are actions of the curves on a punctured surface on a finite dimensional vector space Vr ( , c) indexed by a level r and a coloring c of the marked points taken in a set of r colors.The dimension N = N (r ) of Vr ( , c) will diverge as r → ∞ and r1 can be considered as a phenomenological Planck constant . In [12] we provided the construction of an explicit orthogonal basis of Vr ( , c) and we conjectured that any curve operator is expressed in this basis by a matrix essentially of the form Mγ . More precisely we showed that the conjecture is true in the case where is either the punctured 2-torus or the 4 times punctured sphere, Even more, we proved that (the matrix of) any curve operator belongs to the algebra generated by three matrices of the form M Nr , M Nr , M Nr defined in (1.3) where 0

1

d

⎧ r ⎨ 0 (τ, θ ) = γ0 (τ, r ) r (τ, θ ) = 2γ1 (τ, r ) cos θ I ⎩ r1 d (τ, θ ) = e 2r γ1 (τ, r ) cos (θ + τ ), for two explicit families of functions γ0 , γ1 . We showed in [12] that, for “most” values of the coloring of the marked points of , the functions 0r , 1r , dr are smooth functions on the sphere, and that, indeed, the corresponding curve operators are standard Töplitz operators. This proves also that any curve operator is Töplitz, by the stability result by composition of the Töplitz class. Moreover the leading symbols of any curve operator happen to be the classical trace function associated to the corresponding curve (see [12] for details). Theorem 20 of the present article express the same result for any coloring of the marked points, at the expense of replacing Töplitz quantization by a-Töplitz one. The only difference, unavoidably for the reason of the change of Töplitz paradigm, is the fact that the a-Töplitz leading symbol of the curve operator is not (and cannot) the classical trace function of the curve, but we are able to compute or conjecture it out of the trace function.

2 Hilbert Spaces Associated to New Quantizations of the Sphere 2.1 The Standard Geometric Quantization of the Sphere In this section we will consider the quantization of the sphere in a very down-to-earth way. See [8, 12] for more details. Given an integer N , we define the space H N of polynomials in the complex variable z of order strictly less than N and set

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i P, Q = 2π



 C

P(z)Q(z) dzdz and ϕnN (z) = (1 + |z|2 ) N +1

N! z n (2.1) n!(N − 1 − n)!

The vectors (ϕnN )n=0...N −1 form an orthonormal basis of H N . By the stereographic projection  S 2 (τ, θ ) ∈ [0, 1] × S 1 → z =

τ eiθ ∈ C ∪ {∞}, 1−τ

The space H N can be seen as a space of functions on the sphere (with a specific behaviour at the north pole). Write dμ N =

dzdz i . 2π (1 + |z|2 ) N +1

(2.2)

As a space of analytic functions in L 2 (C, dμ N ), the space H N is closed. For z 0 ∈ C, we define the coherent state ρz0 (z) = N (1 + z 0 z) N −1 .

(2.3)

These vectors satisfy  f, ρz0  = f (z 0 ) for any f ∈ H N and the orthogonal projector π N : L 2 (C, dμ N ) → H N satisfies (π N ψ)(z) = ψ, ρz . For f ∈ C ∞ (S 2 , R) we define the (standard) Töplitz quantization of f as the operator T N [ f ] : HN → HN  T N [ f ] := f (z)|ρz ρz |dμ N (z) C i.e. T N [ f ]ψ := f (z)ρz , ψH N ρz dμ N (z) = π N ( f ψ) for ψ ∈ H N . (2.4) C

A Toeplitz operator on S 2 is a sequence of operators (TN ) ∈ End(H N ) such that there exists a sequence f k ∈ C ∞ (S 2 , R) such that for any integer M the operator R NM defined by the equation TN =

M 

N −k T fk + R NM

k=0

is a bounded operator whose norm satisfies ||R M || = O(N −M−1 ). An easy use of the stationary phase Lemma shows that the (anti-)Wick symbol T ρ ,ρ  (also called Husimi function) of T f , namely ρf z ,ρz z z satisfies

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203

T f ρz , ρz  1 = f + S f + O(N −2 ), ρz , ρz  N

(2.5)

where S = (1 + |z|2 )2 ∂z ∂z is the Laplacian on the sphere.

2.2 The Building Vectors Let a ∈ S(R), ||a|| L 2 (R) = 1 and z ∈ C. We define  ψza =

τ (z)t 

a(t)ei R

dt ρeit z √ 2π

(2.6)

|z| π where τ (z) = 1+|z| 2 and  = N . Although we won’t need it in this paper, let us note that, when z is far away form the origin and the point at infinity, ψza is a lagrangian semiclassical distribution (WKB state) (by a similar construction as in [15]). N −1  N! Since, by (2.3), ρz = z n ϕnN , we get that n!(N −1−n)! 2

n=1

ψza =

N −1  n=0



τ (z) − n  a 



 N −1   N! τ (z) − n z n ϕnN =  a ϕnN (z)ϕnN . n!(N − 1 − n)!  n=0

(2.7) where  a is the Fourier transform of a 1  a (y) := √ 2π

 ei x y a(x)d x. R

Remark 1 Note that, by (2.7), ψza depends only on the values of a˜ on [0, N ]. Therefore one can always restrict the choice of a to the functions whose Fourier transform is supported on [0, N ]. In the sequel of this article we will always do so. Lemma 2

 C

|ψza ψza |dμ N (z)

=

N −1 

CnN |ϕnN ϕnN |

(2.8)

n=0

with CnN =

(N − 1)! n!(N − 1 − n)!



1 0

    n  τ − n 2 τ dτ  a (1 − τ ) N −1 .    1−τ 

(2.9)

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T. Paul

Moreover, as N − 1 =

→∞

1 

1 ), 0 < n < 1. N

(2.10)

√ | a (λ − n)|2 λn e−λ 2π λdλ, 0 ∼ n.

(2.11)

CnN = 1 + O( CnN ∼

1 n!



∞

0

CnN ∼ C NN−1−n , n ∼ 1.

(2.12)

Proof Deriving (2.8) is a straightforward calculus after (2.7). By the asymptotic formula for the binomial we get that, as N , n → ∞, (N − 1)! ∼ n!(N − 1 − n)!

 1

n N −1 − N n−1

−n 

n 1− N −1

(N −1)

.

Moreover since 0 < n < 1 we get since  a is fast decreasing at infinity,   n   n  +∞    1    dτ τ − n 2 τ − n 2 τ τ N −1 dτ ∼   a a (1 − τ ) (1 − τ ) N −1      1−τ   1−τ  0 −∞

  τ −n 2 a   → ||a|| L 2 (R) δ(τ − n) as  = and 1  Definition 3 ψnN :=

1 N −1

→ 0. Therefore we get (2.10).

 CnN ϕnN .

This definition is motivated by (2.8) which actually reads  C

|ψza ψza |dμ N (z)

=

N −1 

|ψnN ψnN |.

(2.13)

n=0

This leads to the following equality:  C

|ψza a ψza |dμ N (z) = 1HaN ,

(2.14)

where HNa is the same space of polynomials as HN but now endowed with the renormalized scalar product ·, ·a fixed by ψmN , ψnN a = δm,n ,

(2.15)

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205

and |ψza a ψza |ψ := ψza , ψa ψza , ψ ∈ HaN .

(2.16)

2.3 The Hilbert Structure The Hilbert scalar product on HaN is obtained out of (2.15) by bi-linearity. Since any polynomial f satisfies f =

N −1 

1 ψ N , f ψbN , CnN n

ϕnN , f ϕbN =

0

we get  f, ga :=

N −1  n=0

N −1  1 1 N N  f, ψ ψ , g ==  f, ϕnN vpnN , g. n n N 2 N (Cn ) C n n=0

Note that , a is not given by an integral kernel. But if we “change” of representation and define F(z) := ψza , f , G(z) := ψza , g then, by (2.13) we have   f, ga = F, G =

C

F(z)G(z)dμ N (z).

Let us remark finally that  F(z) =

a(t)ei R

τ (z)t 

dt f (eit z) √ 2π



and f = namely

f (z ) =

C

F(z)|ψza dz,

 C

F(z)ψza (z )dz.

3 Singular Quantization This section is the heart for the present paper. We will first show how operators on HaN defined as matrices on the basis {ψnM , } act on the building operators ψza by action on a (Sect. 3.3). This will allow us, in Sect. 3.4, to assign to each of these

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matrices symbols whose symbolic calculus is studied in Sect. 3.5. This will lead us finally to Sect. 3.6 where we define the a-Töplitz quantization.

3.1 A Toy Model Case Let us consider the N × N matrix ⎛ ⎞ ⎛ 0 0 1 0 0 ... 0 ⎜1 ⎜1 0 1 0 . . . 0 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ... ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ... ⎟ = ⎜ ⎜ ⎜ ⎟ ⎜ ... ⎜ ⎟ ⎜ ⎝0 ⎝0 . . . 0 1 0 1 ⎠ 0 0 ... 0 0 1 0

0 0 0 0 0 0 ... ... ... ... 0 1 ... 0 0

⎞ ⎛ 0 0 ⎜0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟+⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0⎠ ⎝0 0 1 0

... ...

1 0 0 0 1 0 ... ... ... ... 0 0 ... 0 0

⎞ 0 0⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 1⎠ 0 0

... ...

=: M1 =: M1+ + M1− − a and let us consider the operator M1 = M+ 1 + M1 on H N whose matrix on the N orthonormal basis {ψn , n = 0, . . . , N − 1} is M. That is

⎧ ± N ⎪ ⎨ M1 ψ0 = N M± = 1 ψi ⎪ ⎩ ± N M1 ψ N −1 = Proposition 4

1±1 2

ψ1N N ψi±1 , 1≤i ≤ N −2

1∓1 2

ψ NN−2

M1 ψxa = ψz 1 (z)a

where the operator 1 (z) is given by (3.5) below. Proof By (2.7) we get that, calling DnN = (CnN )− 2 (once again  = 1

   N −2   √ N −1 n N N τ (z) − n a ψz =  a N z Dn ψn . n  n=0 Therefore

1 ), N −1

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207

 = a



M1 ψza

τ (z) √ N D0N ψ1N 

   N −2   N −1 n N N τ (z) − n √ N +  a N z Dn (ψn−1 + ψn+1 ) n  n=1   τ (z) − 1 √ N −1 N + a Nz D N −1 ψ NN−2      τ (z) √ τ (z) − 1 √ N −1 N N N = a N D0 ψ1 +  a Nz D N −1 ψ NN−2      N −3   N − 1 n+1 N τ (z) − (n + 1) √ +  a N z Dn+1 ψnN n + 1  n=0    N −1   N − 1 n−1 N τ (z) − (n − 1) √ +  a N z Dn−1 ψnN n − 1  n=2    N −2   N − 1 n+1 N τ (z) − (n + 1) √ =  a N z Dn+1 ψnN n + 1  n=0    N −1   N − 1 n−1 N τ (z) − (n − 1) √  a N z Dn−1 ψnN + n − 1  n=1

(3.1)

(3.2)

Let us consider the sum in (3.2). On can write it as ψsud

   N −1 n N N τ (z) − (n − 1) √ μ(n) a = N z Dn ψn n z  n=0   N −1  τ (z) − (n − 1) 1 μ(n) a = ϕn (z)ϕn z  n=0 

N −1  1



with μ(n) =

n , N −n

n>0

0, n = 0

We get that ψsud = ψzbsud 

where bsud =

 ix  e C·N τ (z) − i∂x a = 1− a. μ N  z C·−1

(3.3)

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T. Paul C·N N C·−1

Here we have denote by

C·N N C·−1  and 

 : ξ ∈]0, N [→

C·N N μ C·−1

the function defined out of (2.9) by

1   τ −ξ  2  τ ξ a  1−τ (1 − τ ) N −1 dτ 0    ξ ,    N − ξ 1  τ −(ξ −1) 2  τ ξ dτ N −1  a (1 − τ )  1−τ 0    

is meant as the product of the two functions, i.e.

C·N N (ξ )μ(ξ ) C·−1

(3.4)

C·N N μ(ξ ) C·−1

=

using (3.4). Note finally that, by the band limited hypothesis on a

in Remark 1, bsud is well defined. Similarly we get that the sum in (3.1) is    N −1 n N N τ (z) − (n + 1) √ = zν(n) a N z Dn ψn n  n=0 

N −1 

ψnor d



with

N −n−1 , n+1

n < N −1 0, n = N − 1

ν(n) = So

ψnor d = ψzbnor d where  bnor d =

C·N N ν N C·+1



 τ (z) − i∂x ze−i x a = 1+ a. 

We define  1 (z) =

C·N N μ N C·+1

 =



τ (z) − i∂x 

C·N + 1 (z) + N C·+1





ei x + z



C·N N ν N C·−1



 τ (z) − i∂x ze−i x . (3.5) 

C·N − 1 (z) N C·−1

where μ N = χ[ 21 ,N − 21 ] μ, ν N = χ[− 21 ,N − 23 ] ν,

(3.6)

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209

χ ∈ C ∞ (R) satisfies ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨ χ (ξ ) > 0 χ[a,b] (ξ ) = 1 ⎪

⎪ χ (ξ )