Analysis and operator theory. T.Kato's 100 birthday 9783030126605, 9783030126612

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Springer Optimization and Its Applications 146

Themistocles M. Rassias Valentin A. Zagrebnov Editors

Analysis and Operator Theory Dedicated in Memory of Tosio Kato’s 100th Birthday Foreword by Barry Simon

Springer Optimization and Its Applications Volume 146 Managing Editor Panos M. Pardalos

(University of Florida)

Honorary Editor Ding-Zhu Du (University of Texas at Dallas) Advisory Editors J. Birge, Booth School of Business (University of Chicago) S. Butenko, Department of Industrial and Systems Engineering (Texas A & M University) F. Giannessi, Dipto. Matematica (University of Pisa) S. Rebennack, Institute of Operations Research (Karlsruhe Institute of Technology) T. Terlaky, Department of Industrial and Systems Engineering (Lehigh University) Y. Ye, Department of Engineering (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches.

More information about this series at http://www.springer.com/series/7393

Themistocles M. Rassias Valentin A. Zagrebnov

•

Editors

Analysis and Operator Theory Dedicated in Memory of Tosio Kato’s 100th Birthday

Foreword by Barry Simon, IBM Professor of Mathematics and Theoretical Physics, Emeritus; California Institute of Technology, Pasadena, CA 91125, USA

123

Editors Themistocles M. Rassias National Technical University of Athens Zografou Campus Athens, Greece

Valentin A. Zagrebnov Institute of Mathematics Aix-Marseille University Marseille, France

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-030-12660-5 ISBN 978-3-030-12661-2 (eBook) https://doi.org/10.1007/978-3-030-12661-2 Library of Congress Control Number: 2019930373 Mathematics Subject Classification (2010): 46L80, 47A58, 46Txx, 00A79 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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

Foreword

Tosio Kato (1917–1999) profoundly influenced the science of his time. He was the founder of a large field, the theory of Schrödinger operators (which describes the mathematics of the fundamental objects of the physics of ordinary matter at small scale). Indeed, one could say collection of fields—for example, shortly after of writing this, I will be attending a conference on almost periodic and random Schrödinger operators. Kato not only founded the field with his 1951 paper on self-adjointness. He continued with influential contributions for the rest of his life, especially the next 30 years, on a variety of aspects of the theory including perturbation and scattering theory, spectral analysis, properties of eigenfunctions, and adiabatic theory. His work is marked by depth, beauty, and elegance. Like Newton who also made pivotal contribution to Coulomb systems, Kato found his great self-adjointness result while evacuated from his University and its library. Because of the War, he spent the middle part of the 1940s in the Japanese countryside where he also found his basic results on perturbation theory. This disruption explains why Kato only received his doctorate when he was over 30 years of age. Like many of the other founders of modern mathematical physics (e.g., Jost, Thirring, Wightman), Kato’s formal training was in physics at the University of Tokyo, where he served as a Professor of Physics, but he learned to exploit rigorous proof and spent the latter half of his career as a Professor of Mathematics at University of California, Berkeley. Kato’s opus wasn’t limited to Schrödinger operators. He also made seminal contributions to the theory of nonlinear PDEs where he was an early pioneer. Besides these two topics, he has worked in semigroup theory and to a variety of parts of functional analysis. This book, written by a group of admirers of Kato who were all influenced by his work, was begun during the centennial year of Kato’s birth and is dedicated to his memory. Its breadth mirrors his. Speaking for myself, I have been impacted by Kato’s work throughout my career up until today. As a graduate student 50 years ago, I learned an enormous amount from his great book on Perturbation Theory which was central to my first major v

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Foreword

work on the quantum anharmonic oscillator. During the 1970s, we exchanged numerous letters which stimulated research on both our parts. I am very glad to be able to honor Kato with my contribution here and would like to thank the editors for putting this project together, for inviting me to participate and asking me to write this foreword. Los Angeles, USA October 2018

Barry Simon IBM Professor of Mathematics and Theoretical Physics, Emeritus; California Institute of Technology Pasadena, CA 91125, USA e-mail: [email protected]

Preface

Analysis and Operator Theory—Dedicated in Memory of Tosio Kato’s 100th Birthday features a collection of carefully selected research as well as survey articles devoted to a broad spectrum of subjects of Mathematical Analysis and Mathematical Physics, to which Kato contributed monumental results. The breadth and full depth of Kato’s legacy is impossible to exhaust by even more than one volume dedicated to him. A brilliant mathematician, worldrenowned specialist in functional analysis and operator theory, he was also an outstanding teacher. His great book “Perturbation Theory for Linear Operators” first published by Springer in 1966 is an encyclopedia for mathematicians and mathematical physicists. Many of his papers, e.g., on operator theory, evolution equations, or spectral analysis, are so remarkably written that they are still great classics to be read in the original. The book chapters presented here have been written by a number of experts in the various areas of Kato’s interest. They are aiming to present the actual state of the art in the topics, which were developed by Tosio Kato in various branches of analysis and operator theory. Some of his achievements, such as Kato’s inequality, the Kato type matrix limit theorem, the Howland–Kato commutator problem, the Kato class of potentials, or the Trotter–Kato product formulae, are directly manifesting in the titles of the chapters of this volume. Among others, there are related to or inspired by certain ideas of Kato. A special article presents a report of Tosio Kato’s work on nonrelativistic Quantum Mechanics, and one of the chapters is dedicated to an unpublished paper of his. It is hoped that this publication provides an extensive account of research results which will be of usefulness for a wide readership, from graduate students to established researchers in the corresponding domains.

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Preface

We would like to express our gratitude to all the scientists who contributed valuable works in this book as well as to Barry Simon who wrote the foreword. Additionally, we would like to acknowledge the superb assistance of the staff of Springer for the preparation of this publication. Athens, Greece Marseilles, France

Themistocles M. Rassias Valentin A. Zagrebnov

Contents

Complementarity and Stochastic Independence . . . . . . . . . . . . . . . . . . . Luigi Accardi and Yun-Gang Lu

1

Norm Conditions for Separability in Mm Mn . . . . . . . . . . . . . . . . . . . Tsuyoshi Ando

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Kato’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Arendt and A. F. M. ter Elst

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Tosio Kato’s Unpublished Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Claude Bardos and Hisashi Okamoto

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On the Border Lines Between the Regions of Distinct Solution Type for Solutions of the Friedmann Equation . . . . . . . . . . . . . . . . . . . . . . . . Hellmut Baumgärtel Scattering on Leaky Wires in Dimension Three . . . . . . . . . . . . . . . . . . . Pavel Exner and Sylwia Kondej Computing Traces, Determinants, and f-Functions for Sturm–Liouville Operators: A Survey . . . . . . . . . . . . . . . . . . . . . . . Fritz Gesztesy and Klaus Kirsten

65 81

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On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Bernard Helffer and Jean Nourrigat Matrix Limit Theorems of Kato Type Related to Positive Linear Maps and Operator Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Fumio Hiai The Howland–Kato Commutator Problem . . . . . . . . . . . . . . . . . . . . . . . 191 Ira Herbst and Thomas L. Kriete

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Pointwise Exponential Decay of Bound States of the Nelson Model With Kato-Class Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Fumio Hiroshima Regular KMS States of Weakly Coupled Anharmonic Crystals and the Resolvent CCR Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Tomohiro Kanda and Taku Matsui Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Hagen Neidhardt, Artur Stephan and Valentin A. Zagrebnov Exact Solutions to Problems with Perturbed Differential and Boundary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 I. N. Parasidis and E. Providas On a Few Equivalent Statements of a Hilbert-Type Integral Inequality in the Whole Plane with the Hurwitz Zeta Function . . . . . . . 319 Themistocles M. Rassias and Bicheng Yang Tosio Kato’s Work on Non-relativistic Quantum Mechanics: A Brief Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Barry Simon The Factor Type of Conservative KMS-Weights on Graph C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Klaus Thomsen Trotter–Kato Product Formulae in Dixmier Ideal . . . . . . . . . . . . . . . . . 395 Valentin A. Zagrebnov

Complementarity and Stochastic Independence Luigi Accardi and Yun-Gang Lu

Abstract A mathematical approach to the notion of complementarity in quantum physics is described and its historical development is shortly reviewed. After that, the notion of n-complementarity is introduced as a natural extension of complementarity and at the same time as weak form of stochastic independence. Several examples in which n-complementarity is realized but not independence are produced. The construction of these examples is based on the structure of Interacting Fock Space (IFS) that is strictly related to the classical theory of orthogonal polynomials. A brief description of both this notion and this connection is included to make the paper self-contained.

1 Introduction The history of the early development of the notion of complementarity is vividly described in Chapter 7.2 of M. Jammer’s monograph [17] on the development of Quantum Mechanics. N. Bohr first introduced the term complementarity during the Como conference in 1927 as a candidate to unify De Broglie’s idea of wave–particle duality with Heisenberg indeterminacy principle and to capture in great generality the physical essence of both statements. In the same Chapter Jammer recalls the comment, attributed by Wigner to von Neumann: Well, there are many things that do not commute and you can easily find three operators that do not commute, and explains that this comment should be interpreted as an implicit critique to Bohr’s idea that complementarity refers to pairs of attributes (or observables).

L. Accardi Centro Vito Volterra, Università di Roma Tor Vergata, 00133 Rome, Italy e-mail: [email protected] Y.-G. Lu (B) Department of Mathematics, University of Bari Aldo Moro, Campus of University Ernesto Quagliariello, Via Edoardo Orabona 4, 70126 Bari, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_1

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L. Accardi and Y.-G. Lu

In quantum theory the non-commutativity of two Hermitean linear operators implies a lower bound on the product of the variances of these operators, with respect to any state, which is zero on every state if and only if the two operators commute. The fact that there exist states such that this lower bound can be strictly positive is interpreted as a statistical formulation of Heisenberg indeterminacy principle because any attempt to experimentally decrease the variance of one of the two operators implies an increase in the variance of the other one. In terms of the physical observables represented by these Hermitean operators, this means that any increase in the precision of a measurement of one of these observables implies a larger imprecision in the measurement of the other one. Pairs of observables satisfying Heisenberg commutation relations are characterized by the property that this lower bound is strictly positive and independent of the state. This fact suggests a natural way to defend Bohr’s contention on complementarity from von Neumann critique by selecting, among pairs of incompatible observables, the complementary ones as those which are in some sense maximally incompatible. The problem is then reduced to give a mathematical formulation of the notion of maximal incompatibility. On the other hand, Heisenberg commutation relations are a model-dependent statement without a direct physical interpretation while, as explained by von Neumann [29], all statements about quantum mechanics that can be directly compared with experiments can be reduced to the calculation of expectation values or equivalently to values of observables and their transition (conditional) probabilities. The program to find an intrinsic, model-independent property that expresses the maximal incompatibility of two observables was formulated in [1] where it was shown that it leads to a natural problem in operator theory which is still open nowadays even in the finite-dimensional case.

1.1 An Open Problem in Operator Theory The starting point of [1] was the idea that two observables should be considered maximally incompatible if the exact knowledge of the value of one of them gives zero information on the values of the other one. If the possible values of these observables (spectrum of the corresponding operators) have a finite cardinality, the natural way to express the zero information condition on these values is to say that, with the given information, all the values of the given observable are equiprobable. If the two observables have spectrum with the same cardinality, the only case considered in [1], the situation is perfectly symmetric. Definition 1 A pair A, B of N –valued observables (N < +∞) is called complementary if, denoting (aα ), (bβ ) their values, their transition probability matrix corresponds to the maximum indeterminacy, i.e., P(A = aα |B = bβ ) =

1 ; α, β = 1, . . . , N N

(1)

Complementarity and Stochastic Independence

3

we assume that the values (aα ), (bβ ) are mutually different (non-degenerate spectrum). In the quantum model the two observables A, B are represented by Hermitean operators on a complex N -dimensional Hilbert space H ≡ C N and we will use the same symbol for an observable and the corresponding operator. Denoting (ψαA ), (ψβB ) their eigenvectors, the corresponding transition probability matrix is   given by |ψαA , ψβB |2 . This motivates the following definition: Definition 2 Two orthonormal bases in H , (ϕα ), (ψβ ) are called complementary if 1 ; ∀ α, β (2) |ϕα , ψβ |2 = N Remark If A, B satisfy the finite version of the CCR [27], namely ei h A eik B = ei hk eik B ei h A ; h, k = 1, . . . , N

(3)

then they are complementary (for a proof see the end of section 1.2). Condition (2) is equivalent to τ (E A (I ) · E B (J )) = cλ(I ) · λ(J )

(4)

where I , J are sub-sets of {1, . . . , N }, E A (·), E B (·) the spectral projectors of A, B respectively, λ is the uniform measure on {1, 2, . . . , N } (i.e., λ ≡ (1, 1, . . . , 1)), τ the non-normalized trace on B(H ) ≡ N × N C-matrices and c = 1/N . The formulation (4) of the complementarity condition is better suited than (2) for infinitedimensional extension. Definition 3 Let A be a semi-finite von Neumann algebra acting on a Hilbert space H and with a trace τ . A pair of self-adjoint operators A, B on H , affiliated to A is called (τ, λ)-complementary where here and in the following λ denotes the Lebesgue measure on R if, for any pair I , J of bounded Borel subsets of R, condition (4) is satisfied with c a strictly positive constant. Remark The identity τ (|E A (I )E B (J )|2 ) = τ (E B (J )E A (I )E B (J )) = τ (E A (I )E B (J ))

(5)

implies that E Q (I )E P (J ) is trace class if and only if it is Hilbert-Schmidt and in this case its trace coincides with its Hilbert–Schmidt norm. The above definition of complementarity was given in [1] and one of the problems posed there (Problem (I.f)) was Problem (I.f). Classify the complementary pairs in B(H ) (or more generally, in a semi-finite von Neumann algebra) up to the natural unitary equivalence.

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Later developments (see section 2 and the following ones) have shown that Definition 3 has to be extended in order to include all examples available now. Furthermore, these examples show that there is more than one natural equivalence relation among complementary pairs. To explain this issue it is convenient to give a third equivalent formulation of the relation (4) introducing the abelian von Neumann sub-algebras A A , A B of B(H ) generated by the spectral projections of A, B respectively. The condition of non-degeneracy of the spectrum becomes in this case the requirement that the algebras A A and A B are maximal abelian sub-algebras of A . The integral with respect to λ, defines a weight on L ∞ (R, λ) which is finite for f ∈ L ∞ ∩ L 1 (R, λ). Using the same symbol for a measure and the associated integral, the identities λ A ( f (A)) := λ( f ) =: λ B ( f (B)) show that λ induces weights λ A , λ B respectively on A A and A B . With these notations (4) can be re-written in the equivalent form τ ( f (A) · g(B)) = cλ A ( f (A))λ B (g(B)) = cλ( f ) · λ(g) ; f, g ∈ L ∞ ∩ L 1 (R) (6) The identity (6) suggests a natural way to construct new complementary pairs out of a given one. It is natural to consider equivalent two pairs obtained one from the other from this construction. As already mentioned in the infinite-dimensional case, there are several variants of this definition that can also be considered, but we will not discuss them here. Definition 4 Two complementary pairs (A, B), (A , B ) of self-adjoint operators on H , affiliated to A are said to be equivalent if there exist automorphisms u A , u B of A satisfying ; u B (A B ) = A B (7) u A (A A ) = A A λB ◦ u A = λ A

;

λB ◦ u B = λB

(8)

A = u A (A)

;

B = u B (B)

(9)

where we use the same symbols u A , u B to denote their restrictions on A A and A B respectively and u A (A), u B (B) are defined in terms of their spectral decompositions. If u A , u B satisfy conditions (7), (8) and (A , B ) are defined by (9) then, for any f, g ∈ L ∞ ∩ L 1 (R), one has the identity τ ( f (A ) · g(B )) = τ ( f (u A (A)) · g(u B (B))) = τ (u A ( f (A)) · u B (g(B))) But, due to (7), u A ( f (A)) and u B (g(B)) are still in A A and A B respectively. Therefore one can apply (6) and from (8) one has τ ( f (A ) · g(B )) = cλ A (u A ( f (A))λ B (u B (g(B))) = cλ A ( f (A)λ B (g(B))) = cλ( f ) · λ(g)

Complementarity and Stochastic Independence

5

i.e. (A , B ) is a complementary pair. If A = B(H ) and A, B have pure point spectrum, then by maximal abelianity A A and A B are generated by rank-one projectors of the form ϕα ϕα∗ , ψβ ψβ∗ where (ϕα ), (ψβ ) are two orthonormal bases of H and ψβ ψβ∗ (ξ ) := ψβ∗ , ξ ψβ

;

ξ ∈H

In this case, u A and u B act as permutations on the indexes α and β and the complementarity of (A , B ) is simply expression of the invariance of (2) under two separate permutations of these indexes. In the continuous case permutations are replaced by λ-preserving invertible bi-measurable maps. Problem (I.f) above, i.e., the classification of complementary pairs up to the equivalence relation introduced in Definition 4, arises for complementary pairs a problem analogue to that solved by von Neumann on the uniqueness up to isomorphism and irreducibility of continuous representations of the CCR in bounded form for systems with finitely many degrees of freedom. The analogue of the irreducibility condition here is that the spectral projections of the complementary pair generate the whole von Neumann algebra A . It was already clear in [1] that, contrarily to the CCR case, in finite dimensions the problem has a negative solution, i.e., there exist inequivalent complementary pairs. In the infinite-dimensional case there is almost no literature on the equivalence problem with the exception of a paper by Cassinelli and Varadarajan [13]. As shown in section 1.2 below, the position and momentum operators on L 2 (R, λ) form a complementary pair. The problem then is if there exist other complementary pairs in L 2 (R, λ) inequivalent to it in the sense of Definition 4. This problem is an open challenge for experts in operator theory. In [1] it was conjectured that the answer to this problem is negative because the requirements of Definition 4 are effectively very weak as one can see from the finite-dimensional case. Since the Lebesgue measure is not a probability measure, in the continuous case one cannot interpret the left-hand side of (4) as a joint probability. However, following Renyi [26], for a σ -finite measure μ on R × R a compatible system of conditional probabilities defined by P([A ∈ I0 ] ∩ [B ∈ J0 ]|[A ∈ I ] ∩ [B ∈ J ]) :=

μ(I0 ∩ J0 ) μ(I ∩ J )

;

I0 ⊆ I , J0 ⊆ J

where I, J are sets of finite μ-measure. With this definition (4) becomes equivalent to P([A ∈ I0 ] ∩ [B ∈ J ]|[A ∈ I ] ∩ [B ∈ J ]) =

λ(I0 ) ; λ(I )

I0 ⊆ I , ∀J , λ(J ) < +∞

and this is the analogue of the zero information condition (1) for infinite measures.

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1.2 Complementarity of Position and Momentum √ If, √ in the finite version of the CCR (3), √ one introduces √ the rescalings h → h/ n, k → n → ∞, so that h/ n √ → t, k/ n → s and moreover the equality k/ √n and lets √ √ hk ei(h/ n)A ei(k/ n)B = ei n ei(k/ n)B ei(h/ n)A becomes eit A eis B = eist eis B eit A By von Neumann uniqueness theorem one can then identify A ≡ Q Position ;

B ≡ P Momentum

There are at least 2 ways to make these heuristic considerations rigorous: one is based on contractions of Lie algebras (see [10]) and another one on quantum central limit theorems (see [2]). Therefore we know that position and momentum operators in L 2 (R, λ) are approximated by complementary pairs. Intuitively this suggests that they too should be a complementary pair. The following theorem shows that this intuition is correct: Theorem 1 Position and momentum operators in L 2 (R, λ) are a complementary pair with constant c = 1/(2π ). Proof Let Q, P denote position and momentum operators in L 2 (R, λ). Denote E P (B) (resp. E Q (B)) their spectral projection on the Borel set B and F Fourier transform  1 e−ikx f (x)d x fˆ(k) := F f (k) := √ 2π R Then for ψ in the space S (R)

E P (B)ψ( p) = F F −1 E Q (B)ψ( p) = F E Q (B)F −1 ψ( p)

1 = √ 2π =

1 2π





 dq

e−i pq [E Q (B)F −1 ψ](q)dq =

1 2π

1 dp e−i( p− p )q χ B (q)ψ( p ) = √ 2π

1 =√ 2π







e−i pq dq χ B (q) 1 ψ( p )dp √ 2π







ei p q ψ( p )dp

dqe−i( p− p )q χ B (q)

dp ψ( p )χˆ B ( p − p )

Therefore 1 E Q (I )E P (J )ψ(q) = χ I (q) √ 2π



1 dq ψ(q )χˆ J (q − q ) = √ 2π

so that E Q (I )E P (J ) has an integral kernel given by



  dq χ I (q)χˆ J (q − q ) ψ(q )

Complementarity and Stochastic Independence

7

1 E Q (I )E P (J )(q, q ) = √ χ I (q)χˆ J (q − q ) 2π For bounded Borel sets I , J and with  ·  denoting the L 2 -norm, one has   =

1 2π



dqdq |E Q (I )E P (J )(q, q )|2 = 

dqχ I (q)

dq |χˆ J (q − q )|2 =

1 2π

 

dqdq



dqχ I (q)χˆ J 2 =

1 χ I (q)|χˆ J (q − q )|2 2π 1 1 |I | χ J 2 = |I | |J | < +∞ 2π 2π

Having a square-integrable kernel, E Q (I )E P (J ) is an Hilbert–Schmidt operator and its Hilbert–Schmidt norm coincides with the L 2 -norm of its kernel (see [18], Chap. V, sect. 2.4). Finally from (5) we conclude that E Q (I )E P (J ) is trace class and that its trace coincides with the its Hilbert–Schmidt norm.  Remark Theorem 1 and the Stone–von Neumann theorem imply that, also in the infinite dimensional case, any pair A, B of self-adjoint operators satisfying the CCR ei h A eik B = ei hk eik B ei h A

;

h, k ∈ R

(10)

and such that the maps t → eit A , t → eit B are continuous in the weak operator topology, is a complementary pair. A less computational and quicker proof of Theorem 1, which however does not give the explicit form of the constant c, is the following. Using the above-mentioned result that, for bounded Borel sets I, J of Rd , E Q (I )E P (J ) is trace class, and the properties of the trace on B(L 2 (Rd )) one verifies that the map (11) I × J ⊂ R2d → τ (E Q (I )E P (J )) defines a positive σ -finite countably additive measure on R2d finite on bounded Borel sets. Similarly, for N ∈ N and Q, P replacing A, B in (3), one has the measure I × J ⊂ {1, . . . , N } × {1, . . . , N } → τ (E Q (I )E P (J )) From the CCR (10) (resp. (3)) one deduces that, with addition interpreted as addition modulo N in the N -dimensional case, τ (E Q (I + s)E P (J + t)) = τ (e−isp E Q (I )eisp eitq E P (J )e−itq )

(12)

= τ (E Q (I )eisp eitq E P (J )(eisp eitq )∗ ) = τ (E Q (I )eitq eisp E P (J )(eitq eisp )∗ ) = τ (e−itq E Q (I )eitq eisp E P (J )e−isp ) = τ (E Q (I )E P (J )) Therefore the measure (11) is translation invariant and it is known that any such a measure is a multiple of the Lebesgue measure in the infinite-dimensional case, or

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a multiple of the uniform measure in the N -dimensional case, i.e., for some c > 0 one must have τ (E Q (I )E P (J )) = cλ(I )λ(J ) where λ is the Lebesgue measure in the infinite-dimensional case and the counting measure in the N -dimensional case.

1.3 Further Developments of the Theory of Complementary Pairs After 1982 a large literature has accumulated on complementary pairs. Popa [25] studied complementary pairs in finite von Neumann algebras and used them to solve a problem posed by Kadison. Busch and Lahti [12] simplified the proof of Theorem 1 (their proof is used here). Kraus [19] adopted Definition 2 of complementarity and begun a systematic study of the equivalence problem in the finite-dimensional case. He discussed, always in the finite-dimensional case, entropic uncertainty principles, which can be described as follows. In addition to the Boltzmann entropy, many numerical indexes associated to probability measures on R have been introduced and often the name entropy is used also for them. If such an index S is calculated on the probability distribution of an observable A in a state ϕ, the resulting number is denoted Sϕ (A). An entropic uncertainty principle for two observables (A, B) is a lower bound on the sum Sϕ (A) + Sϕ (B). Kraus conjectured an optimal lower bound in the case when S is the Boltzmann entropy and explicitly calculated it for complementary pairs. This conjecture was proved by Maassen and Uffink [21] for a large family of entropic-type indexes including the original Boltzmann entropy. An excellent survey on complementary pairs and entropic uncertainty principles is in Ohya and Petz’s monograph [22] which also contains additional bibliography. The use of complementarity in state estimation was discussed by Parthasarathy in [23] and Petz and Ruppert in [24]. Wootters and Fields [30] discovered that complementary bases in the sense of Definition 2 play an important role in quantum information and re-baptized them mutually unbiased bases, a term now widely used in quantum information where the associated unitary matrices are called complex Hadamard matrices in view of the fact that Hadamard used some matrices with entries −1, 0, 1 whose columns satisfy condition (2) up to normalization (see [28] for a survey). The fact that the eigenvectors of a pair of Hermitean matrices A, B satisfying the finite version of the CCR (3) also satisfies (2) was noted by Schwinger [27] without mentioning neither complementarity nor probabilistic or information theoretical implications, nor the continuous case. In connection with state estimation, Ivonovic [16] extended Schwinger’s result proving that in C N one can construct N + 1 ortho-normal bases any pair of which satisfies (2). Considering, for each of these bases, the algebra generated by the rank-

Complementarity and Stochastic Independence

9

one projectors on its vectors, one obtains N + 1 sub-algebras of the algebra M N (C) of N × N complex matrices that are 2-complementary, with respect to the nonnormalized trace, in the sense of Definition 5 in section 2. The problem of finding the maximal number of sub-algebras of M N (C) with this property was studied by Björck [11]. Haagerup [15] determined this number for the subclass of circulant Hadamard matrices (i.e., those whose (i, j)-entry depends on i − j) in the case N = 2, 3, 5, 7 and gave an upper estimate for general prime N . In the infinite-dimensional case, with the exception of the above mentioned paper of Cassinelli and Varadarajan [13] and Chapter 16 in the monograph of Ohya and Petz, there is essentially nothing on the equivalence problem.

2 n-Complementarity Definition 5 in the present section includes all notions of complementarity discussed up to now and extends them in three directions – inclusion of the infinite-dimensional case, – factorization of expectation values of n-tuples rather than pairs, – consideration of the above two properties for non-tracial states. The motivation of this extension comes both from the theory of orthogonal polynomials and from quantum probability. In fact it is clear from (6) that complementarity is a weak form of stochastic independence, more precisely tensor independence, but it differs from it in two important items. First, the weights appearing in the right-hand side are not the restrictions of the weight on the left-hand side on the two sub-algebras (which, in the case of (6) are identically infinite). Second, the factorization of the expectation value in the left-hand side takes place only for pairs and not for general n-tuples as required by tensor independence. This creates a hybrid situation in which some expectations values factorize according to the rule of tensor independence, but others may factorize according to the rule of other notions of independence. The present paper is a first step in the investigation of this phenomenon, not known up to now in quantum probability. It is easy to construct examples of tensor independence, just by taking tensor products of algebras and of states. So the problem is to construct examples of algebras and states which satisfy the complementarity condition, or its extension given in Definition 5 below, but are not tensor independent. The notion of Interacting Fock Space (IFS) (see section 3 below) was motivated by the stochastic limit of quantum electro-dynamics without dipole approximation and it was immediately clear that it provided a natural tool to construct examples of stochastic independence (which are in fact forms of stochastic dependence easily coded in some algebraic rules, but difficult to read from the joint expectations, see [9] for a discussion of this issue). In view of the connections between independence and complementarity, it is natural to ask oneself if the framework of IFS can also be used to produce new

10

L. Accardi and Y.-G. Lu

nontrivial examples of complementary pairs. We will see in the following that this is indeed the case. The theory of IFS (see section 3) includes the theory of orthogonal polynomials, and in fact is identified with it in the case of polynomials in one variable (see [3] for bibliography and more details). One of the consequences of this inclusion is the notion of quantum decomposition of a classical random field (or random variable) with all moments which allows to associate, in a canonical way, to every classical random field a non-commutative algebra of operators acting on an IFS. This allows to extend classical random fields to some typically quantum notions such as complementarity. Experience coming from the stochastic limit of quantum theory, where this structure was first deduced, shows that in physics nontrivial (i.e., non-tensor product type) IFS arise naturally in connection with nonlinear interactions. This has been verified in a number of models such as: particles interacting with a Boson field (QED, polaron model, . . . ), Anderson model, tri-linear field–field interaction, Stochastic bosonization of Fermi system in dimensions ≥ 3, . . . . These models are described in Part II of the monograph [8] (see also [6, 7, 9, 20]). The connection between IFS and nonlinear interactions is confirmed by the fact that the assignment of a family of orthogonal polynomials, in any even infinite dimensions, is equivalent to the assignment of a special class of symmetric IFS (characterized in [3]) and the property that the associated IFS is of tensor product type (like usual Bose or Fermi fields) characterizes product measures [5]. Motivated by the examples discussed in the previous sections and by those to be discussed in the following we introduce the following definition. Definition 5 Let A be a ∗-algebra, in the following, unless otherwise specified, and let (A j ) j∈I (I a set) be a family of ∗-sub-algebras of A . Given a weight ϕ on A and n ∈ N, the family (A j ) j∈I of ∗-sub-algebras of A is called n–complementary with respect to ϕ, or ϕ-n–complementary, if there exist weights ϕ j on A j ( j ∈ I ) such that, for any k ∈ {1, 2, . . . , n} and any injective function j : h ∈ {1, . . . , k} → jh ∈ I one has: k      ϕ Y j1 · · · Y jk = ϕ jh Y jh ; Y jh ∈ A jh , h ∈ {1, · · · , k} h=1

in the sense that, if either side of the identity exists and is finite, the same is true for the other one and the identity holds. 2–complementarity is simply called complementarity. If the family (A j ) j∈I is ϕ–n–complementary for any n ∈ N with n ≤ |I | (cardinality of I ), it is called completely complementary. Remark In Definition 12 below we introduce the notion of strong complementarity which includes the additional condition that   , j∈I (13) ϕ j := ϕ  Aj

Complementarity and Stochastic Independence

11

  where  denotes restriction. This condition allows to relate in a natural way the examples discussed in section 3 below with the basic original example of position and momentum discussed in section 1.2. In fact, in a general Interacting Fock Space (IFS) one deals with creation and annihilation operators a ± rather than position and momentum. In any IFS one can define a generalized position operator using the standard relation between field operators and creation/annihilation opera√ tors valid in usual quantum mechanics, √ i. e. q := (a + + a)/ 2 and a generalized momentum operator p := (a + − a)/i 2. In IFS coming from orthogonal polynomials and for vacuum–symmetric random variables (i.e., with vanishing odd vacuum moments) one can prove that the generalized position operator coincides with the usual multiplication operator and that the commutation relations between generalized position and momentum operators extend in a natural way the Heisenberg commutation relations between position and momentum. Since condition (13) involves algebras rather than pairs of self–adjoint operators, one can exploit the linear relation expressing generalized position and momentum operators as linear combinations of creation/annihilation operators and conclude that, if the algebras are ϕ-strongly complementary then the corresponding generalized position and momentum operators satisfy condition (6) with τ , λ A , λ B replaced by ϕ. In this case ϕ is a state, hence the normalization condition ϕ(1) = cϕ(1)ϕ(1) ⇐⇒ c = 1 uniquely fixes the constant c to be 1. Note that the analogue condition for usual position and momentum operators, with τ = T r and λ A = λ B = Lebesgue measure, would be difficult to formulate because creation/annihilation operators are not selfadjoint and to our knowledge for them there is no analogue of the trace formula of Theorem 1.

3 Interacting Fock Spaces 3.1 Tensor Algebras Over Vector Spaces Let H1 be a vector space (always complex, unless otherwise specified) and, for each n ∈ N, denote H1⊗n (resp. H1⊗n ) the vector space algebraic tensor product (resp. symmetric tensor product) of n copies of H1 . The tensor algebra (resp. symmetric tensor algebra) over H1 is defined by Tens(H1 ) :=

˙ n∈N

H1⊗n

;

Tenssym (H1 ) :=

˙ n∈N



H1⊗n

(14)

 where ˙ denotes vector space direct sum and, in the right hand sides of (14), by definition

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H1⊗0 = H1⊗0 := C ·

(15)

  where is a norm-one vector. The spaces Hn n∈N are called the n–particle spaces, if n = 0 one speaks of the vacuum space. If K 1 is another vector space, a linear morphism from Tens(H1 ) to Tens(K 1 ) is a linear map A : Tens(H1 ) → Tens(K 1 ) preserving the vector space gradation such that A H1 = K 1

3.2 Actions of H1 and H1∗ on Tens(H1 ) and Tenss ym (H1 ) For a general vector space K , we denote L (K ) the algebra of linear operators of K into itself. For n ∈ N, f, gn , . . . , g1 ∈ H1 , the creation operator ∗f : Tens(H1 ) → Tens(H1 ) is algebraically defined

∗f (gn ⊗ · · · ⊗ g2 ⊗ g1 ) := f ⊗ gn ⊗ · · · g2 ⊗ g1

(16)

Similarly one defines the symmetric creation operators B +f : Tenssym (H1 ) → Tenssym (H1 ) by gn ⊗ ···⊗ g2 ⊗ g1 ) := f ⊗ ···⊗ g2 ⊗ g1 B +f (gn ⊗

(17)

Both maps f → ∗f , B +f are linear. By polarization, Tenssym (H1 ) is linearly generated by the vectors of the form g ⊗n . On these vectors B +f takes the form n ˆ ⊗ := B +f g ⊗n := f ⊗g

1 ⊗k g ⊗ f ⊗ g ⊗(n−k) n + 1 k=0 n

(18)

Denote H1∗ , H1  any duality between H1 and its algebraic dual H1∗ . For n ∈ N, gn , . . . , g1 ∈ H1 and f ∗ ∈ H1∗ , the algebraically defined operator

f ∗ : Tens(H1 ) → Tens(H1 )

f ∗ (gn ⊗ · · · ⊗ g2 ⊗ g1 ) :=  f ∗ , gn gn−1 ⊗ · · · g2 ⊗ g1 ; f ∗ := 0

(19)

is called full annihilation operators (or full annihilator) and the algebraically defined operator B f ∗ : Tenssym (H1 ) → Tenssym (H1 ) ˆ · · · ⊗g ˆ 2 ⊗g ˆ 1 ) := B (gn ⊗

n

f∗

ˆ 1 ; B f ∗ := 0 ˆ · · · ⊗g ˆ k+1 ⊗g ˆ k−1 · · · ⊗g  f ∗ , gk gn ⊗

k=1

(20)

Complementarity and Stochastic Independence

13

This implies B f ∗ h ⊗(n+1) = (n + 1) f ∗ , hh ⊗n

;

h ⊗0 := , B f ∗ = 0

(21)

is called annihilation operator (or annihilator). Whenever it is given an anti-linear embedding f ∈ H1 → f ∗ ∈ H1∗ , for example induced by a semi-scalar product on H1 (see, e.g., [14]) we will write f for f ∗ and B f for B f ∗ . With these notations the annihilation operators are anti-linear in f ∈ H1 . The number operator  is also algebraically defined on both Tens(H1 ) and Tenssym (H1 ) by the following prescription: for any function F : N → C   F()

H1⊗n

  = F()



H1⊗n

:= multiplication by F(n)

(22)

3.3 Commutations Relations In this section, we suppose that it is given an anti-linear embedding f ∈ H1 → f ∗ ∈ H1∗ and we use the notations f and B f for the annihilation operators. For A ∈ L (H1 ) denote 1 (A) : Tens(H1 ) → Tens(H1 ) the linear operator defined by linear extension of 1 (A)(gn ⊗ · · · ⊗ g2 ⊗ g1 ) := (Agn ) ⊗ · · · ⊗ g2 ⊗ g1

(23)

Similarly denote (A) : Tens(H1 ) → Tens(H1 ) the linear operator defined by linear extension of (A)g ⊗n := (Ag)⊗n

(24)

One can prove that (A) is well defined. From (19), (21), (16), (17) it follows that, for n ∈ N, g, gn , . . . , g1 ∈ H1 and f ∗ ∈ H1∗ , one has

f ∗g (gn ⊗ · · · ⊗ g2 ⊗ g1 ) = f (g ⊗ gn ⊗ · · · ⊗ g2 ⊗ g1 ) =  f ∗ , ggn ⊗ · · · ⊗ g2 ⊗ g1

∗g f (gn ⊗ · · · ⊗ g2 ⊗ g1 ) =  f ∗ , gn  ∗g gn−1 ⊗ · · · ⊗ g2 ⊗ g1 =  f ∗ , gn (g ⊗ gn−1 ⊗ · · · ⊗g2 ⊗g1 )

= ((g f ∗ )gn ⊗ gn−1 ⊗ · · · ⊗ g2 ⊗ g1 ) = 1 (g f ∗ )(gn ⊗ gn−1 ⊗ · · · ⊗ g2 ⊗ g1 )

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where 1 is defined by (23) and g ⊗ f ∗ ∈ H1 ⊗ H1∗ → g f ∗ ∈ L (H1 ) := {linear operators on H1 } is the natural embedding defined by g f ∗ (h) :=  f ∗ , hg In conclusion in the full case one has the following multiplication table:

f ∗g =  f ∗ , g · 1

;

∗g f = 1 (g f ∗ )

(25)

In the symmetric case (20), i.e., ˆ · · · ⊗g ˆ 2 ⊗g ˆ 1 ) := B f (gn ⊗

n

ˆ · · · ⊗ ˆ gk ⊗ ˆ · · · ⊗g ˆ 2 ⊗g ˆ 1  f ∗ , gk gn ⊗ k=1

implies that, denoting gn+1 := g and gk := h for any k ∈ {1, · · · , n}, B f Bg+ h ⊗n

ˆ · · · ⊗g ˆ 1= = B f gn+1 ⊗

n+1

ˆ · · · ⊗g ˆ k+1 ⊗g ˆ k−1 ⊗ ˆ · · · ⊗g ˆ 1  f ∗ , gk gn+1 ⊗

k=1 ∗

=  f , gh

⊗n

+

n

∗

∗

ˆ · · · ⊗g ˆ k+1 ⊗g ˆ k−1 ⊗ ˆ · · · ⊗g ˆ 1 =  f , gh  f , gk gn ⊗

⊗n

∗

+  f , h

k=1

n

(26) ˆ ⊗n g ⊗h

k=1

ˆ ⊗n =  f ∗ , gh ⊗n + n f ∗ , hg ⊗h Similarly

So that

ˆ ⊗k Bg+ B f h ⊗n = n f ∗ , hBg+ h ⊗(n−1) = n f ∗ , hg ⊗h B f Bg+ − Bg+ B f =  f ∗ , g

(27)

3.4 Interacting Fock Spaces Over a Semi-Hilbert Space We choose the framework of semi-Hilbert spaces rather than the more usual one of Hilbert spaces because it is useful to have a common space for all the operators involved. This choice is familiar in the theory of orthogonal polynomials where it is used to compare operators associated to mutually singular measures. (see [3] for a discussion of this choice).

Complementarity and Stochastic Independence

15

Definition 6 A semi-scalar product  · , ·  on a vector space H1 is a positive sesquilinear form on H1 . This is equivalent to give an anti-linear embedding f ∈ H1 → f ∗ ∈ H1∗ satisfying

 f ∗, f  ≥ 0

;

∀ f ∈ H1

where  · , ·  is the duality between H1 and its algebraic dual H1∗ (see (19)). A semi-Hilbert space is a pair (H1 ,  · , · ) where H1 is a vector space and  · , ·  semi-scalar product on H1 . A zero-norm vector is a vector f ∈ (H1 ,  · , · ) such that  f  = 0 (notice that  ·  is a semi-norm). These vectors are orthogonal to H1 . If (K 1 ,  · , · 1 ) is another semi-Hilbert space, a semi-Hilbert space linear operator A : (H1 ,  · , · ) → (K 1 ,  · , · 1 ) is a linear operator mapping zero norm vectors into zero norm vectors. Such an operator is called adjointable if there exists another linear operator A∗ : (K 1 ,  · , · 1 ) → (H1 ,  · , · ) satisfying

Aξ, η1 = ξ, A∗ η1

;

∀ξ ∈ H1 , ∀η ∈ K 1

A semi-Hilbert space isometry is an adjointable operator preserving the scalar products. Semi-Hilbert space direct sums are defined as orthogonal sums without completion. Remark Unless otherwise specified we will always suppose in the following that the topology induced by a semi-scalar product is separable. Definition 7 Let H1 be a vector space. An interacting Fock space over H1 is defined by (i) A semi-scalar product  · , ·  on Tens(H1 ) such that the vector space direct sums in (14) become semi-Hilbert space orthogonal sums. (ii) , defined by (15) is a unit vector in (Tens(H1 ) ,  · , · ). (iii) For each f ∈ H1 , the creation operator ∗f , defined by (16), has an adjoint a f with respect to the semi-scalar product  · , ·  on Tens(H1 ) called the annihilation operator on (Tens(H1 ) ,  · , · ) with test function f . If K 1 is another vector space, a morphism V : (Tens(H1 ) ,  · , · ) → (Tens(K 1 ) ,  · , · 1 ) is a linear isometry preserving the vector space gradation and such that V = 1 .

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Replacing everywhere in the above definition the tensor products by symmetric tensor products, Tens(H1 ) becomes Tenssym (H1 ) and, with the definition of creator given by (17), one obtains the definition of symmetric interacting Fock space   over H1 , Tenssym (H1 ) ,  · , · sym , whose creator (resp. annihilator) is denoted B +f (resp. B f ) Notations. When, for each n ∈ N, the restriction of the semi-scalar product on (Tens(H1 ) ,  · , · ) (resp. Tenssym (H1 ) ,  · , · sym ) coincides with the n-th tensor power (resp. symmetric n-th tensor power) of the semi-scalar product  · , · 1 , for the pair ( ∗f , a f ) (resp. (B +f , a f )) we use the notation ( ∗f , f ) (resp. (B +f , B f )). Otherwise we use the notation (a +f , a f ) in both cases. This is motivated by the fact that many interesting IFS arise as deformations of the two special cases described above. Finally, vacuum expectation values will be denoted indifferently ϕ( · ) ≡  , ( · )  ≡ ( · ) Remark There is a more intrinsic definition of IFS (resp. symmetric IFS) from which the concrete representations of Definition 7 can be deduced (see [3]).

3.5 1-Mode Type IFS 1-mode type IFS were introduced in [4]. Here we recall their definition and some basic properties. Definition 8 Let H1 be a vector space. Fix a principal Jacobi sequence (ωn ), i.e., a sequence satisfying ω0 = 0 ; ωn = 0 ⇒ ωn+k = 0

;

∀k ∈ N∗

Define the sequence λ ≡ (λn ) by: λn := ωn ! := ω1 ω2 · · · ωn The 1-mode type IFS (1-MTIFS) over H1 with Jacobi sequence ω, denoted ω, f ull (H1 ), is defined as the IFS with creator defined by a +f := ∗f

(28)

where ∗f is given by (16) and semi-scalar product uniquely determined by the conditions  f n ⊗ · · · ⊗ f 1 , gn ⊗ · · · ⊗ g1 n := λn  f n ⊗ · · · ⊗ f 1 , gn ⊗ · · · ⊗ g1 ⊗n

(29)

Complementarity and Stochastic Independence

= λn

n 

17

 f j , g j  H1 ; ∀n ∈ N

j=1

Definition 9 The full Fock space over H1 , denoted  f ull (H1 ), is the 1-mode type IFS over H1 with principal Jacobi sequence ωn := 1

;

∀n ∈ N

In this section we prove that the structure of a general 1-MTIFS ω, f ull (H1 ) is obtained as a deformation of the structure of  f ull (H1 ). Recall that the creation (annihilation) operators on  f ull (H1 ) are denoted ∗f , f and those on ω, f ull (H1 ), a ±f . From (28) and (29) one deduces the action of the annihilator a f ( f n ⊗ · · · ⊗ f 2 ⊗ f 1 ), gn−1 ⊗ · · · ⊗ g1 n−1 =  f n ⊗ · · · ⊗ f 2 ⊗ f 1 , a +f (gn−1 ⊗ · · · ⊗ g1 )n

= λn  f n ⊗ · · · ⊗ f 1 , gn ⊗ · · · ⊗ g1 ⊗n = λn  f n , f 

n−1 

 f j , g j  H1

j=1

=

λn  f n , f  (λn−1  f n−1 ⊗ · · · ⊗ f 1 , gn−1 ⊗ · · · ⊗ g1 ⊗n ) λn−1

= ωn  f n , f  f n−1 ⊗ · · · ⊗ f 1 , gn−1 ⊗ · · · ⊗ g1 n−1 = ωn  f, f n  f n−1 ⊗ · · · ⊗ f 1 , gn−1 ⊗ · · · ⊗ g1 n−1

and since n ∈ N and f, f 1 , . . . , f n , g1 , . . . , gn−1 ∈ H1 are arbitrary, this is equivalent to (30) a f ( f n ⊗ · · · ⊗ f 2 ⊗ f 1 ) = ωn  f, f n  f n−1 ⊗ · · · ⊗ f 1 ; ∀n ∈ N Comparing (19) with (30) and recalling the notation (22) one finds a f = f ω ; ∀ f ∈ H1

(31)

a f = B f ω = 0

(32)

Notice that and, due to (28) and (31), the operators g ∗f and ∗f g preserve the gradation built with the operators a ±f .

3.6 1-Mode Type Symmetric IFS Definition 10 Let H1 be a vector space and let (ωn ) and (λn ) be as in Definition 8. Fix a principal Jacobi sequence (ωn ), i.e. a sequence satisfying The 1-symmetric 1-MTIFS over H1 with Jacobi sequence ω, denoted ω (H1 ), is defined as the

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L. Accardi and Y.-G. Lu

symmetric IFS over H1 with creator defined by a +f := ∗f

(33)

where ∗f is given by (16) and semi-scalar product uniquely determined by the conditions ···⊗ f 1 , gn ⊗ ···⊗ g1 n := λn  f n ⊗ ···⊗ f 1 , gn ⊗ ···⊗ g1 ⊗  fn ⊗ n

(34)

By polarization, in the symmetric case, (29) is equivalent to n  f ⊗n , g ⊗n n := λn  f ⊗n , g ⊗n ⊗ n = λn  f, g H

1

;

∀n ∈ N

(35)

Definition 11 The Boson Fock space over H1 , denoted (H1 ), is the 1-mode type symmetric IFS over H1 with principal Jacobi sequence ωn := 1

;

∀n ∈ N

In this section we prove that the structure of a general symmetric 1-MTIFS ω (H1 ) is obtained as a deformation of the structure of (H1 ) and we show how this deformation affects the commutation relations. Recall that the creation (annihilation) operators on (H1 ) are denoted B ±f those on ω (H1 ), a ±f . With these notation, one has  f ⊗n , g ⊗n n =  f, g H1  f ⊗(n−1) , g ⊗(n−1) n−1 it follows that  f ⊗n , g ⊗n n =  f, g

 λn  λn−1  f, gn−1 = ωn  f, g f ⊗(n−1) , g ⊗(n−1) n−1 H1 λn−1

so that  f ⊗n 2n =  f ⊗n , f ⊗n n = ωn ! f ⊗n , f ⊗n ⊗n = λn  f, f nH1 = λn  f ⊗n n2⊗ n We know that as vector spaces (H1 ) = ω (H1 ) = Tenssym (H1 ), and that one has a +f = B +f

;

∀ f ∈ H1

(36)

The adjoint of a +f is computed as follows: a +f g ⊗n , h ⊗(n+1) n+1 = λn+1 B +f g ⊗n , h ⊗(n+1) ⊗(n+1) = λn+1  f, hg ⊗n , h ⊗n ⊗n = ωn+1  f, hλn g ⊗n , h ⊗n ⊗n = ωn+1  f, hg ⊗n , h ⊗n n = g ⊗n , a f h ⊗(n+1) n

Complementarity and Stochastic Independence

19

Therefore, recalling the definition (22) of the number operator  on Tenssym (H1 ), one has a f h ⊗(n+1) = ωn+1  f, hh ⊗n =

ωn+1 ω+1 (n + 1) f, hh ⊗n = B f h ⊗(n+1) (37) n+1 +1 = Bf

ω ⊗(n+1) h 

Equivalently af = Bf

ω 

(38)

Notice that, with the convention 0/0 := 1, one has a f = B f ω = 0

(39)

and, due to (36) and (39), the operators Bg B +f and B +f Bg preserve the gradation built with the operators a ±f . Lemma 1 a f F = F+1 a f

;

F a +f = a +f F+1

(40)

[a f , ag+ ] = B f (ω − ω−1 ) Bg+ +  f, gω = B f Bg+ (ω+1 − ω ) +  f, gω (41) Proof For any function F : N → C one has a f F h ⊗n = Fn a f h ⊗n = F+1 a f h ⊗n Therefore a f F = F+1 a f which is the first of (40). Taking adjoints and denoting F¯ the pointwise conjugate of F one finds F¯ a +f = a f F¯+1 which, since F is arbitrary is equivalent to the second of (40). Finally, from (36) and (39) it follows that [a f , ag+ ] = [B f ω , Bg+ ] = B f [ω , Bg+ ] + [B f , Bg+ ]ω and, since

h ⊗n = ωn+1 g ⊗ h ⊗n = ωn+1 Bg+ h ⊗n ω Bg+ h ⊗n = ω g ⊗ Bg+ ω h ⊗n = ωn Bg+ h ⊗n

(42)

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L. Accardi and Y.-G. Lu

it follows that [ω , Bg+ ]h ⊗n = ωn+1 Bg+ h ⊗n − ωn Bg+ h ⊗n = (ωn+1 − ωn ) Bg+ h ⊗n = (ω − ω−1 ) Bg+ h ⊗n

i.e.

[ω , Bg+ ] = (ω − ω−1 ) Bg+ = Bg+ (ω+1 − ω )

where in the last identity we have used (40). Therefore (42) becomes [a f , ag+ ] = B f (ω − ω−1 ) Bg+ +  f, gω = B f Bg+ (ω+1 − ω ) +  f, gω 

and this proves (41).

4 Complementarity in IFS   For any semi-Hilbert space H1 , ·, · H1 and any IFS space over H1  I (H1 ) :=



H1⊗n

, ·, ·n

n∈N

 n∈N

 , a+ ,

we denote P(a + , a) := polynomial ∗ -algebra in the variables {ah+ , ah : h ∈ H1 }   ϕ( · ) :=  , ( · ) 

P (a + ,a)

∈ S (P(a + , a))

the restriction of the vector state  , ( · )  on P(a + , a) induced by the vacuum state in  I (H1 ) and, for f ∈ H1 P(a +f , a f ) := polynomial ∗ –algebra in the variables {a +f , a f } Definition 12 An IFS  I (H1 ) is said to enjoy the strong complementarity property  if, for anyfamily ( f j ) j∈J (J ⊆ N) of mutually orthogonal vectors, the ∗-algebras P(a +f j , a f j ) are strongly complementary with respect to the vacuum state ϕ in j∈J

the sense of Definition 5, with   A := P(a + , a) ; A j := P(a +f j , a f j ) ; ϕ j := ϕ 

P (a +f ,a f j ) j

,

j ∈ J (43)

Complementarity and Stochastic Independence

21

The following theorem provides a useful tool to construct examples of complementary algebras acting on IFS. Theorem 2 If the IFS  I (H1 ) satisfies the following two properties: Property (1) For any m ∈ N, any ε := (εn , . . . , ε1 ) ∈ {−1, +1}n and any f ∈ H1 , the vector a εfm · · · a εf1 belongs to

lin-span–{(a +f )n : n ∈ N}

(a +f )0 := 1

;

where 1 denotes the identity in P(a + , a). Property (2) For any N , n ∈ N, any ε := (ε N , . . . , ε2 , +1) ∈ {−1, +1} N   ϕ agε NN · · · agε11 (a +f )n = 0

(44)

whenever f, g1 , . . . , g N ∈ H1 are such that g j ⊥ f for all j and j = k ⇒ g j ⊥ gk Then  I (H1 ) enjoys the strong complementarity property in the sense of Definition 12. Proof We have to prove that, for any n ∈ N and any f 1 , . . . , f n ∈ H1 such that f j ⊥ f k for j = k, the algebras P(a +f j , a f j ) ( j ∈ {1, . . . , n}) are ϕ–n–complementary in the sense of Definition 5, with ϕ j given by (43), i.e., ϕ(Y f1 · · · Y f j ) =

n 

ϕ(Y f j )

∀Y j ∈ P(a +j , a j )

;

(45)

j=1

By linearity (45) is equivalent to n      ε j,m ε1,m 1 ε j,1 ε n εn,1 ε1,1 j ϕ (a fn,m · · · a ) · · · (a · · · a ) = ϕ a · · · a fn f1 f1 fj fj n j=1

where the ε j,h ∈ {−1, +1} are arbitrary. Re-labeling the indexes we write ε

ε

ε2,m 2

n (a fn,m · · · a fn,1 ) · · · (a f2 n n

ε

· · · a f2,1 ) =: agε NN · · · agε11 2

So that the left-hand side of (46) becomes   ε1,m ε ϕ agε NN · · · agε11 (a f1 1 · · · a f1,1 ) 1

(46)

22

L. Accardi and Y.-G. Lu

By Property (1) there exists a finite sub-set F ⊂ N \ {0} such that ε1,m 1

a f1

ε

· · · a f1,1 = c0 ( f 1 ) + 1



cn ( f 1 )(a +f1 )n

n∈F

for some complex constants c0 ( f 1 ), cn ( f 1 ) (n ∈ F). By the Fock property   ε ε 1,m c0 ( f 1 ) = ϕ a f1 1 · · · a f1,1 1 Therefore the left-hand side of (46) can be written in the form  

  ϕ agε NN · · · agε11 c0 ( f 1 ) + cn ( f 1 )ϕ agε NN · · · agε11 (a +f1 )n n∈F

Since f 1 and the g j satisfy the conditions of Property (2), it follows that   ϕ agε NN · · · agε11 (a +f1 )n = 0

;

∀n ∈ F

Therefore the left-hand side of (46) is equal to    ε  ε2,m 2 ε n εn,1 ε2,1 ε1,1 1,m 1 · · · a ) · · · (a · · · a ) ϕ a · · · a ϕ (a fn,m f f f f f n n 2 2 1 1 

and the thesis follows by induction.

4.1 All 1MTIFS Enjoy the Complementarity Property In this section we prove that the quantum polynomial algebras associated to 1MTIFS satisfy Properties (1) and (2) of Theorem 2. This implies that these algebras enjoy the strong complementarity property with respect to the vacuum state in the sense of Definition 12. Theorem 3 Every 1-mode type IFS or symmetric 1-mode type IFS satisfies Property (1) and Property (2) of Theorem 2. Proof Property (1). We have to prove that for any m ∈ N, ε := (εn , . . . , ε1 ) ∈ {−1, +1}n and f ∈ H1 , the vector a εfm · · · a εf1 belongs to

lin-span–{(a +f )n : n ∈ N}

(47) ;

(a +f )0 := 1

(48)

Complementarity and Stochastic Independence

23

where 1 denotes the identity in P(a + , a). This is clear if either m = 1 or a εf1 = a f because the zero vector belongs to the space (48). So we can suppose that ε1 = +1. Suppose by induction that the statement is true up to m − 1 and define h := max{ j ≤ m : εr = +1 ∀ r ∈ {1, · · · , j}} then h ≥ 1 because of ε1 = +1. If h = m, then the vector (47) is equal to (a +f )m which belongs to the space (48) by definition. Otherwise εh+1 = −1 and the vector (47) has the form (49) a εfm · · · a f (a +f )h If h = 1 this expression becomes, both in the symmetric and the full case  f, f ω1 a εfm · · · a εf3 and the thesis follows by induction on m. If h > 1, in the full case, by (30), (49) is equal to ε

ωm  f, f a εfm · · · a fh+2 (a +f )h−1 and again the thesis follows by induction on m. In the symmetric case, by (41), (49) is equal to h−1

a εfm · · · (a +f ) j [a f , a +f ](a +f )h− j−1 j=0

=

h−1

  a εfm · · · (a +f ) j B f B +f (ω+1 − ω ) +  f, f ω (a +f )h− j−1

j=0

=

h−1

a εfm · · · (a +f ) j B f B +f (ω+1 − ω ) (a +f )h− j−1 +  f, f 

j=0

=

h−1

a εfm · · · (a +f ) j ω (a +f )h− j−1

j=0

h−1





ωh− j − ωh− j−1 a εfm

· · · (a +f ) j B f

B +f (a +f )h− j−1 +  f,

⎛ ⎞ h−1

f⎝ ωh− j−1 ⎠ a εfm · · · (a +f )h−1

j=0

j=0

⎛ = ωh  f, f a εfm · · · (a +f )h−1 B f B +f +  f, f  ⎝

h−1

⎞ ωh− j−1 ⎠ a εfm · · · (a +f )h−1

j=0

and also in this case the thesis follows by induction on m. Property (2). We have to prove that, for any N , n ∈ N, any ε := (ε N , . . . , ε2 , +1) ∈ {−1, +1} N   (50) ϕ agε NN · · · agε11 (a +f )n = 0

24

L. Accardi and Y.-G. Lu

for any f, g1 , . . . , g N ∈ H1 such that g j ⊥ f for all j and j = k ⇒ g j ⊥ gk . The identity (50) is clear if ε j = +1 for any j ∈ {1, · · · , N }, so one can suppose that the set of j ∈ {1, . . . , N } such that ε j = −1 is non-empty. Denote h := min{ j ∈ {1, . . . , N } : ε j = −1} Then the right hand side of (50) has the form   + + + n a a · · · a (a ) ϕ agε NN · · · agεh+1 g h g g f h+1 h−1 1

(51)

and the identity (50) is true if h − 1 = N = 0 by the Fock property. Otherwise in the full case, by (30), (51) is equal to ⎧   + n + + ⎨ωn+h−1 gh , gh−1 ϕ a ε N · · · agεh+1 , a · · · a · · · (a ) h+1 gh−2 gN g1 f   + ⎩ωn g1 , f ϕ a ε N · · · a ε2 (a )n−1 , gN g2 f

if h > 1 if h = 1

and in both cases this quantity is 0 because by assumption g j ⊥ f for all j and gh−1 ⊥ gh . Thus the thesis follows by induction. Under the same assumptions, in the symmetric case using (38), (51) becomes Bgh ω ω Bg+h−1 · · · ω Bg+1 (ω B +f )n   , agε NN · · · agεh+1 h+1 Bgh Bg+h−1 · · · Bg+1 (B +f )n  = ωn+h−1 (ωn+h−1 !) , agε NN · · · agεh+1 h+1 and by the Fock property this is equal to [Bgh , Bg+h−1 · · · Bg+1 (B +f )n ]  = ωn+h−1 (ωn+h−1 !) , agε NN · · · agεh+1 h+1 = ωn+h−1 (ωn+h−1 !)

h−1

ε

+ n + + + + +  , agε NN · · · agh+1 h+1 Bgh−1 · · · Bg j+1 [Bgh , Bg j ]Bg j−1 · · · Bg1 (B f ) 

j=1

+ωn+h−1 (ωn+h−1 !)

n−1

 , agε NN · · · agεh+1 Bg+h−1 · · · Bg+1 (B +f ) j [Bgh , B +f ](B +f )n− j−1  h+1

j=0

with the obvious conventions on the extreme values of the sums. Using (27) with the embedding f ∈ H1 → f ∗ ∈ H1∗ given by the semi-scalar product, this becomes ωn+h−1 (ωn+h−1 !)

h−1

 , agε NN · · · agεh+1 Bg+h−1 · · · Bg+j+1 gh , g j Bg+j−1 · · · Bg+1 (B +f )n  h+1

j=1

+n gh , f  ωn+h−1 (ωn+h−1 !) , agε NN · · · agεh+1 Bg+h−1 · · · Bg+1 (B +f )n−1  h+1

Complementarity and Stochastic Independence

25

which is zero by the orthogonality assumption. Thus again the thesis follows by induction. 

4.2 Complementarity Without Independence In this sub-section, we prove that in 1-MTIFS, in products with repeated test functions, the factorization property of vacuum expectation values in general does not hold. This proves that n-complementarity is a strictly weaker property than independence. Since n-complementarity implies complementarity for each pair of algebras involved, it is sufficient to consider the case of complementarity. In all IFS the field operators are defined by Q f ≡ X f = a +f + a f ∈ P(a +f , a f )

(52)

The vacuum distribution of X f defines a classical ϕ-symmetric random variable, i.e.,  = X 2n+1 =0  , X 2n+1 f f whose variance is

X 2f  = a f a +f  = ω1  f 2

Now consider X f (a +f )n = (a +f + a f )(a +f )n = (a +f )n+1 + a f (a +f )n = (a +f )n+1 +

n−1  

(a +f )k B f a f , a +f B +f (a +f )n−k−1 k=0

= (a +f )n+1 +

n−1

(a +f )k B f (ω+2 − ω+1 ) B +f (a +f )n−k−1

k=0

=

(a +f )n+1

n−1 

+ (ωn−k+1 − ωn−k ) (a +f )n−1 k=0

Since ω0 = 0, it follows that X f (a +f )n = (a +f )n+1 + ωn (a +f )n−1

(53)

26

L. Accardi and Y.-G. Lu

which is the monic Jacobi relation for the symmetric vacuum random variable X f , where the (a +f )n is its n-th orthogonal polynomial. Since the 1MTIFS are of type I in the sense of [6], it follows that the right hand side of (53) is isomorphic to the canonical quantum decomposition of X f (see [6] for the definition) and that in the isomorphism 

C · (a +f )n ≡ L 2pol (R, μ) := closure of the polynomials in L 2 (R, μ)

n∈N

X f goes into the multiplication operator and the (a +f )n (n ∈ N) into the orthogonal polynomials of X f . In the language of [6] this means that the quantum decomposition (52) is isomorphic to the canonical quantum decomposition of X f . Lemma 2 Suppose that f ⊥ g. Then   [X f , X g ] = Bg+ B f − B +f Bg (ω+1 − ω )

(54)

Proof Using (41) and f ⊥ g, one finds [X f , X g ] = [a +f + a f , ag+ + ag ] = [a +f , ag ] + [a f , ag+ ] = [a f , ag+ ] − [ag , a +f ]   = Bg+ B f − B +f Bg (ω+1 − ω ) 

which is (54).

Theorem 4 Let f, g ∈ H1 be such that f ⊥ g. If ω1 = 0 (otherwise one has a trivial IFS), then the operator random variables X f , X g are not ϕ-tensor-independent in ω, f ull (H1 ). If ω1 = ω2 they are also not ϕ-tensor-independent in ω (H1 ). In particular, the algebras P(a +f , a f ) and P(ag+ , ag ) in both cases are ϕ-complementary but not ϕ-tensor-independent Proof In the full case one has X f X g X f X g  = X f , X g X f X g  = a +f , (ag+ + ag )(a +f + a f )ag+  = ag+ a +f , a +f ag+  = a +f , ag a +f ag+  = 0 but, using (31) one finds X 2f X g2  = a +f , a +f ag+ , ag+  = ω ∗f , ω ∗f ω ∗g , ω ∗g  = ω12  ∗f , ∗f  ∗g , ∗g  = ω12  f 2 g2 = 0 In the symmetric case one has from the formula (54)

Complementarity and Stochastic Independence

27

X f X g X f X g  = X f [X g , X f ]X g  + X 2f X g2    = X f Bg+ B f − B +f Bg (ω+1 − ω ) X g  + X 2f X g2    = a f Bg+ B f − B +f Bg (ω+1 − ω ) ag+  + X 2f X g2  and X 2f X g2  = X 2f X g2  because o f f ⊥ g Using (38) and ω0 = 0 this becomes   (ω2 − ω1 )a +f , Bg+ B f −B +f Bg ag+  + X 2f X g2    = (ω2 − ω1 )ω B +f , Bg+ B f − B +f Bg ω Bg+  + X 2f X g2  = ω12 (ω2 − ω1 )B +f , Bg+ B f Bg+  − ω12 (ω2 − ω1 )B +f , B +f Bg Bg+  + X 2f X g2  = ω12 (ω2 − ω1 ) Bg B +f , B f Bg+  − ω12 (ω2 − ω1 ) B f B +f , Bg Bg+  + X 2f X g2  = ω12 (ω2 − ω1 )  f, g2 − ω12 (ω2 − ω1 )  f 2 g2 = −ω12 (ω2 − ω1 )  f 2 g2 + X 2f X g2 

Thus from Theorem 3 we conclude that X f X g X f X g  = − (ω2 − ω1 )  f 2 g2 + X 2f X g2   = X 2f X g2 

if ω2 = ω1 .



5 Weak Complementarity in Monotone IFS Monotone IFS arise from functional central limit theorems of monotone-independent operator random variables. In this section, we prove that monotone IFS do not enjoy the strong complementarity property in the sense of Definition 12. Then we prove that they enjoy a weak form of the strong complementarity property.

5.1 The Monotone IFS Theorem 5 There exists a unique IFS over L 2 (R) with the following properties a +f Fn := f ⊗ Fn

(55)

28

L. Accardi and Y.-G. Lu

a f a +f1 · · · a +fn =



( f¯ f 1 )(t)a +f2 χ[0,t] · · · a +fn χ[0,t] dt

(56)

af = 0

(57)

Proof We prove that the 3 conditions above uniquely determine the semi-scalar product on Tens(H1 ). For each n ∈ N, define the sesqui-linear form κn ( f 1 ⊗ · · · ⊗ f n ; g1 ⊗ · · · ⊗ gn ) := a +f1 · · · a +fn , ag+1 · · · ag+n  =  =  =

dt ( f¯1 g1 )(t)a +f2 χ[0,t] · · · a +fn χ[0,t] , ag+2 χ[0,t] · · · ag+n χ[0,t] 

  dt ( f¯1 g1 )(t)κn−1 f 2 χ[0,t] ⊗ · · · ⊗ f n χ[0,t] ; g2 χ[0,t] ⊗ · · · ⊗ gn χ[0,t]

Clearly κ1 is positive definite. Suppose by induction that κn−1 is positive definite. Then since, for each t ∈ R the kernel ( f 1 ⊗ f 2 ⊗ · · · ⊗ f n , g1 ⊗ g2 ⊗ · · · ⊗ gn )  

→ ( f¯1 g1 )(t)κn−1 f 2 χ[0,t] ⊗ · · · ⊗ f n χ[0,t] ; g2 χ[0,t] ⊗ · · · ⊗ gn χ[0,t] is positive definite by the induction assumption and Schur’s Lemma, κn is positive definite as an integral of kernels with this property with respect to a positive measure. Uniqueness of the IFS follows from the fact that the scalar product is uniquely determined. 

5.2 Weak Complementarity in Monotone IFS The following Lemma shows that monotone IFSs do not enjoy the strong complementarity property. Lemma 3 In the monotone Fock space over L 2 ([0, 1]): (i) Property (2) of Theorem 2 is not satisfied, (ii) There exist orthogonal test functions f 1 , f 2 ∈ L 2 ([0, T ]) such that the associated polynomial algebras P(a +f1 , a f1 ), P(a +f2 , a f2 ) are not complementary with respect to the vacuum state. Proof Consider the functions f 1 = χ[0, 21 ] − χ[ 21 ,1] They satisfy  f 1 , f 2  = 0 and

,

f 2 = χ[0,1]

Complementarity and Stochastic Independence

29

a f1  = a f1 a +2 f2  = 0 However a f1 a f2 a +2 f2  =

 =

0

1 2

(1 − s)ds−

 1 1 2



1

0

(1 − s)ds=

| f 2 |2 (t)a f1 a +f2 χ[0,t] dt =



1

f 1 (s)(1 − s)ds

0

    1  1 1 1 1 1 1 1 1 2 1 1 2 1 − s 2  2 − + s 2  1 = 12 − − = = 0 2 2 0 2 2 2 2 2 2 2 4 2

i.e. even if  f 1 , f 2  = 0, one has:   1 +2 = a f1 a f2 a +2 f 2   = a f 1 a f 2 a f 2  = 0 4 From this both statements of the theorem follow.



Lemma 4 On the monotone Fock space over L 2 ([0, T ]) with T ∈ [1, +∞] ([0, +∞] understood as [0, +∞)), for any n ≥ 0, m ≥ 1 and ε ∈ {−1, 1}n , for any { f k }nk=1 and {gh }m h=1 such that ∀k, h f k gh =a.s. 0 ; one has

ε(n) + + a ε(1) f 1 · · · a f n ag1 · · · agm  = 0

(58)

Proof (58) is trivial for n = 0. For n = 1, (58) is trivial if ε(1) = +1 and, if ε(1) = −1, then  ε(1) + + + + f¯1 g1 (t)ag+2 χ[0,t] · · · ag+m χ[0,t] a f1 ag1 · · · agm = a f1 ag1 · · · agm = which is equal to zero thanks to the fact that f 1 g1 =a.s. 0. Thus (58) holds for n = 0, 1. Suppose that the thesis is true for all n ≤ N and consider the case n = N + 1. If ε(k) = +1 for any k, (58) is trivial. So we examine (58) for ε such that ε−1 ({−1}) = ∅. If ε(N + 1) = −1 , then +1) + + + + a ε(N f N +1 ag1 · · · agm = a f N +1 ag1 · · · agm

which is equal to zero in virtue of the fact f N +1 g1 =a.s. 0 So we consider such ε that ε−1 ({−1}) = ∅ and k := max ε−1 ({−1}) < N + 1

30

L. Accardi and Y.-G. Lu

In this case ε(N +1) + ε(1) ε(k−1) + + + + + a ε(1) f 1 · · · a f N +1 ag1 · · · agm  = a f 1 · · · a f k−1 a f k a f k+1 · · · a f N +1 ag1 · · · agm 

 =

ε(k−1) + + + + f¯k f k+1 (t)a ε(1) f 1 · · · a f k−1 a f k+2 χ[0,t] · · · a f N +1 χ[0,t] ag1 χ[0,t] · · · agm χ[0,t] 

Now notice that for any r ∈ {1, · · · , k − 1} ∪ {k + 2, · · · , N + 1} and for any h ∈ {1, 2, · · · , m}, it follows from the fact gh fr =a.s. 0 that gh χ[0,t] fr = (gh χ[0,t] )( fr χ[0,t] ) =a.s. 0 So the induction assumption gives ε(k−1) + + + + a ε(1) f 1 · · · a f k−1 a f k+2 χ[0,t] · · · a f N +1 χ[0,t] ag1 χ[0,t] · · · agm χ[0,t]  = 0 ; ∀t

hence

ε(N +1) + + a ε(1) f 1 · · · a f N +1 ag1 · · · agm  = 0



Therefore the thesis follows by induction.

Analysis of the proof of Lemma 3, suggests the following weaker form of Definition 12. Definition 13 Let H1 be a semi-Hilbert space of functions defined on a measure space (S, B, μ) with any scalar product such that any two functions with disjoint support are orthogonal. An IFS  I (H1 ) is said to enjoy the weak complementarity property if any family ( f j ) j∈J  (J ⊆ N) ofvectors in H1 with mutually disjoint supports is such that

the algebras P(a +f j , a f j )

j∈I

are complementary with respect to the vacuum state

ϕ in the sense of Definition 5, with   ϕ j := ϕ 

P (a +f ,a f j )

;

j∈I

j

Theorem 6 The monotone Fock space over L 2 ([0, T ]) enjoys the weak complementarity property in the sense of Definition 13. Proof It is sufficient to prove that, for any n ≥ 0, m ≥ 1 and { f k }nk=1 , {gh }m h=1 such that (59) f k gh =a.s. 0 the identity ε(N ) ε (1) ε(1) ε(N ) ε (m) ε (1) ε (m) a ε(1) f 1 · · · a f n ag1 · · · agm  = a f 1 · · · a f n ag1 · · · agm 







(60)

Complementarity and Stochastic Independence

31

holds on the monotone Fock space over L 2 ([0, T ]) for any ε ∈ {−1, 1}n , ε ∈ {−1, 1}m . Notice that, if either ε (m) = −1 or ε(1) = +1, then (60) holds trivially because both sides are zero. If ε (h) = +1 for all h, then the same happens because, in this case, Lemma 3 implies that the left-hand side of (60) is zero and the right-hand side is zero because ag+1 · · · ag+m  = 0 Therefore it is sufficient to prove (60) in the case when {h : ε (h) = −1} = ∅ ; ε (m) = +1 ; ε(1) = −1

(61)

For m = 1, no ε can satisfy (61). For m = 2, (61) implies that, (ε (1), ε (2)) = (−1, +1) In this case (60) follows from







agε1(1) agε2(2) = ag1 ag+2 = g1 , g2  = ag1 ag+2  = agε1(1) agε2(2)  Suppose by induction that the thesis holds for m ≤ N and let ε ∈ {−1, 1} N +1 satisfy (61). Denote m 0 := max {h : ε (h) = −1} Then (61) implies that m 0 < N + 1. In this case





+1) = agm0 ag+m +1 · · · ag+N +1 agεm(m 0 ) agεm(m+10 +1) · · · agε N(N +1 0 0 0    + + = g¯ m 0 gm 0 +1 (t)agm +2 χ[0,t] · · · ag N +1 χ[0,t]

(62)

0

Since, for all t ∈ R, condition (59) remains true if all the gh are multiplied by χ[0,t] , one can use the induction assumption and (62) to obtain ε(n) ε (1) ε (N +1)  a ε(1) f 1 · · · a f n ag1 · · · ag N +1





  ε(n) ε (1) ε (m 0 −1) + = g¯ m 0 gm 0 +1 (t)a ε(1) agm +2 χ[0,t] · · · ag+N +1 χ[0,t]  f 1 · · · a f n ag1 · · · agm 0 −1 0    ε(n) ε (1) ε (m 0 −1) + = g¯ m 0 gm 0 +1 (t)a ε(1) agm +2 χ[0,t] · · · ag+N +1 χ[0,t]  f 1 · · · a f n ag1 · · · agm 0 −1 0    ε(n) = a ε(1) g¯ m 0 gm 0 +1 (t)agε1(1) · · · agεm(m−10 −1) ag+m +2 χ[0,t] · · · ag+N +1 χ[0,t]  f1 · · · a fn  · 0

ε(n) ε (1) ε (m 0 −1) agm0 ag+m = a ε(1) f 1 · · · a f n ag1 · · · agm −1

0

and this proves the statement.

0



0 +1

· · · ag+N +1  

32

L. Accardi and Y.-G. Lu

References 1. Accardi, L.: Some trends and problems in quantum probability. In: Accardi, L., Frigerio, A., Gorini, V. (eds.) Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, Proceedings of 2-d Conference: Quantum Probability and Applications to the Quantum Theory of Irreversible Processes Villa Mondragone (Rome), vol. 9, pp. 6–11 (1982), Springer LNM, vol. 1055, pp. 1–19. Springer, Berlin (1984) 2. Accardi, L., Bach, A.: Central limits of squeezing operator. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications IV. Springer LNM, vol. 1396, pp. 7–19. Springer, Berlin (1987) 3. Accardi, L., Barhoumi, A., Dhahri, A.: Identification of the theory of orthogonal polynomials in d-indeterminates with the theory of 3-diagonal symmetric interacting Fock spaces on Cd . Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IDA-QP) 20(1), 1–55 (2017) August (2015), revised 25 Jan 2016 4. Accardi, L., Crismale, V., Lu, Y.G.: Constructive universal central limit theorems based on interacting Fock spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IDA-QP) 8(4), 631–650 (2005). Preprint Volterra vol. 591 (2005) 5. Accardi, L., Kuo, H.-H., Stan, A.: Characterization of probability measures through the canonically associated interacting Fock spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IDA-QP) 7, 485–505 (2004) 6. Accardi, L., Lu, Y.G.: The quantum moment problem for a classical random variable and a classification of interacting Fock spaces, to be submitted to (2017) 7. Accardi, L., Lu, Y.G., Mastropietro, V.: Stochastic bosonization for a d ≥ 3 Fermi system. Annales de l’Inst. Henri Poincaré 66(2), 185–213 (1997). Preprint Volterra vol. 205 (1995) 8. Accardi, L., Lu, Y.G., Volovich, I.V.: Quantum Theory and Its Stochastic Limit. Springer, Berlin (2002) 9. Accardi, L., Lu, Y.G., Volovich, I.: The QED Hilbert Module and Interacting Fock Spaces. Publications of IIAS (Kyoto), N1997–008 (1997) 10. Arecchi, F.T., Courtens, E., Gilmore, R., Thomas, H.: Atomic coherent states in quantum optics. Phys. Rev. A 6, 2211–2237 (1972) 11. Björck, G.: Functions of modulus 1 on Z n , whose Fourier transforms have constant modulus, and cyclic n–roots. Recent Advances in Fourier Analysis and Its Applications. NATO, Advance Science Institutes Series C, Mathematical and Physical Sciences, vol. 315, pp. 131–140. Kluwer Academic Publisher, Netherlands (1990) 12. Busch, P., Lahti, P.J.: To what extent do position and momentum commute? Phys. Lett. A 115(6), 259–264 (1986) 13. Cassinelli, G., Varadarajan, V.S.: On Accardi’s notion of complementary observables. Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IDAQP) 5(2), 135–144 (2002) 14. Conway, J.B.: A Course in Functional Analysis. Springer, Berlin (1985) 15. Haagerup, U.: Cyclic p–roots of prime length p and related complex Hadamard matrices (2008). arXiv:0803.2629 [math.AC]. Revised version arXiv:0803.2629v1 [math.AC] 16. Ivonovic, I.D.: Geometrical description of quantal state determination. J. Phys. A Math. Gen 14, 3241 (1981) 17. Jammer, M.: The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York (1966) 18. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1967) 19. Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987) 20. Lu, Y.G.: Interacting Fock space related to the Anderson model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IDAQP) 1(2), 247–283 (1998) 21. Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103–1106 (1988) 22. Ohya, M., Petz, D.: Quantum Entropy and Its Use. Texts and Monographs in Physics. Springer, Berlin (1993)

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23. Parthasarathy, K.R.: On estimating the state of a finite level quantum system. Infin. Dimens. Anal. Quantum. Probab. Relat. Top. (IDAQP) 7(4), 607–617 (2004) 24. Petz, D., Ruppert, L.: Efficient quantum tomography needs complementary and symmetric measurements. Rep. Math. Phys. 69(2), 161–177 (2012) 25. Popa, S.: Orthogonal pairs of ∗-sub-algebras in finite von Neumann algebras. J. Oper. Theory 9, 253–268 (1983) 26. Renyi, A.: Foundations of Probability. Holden-Day, San Francisco (1970) 27. Schwinger, J.: Unitary operator bases. Proc. Natl. Acad. Sci. 46, 570–579 (1960) 28. Tadej, W., Zyczkowski, K.: A concise guide to complex Hadamard matrices. Open Syst. Inf. Dyn. 13(2), 133–177 (2006) 29. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955) 30. Wootters, W.K., Fields, B.D.: On the U.V. mutually unbiased bases. Ann. Phys. 191, 363 (1989)

Norm Conditions for Separability in Mm ⊗ Mn Tsuyoshi Ando

Abstract An element S of the tensor product Mm ⊗ Mn is said to be separable if it admits a (separable) decomposition S =



X p ⊗ Y p ∃ 0 ≤ X p ∈ Mm , ∃ 0 ≤ Y p ∈ Mn .

p

This decomposition is not unique. We present some conditions on suitable norms of S which guarantee its separability. Even when separability of S is guaranteed by some method, its separable decomposition itself is difficult to construct. We present a general condition which makes it possible to find a way of an explicit separable decomposition.

1 Introduction Let Mn denote the space of n × n (complex) matrices for each n = 1, 2, . . .. Each element of Mn is considered as a linear map from Cn to itself. Here an element x of Cn is understood as a column n-vector, and correspondingly x ∗ is a row n-vector. Then given x, y ∈ Cn , according to the rule of matrix multiplication, x ∗ y is the inner product of x and y, that is, x ∗ y = x|y while yx ∗ is a matrix of rank ≤ 1 in Mn . Notice here that the inner product √ x|y is linear in y and anti-linear in x. With this inner product and norm x = x|x, Cn becomes a Hilbert space. Correspondingly the space Mn becomes a Hilbert space with the inner product T |S := Tr(T ∗ S) ∀ S, T ∈ Mn

T. Ando (B) Hokkaido University (Emeritus), Hokkaido, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_2

35

36

T. Ando

where Tr(S) is the trace of S, and the Hilbert–Schmidt (or Frobenius) norm S 2 :=

 S|S ∀ S ∈ Mn .

For self-adjoint S, T ∈ Mn , the order relation S ≥ T or equivalently T ≤ S is defined as S − T is positive semi-definite. Therefore S ≥ 0 or 0 ≤ S means simply that S is positive semi-definite. The operator (or spectral) norm S of S ∈ Mn is defined by S := sup{ Sx ; x = 1}.

(1)

When S is selfadjoint, the norm S is calculated simply as S = S∗

=⇒

S = sup{|x|Sx|; x = 1}.

(2)

As an immediate consequence of (2), S = S ∗ , I − S ≤ 1

=⇒

S ≥ 0.

(3)

In fact, the assumption guarantees that x 2 − x|Sx ≤ x 2 ∀ x. Next let us turn to the tensor product Mm ⊗ Mn of two matrix spaces Mm and Mn . There are canonical identifications Mm ⊗ Mn ∼ Mm (Mn ) ∼ Mmn . Here Mm (Mn ) denotes the space of m × m block matrices with entries in Mn . The first identification is understood in the following way: X ⊗ Y ∼ [ξ j,k Y ] j,k ∀ X = [ξ j,k ] j,k ∈ Mm , ∀ Y ∈ Mn .

(4)

The second identification is understood by cutting each mn × mn matrix into m × m block matrices in a natural way. Therefore by (4) 

E j,k ⊗ S j,k ∼ S = [S j,k ] j,k

j,k

where E j,k ( j, k = 1, 2, . . . , m) is the matrix unit in Mm , that is, E j,k = e j ek∗ ( j, k = 1, 2, . . . , m) with the canonical orthonormal basis e j ( j = 1, 2, . . . , m) of Cm .

Norm Conditions for Separability in Mm ⊗ Mn

37

In the following we abuse the notation: S =



E j,k ⊗ S j,k ∀ S = [S j,k ] j,k ∈ Mm (Mn ).

(5)

j,k

According to the general rule, the space Mm (Mn ) becomes a Hilbert space with inner product and norm; S|T = Tr(S∗ T) =



Tr(S ∗jk T jk ) and S 2 =

 S|S.

j,k

Further the spectral norm (1) for S = [S jk ] j,k takes the form     S = sup { S j,k xk 2 }1/2 ; xk 2 = 1 ∀ xk ∈ Cn (k = 1, 2, . . . , n) . j

k

k

(6) In the following, M(m,n) will denote the real subspace of Mm (Mn ) consisting of ∗ selfadjoint elements, that is, the subspace of S = [S j,k ] j,k with S j,k = Sk, j ( j, k = 1, 2, . . . , m). Let P denote the cone of positive semi-definite elements in M(m,n) . We shall use S ≥ 0 as usual when S ∈ P, and S > 0 when S ≥ 0 and is invertible. In the tensor product theory a fact of key importance is the following [3, p.23]; 0 ≤ X ∈ Mm , 0 ≤ Y ∈ Mn

=⇒

X ⊗ Y ∈ P.

(7)

The cone, generated by X ⊗ Y with 0 ≤ X ∈ Mm and 0 ≤ Y ∈ Mn , will be denoted by P+ . The cone P+ is contained in the cone P by (7). A block matrix in P+ is called separable. By definition, S is separable if and only if it admits a decomposition S =



X p ⊗ Y p ∃ 0 ≤ X p ∈ Mm , ∃ 0 ≤ Y p ∈ Mn .

(8)

p

Difficulty is in the fact that decomposition (8) is not unique. A nontrivial fact is that the cone P+ is topologically closed. This comes from the finite dimensionality of Mm ⊗ Mn . (See [1, p. 8]).

2 Norm Conditions for Separability Let us begin with an inner characterization of the cone P+ . A linear map ϕ : Mn → M N is said to be positive if ϕ(S) ≥ 0 whenever S ≥ 0. It is called unital if ϕ(In ) = I N where In and I N are the identity matrices in Mn and M N respectively. It is known as Russo-Dye theorem [4, Theorem 2.3.7] that a unital, positive linear map ϕ is

38

T. Ando

contractive in the sense ϕ(S) ≤ S ∀ S ∈ Mn .

(9)

A linear map ϕ : Mn → M N gives rise to a linear map ϕ˜ : Mm (Mn ) → Mm (M N ) by

   ϕ˜ [S j,k ] j,k := ϕ(S j,k )

j,k

∀ S = [S j,k ] j,k .

(10)

˜ But positivity of ϕ The map ϕ˜ is often denoted by idm ⊗ ϕ. If ϕ is unital, so is ϕ. does not imply positivity of ϕ˜ in general. Horodecki’s [7] established an inner characterization of separability of S ∈ M(m,n) in the following form. Theorem 1 (Horodecki’s [7]) Let S ∈ M(m,n) . Then S ∈ P+

⇐⇒ ϕ(S) ˜ ≥ 0 ∀ N , ∀ unital positive linear map ϕ : Mn → M N .

In view of (3), an actual form of Theorem 1, useful in establishing separability of S, is the following. Corollary 2 Let S ∈ M(m,n) . Then, ∀ N , with I = Im ⊗ I N , I − ϕ(S) ˜ ≤ 1 ∀ unital positive linear ϕ : Mn → M N

=⇒ S ∈ P+ .

A norm ||| · ||| on Mm (Mn ) will be said to have property () if ∀ N ϕ(S) ˜ ≤ |||S||| ∀ unital positive linear ϕ : Mn → M N , ∀ S ∈ Mm (Mn ). () Corollary 3 If a norm ||| · ||| on Mm (Mn ) has property (), then for S ∈ M(m,n) |||I − S||| ≤ 1

=⇒ S ∈ P+ .

In fact, since for any unital positive linear map ϕ : Mn → M N , I − ϕ(S) ˜ = ϕ(I ˜ − S) ≤ |||I − S||| ≤ 1, where I on the left-hand term is the identity in Mm (M N ) while I on the middle and ˜ ≥ 0 and the right hand terms are the identity in Mm (Mn ), appeal to (3) to see ϕ(S) then to Theorem 1 to see S ∈ P+ . Define a functional · (+) for S = [S j,k ] j,k ∈ Mm (Mn ) by



S (+) := S j,k  where S j,k 1, 2, . . . , m).

j,k

j,k



,

(11)

is the element of Mm whose ( j, k)-entry is S j,k ( j, k =

Norm Conditions for Separability in Mm ⊗ Mn

39

Lemma 4 The functional · (+) is a norm with property (). Proof That · (+) becomes a norm on Mm (Mn ) is obvious. Let us show that S ≤ S (+) ∀ S = [S jk ] j,k ∈ Mm (Mn ). This results from (6) and definition (11) by the inequalities;



S j,k xk ≤

k



S j,k · xk ∀ xk ∈ Cn ( j = 1, 2, . . . , n).

k

Take a unital positive linear map ϕ : Mn → M N . Since ϕ is contractive by (9) ϕ(S j,k ) ≤ S j,k ∀ j, k we can see again from (6) ϕ(S) ˜ ≤ ϕ(S) ˜ (+) ≤ S (+) . This completes the proof. Corollary 5 Every norm ||| · ||| on Mm (Mn ) for which S (+) ≤ |||S||| ∀ S ∈ Mm (Mn ) has property (). The following are among simple examples of norms with property (): (A) (B)

|||S||| := m · S . 2 |||S||| := S 2 = j,k S j,k 2

(A)

(Hilbert–Schmidt norm).

Since S j,k ≤ S ∀ j, k, by (6) S (+)

⎡ ⎤

1



≤ S · ⎣· · ·⎦ · [1 . . . 1] = m · S . 1

(B) Since S j,k ≤ S j,k 2 ∀ j, k, and the operator norm ≤ the Hilbert–Schmidt norm in Mm , S (+) ≤



S j,k 2 ≤

j,k



S j,k 22 = S 2 .

j,k

Corollary 6 Let S ∈ M(m,n) . I − S ≤ 1/m or

I − S 2 ≤ 1

=⇒

S ∈ P+ .

40

T. Ando

The first assertion can be found in [1] while the second in [6].

3 Actual Conditions for Separability Notice that S is separable if and only if t · S is separable ∀ t > 0 or ∃ t > 0. It is easy to see that for 0 < S ∈ Mn S = λmax (S) = the maximum eigenvalue of S and

S −1 −1 = λmin (S) = the minimum eigenvalue of S.

Theorem 7 Let 0 < S ∈ M(m,n) . Then S · S−1 ≤

2 m+1 = 1+ m−1 m−1

=⇒

S ∈ P+ .

Proof The assumption means that there is 0 < t such that m−1 m+1 · S−1 ≤ t ≤ · S −1 . m m Then for such t > 0 tS − I ≤ {t · λmax (S) − 1} · I 1 m+1 − 1} · I ≤ · I, ≤ { m m and I − tS ≤ {1 − tλmin (S)} · I ≤ Therefore I − tS ≤

1 . m

Now appeal to Corollary 6 to see that S ∈ P+ . Notice here the obvious relation; 1 ≤ S · S−1 . Theorem 8 Let 0 ≤ S ∈ M(m,n) . Then

1 · I. m

Norm Conditions for Separability in Mm ⊗ Mn

S 2 ≤ √ Proof With t :=

Tr(S) S 22

1 mn − 1

41

· Tr(S)

=⇒

S ∈ P+ .

> 0,

I − t · S 22 − 1 = t 2 · S 22 − 2t · Tr(S) + mn − 1 √ Tr(S)2 Tr(S)  √ Tr(S)  · ≤ 0. = mn − 1 − = mn − 1 + mn − 1 − 2 S 2 S 2 S 2 Now appeal to Corollary 6 to see that S ∈ P+ . Notice here the obvious Schwartz inequality; 1 S 2 ≥ √ · Tr(S) ∀ 0 ≤ S ∈ M(m,n) . mn

4 Explicit Separable Decomposition Let us begin with some examples of S’s which admit explicit separable decompositions. Lemma 9 For 0 ≤ S j ∈ Mn ( j = 1, 2, . . . , m) the block matrix ⎡

S1 ⎢0 ⎢ S := Diag(S1 , S2 , . . . , Sm ) = ⎢ . ⎣ ..

0 S2 .. .

0 0 .. .

... ... .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

0 0 0 . . . Sm

admits an explicit separable decomposition. This is trivial because S = j E j, j ⊗ S j by (5). Lemma 10 If S ∈ Mn and α · β ≥ S 2 ∃ α, β > 0 then for any 1 ≤ j < k ≤ m E j, j ⊗ α · I + E j,k ⊗ S + E k, j ⊗ S ∗ + E k,k ⊗ β · I admits an explicit separable decomposition in Mm ⊗ Mn ∀ m ≥ 2. Proof It suffices to prove that when 1 ≥ S 2 the block matrix S = an explicit separable decomposition in M2 ⊗ Mn .



 I S admits S∗ I

42

T. Ando

It is immediate to see that S≥0

⇐⇒

1 ≥ S 2 .

Take a unital positive linear map ϕ : Mn → M N . Since ϕ is contractive by (9), definition (10) shows ϕ˜

      I S  ϕ(I ) ϕ(S) I ϕ(S) = = ≥ 0, S∗ I ϕ(S ∗ ) ϕ(I ) ϕ(S)∗ I

so that S is separable in M2 ⊗ Mn by Theorem 1. Let us present an explicit separable decomposition of S. It is an exercise that the contraction S is written as S = 21 {U + V } ∃ unitary U, V. By the spectral theorem for unitary matrices in Mn U=

n 

eiθ j P j and V =

j=1

n 

eiϕ j Q j ∃ 0 ≤ θ j , ϕ j < 2π ( j = 1, 2, . . . , n)

j=1

where P j , Q j are orthoprojections with n  j=1

Then S=

1 2

Pj =

n 

Q j = I.

j=1

  n  n     1 eiθ j 1 eϕ j + ⊗ P ⊗ Qj . j −iθ j −ϕ j e 1 e 1 j=1

j=1



 1 eiθ Since 0 ≤ −iθ in M2 ∀ 0 ≤ θ < 2π , each summand belongs to P+ , so does e 1 S. Now let us return to the case of the norm · (+) . Lemma 11 Let 0 ≤ S = [S j,k ] j,k ≤ I. Then ⎤ λmin (S1,1 ) − S1,2 − S1,3 . . . − S1,m ⎢ − S2,1 λmin (S2,2 ) − S2,3 . . . − S2,m ⎥ ⎥ ⎢ ⇐⇒ ⎢ ⎥ ≥ 0. .. .. .. .. .. ⎦ ⎣ . . . . . − Sm,1 − Sm,2 − Sm,3 . . . λmin (Sm,m ) ⎡

I − S (+) ≤ 1

Norm Conditions for Separability in Mm ⊗ Mn

43

Proof Since the matrix ⎡

⎤ I − S1,1 S1,2 S1,3 . . . S1,m ⎢ S2,1 I − S2,2 S2,3 . . . S2,m ⎥ ⎢ ⎥ A := ⎢ ⎥ .. .. .. .. .. ⎣ ⎦ . . . . . Sm,2 Sm,3 . . . I − Sm,m Sm,1 is entrywise nonnegative and symmetric, A ≤ 1 simply means that I − A ≥ 0, that is ⎡ ⎤ 1 − I − S1,1 − S1,2 − S1,3 . . . − S1,m ⎢ − S2,1 1 − I − S2,2 − S2,3 . . . − S2,m ⎥ ⎢ ⎥ ⎢ ⎥ ≥ 0. .. .. .. .. .. ⎣ ⎦ . . . . . − Sm,1

− Sm,2

− Sm,3 . . . 1 − I − Sm,m

Further when 0 ≤ S ≤ I, 1 − I − S j, j = λmin (S j, j ) ∀ j = 1, 2, . . . , m. This completes the proof. There are many interesting results, based on the Perron–Frobenius theory on entrywise nonnegative matrices. (See, for instance, [2].) The most basic is the following assertion. If A ∈ Mm is entrywise nonnegative, then the eigenvalue with maximum modulus is real nonnegative and a corresponding eigenvector is entrywise nonnegative. When applied to the case when A is symmetric in addition, this produces the following result. Theorem 12 (Drew, Johnson and Loewy [5]) Let A = [α j,k ] j,k ∈ Mm be entrywise nonnegative and symmetric. If its comparison matrix ⎡

α1,1 −α1,2 −α1,3 ⎢ −α2,1 α2,2 −α2,3 ⎢ M(A) := ⎢ . .. .. ⎣ .. . . −αm,1 −αm,2 −αm,3

⎤ . . . −α1,m . . . −α2,m ⎥ ⎥ .. ⎥ ... . ⎦ . . . αm,m

is positive semi-definite, then there is a decomposition of each diagonal entry α j, j =



λ j,k with λ j,k ≥ 0 ∀ k = j

k= j

such that λ j,k λk, j ≥ α j,k αk, j = |α j,k |2 ∀ j = k.

44

T. Ando

Theorem 13 If 0 ≤ S ≤ I and I − S (+) ≤ 1 then S admits an explicit separable decomposition. Proof Since by Lemma 11 ⎤ λmin (S1,1 ) − S1,2 − S1,3 . . . − S1,m ⎢ − S2,1 λmin (S2,2 ) − S2,3 . . . − S2,m ⎥ ⎥ ⎢ ⎥ ≥ 0, ⎢ .. .. .. .. .. ⎦ ⎣ . . . . . − Sm,1 − Sm,2 − Sm,3 . . . λmin (Sm,m ) ⎡

by Theorem 12 each λmin (S j, j ) admits a decomposition λmin (S j, j ) =



λ j,k ∃ λ j,k ≥ 0 ∀ k = j

k= j

such that λ j,k λk, j ≥ S j,k 2 ∀ k = j. Then by Lemma 10 E j, j ⊗ λ j,k · I + E j,k ⊗ S j,k + E k, j ⊗ Sk, j + E k,k ⊗ λk, j · I admits an explicit separable decomposition in Mm ⊗ Mn . Finally by Lemma 9 S ≡ +





  E j, j ⊗ S j, j − λmin (S j, j ) · I

j

E j, j ⊗ λ j,k · I + E j,k ⊗ S j,k + E k, j ⊗ Sk, j + E k,k ⊗ λk, j · I



j 0 such that d 

Re

ckl (x) ξk ξk ≥ μ |ξ |2

k,l=1

for all x ∈ Rd and ξ ∈ Cd . We emphasise that we do not assume that ckl = clk for all k, l ∈ {1, . . . , d}. Consider the operator A=−

d 

∂k ckl ∂l : L 1,loc (Rd ) → D  (Rd ).

k,l=1

Define the operator A1 on L 1 (Rd ) with maximal domain D(A1 ) = {u ∈ L 1 (Rd ) : Au ∈ L 1 (Rd )} given by A1 u = Au. Then −A1 is the generator of a contractive C0 -semigroup, see for example [8] Theorem 3.1 and Lemma 3.7. The domain of A1 is larger than the Sobolev space W 2,1 (Rd ). Next, define the operator V1 in L 1 (Rd ) by

50

W. Arendt and A. F. M. ter Elst

D(V1 ) = {u ∈ L 1 (Rd ) : V u ∈ L 1 (Rd )} and V1 u = V u. Then −V1 generates a positive contraction semigroup T on L 1 (Rd ) given by (Tt u)(x) = e−t V (x) u(x). It is clear that both A1 and V1 are m-accretive and densely defined. Surprising is that A1 + V1 is m-accretive, with the natural domain D(A1 ) ∩ D(V1 ). Moreover, it is also surprising that the semigroup generated by −(A1 + V1 ) is holomorphic on L 1 . These two regularity results are still of big actuality. For the Laplacian they were proved by Kato with very elegant arguments involving Kato’s inequality. We now extend these results to (possibly non-symmetric) elliptic operators. Theorem 2.1 Consider the operator A1 + V1 with natural domain D(A1 + V1 ) = D(A1 ) ∩ D(V1 ) = {u ∈ L 1 (Rd ) : Au ∈ L 1 (Rd ) and V u ∈ L 1 (Rd )} given by (A1 + V1 )u = Au + V u. Then −(A1 + V1 ) generates a positive contractive C0 -semigroup on L 1 (Rd ), which is a holomorphic semigroup. Moreover, D(Rd ) is a core for A1 + V1 . We need two lemmas. The first is an extension of Kato’s inequality. Kato proved this for real symmetric ckl in [14] Lemma A. We allow non-symmetric coefficients and need an additional argument for that. Lemma 2.2 If u ∈ L 1,loc (Rd ) and Au ∈ L 1,loc (Rd ), then Re(sgnu Au) ≥ A|u| in D  (Rd ).

 Proof First suppose in addition that u ∈ C ∞ (Rd ). Let ε > 0. Define u ε = |u|2 + ε. If l ∈ {1, . . . , d} then u ε ∂l u ε = Re(u ∂l u). Multiplication with the real-valued ckl and taking a partial derivative ∂k gives d 

(∂k u ε ) ckl ∂l u ε +

k,l=1

d 

u ε ∂k ckl ∂l u ε = Re

k,l=1

d  k,l=1

∂k u ckl ∂l u + Re

d 

u ∂k ckl ∂l u.

(2)

k,l=1

We next need an elementary matrix inequality. If C is a real-valued positive definite d × d matrix, Cs = 21 (C + C ∗ ) and ξ, η ∈ d R , then Re C(ξ + i η), ξ + i η = Cs (ξ + i η), ξ + i η = Cs ξ, ξ + Cs η, η ≥ Cs ξ, ξ = Cξ, ξ , where ·, · is the inner product on Cd .

Kato’s Inequality

51

If x ∈ Rd , choose ξ = Re(u ∇u)(x) = (u ε ∇u ε )(x) and η = Im(u ∇u)(x). Then |u ε |2

d 

(∂k u ε ) ckl ∂l u ε ≤ |u|2 Re

k,l=1

d 

∂k u ckl ∂l u

k,l=1

≤ |u ε | Re 2

d 

∂k u ckl ∂l u.

k,l=1

Dividing by |u ε |2 and using (2) gives d 

u ε ∂k ckl ∂l u ε ≥ Re

k,l=1

d 

u ∂k ckl ∂l u.

k,l=1

Dividing by u ε and taking the limit ε ↓ 0 gives A|u| ≤ Re(sgnu Au) in D  (Rd ). The condition u ∈ C ∞ (Rd ) can be dropped by regularising u and is left to the reader. Occasionally we need duality arguments. We denote by At the operator with coefficients ckl replaced by the transposed coefficients clk . Lemma 2.3 Let λ ∈ C with Re λ ≥ 0. Moreover, let u ∈ D(A1 + V1 ). Then V1 u L 1 (Rd ) ≤ (λ I + A1 + V1 )u L 1 (Rd ) . Proof Let u ∈ D(A1 + V1 ). Write f = (λ I + A1 + V1 )u. Then λ |u| + sgnu Au + V |u| = sgnu f . Taking the real part, it follows from Kato’s inequality (1) that 

 Rd

ϕ V |u| = ≤

R

d

Rd

  ϕ Re(sgnu f ) − ϕ Re(sgnu Au) − ϕ |u| Reλ Rd Rd  ϕ|f|− |u| At ϕ

(3)

Rd

for all 0 ≤ ϕ ∈ D(Rd ). Fix τ ∈ D(Rd ) such that 0 ≤ τ ≤ 1 and τ | B1 (0) = 1. For all n ∈ N define τn ∈ D(Rd ) by τn (x) = τ (n −1 x). Then limn→∞ τn (x) = 1 for all d x ∈ Rd and limn→∞ (Aτn )(x) = 0 uniformly  for all x ∈R . Moreover, choosing ϕ = τn in (3) and taking the limit n → ∞ gives Rd V |u| ≤ Rd | f |, that is V1 u L 1 (Rd ) ≤ (λ I + A1 + V1 )u L 1 (Rd ) . Proof (of Theorem 2.1) Since D(Rd ) ⊂ D(A1 ) ∩ D(V1 ) = D(A1 + V1 ), the operator A1 + V1 is densely defined and accretive. Let Amin = A|D (Rd ) . We shall show that ran(I + Amin + V ) is dense in L 1 (Rd ) and A1 + V1 is closed. Then it follows from the Lumer–Phillips theorem [17] Theorem 3.1 that −(A1 + V1 ) generates a contraction C0 -semigroup on L 1 (Rd ) and D(Rd ) is a core. d d We first prove  that ran(I + Amin + V ) is dense in L 1d(R ). Let f ∈ Lt ∞ (R ) and suppose that (ϕ + Aϕ + V ϕ) f = 0 for all ϕ ∈ D(R ). Then f + A f + V f = 0 in D  (Rd ). Note that this implies that −At f = f + V f ∈ L 1,loc (Rd ). Moreover,

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| f | + sgn f At f + V | f | = sgn f ( f + At f + V f ) = 0. Therefore | f | + V | f | = −Re(sgn f At f ) ≤ −At | f | by Kato’s inequality in Lemma 2.2 and (I + At )| f | ≤ −V | f | ≤ 0. Hence  Rd

| f | (I + A1 )ϕ ≤ 0

(4)

for all 0 ≤ ϕ ∈ D(Rd ). It is well known that D(Rd ) is a core for A1 (see for example [10] Theorem 9.8). We will improve this by showing that the positive test functions  are dense in the positive cone of the domain. Fix ψ, τ ∈ D(Rd ) with ψ ≥ 0, ψ = 1, 0 ≤ τ ≤ 1 and τ | B1 (0) = 1. For all n ∈ N define ψn , τn ∈ D(Rd ) by ψn (x) = n d ψ(n x) and τn (x) = τ (n −1 x). Let 0 ≤ u ∈ D(A1 ). Then ψn ∗ u ∈ W 2,1 (Rd ) ⊂ D(A1 ) and limn→∞ ψn ∗ u = u in D(A1 ). Moreover, limm→∞ τm (ψn ∗ u) = ψn ∗ u in D(A1 ) for all n ∈N. Now 0 ≤ τm (ψn ∗ u) ∈ D(Rd ) for all n, m ∈ N. Using (4) one deduces that | f | (I + A1 )u ≤ 0. This is for all 0 ≤ u ∈ D(A1 ). Next let 0 ≤ u ∈ L 1 (Rd ). Since the semigroup generated  by −A1 is a positive semigroup, one deduces that (I + A1 )−1 u ∈ D(A1 )+ . So | f | u ≤ 0. Therefore | f | = 0 and f = 0. Consequently ran(I + Amin + V ) is dense in L 1 (Rd ). Next we show that the operator A1 + V1 is closed. It follows from Lemma 2.3 that V1 u L 1 (Rd ) ≤ (A1 + V1 )u L 1 (Rd ) for all u ∈ D(A1 + V1 ). Since both A1 and V1 are closed, also A1 + V1 is closed. We proved that −(A1 + V1 ) generates a contraction C0 -semigroup on L 1 (Rd ) and that D(Rd ) is a core. Next we show that the semigroup generated by −(A1 + V1 ) is positive. Let 0 ≤ f ∈ L 1 (Rd ), λ > 0 and define u = (λ I + A1 + V1 )−1 f . Then both (λ I + A1 + V1 )u = f and (λ I + A1 + V1 )u = f = f , so u = u and u is realvalued. Moreover, Lemma 2.2 implies that (λ I + A + V )|u| ≤ λ |u| + (sgnu) Au + V |u| = (sgnu) f ≤ f = (λ I +A+V )u. Therefore (λ I + A + V )(|u| − u) ≤ 0 and hence (λ I + A)(|u| − u) ≤ 0 since V ≥ 0. So  (|u| − u)(λI + At )ϕ ≤ 0 Rd

for all 0 ≤ ϕ ∈ D(Rd ). Let B∞ be the adjoint of the operator A1 on the space L ∞ (Rd ) with weak∗ -topology. Then  Rd

(|u| − u)(λI + B∞ )ϕ ≤ 0

(5)

for all 0 ≤ ϕ ∈ D(Rd ). Note that |u| − u ∈ L 1 (Rd ). Using a same approximation as above, this time in the weak∗ -topology, one deduces that (5) is valid for all 0 ≤ ϕ ∈ D(B∞ ). Now −B∞ is the generator of a positive semigroup on L ∞ (Rd ) with the

Kato’s Inequality

53

weak∗ -topology. Therefore if 0 ≤ ψ ∈ L ∞ (Rd ), then (λI + B∞ )−1 ψ ∈ D(B∞ )+ and (|u| − u) ψ ≤ 0. Hence |u| − u = 0 and u = |u| ≥ 0. So (λ I + A1 + V1 )−1 is a positive operator for all λ > 0. Therefore the semigroup generated by −(A1 + V1 ) is positive. Finally we show that −(A1 + V1 ) generates a holomorphic semigroup on L 1 (Rd ). Since −(I + A1 ) generates a bounded holomorphic semigroup on L 1 (Rd ) by [8] Theorem 5.4, there exists a c > 0 such that (λ I + I +A1 )u L 1 (Rd ) ≥ c |λ| u L 1 (Rd ) for all u ∈ D(Δ1 ) and λ ∈ C with Reλ > 0. Then it follows from Lemma 2.3 that c |λ| u L 1 (Rd ) ≤ (λ I + I + A1 )u L 1 (Rd ) ≤ 2(λ I + I + A1 + V1 )u L 1 (Rd ) for all u ∈ D(A1 + V ) and λ ∈ C with Reλ > 0. Hence −(I + A1 + V1 ) generates a bounded holomorphic semigroup on L 1 (Rd ). This completes the proof of Theorem 2.1. Remark 2.4 Holomorphy on L 1 is always difficult to prove. Kato’s trick, which we used in the above proof and which is based on Kato’s inequality, allows one to deduce holomorphy of the perturbed semigroup generated by −(A1 + V1 ) from the holomorphy of the semigroup generated by −A1 . In the case of the Laplacian, the latter is the Gaussian semigroup and holomorphy on L 1 is obvious. In our more general case, holomorphy on L 1 (Rd ) is not so easy to prove. It follows from Gaussian estimates of the semigroup generated by −A1 , as we say in the proof. In this case, however, we could have used Gaussian estimates directly. Since 0 ≤ e−t (A1 +V1 ) ≤ e−t A1 for all t > 0 (remember that V ≥ 0), also the perturbed semigroup satisfies Gaussian estimates. The Gaussian kernel bounds imply that there exists a C0 -semigroup on L 2 (Rd ) which is consistent with (e−t (A1 +V1 ) )t>0 . But it is unclear whether this semigroup on L 2 (Rd ) is holomorphic since merely V ∈ L 1,loc (Rd ). Kato’s beautiful trick avoids this detour and problem. For further applications of Kato’s inequality we refer to Kato’s papers [14, 16], and [19] Section X.4. In [3] the Kato inequality has been used to show that the Dirichlet Laplacian generates a holomorphic semigroup on L 1 (Ω), where Ω ⊂ Rd is open. One may use Kato’s inequality in order to give a characterisation of the positivity of the semigroup in terms of its generator, see [1, 18]. Remark 2.5 Most of Theorem 2.1 can be extended to complex valued V ∈ L 1,loc (Rd ) for which there exists an M > 0 such that |ImV | ≤ M ReV . One can define similarly the operator A1 + V1 . It follows that −(A1 + V1 ) is an m-accretive operator, that is, it is the generator of a contraction C0 -semigroup on L 1 (Rd ). Moreover, this semigroup is holomorphic. The main difference is in Lemma 2.3. If u ∈ D(A1 + V1 ), then (ReV1 )u L 1 (Rd ) ≤ (λ I + A1 + V1 )u L 1 (Rd ) for all λ ∈ C with Reλ ≥ 0. Since V1 u L 1 (Rd ) ≤ (1 + M) (ReV1 )u L 1 (Rd ) , all other estimates follow with obvious modifications. Of course, the semigroup is no longer positive if V is not real-valued.

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A similar extension on L 2 (Rd ) for the Laplacian was done by Kato in [15]. Without the ellipticity condition, but merely with a sectoriality condition on the coefficients, it is an open problem whether D(Rd ) is a core for the generator, even if V = 0. In one dimension this is the case on L 1 (R) by [12] Corollary 1.6.

3 Kato’s Inequality and the Maximum Principle There are other versions of Kato’s inequality. For simplicity we restrict ourselves to the Laplacian in the rest of this paper and we restrict to real-valued functions and merely consider real potentials. Instead of j (r ) = |r | one may consider an arbitrary convex, lower semicontinuous function j : R → (−∞, ∞] satisfying j (0) = 0. Denote by ∂ j (r ) = {w ∈ R : j (s) − j (r ) ≥ w (s − r ) for all s ∈ R} the subdifferential of j at r . Theorem 3.1 Let Ω ⊂ Rd be open. Let u ∈ L 1,loc (Ω, R) and suppose that Δu ∈ L 1,loc (Ω). Let w : Ω → R be a measurable function such that w(x) ∈ ∂ j (u(x)) for almost all x ∈ Ω and w Δu ∈ L 1,loc (Ω). Then j ◦ u ∈ L 1,loc (Ω) and w Δu ≤ Δ( j ◦ u)

(6)

in D  (Ω). Proof See [5] Proposition 5. The inequality (6) means that  Ω

 w (Δu) ϕ ≤

Ω

( j ◦ u) Δϕ

for all 0 ≤ ϕ ∈ D(Ω). A concrete example is j (r ) = r + , which gives the next special case of Theorem 3.1. Corollary 3.2 Let u ∈ L 1,loc (Ω, R) be such that Δu ∈ L 1,loc (Ω). Then 1[u>0] Δu ≤ Δ(u + )

(7)

in D  (Ω). This inequality implies Kato’s inequality (1), but the converse implication is not clear since Δ is not known to be local on L 1,loc (Ω). An interesting version of the maximum principle which can be proved with the help of Kato’s inequality is the following.

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55

Theorem 3.3 Let Ω ⊂ Rd be open. Further let u ∈ L 1,loc (Ω, R) be such that Δu ∈ L 1,loc (Ω) and −Δu ≤ 0 in D  (Ω). Let c ∈ R be such that (u − c)+ ∈ H01 (Ω). Then u ≤ c. Note that (u − c)+ ∈ H01 (Ω) is a weak formulation to express that u ≤ c at the boundary. If u ∈ H 1 (Ω), then the result is a special case of [13] Theorem 8.1. But here we merely suppose that u, Δu ∈ L 1,loc (Ω). Proof (of Theorem 3.3) It follows from Kato’s inequality (7) that −Δ((u − c)+ ) ≤ −1[u>c] Δ(u − c) = −1[u>c] Δu ≤ 0. Hence

 Ω

(∇((u − c)+ )) ∇v = −

 Ω

(Δ((u − c)+ )) v ≤ 0

 for all 0 ≤ v ∈ D(Ω). Therefore Ω (∇((u − c)+ )) ∇v ≤ 0 for all 0 ≤ v ∈ H01 (Ω) by (u − c)+ ∈ H01 (Ω) one can choose v = (u − c)+ to deduce that  density. Since 1 + 2 + Ω |∇((u − c) )| ≤ 0. Using once again that (u − c) ∈ H0 (Ω) one concludes that + (u − c) = 0, that is u ≤ c almost everywhere on Ω.

4 The Dirichlet Problem for −Δ + V Let Ω ⊂ Rd be open bounded and connected with boundary Γ = ∂Ω. We assume that Ω is Wiener regular, that is for all ϕ ∈ C(Γ ) there exists a unique u ∈ C(Ω) such that Δu = 0 and u|Γ = ϕ. So the Dirichlet problem is well-posed. Here Δu ∈ D  (Ω) is understood as a distribution. Now let V ∈ L ∞ (Ω, R). We consider the Dirichlet problem ⎧ ⎨ u ∈ C(Ω), −Δu + V u = 0 in D  (Ω), ⎩ u|Γ = ϕ. We denote by Δ D the Dirichlet Laplacian, that is D(Δ D ) = {u ∈ H01 (Ω) : Δu ∈ L 2 (Ω)} and Δ D u = Δu. Then −Δ D is a self-adjoint operator with compact resolvent. Consequently also −Δ D + V is self-adjoint with compact resolvent. We denote by σ (−Δ D + V ) its spectrum. Now we can describe well-posedness. The following results are special cases of [9], where elliptic second-order operators in divergence form with complex first-order terms and potentials are considered instead of the Laplacian. The proofs in our case are much simpler.

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Theorem 4.1 The following are equivalent. (i) For all ϕ ∈ C(Γ ) there exists a unique u ∈ C(Ω) such that −Δu + V u = 0 and u|Γ = ϕ. (ii) 0 ∈ / σ (−Δ D + V ). For the proof we need two lemmas. Lemma 4.2 Let f ∈ L ∞ (Ω). (a) If u ∈ H01 (Ω) and −Δu + V u = f , then u ∈ C0 (Ω). (b) If u ∈ C0 (Ω) and −Δu + V u = f , then u ∈ H01 (Ω). Proof ‘(a)’ Since f ∈ L ∞ (Ω) ⊂ L 2 (Ω), it follows that u = (−Δ D + V )−1 f . Then the Gaussian kernel bounds of [8] Theorem 3.1 of the semigroup generated by D the part of Δ D in L ∞ (Ω), −(−Δ D + V ) imply that u ∈ L ∞ (Ω). Denoting by Δ∞ D D + V ) = D(−Δ∞ ). Now the statement folthe assumption implies that u ∈ D(−Δ∞ lows from [6] Theorem 2.4(i)⇒(iii). ‘(b)’ This follows from [6] Lemma 2.2a. / σ (−Δ D + V1 ). Let ϕ ∈ Lemma 4.3 Let V1 , V2 ∈ L ∞ (Ω) and suppose that 0 ∈ C(Γ ). Let u 2 ∈ C(Ω) and assume that −Δu 2 + V2 u 2 = 0 and u 2 |Γ = ϕ. Define u 1 = u 2 + (−Δ D + V1 )−1 (V2 − V1 )u 2 . Then u 1 ∈ C(Ω). Moreover, −Δu 1 + V1 u 1 = 0 and u 1 |Γ = ϕ. Proof It follows from Lemma 4.2(a) that (−Δ D + V1 )−1 (V2 − V1 )u 2 ∈ C0 (Ω). Therefore u 1 ∈ C(Ω) and u 1 |Γ = ϕ. Moreover, −Δu 1 + V1 u 1 = −Δu 2 + V1 u 2 + (V2 − V1 ) u 2 = −Δu 2 + V2 u 2 = 0 as required. Proof (of Theorem 4.1) ‘(i)⇒(ii)’ Suppose that 0 ∈ σ (−Δ D + V ). Since −Δ D + V has compact resolvent, there exists a u ∈ ker(−Δ D + V ) \ {0}. Then u ∈ H01 (Ω) and −Δu + V u = 0. It follows from Lemma 4.2(a) that u ∈ C0 (Ω). Hence the uniqueness in Condition (i) of Theorem 4.1 fails if ϕ = 0. ‘(ii)⇒(i)’ Let ϕ ∈ C(Γ ). Since Ω is Wiener regular there exists a unique u 0 ∈ / σ (−Δ D + V ). Then C(Ω) such that −Δu 0 = 0 and u 0 |Γ = ϕ. By assumption 0 ∈ one can apply Lemma 4.3 with V1 = V and V2 = 0 to obtain the existence of a solution of the Dirichlet problem. For the uniqueness part, suppose that u ∈ C0 (Ω) and −Δu + V u = 0. Then u ∈ H01 (Ω) by Lemma 4.2(b) and Δu = V u ∈ L 2 (Ω). / σ (−Δ D + V ), Consequently u ∈ D(−Δ D + V ) and −Δ D u + V u = 0. Since 0 ∈ one deduces that u = 0. If 0 ∈ / σ (−Δ D + V ), then by Theorem 4.1 one can define the map BV : C(Γ ) → C(Ω) by BV ϕ = u, where u ∈ C(Ω) is the unique element such that −Δu + V u =

Kato’s Inequality

57

0 and u|Γ = ϕ. Then BV is linear and bounded by the closed graph theorem. If V is constant, say V = λ 1Ω with λ ∈ R, then for simplicity we write Bλ = Bλ 1Ω . We obtain from Lemma 4.3 the following useful formula: Lemma 4.4 Let V1 , V2 ∈ L ∞ (Ω). Assume that 0 ∈ / σ (−Δ D + V1 ) and also 0 ∈ / D σ (−Δ + V2 ). Then BV1 − BV2 = (−Δ D + V1 )−1 (V2 − V1 ) BV2 .

5 Kato’s Inequality and the Dirichlet Problem Let Ω ⊂ Rd be open bounded connected and Wiener regular, with boundary Γ = ∂Ω. Let V ∈ L ∞ (Ω). Then −Δ D + V is a lower bounded self-adjoint operator with a discrete spectrum. Define λ1 (−Δ D + V ) = min σ (−Δ D + V ). Then λ1 (−Δ D + V ) is the smallest eigenvalue of the operator −Δ D + V . Note that λ1 (−Δ D + V ) ≥ 0 if and only if −Δ D + V is a positive self-adjoint operator. Theorem 5.1 Let V ∈ L ∞ (Ω). Suppose that λ1 (−Δ D + V ) > 0. Then the operator BV is positive, that is ϕ ≥ 0 implies BV ϕ ≥ 0. Theorem 5.1 follows from the following more general assertion. Proposition 5.2 Let V ∈ L ∞ (Ω) and assume that λ1 (−Δ D + V ) > 0. Let u ∈ C(Ω) and assume that f = −Δu + V u ∈ L 2,loc (Ω). Further assume that f ≤ 0 and u|Γ ≤ 0. Then u ≤ 0. 2 Proof Since Δu ∈ L 2,loc (Ω) it follows from elliptic regularity that u ∈ Hloc (Ω), see 1 + [11] Proposition II.3.8. Let ε > 0. Then (u − ε) ∈ Cc (Ω) ∩ Hloc (Ω) ⊂ H01 (Ω). Moreover, −Δ(u − ε) + V (u − ε) = f − ε V ≤ −ε V ≤ ε V − .

Hence −1[u>ε] Δ(u − ε) + V (u − ε)+ ≤ ε V − . By Kato’s inequality (7) it follows that −Δ((u − ε)+ ) + V (u − ε)+ ≤ ε V − . Therefore   + + (∇((u − ε) )) ∇v + V (u − ε) v ≤ ε V− v Ω

Ω

first for all 0 ≤ v ∈ D(Ω) and then bydensity for all 0 ≤ v ∈ H01 (Ω). Choosing v = (u − ε)+ gives Ω |∇((u − ε)+ )|2 + Ω V |(u − ε)+ |2 ≤ ε Ω V − (u − ε)+ . Since  λ1 (−Δ + V ) = min{ D

 |∇w| + 2

Ω

Ω

V |w|2 : w ∈ H01 (Ω) and w2 = 1}

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it follows that +



(u − ε) 2 λ1 (−Δ + V ) ≤

+

D

≤ε

Ω





|∇((u − ε) )| +

Ω

2

Ω

V |(u − ε)+ |2

V − (u − ε)+ .

Taking the limit ε ↓ 0 one deduces that u + 2 λ1 (−Δ D + V ) ≤ 0. Hence u + = 0. Next we prove a partial converse of Theorem 5.1. Define λ1D = λ1D (−Δ D ). Theorem 5.3 Let V ∈ L ∞ (Ω) and suppose that 0 ∈ / σ (−Δ D + V ). Assume that D V ≤ −λ1 . Then the operator BV is not positive. For the proof we use the following facts. Lemma 5.4 Let α ∈ R ∩ ρ(Δ D ). (a) The operator (α I − Δ D )−1 is positive if and only if α > −λ1D . (b) limα↓−λ1D (α I − Δ D )−1 1Ω  L ∞ (Ω) = ∞. Proof Statement (a) follows from [4] Proposition 3.11.2. D D ‘(b)’ Let Δ∞ denote the part of the operator Δ D in L ∞ (Ω). Then σ (−Δ∞ )= D −1 ) L (L ∞ (Ω)) = ∞. If σ (−Δ D ) by [6] Proposition 1.4. Hence limα↓−λ1D (α I − Δ∞ D −1 ) is a positive operator, because the α ∈ (−λ1D , ∞), then the operator (α I − Δ∞ D kernel of the semigroup generated by Δ∞ is positive. Finally, if S ∈ L (L ∞ (Ω)) is a positive operator, then SL (L ∞ (Ω)) = S1Ω  L ∞ (Ω) and Statement (b) follows. Proof (of Theorem 5.3) Assume that BV is positive. Let α ∈ (−λ1D , 0). Then α ∈ ρ(Δ D ) and the operator (α I − Δ D )−1 is positive by Lemma 5.4(a). Since V ≤ −λ1D ≤ α one has (V − α) ≤ 0. Hence by Lemma 4.4 one deduces Bα = BV + (α I − Δ D )−1 (V − α) BV ≤ BV . In particular, 0 ≤ Bα 1Γ ≤ BV 1Γ . On the other hand, using again Lemma 4.4 one deduces Bα 1Γ = B0 1Γ + (α I − Δ D )−1 (−α) B0 1Γ ≥ (−α)(α I − Δ D )−1 1Ω . Since limα↓−λ1D (α I − Δ D )−1 1Ω  L ∞ (Ω) = ∞ by Lemma 5.4(b), this leads to a contradiction. Next we discuss comparison. Let V1 , V2 ∈ L ∞ (Ω) be such that V1 ≤ V2 . Then obviously λ1 (−Δ D + V1 ) ≤ λ1 (−Δ D + V2 ). If Ω is connected and V1 = V2 , then λ1 (−Δ D + V1 ) < λ1 (−Δ D + V2 ) by [2] Theorem 1.3. Proposition 5.5 Let V1 , V2 ∈ L ∞ (Ω) be such that V1 ≤ V2 . Moreover, suppose λ1 (−Δ D + V1 ) > 0 and the operator (−Δ D + V1 )−1 is positive. Then BV2 ≤ BV1 .

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59

Proof Since λ1 (−Δ D + V2 ) ≥ λ1 (−Δ D + V1 ) > 0 the operator BV2 is positive by Theorem 5.1. Then Lemma 4.4 gives that BV1 − BV2 = (−Δ D + V1 )−1 (V2 − V1 ) BV2 is positive. We next determine the asymptotic behaviour of the solution of the Dirichlet problem when the potential tends to infinity. Proposition 5.6 If ϕ ∈ C(Γ ), then lim Bα ϕ = 0. α→∞

Proof Using again Lemma 4.4 one deduces that lim Bα ϕ = lim B0 ϕ − α (α I − Δ D )−1 B0 ϕ = 0

α→∞

α→∞

as required. Finally we discuss the maximum principle. The next lemma is easy to prove. Lemma 5.7 Let V ∈ L ∞ (Ω) and assume that 0 ∈ / σ (−Δ D + V ). Then the following conditions are equivalent. (i) If c > 0, ϕ ∈ C(Γ ) and ϕ ≤ c 1Γ , then BV ϕ ≤ c 1Ω . (ii) The operator BV is positive and BV 1Γ ≤ 1Ω . Condition (ii) in Lemma 5.7 means that the operator BV is submarkovian. Theorem 5.8 (a) Let 0 ≤ V ∈ L ∞ (Ω). Then BV is submarkovian. Moreover, BV 1Γ = 1Ω if and only if V = 0. (b) Let α ∈ R \ σ (Δ D ). Then Bα is submarkovian if and only if α ≥ 0. Proof ‘(a)’ We already know from Theorem 5.1 that the operator BV is positive. Lemma 4.4 gives

BV 1Γ = B0 1Γ + (−Δ D + V )−1 (0 − V ) B0 1Γ = 1Ω − (−Δ D + V )−1 V. Hence BV 1Γ ≤ 1Ω and BV is submarkovian. Moreover, BV 1Γ = 1Ω if and only if the function (−Δ D + V )−1 V = 0, which is equivalent with V = 0. ‘(b)’ The implication ‘⇐’ follows from Statement (a). So it remains to show ‘⇒’. Suppose Bα is submarkovian. Then Bα is positive and it follows from Theorem 5.3 that α > −λ1D . If α < 0, then Lemma 4.4 gives Bα 1Γ = B0 1Γ + (α I − Δ D )−1 (−α) B0 1Γ = 1Ω + (−α) (α I − Δ D )−1 1Ω > 1Ω ,

where the last inequality follows from Lemma 5.4(a). This gives a contradiction. Acknowledgements Part of this work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

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Tosio Kato’s Unpublished Paper Claude Bardos and Hisashi Okamoto

Abstract An unpublished manuscript of Tosio Kato was recently found. His paper is published here, after nearly 30 years, together with some comments by the authors.

When Tosio Kato died in Oakland, California, in 1999, he left very many research notes, which were later sent to Japan. According to the will of Mizue Kato (Tosio’s widow) who died in 2011, Hiroshi Fujita, Shige Toshi Kuroda and Makio Ishiguro were endowed with these materials and they subsequently asked the second author of the present paper to help them by ordering and inspecting these materials for proper uses in future. It is only in 2017 that the second author found that one of his notes remained unpublished. It was submitted to Themistocles M. Rassias in 1990 and was later returned to Tosio Kato in the same year. Since then it remained unpublished until now. With the permission of H. Fujita and S.T. Kuroda, we now let Kato’s paper be open to public. We are very happy with this publication, since this paper of Kato seems to have historical merit, as will be seen from our comments right after his paper. The original manuscript was typeset by a word processor by Kato. It is the second author who typeset it by LATEX and any typo, if there is any, is responsible to him. On this occasion the authors would like to direct the reader’s attention to a volume which was authored by Tosio Kato and edited by Shige Toshi Kuroda entitled “Mathematical Theory of Quantum Mechanics” in Japanese. This volume was finished in 1945 but published only in 2017.1 The reader of this volume will surely note that 1 By

Kindaikagakusya: ISBN 978-4-7649-0545-0.

C. Bardos Laboratoire J.-L. Lions, Paris Cedex 05, France H. Okamoto (B) Department of Mathematics, Gakushuin University, Tokyo 171-8588, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_4

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some of Kato’s innovative ideas are already there as early as in 1945. ∗∗∗ A remark on a theorem of C. Bardos on the 2D−Euler equation Tosio Kato Department of Mathematics, University of California, Berkeley, CA 94720, USA The object of this note is to complete the proof of a uniqueness theorem due to Bardos [1] Consider the Euler equation for an incompressible fluid in a bounded domain Ω ⊂ R2 with C 2 -boundary Γ : ∂t u + (u · ∂)u = f − ∂π, u·n =0

∂ ·u =0

on Γ × (0, T ),

in Ω × (0, T ), u(x, 0) = a(x);

(1)

here u is the velocity field, π the pressure, f the external force, and n is the unit normal to Γ . Denote by H the space of u ∈ L 2 (Ω)2 such that ∂ · u = 0 in Ω and u · n = 0 on Γ , and define the space V similarly with L 2 replaced by H 1 (the Sobolev space). We denote by   p the L p (Ω)-norm, and write   for  2 . In [1] it is proved, among other things, that if a ∈ V , ∂ ∧ a ∈ L ∞ (Ω), f ∈ 2 L ((0, T ); H ) and ∂ ∧ f ∈ L ∞ (Ω × (0, T )), then (1) has a solution u such that u ∈ L ∞ ((0, T ); V ) ∩ C([0, T ); H ),

(2)

and ∂u(t) p ≤ cp,

t ∈ (0, T ), for 2 ≤ p < ∞,

(3)

where c is a constant depending on a and f but not on p. (In what follows we denote similar constants indiscriminately by c.) Moreover, u is the unique solution in the class (2). The uniqueness proof in [1] aims at showing that w ≡ v − u = 0 if v is another solution of (1) in the class (2). Using (1) and (3), one first deduces the estimate ∂t w2 ≤ cpw22 p

for 2 ≤ p < ∞,

(4)

where 1/ p + 1/ p  = 1. Then one wants to show that ∂t w2 ≤cp(M(t))2/ p w2−2/ p , where M(t) = w(t)∞ . Here is a difficulty, however, since it is not known that w(t) ∈ L ∞ . In fact (3) implies that u(t)∞ ≤ c (because Ω is bounded), but v need not have this property. In what follows we shall show how one may overcome this difficulty. For this we need a lemma. Lemma 1 ϕ p ≤ cp 1/2 ϕ H 1

for

2 ≤ p < ∞, ϕ ∈ H 1 (Ω).

Tosio Kato’s Unpublished Paper

63

Proof By the extension theorem we may assume that Ω = R2 . If ϕˆ is the Fourier transform of ϕ, it follows from the Hölder inequality that     ϕˆ   ≤  · −1 q  · ϕˆ  , p

2 ≤ p < ∞,

1 1 1 1 1 = − = − , q 2 p p 2

 1/2 where ξ = 1 + |ξ |2 . But it is easy to see that  · −1 q ≤ c(q − 2)−1/q = ˆ = ϕ H 1 , the desired result folc( p/2q)1/q ≤ cp 1/2 because q > 2. Since  · ϕ lows by the Hausdorff–Young theorem. To complete the uniqueness proof, we use the Hölder inequality to obtain w2 p ≤ w1−λ wλp ,

λ = 1/( p − 2) ≤ 1,

3 ≤ p < ∞.

Hence (4) gives ∂t w2λ = λw2λ−2 ∂t w2 ≤ λw2λ−2 cpw22 p ≤ λcpw2λ p .

(5)

But w p ≤ (cp)1/2 by Lemma 1, since the H 1 -norm of w = v − u is bounded due to u, v, ∈ L ∞ ((0, T ); V ). Since λ = 1/( p − 2) and w(0) = 0, integration of (5) leads to the estimate w(t)2/( p−2) ≤ ct ( p/( p − 2))(cp)1/( p−2) , or

w(t) ≤ (ct)( p−2)/2 ( p/( p − 2))( p−2)/2 (cp)1/2 .

Suppose now that t is so small that ct < 1. If we let p → ∞, then (ct)( p−2)/2 (cp)1/2 → 0 while ( p/( p − 2))( p−2)/2 stays bounded. Hence w(t) = 0 if ct < 1. Since the argument can be repeated, we have proved the required uniqueness. Remarks (a) The theorem of Bardos is a refinement of a similar one due to Judoviˇc [2], in which the stream function is used. In [2] the uniqueness is proved, roughly, in the class ∂ ∧ u ∈ L ∞ (Ω × (0, T )), which is (essentially) more restrictive than (2). (b) The same proof applies without change to the case that Ω = R2 . Also, it is not difficult to extend the existence theorem of [1] to this case. 1. Bardos, C.: Existence et unicité de la solution de l’équation d’Euler en dimension deux. J. Math. Anal. Appl. 40, 769–790 (1972) 2. Judoviˇc, V.I.: Non-stationary flows of an ideal incompressible fluid (Russian). Ž. Vyˇcisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963) ∗∗∗ The first author is very well aware of this short note of Tosio Kato. In the summer of 1970, he had a fantastic chance to spend 2 weeks with Kato in a workshop organized by Aronszajn at the University of Kansas devoted to general properties of evolution equations in functional analysis. We had a lot of time to discuss the

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Euler equations and zero viscosity limit of the Navier–Stokes equations in presence of different types of boundary conditions. Back in France he prepared the paper of 1972 which was sent to Kato and then he later wrote the present unpublished short note. The first result on the subject goes back to Judoviˇc and several refinements were made (first by J.-L. Lions, described in his book “Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires”, p. 88) then the first author’s contribution and the above note of Kato. The improvement of Kato was in some sense a forerunner of the modern approach which one could describe as weak–strong uniqueness. The first author takes this opportunity to add a few lines about his scientific debt to Kato: He wrote his thesis in 1969 and by that time the best and the most userfriendly book available for young researchers was Kato’s book “Perturbation Theory for Linear Operators” published by Springer. It was bread and butter of the authors’ generation. The first scientific workshop that the first author was attendant was 1969 meeting on Scattering Theory organized by C. H. Wilcox in Flagstaff. Kato was (with Lax and Phillips) among the main lecturers. Not only science but also the super-friendly atmosphere of this meeting marked a corner stone in the research activities of his generation. The authors guess that most of the contributions of Kato to the mathematical theory of fluid mechanics is quite well documented. However we would like to underline the importance of the Berkeley Note: “Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary.2 His paper written in Japanese, “On well-posedness”,3 is also related to the present note and highly recommended. The present note was for many years ignored by our community, probably since it was not published in a genuine journal with good circulation. However in view of recent developments carried out by many authors including the first author, we may say that the present note is a visionary forerunner for the understanding of problems unavoidable both in theory and also in engineering science. Although Kato did not take action to submit this note after 1990 to any journal, we hope that publishing it in this volume is quite appropriate. In particular with the above comments it gives us an opportunity to emphasize the importance of his contributions in diverse fields and up to the present time. Hence we believe that he would pardon our use of his note in this form. Acknowledgements With great pleasure the second author acknowledges that he owes H. Fujita and S. T. Kuroda a debt of gratitude for their very useful comments. He is supported by JSPS Kakenhi 18H01137.

2 Seminar on nonlinear partial differential equations, ed. S. S. Chern, Springer (1984), pages 85–98. 3 Sugaku,

48 (1996), 298–300.

On the Border Lines Between the Regions of Distinct Solution Type for Solutions of the Friedmann Equation Hellmut Baumgärtel

Abstract It is well known that there are four distinct basic types (two Big Bang types, Lemaitre and Big Crunch type) for solutions of the general Friedmann equation with positive cosmological constant, where radiation and matter do not couple (see e.g. Baumgärtel in Journal of Mathematical Physics 122505, 2012, [2]). In that paper, the system of case distinction parameters contains a “critical radiation parameter” σcr . The present note contains the constructive description of the so-called border lines between Big Bang/Big Crunch type and Big Bang/Lemaitre type for the so-called Hubble solutions of the Friedmann equation by two smooth function branches, expressing the cosmological constant as unique functions of the matter and radiation density (which is considered as a parameter). These functions satisfy simple asymptotic relations with respect to the matter density. They are constructed as the solutions of the equation σ = σcr .

1 Introduction Einstein’s famous field equation of the general relativity theory, linking together the gravitational field with the stress-energy tensor, is also a basic tool in cosmology to solve problems of the large-scale structure of the cosmos. The idea of the existence of a definite large-scale structure of the cosmos is expressed by the “cosmological principle” (E.A. Milne 1933). This principle appears as a boundary condition of Einstein’s field equation. This boundary condition consists of a special ansatz of the gravitational field given by a so-called Robertson–Walker(RW-)metric, and of the special form of of the stress-energy tensor as that of a perfect fluid ([1], p. 69). Moreover, it is assumed that matter and radiation do not couple. Then Einstein’s field equation reduces to the so-called Friedmann equation   σ α 1 dR 2 = + 2 + Λc2 R 2 − εc2 (1) dt R R 3 H. Baumgärtel (B) Department of Mathematics, University of Potsdam, Potsdam, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_5

65

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H. Baumgärtel

for the so-called scale-factor R(t) > 0, a dimensionless number. The constant parameters are: velocity of light c, cosmological constant Λ, curvature constant ε = 0, ±1, matter invariant 8π G (2) α := ρmat (τ )R(τ )3 , 3 and radiation invariant σ :=

8π G ρrad (τ )R(τ )4 , 3

(3)

where ρmat and ρrad denote mass and radiation density. Note that the time invariance of the right-hand sides of (2) and (3) is an implication of the assumption that matter and radiation do not couple. Admissible initial conditions t0 , R0 for a solution of Eq. (1) have the property that the right- hand side of (1) is positive for R := R0 . For an admissible solution R(·), the so-called Hubble parameter t → H (t) is defined by  H (t) :=

1 dR R dt

 (t).

(4)

Solutions of (1) whose Hubble parameter at the present time T coincides with the famous Hubble constant H0 , H (T ) = H0 , are called Hubble solutions in this note. Using the equations (2), (3), the Hubble solutions satisfy the equation H02 =

8π G 8π G 1 c2 ρmat (T ) + ρrad (T ) + Λc2 − ε . 3 3 3 R(T )2

(5)

In the special case, Λ = 0, σ = 0, ε = 0 it reads H02 =

8π G ρmat (T ). 3

(6)

This case is the Euclidean–Einstein–de Sitter model of 1932. Since that time the term ρcr :=

3 H2 8π G 0

is called the critical density. Introducing instead of ρmat (T ), ρrad (T ), Λ the variables x :=

ρmat (T ) , ρcr

y :=

1 Λc2 3 , H02

z :=

ρrad (T ) , ρcr

(7)

then Eq. (5) can be written as 1= x +y+z−ε

c2 H02 R(T )2

(8)

On the Border Lines Between the Regions of Distinct Solution Type …

67

and the constants α, σ, Λ are given by α = x H02 R(T )3 , σ = z H02 R(T )4 , Λ = y

3H02 . c2

(9)

If x + y + z − 1 = 0 then  R(T )2 =

c H0

2

ε , x +y+z−1

(10)

i.e., in this case the scale-factor R(T ) in the present time is uniquely determined. Moreover, one obtains If x + y + z > 1 then ε = +1, if x + y + z < 1 then ε = −1. That is, x + y + z = 1 iff ε = 0.

(11)

This means: the Euclidean case is quite singular, realized only on the plane (11) and in this case the scale-factor is not uniquely determined. In any case, the geometric structure of the cosmos is uniquely determined by the expression x + y + z − 1. Sometimes the term Hc0 is called the Hubble radius.

2 Discriminant and Critical Radiation Parameter The right-hand side of equation (1) can be written in the form Λc2 p(R), 3R 2 where p(R) := Rq(R) + and q(R) := R 3 −

3 σ, Λc2

3α 3ε R+ . Λ Λc2

is a polynomial of the third degree with discriminant Δ :=

 1  2 9α Λ − 4εc4 . 4Λ3 c4

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H. Baumgärtel

Obviously, the type of a solution of (1) depends on the mutual position of R(T ) and the positive zeros of p. The shape of p is independent of σ . This fact suggests to use the polynomial q and its discriminant as a first parameter to distinguish different types of solutions. If R(·) is a Hubble solution, then Δ can be written in the form 1 Δ= 4Λ3



R(T )H0 c

6 D,

where D = D(x, y; z) := 27x 2 y − 4(x + y + z − 1)3 , x ≥ 0, y ≥ 0, z ≥ 0.

(12)

Note that for z = 1, one has D(x, y; 1) = 27x 2 y − 4(x + y)3 = −(x − 2y)2 (4x + y) ≤ 0, hence D(x, y; 1) = 0 if and only if x = 2y. For z > 1, one has −D(x, y; z) = 4(x + y + z − 1)3 − 27x 2 y > 4(x + y)3 − 27x 2 y = −D(x, y; 1) ≥ 0,

and D(x, y; z) < D(x, y; 1) ≤ 0, i.e. D(x, y; z) < 0 for all z > 1. This means: only in the case 0 ≤ z < 1 there are regions D > 0 in the first quadrant x ≥ 0, y ≥ 0. If D(x, y; z) < 0 then we introduce the angle φ, π2 < φ < π by √

1

xy 2 27 , cos φ = − 2 (x + y + z − 1) 32

π < φ < π, 2

(13)

which is associated to the triple (x, y; z). The boundary value φ = π characterizes the triples (x, y; z), where D = 0, i.e., the boundary of the region D < 0, for example, in the case z = 1 the line 2x = y. That is: fixing a parameter z = z 0 , then by equation (13) to each (x, y) of the first quadrant x > 0, y > 0 with D < 0 is associated a uniquely determined angle φ, π2 < φ < π and the algebraic curves φ = const exhaust the region D < 0 within the first quadrant x > 0, y > 0. In the following these curves are called the φ-curves. The second parameter to distinguish different types of solutions of (1) is the critical radiation parameter σcr , introduced in [2], defined for triples, where D < 0, by σcr

      ψ 2 ψ 2 c4 2 −1 1 cos 2 cos = 2 y − 1 , cos ψ = √ cos φ, 3 3 H0 3 2

3π π (14) >ψ > 4 2

On the Border Lines Between the Regions of Distinct Solution Type …

69

a slight modification compared with the expression (12) in [2]. It should be emphasized that σcr is—according to (12) and (13)—a function of x, y; z, just as σ is according to (9) and (10). D and σcr are sufficient to describe the two essential statements on the zeros of the polynomial p ([2], Sect. III) (i) If D < 0 and 0 ≤ σ < σcr , then there are two positive zeros R1 < R2 (if x = 0 then R1 = 0). (ii) If D ≥ 0 and σ > 0 or D < 0 and σ > σcr then there is no positive zero. Hence, in the case D < 0 the σ -value σ := σcr separates the cases (i) and (ii), i.e., the solutions of the equation σ = σcr are the corresponding border lines within the region D < 0. In the case D < 0 and σ = 0, i.e., z = 0, one obtains the explicit expressions R1 (x, y; 0) :=

  1 c 2 −1 (2π − φ) , √ y 2 cos H0 3 3

R2 (x, y; 0) :=

c 2 −1 1 √ y 2 cos φ. H0 3 3

(15)

Since in the case (i), one has p(R) < 0 for R ∈ (R1 , R2 ), this is an forbidden interval for initial conditions t, R. However, for Hubble solutions, R(T ) = R(x, y; z) given by equation (10) is admissible, i.e., one obtains that either R(x, y; z) > R2 (x, y; z)

(16)

R(x, y; z) < R1 (x, y; z).

(17)

or

3 The Equation σ = σcr Also, in the following z is considered as a parameter and the equation is considered in the first quadrant x ≥ 0, y ≥ 0. As already mentioned, the terms σ and σcr are functions of (x, y) and of the parameter z. The function σcr is given by (13). According to equations (9) and (10), one obtains for σ σ =

z c4 . 2 (x + y + z − 1)2 H0

(18)

Then, the equation σ = σcr reads       ψ 2 4zy 8 ψ 2 cos = −1 , 2 cos (x + y + z − 1)2 3 3 3

(19)

where ψ is a function of x, y, and z, according to Eqs. (13) and (14). Solution of equation (19) means the construction of functions x → Y (x; z 0 ), x ≥ 0, for every

70

H. Baumgärtel

parameter z 0 > 0, which satisfy this equation. As a function of φ, the right-hand side of Eq. (19) has a simple structure. In the following, we put       ψ 2 8 ψ 2 cos − 1 =: F(φ), 2 cos 3 3 3

π ≤ φ ≤ π. 2

(20)

One obtains F( π2 ) = 1, F(π ) = 0 and F(·) is strongly monotonically decreasing. The proofs for the solution of Eq. (19) are considered separately for the parameter regions 0 ≤ z < 1, z = 1 and z > 1. The case z = 1 is considered first. On the one hand, it is a rather singular case, but on the other hand it is explicitly solvable.

4 Results 4.1 Theorem (i) Let z = 1. Then the solution of Eq. (19) consists of two functions (branches) (0, ∞) x → Y± (x), given by the parameter representation x := x± (φ), Y± (x) := μ± (φ)x± (φ),

π < φ < π, 2

where μ± (φ) := 3ν± (φ) − 1, ν+ (φ) := and x± (φ) :=

cos 13 φ cos 13 (2π − φ) , ν− (φ) := | cos φ | | cos φ |

4 1 9 F(φ)



3 1 − ν± (φ) ν± (φ)2

(21)

 .

(22)

The solutions Y± (·) have the properties 0 < Y− (x) < such that Y± (x) =

1 x < Y+ (x), x > 0, 2

1 x ± u ± (x), u ± (x) > 0, 2

lim Y+ (x) = 4,

x→0

(23)

lim Y− (x) = 0,

x→0

(24)

On the Border Lines Between the Regions of Distinct Solution Type …

and lim

x→∞

71

u ± (x) = 0. x

(25)

(ii) Let z > 1. Then, the solution of Eq. (19) consists of two functions (branches) (0, ∞) x → Y± (x) given by the parameter representation x = x(φ, ρ± (φ)), Y± (x) = ρ± (φ)x(φ, ρ± (φ)),

π < φ < π, 2

(26)

where x(φ, μ) := (z − 1)

3μ1/3

22/3 (cos φ)2/3 , μ− (φ) < μ < μ+ (φ), (27) − 22/3 (cos φ)2/3 (μ + 1)

and the corresponding φ-curve is given by y± (x, φ) := μx(φ, μ). The terms μ := ρ± (φ) are uniquely determined solutions of the equation (cos φ)2/3 μ1/3 (3μ1/3 − 22/3 (cos φ)2/3 (μ + 1)) = where

32 z − 1 F(φ), 28/3 z

(28)

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H. Baumgärtel

μ− (φ) < ρ− (φ) < ρ+ (φ) < μ+ (φ).

(29)

The solutions Y± (·) have the properties 0 < Y− (x) < Y+ (x), x > 0, 0 < y− (x; φ) < Y− (x)
0, x sufficiently large, 2 √ lim Y± (x) = (1 ± z)2 , x→0

lim

x→∞

u ± (x) = 0. x

(31)

(32)

(iii) Let 0 < z < 1. In this case every φ-curve, π2 < φ < π has two branches, a lower one x → y− (x; φ) for x ≥ a, where a := 1 − z, starting at (a, 0), and an upper one x → y+ (x; φ) for x ≥ 0, starting at (0, a). The branches y± (·; π ) of the π -curve form the boundary of the region defined by D < 0. It corresponds to D = 0 (cf. Sect. 2). The solution of Eq. (19) consists of two functions (branches) (0, ∞) x → Y+ (x), (a, ∞) x → Y− (x).

On the Border Lines Between the Regions of Distinct Solution Type …

73

The branch Y+ (·) is given by the parameter representation π < φ < π, 2

(33)

22/3 (cos φ)2/3 , μ > μ+ (φ), 22/3 (cos φ)2/3 (μ + 1) − 3μ1/3

(34)

x = x(φ, ρ+ (φ)), Y+ (x) = ρ+ (φ)x(φ, ρ+ (φ)), where x(φ, μ) := (1 − z)

and the term μ := ρ+ (φ) is the uniquely determined solution of the equation (cos φ)2/3 μ1/3 (22/3 (cos φ)2/3 (μ + 1) − 3μ1/3 ) =

32 1 − z F(φ), 28/3 z

(35)

where ρ+ (φ) > μ+ (φ). This means: every φ-curve has exactly one intersection with the branch Y+ (·), realized by the upper branch y+ of the φ-curve. In contrast to this property of Y+ (·), the branch Y− (·) has either exactly two intersections with a φ-curve or there is no intersection. More precisely: To every a, 0 < a < 1, there is an angle φ(a), π2 < φ(a) < π, such that the parameter representation of Y− (·) is given by x = x(φ, ρ−± (φ)), Y− (x) = ρ−± (φ)x(φ, ρ−± (φ)), φ(a) < φ < π, where

0 < ρ−− (φ) < μmax < ρ−+ (φ) < μ− (φ),

(36)

(37)

and the left-hand side of Eq. (35) takes its maximum at μmax . Note that for φ := φ(a) the terms ρ−− (φ(a)) and ρ−+ (φ(a)) coincide, ρ−+ (φ(a)) = ρ−− (φ(a)). If φ < φ(a) then there is no intersection with this φ-curve. The solutions Y± (·) have the properties 0 < Y− (x) < 0 < y− (x, φ) < Y− (x)
a, 0 < Y+ (x), x > 0 2 1 x < Y+ (x; ) < y+ (x, φ), x sufficiently large, (38) 2

Y± (x) =

1 x ± u ± (x), u ± (x) > 0, 2 lim Y− (x) = 0,

(39)

x→a

lim Y+ (x) = (1 +

x→0

√

z)2 ,

(40)

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H. Baumgärtel

u ± (x) = 0. x→∞ x

(41)

lim

(iv) Let z = 0. In this case, one has σ = 0, i.e., the solution branches of the equation σ = σcr coincide with the branches y± (·; π ) of the π -curve 4(x + y − 1)3 = 27x 2 y, i.e., they coincide with the curve where D = 0.

4.2 The Regions G ± (z) In the case D(x, y; z) < 0, these regions are defined for z ≥ 1 by G + (z) := {(x, y) : x > 0, y > Y+ (x; z)}, G − (z) := {(x, y) : x > 0, y < Y− (x; z)},

and for 0 ≤ z < 1 by G + (z) := {(x, y) : x > 0, y > Y+ (x : z)}, G − (z) := {(x, y) : x > 1 − z, y < Y− (x; z)},

such that the region D(x, y; z) < 0 is the union G + (z) ∪ G − (z).

On the Border Lines Between the Regions of Distinct Solution Type …

75

These regions can be also characterized by the mutual position of R(T ) = R(x, y; z) and the roots R1 (x, y; z) < R2 (x, y; z) of the polynomial p(·). Actually, one obtains.

4.3 Lemma The regions G ± can be characterized as follows: G + (z) = {(x, y) : R(x, y; z) > R2 (x, y; z)} G − (z) := {(x, y) : R(x, y; z) < R1 (x, y; z)}.

(42)

5 Proofs 5.1 Proof of the Theorem (i) Let z = 1. First note that μ+ (φ) → ∞ and μ− (φ) → 0 for φ →

π , 2

further μ− (φ)
0, μ− (φ) < μ < μ+ (φ).

(45)

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H. Baumgärtel

Therefore, the φ-curve consists of the two rays y = μ± (φ)x, x > 0,

π < φ < π. 2

In the case φ = π , there is only one ray and for φ = axes x = 0 and y = 0. The equation σ = σcr reads

π 2

(46)

the limit rays are the half

4y = (x + y)2 F(φ). For a point (x, y) on a φ-ray (46), this means 4μ± (φ)x = F(φ)(1 + μ± (φ))2 x 2 . The solution of this equation is given by (22), which implies x− (φ) < x+ (φ) and x± (φ) → 0 if φ →

π , x± (φ) → ∞ if φ → π. 2

Then μ− (φ)x− (φ) → 0 and μ+ (φ)x+ (φ) → 4 for φ → π2 . This proves (24). If x = x+ (φ+ ) = x− (φ− ) then x → ∞ iff φ± → π . In this case one has Y± (x) = μ± (φ± )x. Because of (43), this proves (23). Finally, if x → ∞ then μ± (φ± ) → μ± (π ) = 21 . This proves (25). (ii) Let z > 1. For convenience put a := z − 1, a > 0. The equation for the φcurves reads π < φ < π. (47) 4(x + y + a)3 (cos φ)2 = 27x 2 y, 2 Note that in this case the limit case φ = π is excluded because D < 0 everywhere. The limit case φ = π2 corresponds to the half axes x = 0, y = 0. The parameter representation for the φ-curve using μ-rays y = μx, μ > 0 yields the term x = x(φ, μ) given by Eq. (27). According to equations (44) and (45), one has x → ∞ for μ → μ± (φ). That is, in this case the φ-curve (47) has only a single branch, where |cos φ| for fixed φ. Moreover, the expression 3μ1/3 − 22/3 (1 + μ)(cos φ)2/3 x ≥ a 1−|cos φ| takes its maximum at μ := μ+ , μ− such that

1 , 2(cos φ)2/3

μ−
a 1−|cos there are parameters φ|

1 < μ+ 2(cos φ)2/3

and x = x(φ, μ− ) = x(φ, μ+ ). The corresponding values for y are y± (x, φ) = μ± x, i.e., one obtains y± (x, φ) = μ± (φ). (48) lim x→∞ x

On the Border Lines Between the Regions of Distinct Solution Type …

77

The equation (19) for a point (x, y) = (x, μx) on a fixed φ-curve leads the equation (28). The left-hand side of this equation is positive for parameters μ satisfying μ− (φ) < μ < μ+ (φ) and vanishes for μ = μ± (φ). It takes its maximal value in this interval at 21/6 ψ cos , μ1/3 max := 1/3 | cos φ | 3 where the angle ψ is given by Eq. (14). The corresponding maximum is given by 21/3 cos ψ3 (2(cos ψ3 )3 − 3 cos ψ). Now the inequality 2

1/3

ψ cos 3

    32 ψ 3 − 3 cos ψ > 8/3 F(φ) 2 cos 3 2

√ √ is true, it is equivalent with the inequality 21 3 > cos ψ3 and 21 3 is the maximal value of cos ψ3 in the admissible interval for ψ (see Eq. (14)), which is taken at the limit case φ = π2 . The consequence is that for every φ ∈ ( π2 , π ) there are exactly two solutions μ := ρ± (φ) of Eq. (28), where the inequality (29) is satisfied. This proves (26). The relation (30) follows from (29). The relations (31) can be obtained by solving Eq. (19) directly for x = 0 which implies φ = π2 . Concerning relation (32) note that x → ∞ on a fixed φ-curve corresponds to μ → μ± (φ) and for each φ < π there is a solution x = x(φ, μ) with μ := ρ± (φ), according to Eq. (26). Taking the limit φ → π then with Eqs. (48) and (30) one obtains (32). (iii) Let 0 < z < 1. The equation for the φ-curve reads 4(x + y − a)3 (cos φ)2 = 27x 2 y,

π < φ < π. 2

(49)

The limit case φ = π2 corresponds to the half axes x = 0, y ≥ a and y = 0, x ≥ a. The case φ = π corresponds to D = 0. The parameter representation using μ-rays y = μx, μ > 0, yields the term 22/3 (cos φ)2/3 , μ > μ+ (φ), μ < μ− (φ) 22/3 (cos φ)2/3 (μ + 1) − 3μ1/3 (50) for the x-coordinate of the point of the φ-curve. The parameter values μ > μ+ (φ), μ < μ− (φ) describe the upper and lower branch y± (·; φ) of the φ-curve, respectively. Inserting (50) with μ > μ+ (φ) and y = μx into Eq. (19) then the resulting Eq. (35) is the condition for those μ, where the intersection point of the μ-ray with the φ-curve is simultaneously a solution point of Eq. (19). The left-hand side of Eq. (35) vanishes for μ := μ+ (φ), it tends to infinity for μ → ∞ and it is strongly monotonically F(φ) < increasing for μ > μ+ (φ). If π2 < φ < π then 0 < F(φ) < 1 and 0 < 1−z z ∞ because of 0 < z < 1. That is, for every pair {φ, z} there is exactly one solution ρ+ (φ) of Eq. (35) and one has μ+ (φ) < ρ+ (φ). x = x(φ, μ) = a

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For the investigation of solutions of Eq. (35) in the interval (0, μ− (φ)), we write this equation in the form (1 − a)(cos φ)2/3 μ1/3 (22/3 (cos φ)2/3 (μ + 1) − 3μ1/3 ) =

32 a F(φ). 28/3

(51)

The extrema of the left-hand side of Eq. (51) as a function of μ are 1/3

μmin :=

21/6 ψ 21/6 1 1/3 , μ cos := cos (2π − ψ). max 1/3 1/3 | cos φ | 3 | cos φ | 3

First one obtains μ− (φ) < μmin < μ+ (φ). This term has been used for the solution of Eq. (19) in the case z > 1. Further, 0 < μmax < μ− (φ). At this term, the lefthand side takes its maximum. Inserting this value into the left-hand side one gets (1 − a)G(φ), where G(φ) := 3 · 2

1/3

  2  2  1 1 cos (2π − ψ) 1 − 2 cos (2π − ψ) . 3 3

5 π < 13 (2π − ψ) < Recall Eq. (14) which implies π4 > ψ3 > π6 and 12 π tion φ → G(φ), 2 < φ < π, is monotonically increasing and

π . 2

(52)

The func-

2  2    5 π 5 1/3 cos π G( ) = 0, G(π ) = 3 · 2 1 − 2 cos π > 0. 2 12 12

(53)

Comparing the maximum value G(φ) of the left-hand side of Eq. (51) with the righthand side a · 32 2−8/3 F(φ) and taking into account the monotony properties of G(·) and F(·), further (51) and F( π2 ) = 1, F(π ) = 0 then one obtains: There is exactly one angle φ(a), π2 < φ(a) < π , such that (1 − a)G(φ(a)) = a ·

32 F(φ(a)). 28/3

Further, if π2 < φ < φ(a) then the left-hand side of Eq. (51) at the maximum point μmax is smaller than the right-hand side at φ; however, if φ(a) < φ < π then the left-hand side of Eq. (51) at the maximum point is larger than the right-hand side at φ. This implies: In the case π2 < φ < φ(a), there is no solution μ of Eq. (51). If φ(a) < φ < π then there are exactly two solutions ρ−± (φ) of Eq. (51) such that Eq. (36) is true. From Eq. (50), it follows that limμ→0 x(φ, μ) = a for all φ, π2 < φ < π, i.e., the lower branch of the φ-curve starts at (a, 0). This implies (39), according to (37). Relation (40) can be obtained by solving Eq. (19) directly for x = 0, which implies √ φ = π2 . Note that in this case the second (formal) solution (1 − z)2 is excluded

On the Border Lines Between the Regions of Distinct Solution Type …

79

√ √ because of (1 − z)2 < 1 − z which implies D(0, (1 − z)2 , z) > 0, i.e. the pair √ 2 (0, (1 − z) ) belongs to the region D > 0. Concerning relation (41) note that it follows from (50) that for μ > μ+ (φ), μ → μ+ (φ) or μ < μ− (φ), μ → μ− (φ) one has x(φ, μ) → ∞. This means that for the two branches y± (·; φ) of the φ-curves one has lim

x→∞

y± (x; φ) = μ± (φ). x

(54)

However, for each φ, φ(a) < φ < π , there is a solution x := x(φ, μ) of equation (19), either with μ := ρ+ (φ) according to (35) or with μ := ρ−+ (φ) according to (36) and (37). Taking the limit φ → π then with Eq. (52) one obtains (41).

5.2 Proof of the Lemma First note that, according to equations (10) and (15), in the case z 0 = 0 one has to show that   φ 2 4 −1 −1 cos (x + y − 1) > y , (x, y) ∈ G + , 3 3 and (x + y − 1)−1
0. First there are special points √ satisfying this property, for example, (0, y), where y > (1 + z)2 , for G + (z), and, on the other hand, (x, y), where x > 1 large and 0 < y < 1, for G − (z). Further both regions are simply connected, i.e., an arbitrary point of a region can be smoothly joined within the region with such a special point. Choose a smooth path joining the special point (x0 , y0 ), say, of G + (z) with an arbitrary point (x1 , y1 ) of this region. Consider the function R(x, y; z) − R2 (x, y; z) for points of the path. At (x0 , y0 ) it is positive. Assume that R(x1 , y1 ; z) < R1 (x1 , y1 ; z), then the value of the function at this point is smaller than R1 (x1 , y1 ; z) − R2 (x1 , y1 ; z) < 0, i.e. then there is a point on the path such that value of the function at this point is zero which is a contradiction.

6 Remark The asymptotic relation for the functions u ± points to a logarithm-like behavior. The asymptotics of these functions could be investigated more precisely. The ray y = 21 x in the description of the border lines characterizes in the case of vanishing radiation the the transition points from deceleration to acceleration.

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References 1. Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984) 2. Baumgärtel, H.: On a critical radiation density in the Friedmann equation. JMP 53, 122505 (2012)

Scattering on Leaky Wires in Dimension Three Pavel Exner and Sylwia Kondej

Abstract We consider the scattering problem for a class of strongly singular Schrödinger operators in L 2 (R3 ) which can be formally written as Hα,Γ = −Δ + δα (x − Γ ), where α ∈ R is the coupling parameter and Γ is an infinite curve which is a local smooth deformation of a straight line Σ ⊂ R3 . Using Kato–Birman method, we prove that the wave operators Ω± (Hα,Γ , Hα,Σ ) exist and are complete.

1 Introduction It is often said that when a great scientist says that something can be done, it can be done, while if the claim is that it cannot be done, he or she is usually wrong; sooner or later a younger one will come and do the impossible work earning a deserved fame. A nice illustration of this effect can be found in the biography of Tosio Kato to the centenary of whom the present volume is dedicated. There are various testimonies [23] that John von Neumann who otherwise did so much for the mathematical foundations of quantum mechanics discouraged people from attempts to prove the self-adjointness of atomic Hamiltonians because he considered the task hopelessly beyond reach. Kato’s elegant and, in the matter of fact, simple proof [14] was a starting point of the rigorous theory of Schrödinger operators which in the subsequent decades brought a plethora of results and managed to address fundamental questions such as those concerning the stability of matter [18].

P. Exner (B) Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, ˇ near Prague, Czechia 25068 Rež e-mail: [email protected] Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Bˇrehová 7, 11519 Prague, Czechia S. Kondej Institute of Physics, University of Zielona Góra, ul. Szafrana 4a, 65246 Zielona Góra, Poland e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_6

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While this may be arguably the most important result of Tosio Kato, from the other point of view, it is just one item in the broad spectrum of his achievements which extended also to some less well-known directions [12]. We believe that other contributions to this volume, in combination with recent reviews [23, 24] will provide a full picture showing how much mathematical physics owes to him. Different people may have different preferences but his opus magnum, the monograph [13], will probably come to everybody’s mind first. As anybody in the field, the present authors use it regularly and employed also other Kato’s results, for instance, his contribution to the theory of product formulæ [17] that inspired us in the discussion of quantum Zeno dynamics [6–8]. The result we are going to present in this note is based on a method at the origin of which Kato left his footprint and which bears his name together with that of Mikhail Birman. The starting point here was two of his 1957 papers [15, 16], which together with the paper by Rosenblum [22] were the origin of the trace-class perturbation theory. Later substantial contributions were made by the others, the said Birman, Kuroda, Putnam, and Pearson, to name just the main ones—for a description of the history we refer to the review [24] or the notes to Sect. XI.3 in [21]—but the starting point was here. In this note, we are interested in Schrödinger operators with singular “potentials” supported by zero measure sets. In recent years, they were studied as models of “leaky” quantum wires and networks made of them, cf. [11, Chap. 10] for an introduction to the subject and a bibliography. From the mathematical point of view, the parameter which matters is the codimension of the interaction support. If the latter is one, the operators can be treated naturally using the associated quadratic forms in the spirit of [3], for codimension two, the problem is more subtle. The scattering problem in the codimension one case was discussed in [10] where we considered the situation where the singular interaction support is a curve Γ in the plane, or more generally a family of curves, which can be regarded as a local deformation of a single straight line Σ. Under suitable regularity assumptions, we proved there the existence and completeness of the wave operators. In the present note, we address a similar question in the codimension two case, and for simplicity, we restrict ourselves to the situation when Γ is a single curve in R3 being a smooth local deformation of a straight line Σ. Note that the scattering problem with singular interactions supported by curves in R3 has been considered recently1 in [2], however, our task here is different. The curves in the said paper were supposed to be finite and the Hamiltonian was compared to the one describing the free motion in the three-dimensional space, −Δ with the usual mathematical-physics license concerning the units. In our case, the comparison operator can be formally written as Hα,Σ = −Δ + δα (x − Σ), where α ∈ R is the parameter characterizing the interaction strength. Using separation of variables and the well-known result about two-dimensional point interactions [1], we find easily that the spectrum is [ξα , ∞) where ξα < 0 is given by (3) below; in addition to motion at positive energies, the system has now a guided mode in which the particle can move remaining localized 1 See

also recent related results in [5, 19].

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in the vicinity of Σ. The scattering will now mean a comparison between Hα,Σ and the “full” Hamiltonian formally written as Hα,Γ = −Δ + δα (x − Γ ) ;

(1)

and a rigorous definition of these operators will be given in the next section. Our aim is to show that the scattering is well defined in this setting, in other words, that the wave operators for the given pair exist and are complete. In fact, the wave operators are also asymptotically complete as one has σsc (Hα,Γ ) = ∅ under our assumptions, but we leave the proof of this property together with extensions of the result to a wider class of the interaction supports Γ to a subsequent paper.

2 Preliminaries: The Operator First, we have to introduce the main notions. Let Γ ⊂ R3 be an infinite curve of class C 1 and piecewise C 2 which coincides asymptotically with a straight line Σ in a sense to be made precise below. With the usual abuse of notation we regard Γ both as a map R → R3 and its image. Furthermore, without loss of generality, we may fix Σ := {(x1 , 0, 0) : x1 ∈ R} and to parameterize Γ by its arc length s so that we have |Γ˙ | = 1 and |Γ (s) − Γ (s  )| ≤ |s − s  |. We will also suppose that ∃c ∈ (0, 1) such that |Γ (s) − Γ (s  )| ≥ c|s − s  | for ∀s, s  ∈ R ,

(1)

which means, in particular, that the curve Γ has no self-intersections and that it cannot be of a U-shaped form. Our next task is to introduce the Hamiltonian, which will be a singular Schrödinger operator with an interaction supported by the curve Γ , in other words, a singular perturbation of the “free” operator H0 which is the Laplacian in L 2 (R3 ) with the natural domain. There are various ways to do that using, for instance, quadratic forms or a Krein-type formula [2, 4, 20, 25]. For the purpose of the present paper, we recall the method employed in [9] inspired by the classical definition of the twodimensional δ interaction [1]; its advantage is that it has a local character. The curve regularity allows us to associate with Γ , apart from a discrete set, the Frenet’s frame, i.e., the triple (t (s), b(s), n(s)) of the tangent, binormal, and normal vectors, which are by assumption piecewise continuous functions of s. Moreover, at the discontinuity points of Γ¨ , the Frenet frame limits from the two sides differ by a rotation around the tangent vector, and hence one can construct a globally smooth coordinate system and, with an abuse of notation, employ the symbols b(s), n(s) for the rotated binormal and normal, respectively. Using this system, we may further introduce a family of “shifted” curve: given ξ, η ∈ R we denote r = (ξ 2 +η2 )1/2 and set Γrξ η := { Γrξ η (s) := Γ (s) + ξ b(s) + ηn(s) , s ∈ R} ,

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P. Exner and S. Kondej ξη

in view of (1) and the smoothness of Γ there is an r0 > 0 such that Γr ∩ Γ = ∅ holds for all r < r0 . This allows us to define generalized boundary values of a function ξη 2 (R3 \ Γ ) using its restriction f Γrξ η (s) to Γr which is by assumption well f ∈ Hloc 2 defined for 0 < r < r0 . We shall say that a function f ∈ Hloc (R3 \ Γ ) ∩ L 2 (R3 ) belongs to Υ if the limits 1 f Γrξ η (s), Ξ ( f )(s) := − lim r →0 ln r   Ω( f )(s) := lim f Γrξ η (s) + Ξ ( f )(s) ln r r →0

exist a.e. in R independently of the direction r1 (ξ, η) in which they are taken and belong to L 2 (R) as functions of s. This makes it possible to define the sought singular Schrödinger operator as the restriction of the Laplacian acting on R3 \ Γ to a suitable subset of Υ . To be specific, we fix an α ∈ R and define the operator Hα,Γ as follows: D(Hα,Γ ) = Υα := { g ∈ Υ : 2π αΞ (g)(s) = Ω(g)(s) }, Hα,Γ f = −Δf for x ∈ R \ Γ. 3

(2a) (2b)

It was shown in [9] that such an operator is self-adjoint. Note that the absence of a singular interaction means that the singular boundary value Ξ ( f ) vanishes identically, in other words, the free operator H0 corresponds to α = ∞. This fact leads some authors to write the operator in question as −Δ − α1 δ(· − Γ ), see e.g. [2]. This, however, does not fit well with the fact that the two-dimensional δ interaction is “always attractive”, and hence we avoid such formal expressions showing the interaction “strength” and restrict ourselves to the definition (2) in the spirit of [1, Sect. I.5]. Before proceeding further, let us say a few words about the spectrum of Hα,Γ . If the interaction support is a straight line, Γ = Σ, one finds it easily by separation of variables: it is absolutely continuous and equal to σ (Hα,Γ ) = [ξα , ∞), where ξα = −4 e2(−2πα+ψ(1))

(3)

is the eigenvalue of the corresponding one-center two-dimensional δ interaction, with −ψ(1) ≈ 0.57721 being the Euler–Mascheroni constant. For a non-straight Γ , the spectrum of Hα,Γ may be different and depends, in general, on the geometry of Γ . One of the interesting situations concerns curves that are asymptotically straight. In [9], for instance, we assumed that there are ω ∈ (0, 1), μ ≥ 0 and ε, d > 0 such that for all (s, s  ) ∈ Sω,ε , we have

Scattering on Leaky Wires in Dimension Three

1−

85

|Γ (s) − Γ (s  )| d|s − s  | , ≤   |s − s | (|s − s |+1)(1 + (s 2 +s 2 )μ )1/2

(4)

where Sω,ε is the subset of R2 consisting of points (s, s  ) satisfying ω < ss < ω−1 if and |s − s  | < ε if |s + s  | < ε 1+ω . If this assumption is satisfied |s + s  | > ε 1+ω 1−ω 1−ω 1 with some μ > 2 , together with (1), then the essential spectrum is preserved, σess (Hα,Γ ) = [ξα , ∞), and, in addition, the operator Hα,Γ has a non-void discrete spectrum whenever the deformation is nontrivial, Γ = Σ.

3 Preliminaries: The Resolvent In what follows, we adopt a more restrictive assumption about the curve, namely, we suppose that there exists a compact set M ⊂ R3 , such that Γ \Σ ⊂ M.

(1)

To analyze the scattering problem for the pair (Hα,Γ , Hα,Σ ), we need to know more about the resolvent of singular Schrödinger operator (2). In analogy with the considerations of [9], we begin from the embedding of the free resolvent R z := (−Δ − z)−1 : L 2 (R3 ) → W 2,2 (R3 ) to L 2 (Γ ). It is sufficient to restrict the spectral parameter z to negative real values, and hence we consider z = −κ 2 with 2 κ > 0 and denote Rκ = R −κ . It is well known that Rκ is integral operator with the kernel determined by the function G κ (x) :=

e−κ|x| . 4π |x|

 ˘ Γκ :L 2 (Γ ) → L 2 (R3 ) acts as R ˘ Γκ f := 3 G κ (· − x) f (x)δ(x − Γ )dx, Specifically, R R ˘ Γκ . To find and furthermore, we define RΓκ : L 2 (R3 ) → L 2 (Γ ) as the adjoint of R out the resolvent of Hα,Γ , we define the operator Tˇ κ : L 2 (Γ ) → L 2 (Γ ) by (Tˇ κ f )(s) = −

1 (2π )2

 R

ln( p 2 + κ 2 )1/2 ei ps fˆ( p) d p,

(2)

where fˆ stands for the Fourier transform of f and the maximal domain of this operator is D(Tˇ κ ) = { f : Tˇ κ f ∈ L 2 (Γ )}. Furthermore, we set 1 (ln 2 + ψ(1)), Tκ = Tˇ κ + 2π

(3)

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where ψ(1) is, up to the sign, the Euler–Mascheroni constant mentioned above. Finally, we define the integral operator Bκ : L 2 (Γ ) → L 2 (Γ ) with the kernel of the form Bκ (s, s  ) := G κ (Γ (s) − Γ (s  )) − G κ (s − s  ), and the operator ˇ κ ) → L 2 (Γ ) . Qκ := Tκ + Bκ : D := D(T

(4)

Note that the operator Bκ is positive because the function G κ is monotonous and by assumption, we have |Γ (s) − Γ (s  )| ≤ |s − s  |; this fact was crucial in [9] to prove that a curve deformation gives rise to the existence of a discrete spectrum of Hα,Γ . By [9, Thm. 2.1] the operator α − Qκ : L 2 (Γ ) → L 2 (Γ ) is invertible for all κ large enough and κ ˘ Γκ (α − Qκ )−1 RΓκ = Rκ + R (5) Rα,Γ is the resolvent of Hα,Γ . It is not by a chance that this expression has a Kreinlike form because Hα,Γ is a self-adjoint extension of the symmetric operator −Δ : C0∞ (R3 \ Γ ) → L 2 (R3 ). Note also that the geometric perturbation is encoded in the part Bκ of (4), we have Bκ = 0 if Γ = Σ, and consequently, Qκ = Tκ holds in this case. The resolvent expression (5) is a tool to prove the spectral properties of Hα,Γ mentioned at the end of the preceding section. Lemma 1 The operator (α − Qκ )−1 is bounded for all κ large enough. Proof It follows from (2) that Tˇ κ f 2 =

1 4(2π )3



 R

2 ln( p 2 + κ 2 ) | fˆ( p)|2 d p

and therefore for all κ large enough, we have Tκ f 2 ≥ C(ln κ)2  f 2 with a suitable constant C. For the sake of simplicity, we use the symbol C as a generic positive constant which may vary case from case. Furthermore, by [9, Lemma 5.3], the operator Bκ belongs to the Hilbert–Schmidt class under assumption (4), and therefore, a fortiori, if we assume (1). This allows to conclude that (α − Qκ ) f 2 = ((α − Tκ − Bκ ) f, (α − Tκ − Bκ ) f ) ≥ C(ln κ)2  f 2 with another constant C. On the other hand, we know that operator (α − Qκ )−1 exists and from the above inequality we can conclude that it is bounded. 

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4 Existence and Completeness of Wave Operators Now, we are able to pass to our main task in this note, namely, the existence and completeness of the wave operators given by Ω± (Hα,Γ , Hα,Σ ) := s - lim ei Hα,Γ t e−i Hα,Σ t , t→±∞

where we can skip the projection Pac (Hα,Σ ) usually appearing in the definition because the spectrum of Hα,Σ is purely absolutely continuous as we have recalled above. For notational convenience, we decompose the line Σ into three parts, Σ = Σ M ∪ Σ+ ∪ Σ− , where Σ M := M ∩ Σ and Σ± are the straight components of Σ \ M which correspond, respectively, to x1 → ±∞ in the chosen coordinate system. Without loss of generality, we may assume that (0, 0, 0) ∈ M and Σ± = {x : x = (x1 , 0, 0), x1 ∈ (−∞, x− ) ∪ (x+ , ∞) and x± ≷ 0} . In a similar way, one can dissect the curve Γ into three parts, Γ M and Γ± = Σ± . The inverses of the “full” and “free” Birman–Schwinger operators, (α − Qκ )−1 and (α − Tκ )−1 , act, respectively, in L 2 (Γ ) and L 2 (Σ). To compare the resolvents of Hα,Γ and Hα,Σ , we introduce the following embeddings: κ −1 2 2 (α − Qκ )−1 Γi Γ j := χΓi (α − Q ) χΓ j : L (Γ j ) → L (Γi ) ,

where i, j = ±, M, χΓi is the characteristic functions of Γi , and in the analogous way we define (α − Tκ )−1 Σi Σ j . Let us now consider the resolvent difference κ κ κ κ ˘ Γκ (α − Qκ )−1 RΓκ − R ˘Σ − Rα,Σ =R (α − Tκ )−1 RΣ . Rα,Γ

Using the obvious fact κ −1 (α − Qκ )−1 Γ± Γ± = (α − T )Σ± Γ± ,

we are coming to the conclusion that κ κ − Rα,Σ = Rα,Γ

 i, j∈X

κ ˘ Γκ (α − Qκ )−1 R Γi Γ j RΓ j − i



κ κ ˘Σ R (α − Tκ )−1 Σi Σ j RΣ j , i

(1)

i, j∈X

where X := {(i, j) : i, j = +, −, M ∧ (i, j) = (+, +), (−, −)}. This will be used to prove the following result:

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κ κ Theorem 1 The operator Rα,Γ − Rα,Σ belongs to the trace class for all κ large enough.

Let us start from an auxiliary claim: Lemma 2 We have  R3

G κ (y − x)G κ (x − z) dx =

1 −κ|y−z| e . 8π κ

(2)

Proof We use the Fourier representation of the Green function  R3

G κ (y − x) f (x) dx =

1 (2π )3/2



ei py fˆ( p) d p p2 + κ 2

R3

and apply it to the Green function itself, 

1 G (y − x)G (x − z) dx = 3 (2π )3 R κ

κ

 R3

ei p(y−z) dp . ( p 2 + κ 2 )2

Performing the integration over angles in the integral on the right-hand side, we get  I =

R3

ei py 4π dp = 2 ( p + κ 2 )2 |y|

 0

∞

p sin p|y| 2π dp = ( p 2 + κ 2 )2 i|y|

 R

p ei p|y| dp . ( p 2 + κ 2 )2

Next, we extend in a standard way the integration forming the contour by adding upper semicircle and using the Jordan’s lemma which implies that the integral over the semicircle vanishes in the limit of infinite radius. This gives I =

2π i|y|



4π 2 z ei z|y| dz = 2 2 +κ ) |y|

(z 2



Res

upper hal f plane

zei z|y| . + κ 2 )2

(z 2

Using now the generalized Cauchy integral formulae, one gets I =

4π 2 |y|



d zei z|y| dz (z + iκ)2



z=iκ

=

π 2 −κ|y| e . κ

On putting these results together, we arrive at the formula (2).



κ κ Proof of Theorem 1. Let us pick one of the components of Rα,Γ − Rα,Σ , for κ κ κ −1 ˘ instance, RΓ+ (α − Q )Γ+ Γ− RΓ− . The symbol Bδ will conventionally denote the ball of radius δ centered at the origin and χB δ stands for the characteristic function of the ball Bδ . We define the “cut-off” operator family

 κ ˘ Γκ (α − Qκ )−1 Sδ+− ≡ Sδ := χB δ R Γ+ Γ− RΓ− χB δ +

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and ask for its weak limit as δ → ∞. We have, in particular, 

 R3

Sδ (x, x)dx=

R3

 κ G κ (· − x)χB δ (x), (α−Qκ )−1 Γ+ Γ− G (· − x)χB δ (x)

L 2 (Γ+ )

dx .

(3) Using now the Lebesgue’s dominated convergence theorem in combination with Lemma 2 and the equivalence that e−κ|x−y| = e−κ(|x|+|y|) holds, in view of that fact that Γ± = Σ± = {x : x = (x1 , 0, 0), x1 ∈ (−∞, x− ) ∪ (x+ , ∞) and x± ≷ 0}, we obtain  

 π4 −κ|·| (s) ds . (4) Sδ (x, x) dx = 2 e−κ|s| (α − Qκ )−1 lim Γ+ Γ− e δ→∞ R3 κ Γ+ Using further the boundedness of (α − Qκ )−1 Γ+ Γ− in combination with Schwarz inequality, we get from (4) the following estimate:  lim

δ→∞ R3

Sδ (x, x)dx ≤ C

π 4 −κ|·| e  L 2 (Γ+ ) e−κ|·|  L 2 (Γ− ) , κ2

where the constant C is the norm of (α − Qκ )−1 Γ+ Γ− . We want to conclude that  R3

κ ˘ Γκ (α − Qκ )−1 Sδ (x, x)dx → Tr R Γ+ Γ− RΓ− +

κ ˘ Γκ (α − Qκ )−1 as δ → ∞ and to show in this way that the operator R Γ+ Γ− RΓ− belongs + to the trace class. According to the lemma following Theorem XI.31 in [21], the trace can be expressed through the integral of the kernel diagonal provided the latter is continuous in both arguments and the operator is positive. The continuity was mentioned already, the positivity follows from the fact that the operator α − Qκ is positive from all κ large enough, cf. [9, Lemma 5.5]. Let us next consider the component of (1) referring to the operator acting between the spaces L 2 (Γ+ ) and L 2 (Γ M ). We put

 κ ˘ Γκ (α − Qκ )−1 R SδM+ ≡ Sδ := χB δ R Γ M Γ+ Γ + χ B δ . M Applying again Lemma 2 in combination with the Lebesgue’s dominated convergence theorem, one obtains  lim

δ→∞ R3

Sδ (x, x) dx=

π4 κ2

 ΓM

 −κ|Γ (·)−Γ (s)| (α − Qκ )−1 ds . e Γ M Γ+

(5)

Using further the boundedness of (α − Qκ )−1 Γ M Γ+ and the continuous imbedding of spaces L 2 (Γ M ) → L 1 (Γ M ) together with the Fubini’s theorem and Schwarz inequality, we infer that

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 lim

δ→∞ R3

π4   −κ|Γ (·)−Γ (s)| Sδ (x, x) dx ≤ 2 ds (α − Qκ )−1 Γ M Γ+ e κ ΓM

 2 1/2 π4 −κ|Γ (·)−Γ (s)| ≤ 2 |Γ M | e . (α − Qκ )−1 ds Γ M Γ+ κ ΓM

(6)

The integral on the right-hand side of (6) is finite because the integrated function κ ˘ Γκ (α − Qκ )−1 belongs to L 2 (Γ M ). This implies that R Γ M Γ+ RΓ+ belongs to the trace M class in the same way as above. κ κ − Rα,Σ contributing to formula (1) can be The remaining components of Rα,Γ dealt with in the analogous way. The only terms which do not allow for such a treatment are those containing (α − Qκ )−1 Γ± Γ± , however, they cancel when we subtract κ κ from Rα,Σ . Concluding the above discussion, we thus find that the difference Rα,Γ κ κ − Rα,Σ is a trace-class operator for all κ large enough what we have set out to Rα,Γ prove.  Now, we are in a position to present the result indicated in the introduction: Corollary 1 In the stated assumptions, the wave operators Ω± (Hα,Γ , Hα,Σ ) exist and are complete. Proof In view of Theorem 1, the claim follows immediately from Kuroda–Birman theorem, cf. [21, Thm. XI.9].  ˇ within Acknowledgements The research was supported by the Czech Science Foundation (GACR) the project 17-01706S and the EU project CZ.02.1.01/0.0/0.0/16_019/0000778.

References 1. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing, Providence (2005) 2. Behrndt, J., Frank, R.L., Kühn, Ch., Lotoreichik, V., Rohleder, J.: Spectral theory for Schrödinger operators with δ-interactions supported on curves in R3 . Ann. H. Poincaré 18, 1305–1347 (2017) 3. Brasche, J.F., Exner, P., Kuperin, YuA, Šeba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994) 4. Brasche, J.F., Teta, A.: Spectral analysis and scattering for Schrödinger operators with an interaction supported by a regular curve. In: Albeverio, S., Fenstadt, J.E., Holden, H., Lindstrøm, T. (eds.) Ideas and Methods in Quantum and Statistical Physics, pp. 197–211. Cambridge University Press, Cambridge (1992) 5. Cacciapuoti, A., Fermi, D., Posilicano, A.: Scattering from local deformations of a semitransparent plane. J. Math. Anal. Appl. 473, 215–257 (2019) 6. Exner, P., Ichinose, T.: A product formula related to quantum Zeno dynamics. Ann. H. Poincaré 6, 195–215 (2005) 7. Exner, P., Ichinose, T., Kondej, S.: On relations between stable and Zeno dynamics in a leaky graph decay model. In: Proceedings of the Conference “Operator Theory and Mathematical Physics” (B¸edlewo 2004); Operator Theory: Advances and Applications, vol. 174, pp. 21–34. Basel, Birkhäuser (2007)

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8. Exner, P., Ichinose, T., Neidhardt, H., Zagrebnov, V.A.: Zeno product formula revisited. Int. Eq. Oper. Theory 57, 67–81 (2007) 9. Exner, P., Kondej, S.: Curvature-induced bound states for a delta interaction supported by a curve in R 3 . Ann. H. Poincaré 3, 967–981 (2002) 10. Exner, P., Kondej, S.: Scattering by local deformations of a straight leaky wire. J. Phys. A: Math. Gen. 38, 4865–4874 (2005) 11. Exner, P., Kovaˇrík, H.: Quantum Waveguides, p. xxii+382. Springer International, Heidelberg (2015) 12. Fujita, H., Okamoto, H.: Tosio Kato as an applied mathematician: a historical study of a Japanese mathematician, pp. 15–16. ICIAM Newsletter, October (2018) 13. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976) 14. Kato, T.: Fundamental properties of Hamiltonian of the Schrödinger type. Trans. Am. Math. Soc. 70, 195–211 (1951) 15. Kato, T.: On finite dimensional perturbations of self-adjoint operators. J. Math. Soc. Jpn. 9, 239–249 (1957) 16. Kato, T.: Perturbations of continuous spectra by trace class operators. Proc. Jpn. Acad. 33, 260–264 (1957) 17. Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups. In: Topics in Functional Analysis (Essays Dedicated to M.G. Krein on the Occasion of his 70th Birthday). Advances in Mathematics: Supplementary Studies, vol. 3, pp. 185–195. Academic, New York (1978) 18. Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010) 19. Mantile, A., Posilicano, A.: Asymptotic completeness and S-matrix for singular perturbations. J. Math. Pures Appl., to appear; arXiv:1711.07556 20. Posilicano, A.: A Krein-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183, 109–147 (2001) 21. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory IV. Analysis of Operators. Academic Press, New York (1979) 22. Rosenblum, M.: Perturbation of the continuous spectrum and unitary equivalence. Pac. J. Math. 7, 997–1010 (1957) 23. Simon, B.: Tosio Kato’s work on non-relativistic quantum mechanics: part 1. Bull. Math. Sci. 8, 121–232 (2018) 24. Simon, B.: Tosio Kato’s work on non-relativistic quantum mechanics: part 2. Bull. Math. Sci., to appear 25. Teta, A.: Quadratic forms for singular perturbations of the Laplacian. Publ. RIMS 26, 803–817 (1990)

Computing Traces, Determinants, and ζ-Functions for Sturm–Liouville Operators: A Survey Fritz Gesztesy and Klaus Kirsten

Dedicated, with deep admiration, to the memory of Tosio Kato (1917–1999)

Abstract The principal aim of this contribution is to survey an effective and unified approach to the computation of traces of resolvents (and resolvent differences), (modified) Fredholm determinants, ζ-functions, and ζ-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and ζ-function regularized determinants. Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm–Liouville operators on bounded intervals with various (separated and coupled) boundary conditions, and Schrödinger operators on a half-line, are provided and further illustrated with an array of examples. In addition, we consider a class of half-line Schrödinger operators (−d 2 /d x 2 ) + q on (0, ∞) with purely discrete spectra. Roughly speaking, the class considered is generated by potentials q that, for some fixed C0 > 0, ε > 0, x0 ∈ (0, ∞), diverge at infinity of the type q(x) ≥ C0 x (2/3)+ε0 for all x ≥ x0 . We treat all self-adjoint boundary conditions at the left endpoint 0. This manuscript surveys our recent two papers [19, 20].

K.K. was supported by the Baylor University Summer Sabbatical Program. F. Gesztesy (B) 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 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 © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_7

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2010 Mathematics Subject Classiffication Primary: 47A10 · 47B10 · 47G10 · Secondary: 34B27 · 34L40

1 Introduction This survey of our recent papers [19, 20] details the relationship between Fredholm and ζ-function regularized determinants of Hilbert space operators and provides a unified approach to these determinants as well as traces of resolvents. The case of traces of resolvents (and resolvent differences), (modified) Fredholm determinants, ζ-functions, and ζ-function regularized determinants associated with linear operators in a Hilbert space is described in detail in Section 2. In particular, under an appropriate trace class hypothesis on the resolvent of a self-adjoint operator S in H we define its ζ-function and derive a formula for it in terms of the selfadjoint functional calculus and the resolvent of S. In the case of trace class resolvent differences for a pair of self-adjoint operators (S1 , S2 ) in H, we also describe the underlying relative spectral ζ-function for such a pair and relate the corresponding ζfunction regularized relative determinant and a symmetrized Fredholm perturbation determinant. In Section 3, we provide an exhaustive discussion of regular (three-coefficient) self-adjoint Sturm–Liouville operators, that is, operators in L 2 ((a, b); r d x) associated with self-adjoint realizations of differential expressions of the type τ = −r −1 (d/d x) p(d/d x) + q, with arbitrary (separated and coupled) self-adjoint boundary conditions on compact intervals [a, b]. Their traces of resolvents and associated perturbation determinants are calculated for all self-adjoint boundary conditions in terms of concrete expressions involving a canonical system of fundamental solutions φ(z, · , a) and θ(z, · , a) of τ ψ = zψ, and special examples such as Floquet boundary conditions and the Krein–von Neumann extension are highlighted. The ζ-function regularized determinants are determined for all self-adjoint boundary conditions and a variety of concrete examples complete this section. Section 4 then illustrates some of the abstract notions in Section 2 with the help of self-adjoint Schrödinger operators associated with differential expressions of the type −(d 2 /d x 2 ) + q on the half-line R+ = (0, ∞) with short-range potentials q (i.e., we treat the scattering theory situation). Again, we study all self-adjoint boundary conditions at x = 0. The assumed short-range nature of q then necessitates a comparison with the case q = 0 and invoking the concept of Jost functions and Jost solutions, illustrates the case of relative perturbation determinants, relative ζ-functions, and relative ζ-function regularized determinants abstractly discussed in Section 2. Our final Section 5 again treats the case of half-line Schrödinger operators on R+ , but this time with potentials diverging at infinity, and hence with purely discrete spectra (i.e., empty essential spectra). In particular, we assume that q satisfies q ∈

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1 L loc (R+ ; d x), q real-valued a.e. on R+ , and that for some ε0 > 0, C0 > 0, and sufficiently large x0 > 0,

q(x) ≥ C0 x (2/3)+ε0 , x ∈ (x0 , ∞).

(1.1)

Applying modified Fredholm determinants det 2 ( · ) associated to Hilbert–Schmidt operators, we compute traces of resolvent differences and modified Fredholm determinants for half-line Schrödinger operators in terms of a natural analog of the Jost function and the Jost solution when compared with the short-range case in Section 4. Again our discussion involves all self-adjoint boundary conditions at x = 0. The theory is illustrated with the help of the explicitly solvable example q(x) = x, x ∈ R+ , recently also studied in [48]. Finally, we summarize some of the basic notation used in this paper (especially, in Section 2): Let H and K be separable complex Hilbert spaces, ( · , · )H and ( · , · )K the scalar products in H and K (linear in the second factor), and IH and IK the identity operators in H and K, respectively. Next, let T be a closed linear operator from dom(T ) ⊆ H to ran(T ) ⊆ K, with dom(T ) and ran(T ) denoting the domain and range of T . The closure of a closable operator S is denoted by S. The kernel (null space) of T is denoted by ker(T ). The spectrum, point spectrum, and resolvent set of a closed linear operator in H will be denoted by σ(·), σ p (·), and ρ(·); the discrete spectrum of T (i.e., points in σ p (T ) which are isolated from the rest of σ(T ), and which are eigenvalues of T of finite algebraic multiplicity) is abbreviated by σd (T ). The algebraic multiplicity m a (z 0 ; T ) of an eigenvalue z 0 ∈ σd (T ) is the dimension of the range of the corresponding Riesz projection P(z 0 ; T ), m a (z 0 ; T ) = dim(ran(P(z 0 ; T ))) = tr H (P(z 0 ; T )), where (with the symbol



(1.2)

denoting counterclockwise-oriented contour integrals)

P(z 0 ; T ) =

−1 2πi

‰ C(z 0 ;ε)

dζ (T − ζ IH )−1 ,

(1.3)

for 0 < ε < ε0 and D(z 0 ; ε0 )\{z 0 } ⊂ ρ(T ); here D(z 0 ; r0 ) ⊂ C is the open disk with center z 0 and radius r0 > 0, and C(z 0 ; r0 ) = ∂ D(z 0 ; r0 ) the corresponding circle. The geometric multiplicity m g (z 0 ; T ) of an eigenvalue z 0 ∈ σ p (T ) is defined by m g (z 0 ; T ) = dim(ker((T − z 0 IH ))).

(1.4)

The essential spectrum of T is defined by σess (T ) = σ(T )\σd (T ). The Banach spaces of bounded and compact linear operators in H are denoted by B(H) and B∞ (H), respectively. Similarly, the Schatten–von Neumann (trace) ideals will subsequently be denoted by B p (H), p ∈ [1, ∞), and the subspace of all finite rank operators in B1 (H) will be abbreviated by F(H). Analogous notation B(H1 , H2 ), B∞ (H1 , H2 ), etc., will be used for bounded, compact, etc., operators

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between two Hilbert spaces H1 and H2 . In addition, tr H (T ) denotes the trace of a trace class operator T ∈ B1 (H), det H (IH − T ) the Fredholm determinant of IH − T , and for p ∈ N, p ≥ 2, det H, p (IH − T ) abbreviates the pth modified Fredholm determinant of IH − T . Finally, we find it convenient to abbreviate N0 = N ∪ {0}.

2 Traces, (Modified) Fredholm Determinants, and Zeta Functions of Operators In this section, we recall some well-known formulas relating traces and (modified) Fredholm determinants and also discuss the notion of ζ-functions of self-adjoint operators. For background on the material used in this section see, for instance, [30, 31], [32, Chap. XIII], [29, Chap. IV], [42], [55, Chap. 17], [58], [59, Chap. 3]. To set the stage, we start with densely defined, closed, linear operators A in H having a trace class resolvent, and hence introduce the following assumption: Hypothesis 2.1 Suppose that A is densely defined and closed in H with ρ(A) = ∅, and (A − z IH )−1 ∈ B1 (H) for some (and hence for all ) z ∈ ρ(A). Given Hypothesis 2.1 and z 0 ∈ ρ(A), consider the bounded, entire family A( · ) defined by A(z) := IH − (A − z IH )(A − z 0 IH )−1 = (z − z 0 )(A − z 0 IH )−1 , z ∈ C. (2.1) Employing the formula (cf. [29, Sect. IV.1], see also [42], [65, Sect. I.7]),   tr H (IH − T (z))−1 T (z) = −(d/dz)ln(det H (IH − T (z))),

(2.2)

valid for a trace class-valued analytic family T ( · ) on an open set  ⊂ C such that (IH − T ( · ))−1 ∈ B(H), and applying it to the entire family A( · ) then results in      tr H (A − z IH )−1 = −(d/dz)ln det H IH − (z − z 0 )(A − z 0 IH )−1    = −(d/dz)ln det H (A − z IH )(A − z 0 IH )−1 ,

(2.3)

z ∈ ρ(A). One notes that the left- and hence the right-hand side of (2.3) is independent of the choice of z 0 ∈ ρ(A). Next, following the proof of [59, Theorem 3.5 (c)] step by step, and employing a Weierstrass-type product formula (see, e.g., [59, Theorem 3.7]), yields the subsequent result (see also [25]). Lemma 2.2 Assume Hypothesis 2.1 and let λk ∈ σ(A) then

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  detH IH − (z − z 0 )(A − z 0 IH )−1 = (λk − z)m a (λk ) [Ck + O(λk − z)], Ck = 0, z→λk

(2.4)  I − (z − that is, the multiplicity of the zero of the Fredholm determinant det H H  z 0 )(A − z 0 IH )−1 at z = λk equals the algebraic multiplicity of the eigenvalue λk of A. In addition, denote the spectrum of A by σ(A) = {λk }k∈N , λk = λk for k = k . Then  m (λ ) det H (IH − (z − z 0 )(A − z 0 IH )−1 ) = 1 − (z − z 0 )(λk − z 0 )−1 a k k∈N

  λk − z m a (λk ) = , λk − z 0

(2.5)

k∈N

with absolutely convergent products in (2.5). The case of trace class resolvent operators is tailor-made for a number of onedimensional Sturm–Liouville operators (e.g., finite interval problems). But for applications to half-line problems with potentials behaving like x, or increasing slower than x at +∞, and similarly for partial differential operators, traces of higher order powers of resolvents need to be involved which naturally lead to modified Fredholm determinants as follows. Hypothesis 2.3 Let p ∈ N, p ≥ 2, and suppose that A is densely defined and closed in H with ρ(A) = ∅, and (A − z IH )−1 ∈ B p (H) for some (and hence for all ) z ∈ ρ(A). Applying the formula   tr H (IH − T (z))−1 T (z) p−1 T (z) = −(d/dz)ln(det H, p (IH − T (z))),

(2.6)

valid for a B p (H)-valued analytic family T ( · ) on an open set  ⊂ C such that (IH − T ( · ))−1 ∈ B(H), [29, Sect. IV.2] (see also [65, Sect. I.7]) to the entire family A( · ) in (2.1), assuming Hypothesis 2.3, then yields   (z − z 0 ) p−1 tr H (A − z IH )−1 (A − z 0 IH )1− p    = −(d/dz)ln det H, p IH − (z − z 0 )(A − z 0 IH )−1    = −(d/dz)ln det H, p (A − z IH )(A − z 0 IH )−1 , z ∈ ρ(A).

(2.7)

In the special case, p = 2 this yields   tr H (A − z IH )−1 − (A − z 0 IH )−1    = −(d/dz)ln det H,2 IH − (z − z 0 )(A − z 0 IH )−1 .

(2.8)

We refer to Section 5 for an application of (2.8) to half-line Schrödinger operators with potentials diverging at infinity.

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In many cases of interest not a single resolvent, but a difference of two resolvents is trace class and hence one is naturally led to the following generalization discussed in detail in [23, 26, 27] (see also [22]). To avoid technicalities, we will now consider the case of self-adjoint operators A, B below, but note that [27] considers the general case of densely defined, closed linear operators with nonempty resolvent sets: Hypothesis 2.4 Suppose A and B are self-adjoint operators in H with A bounded from below. (i) Assume that B can be written as the form sum (denoted by +q ) of A and a self-adjoint operator W in H (2.9) B = A +q W, where W can be factorized into W = W1∗ W2 , such that

  dom(W j ) ⊇ dom |A|1/2 ,

(2.10) j = 1, 2.

(2.11)

(ii) Suppose that for some (and hence for all ) z ∈ ρ(A), W2 (A − z IH )−1/2 , (A − z IH )−1/2 W1∗ ∈ B2 (H).

(2.12)

Given Hypothesis 2.4, one observes that     dom |B|1/2 = dom |A|1/2 ,

(2.13)

and that the resolvent of B can be written as (cf., e.g., the detailed discussion in [27] and the references therein) (B − z IH )−1 = (A − z IH )−1

 −1 − (A − z IH )−1 W1∗ IH + W2 (A − z IH )−1 W1∗ W2 (A − z IH )−1 , z ∈ ρ(B) ∩ ρ(A). (2.14)

We also note the analog of Tiktopoulos’ formula (cf. [57, p. 45]),  −1 (B − z IH )−1 = (A − z IH )−1/2 IH + (A − z IH )−1/2 W (A − z IH )−1/2 × (A − z IH )−1/2 , z ∈ ρ(B) ∩ ρ(A).

(2.15)

Here the closures in (2.14) and (2.15) are well-defined employing (2.12). In addition, one observes that the resolvent formulas (2.14) and (2.15) are symmetric with respect to A and B employing A = B −q W .

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As a consequence, B is bounded from below in H and one concludes that for some (and hence for all ) z ∈ ρ(B) ∩ ρ(A), 

 (B − z IH )−1 − (A − z IH )−1 ∈ B1 (H).

(2.16)

Moreover, one infers that (cf. [26])   tr H (B − z IH )−1 − (A − z IH )−1

d = − ln det H (B − z IH )1/2 (A − z IH )−1 (B − z IH )1/2 dz

d = − ln det H IH + W2 (A − z IH )−1 W1∗ , z ∈ ρ(B) ∩ ρ(A). dz

(2.17) (2.18)

Here any choice of branch cut of the normal operator (B − z IH )1/2 (employing the spectral theorem) is permissible. The first equality in (2.18) follows as in the proof of [26, Theorem 2.8]. (The details are actually a bit simpler now since A, B are selfadjoint and bounded from below, and hence of positive type after some translation, which is the case considered in [26].) For completeness, we mention that the second equality in (2.18) can be arrived at as follows: Employing the commutation formula (cf., [11]), (2.19) C[IH − DC]−1 D = −IH + [IH − C D]−1 for C, D ∈ B(H) with 1 ∈ ρ(DC) (and hence 1 ∈ ρ(C D) since σ(C D)\{0} = σ(DC)\{0}), the resolvent identity (2.15) with A and B interchanged yields (B − z IH )1/2 (A − z IH )−1 (B − z IH )1/2  −1 = IH − (B − z IH )−1/2 W (B − z IH )−1/2     −1 = IH − (B − z IH )−1/2 W1∗ W2 (B − z IH )−1/2 .

(2.20)

On the other hand, it is well-known (cf. [27, 37]) that IH + W2 (A − z IH )−1 W1∗ −1  = IH − W2 (B − z IH )−1 W1∗    −1  = IH − W2 (B − z IH )−1/2 (B − z IH )−1/2 W1∗ ,

(2.21)

and hence using the fact det H (IH − ST ) = det H (IH − T S)

(2.22)

for S, T ∈ B(H) with ST, T S ∈ B1 (H) (again, since σ(ST )\{0} = σ(T S)\{0}) then proves the second equality in (2.18).

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While this approach based on Hypothesis 2.4 is tailor-made for a discussion of perturbations of the potential coefficient in the context of Schrödinger and, more generally, Sturm–Liouville operators, we also mention the following variant that is best suited for changes in the boundary conditions: Hypothesis 2.5 Suppose that A and B are self-adjoint operators in H bounded from below. In addition, assume that     dom |A|1/2 ⊆ dom |B|1/2 ,   (B − t IH )1/2 (B − t IH )−1 − (A − t IH )−1 (B − t IH )1/2 ∈ B1 (H) for some t < inf(σ(A) ∪ σ(B)).

(2.23) (2.24)

Given Hypothesis 2.5 it has been proven in [26] (actually, in a more general context) that for z ∈ C\[inf(σ(A) ∪ σ(B)), ∞),   (B − z IH )1/2 (B − z IH )−1 − (A − z IH )−1 (B − z IH )1/2 = IH − (B − z IH )1/2 (A − z IH )−1 (B − z IH )1/2 (2.25)    1/2 −1/2 1/2 −1/2 ∗ = IH − (B − z IH ) (A − z IH ) (B − z IH ) (A − z IH )   implying that det H (B − z IH )1/2 (A − z IH )−1 (B − z IH )1/2 is well-defined. Moreover,   tr H (B − z IH )−1 − (A − z IH )−1

d = − ln det H (B − z IH )1/2 (A − z IH )−1 (B − z IH )1/2 , dz z ∈ C\[inf(σ(A) ∪ σ(B)), ∞).

(2.26)

Next, we briefly turn to spectral ζ-functions of self-adjoint operators S with a trace class resolvent (and hence purely discrete spectrum). Hypothesis 2.6 Suppose S is a self-adjoint operator in H, bounded from below, satisfying (2.27) (S − z IH )−1 ∈ B1 (H) for some (and hence for all ) z ∈ ρ(S). We denote the spectrum of S by σ(S) = {λ j } j∈J (with J ⊂ Z an appropriate index set ), with every eigenvalue repeated according to its multiplicity. Given Hypothesis 2.6, the spectral zeta function of S is then defined by ζ(s; S) :=

λ−s j

j∈J λ j =0

for Re(s) > 0 sufficiently large such that (2.28) converges absolutely.

(2.28)

Computing Traces, Determinants, and ζ-Functions … Fig. 1 Contour γ in the complex z-plane

101

The cut Rθ for z −s AA A

6 A

A A

z-plane A A A

γ A   A  s s s A s s s s s s s s s s 

Next, let P(0; S) be the spectral projection of S corresponding to the eigenvalue 0 and denote by m(λ0 ; S) the multiplicity of the eigenvalue λ0 of S, in particular, m(0; S) = dim(ker(S)).

(2.29)

(One recalls that since S is self-adjoint, the geometric and algebraic multiplicity of each eigenvalue of S coincide and hence the subscript “g” or “a” is simply omitted from m( · ; S).) In addition, we introduce the simple contour γ encircling σ(S)\{0} in a counterclockwise manner so as to dip under (and hence avoid) the point 0 (cf. Figure 1). In fact, following [40] (see also [39]), we will henceforth choose as the branch cut of z −s the ray    Rθ = z = teiθ t ∈ [0, ∞) θ ∈ (π/2, π),

(2.30)

and note that the contour γ avoids any contact with Rθ (cf. Figure 1). We note in passing that one could also use a semigroup approach via ζ(s; S) = (s)

−1

−1

= (s)

ˆ

∞

ˆ0 ∞

  dt t s−1 tr H e−t S [IH − P(0; S)]     dt t s−1 tr H e−t S − m(0; S) ,

(2.31)

0

for Re(s) > 0 sufficiently large, but we prefer to work with resolvents in this paper. Lemma 2.7 In addition to Hypothesis 2.6 and the counterclockwise oriented con  tour γ just described (cf. Figure 1), suppose that tr H (S − z IH )−1 [IH − P(0; S)]  is polynomially bounded on γ. Then

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ζ(s; S) = −(2πi)−1

‰

    dz z −s tr H (S − z IH )−1 + z −1 m(0; S)

γ

(2.32)

for Re(s) > 0 sufficiently large. Proof Assuming Re(s) > 0 sufficiently large, a contour integration argument yields ζ(s; S) = −(2πi)−1 = −(2πi)−1 = −(2πi)−1

‰ γ

‰

γ

‰

γ

  dz z −s tr H (S − z IH )−1 [IH − P(0; S)]   dz z −s tr H (S − z IH )−1 + z −1 P(0; S)

(2.33)

    dz z −s tr H (S − z IH )−1 + z −1 m(0; S) ,

  taking into account that tr H (S − z IH )−1 [IH − P(0; S)] is meromorphic with poles precisely at the nonzero eigenvalues λ j = 0 of S, with residues given by (self-adjoint)  spectral projections of S of rank equal to m(λ j ; S). It is very tempting to continue the computation leading to (2.32) and now deform the contour γ soas to “hug” the branch cut Rθ, but this requires the right asymptotic behavior of tr H (S − z IH )−1 [IH − P(0; S)] as |z| → ∞ as well as |z| → 0, and we will investigate this in the context of relative ζ-functions next. In cases where (S − z IH )−1 is not trace class, but one is dealing with a pair of operators (S1 , S2 ) such that the difference of their resolvents lies in the trace class, (2.33) naturally leads to the notion of a relative ζ-function as follows. For pertinent background information on this circle of ideas we refer, for instance, to Forman [16, 17], Müller [49]. Hypothesis 2.8 Suppose S j , j = 1, 2, are self-adjoint operators in H, bounded from below, satisfying   (S2 − z IH )−1 − (S1 − z IH )−1 ∈ B1 (H)

(2.34)

for some (and hence for all ) z ∈ ρ(S1 ) ∩ ρ(S2 ). In addition, assume that S j , j = 1, 2, have essential spectrum contained in (0, ∞), that is, for some λ1 > 0, σess (S j ) ⊆ [λ1 , ∞),

j = 1, 2.

(2.35)

We note, in particular, the essential spectrum hypothesis (2.35) includes the case of purely discrete spectra of S j (i.e., σess (S j ) = ∅, j = 1, 2.) Since S j were assumed to be bounded from below, adding a suitable constant to S j , j = 1, 2, will shift their (essential) spectra accordingly. Given Hypothesis 2.8, and again choosing a counterclockwise-oriented simple contour γ that encircles σ(S1 ) ∪ σ(S2 ), however, with the stipulation that 0 does not lie in the interior of γ, and 0 ∈ / γ (cf. Figure 1), the relative spectral ζ-function for the pair (S1 , S2 ) is defined by

Computing Traces, Determinants, and ζ-Functions …

ζ(s; S1 , S2 ) = −(2πi)−1

‰ γ

103

 dz z −s tr H (S2 − z IH )−1 [IH − P(0; S2 )]  − (S1 − z IH )−1 [IH − P(0; S1 )] (2.36)

for Re(s) > 0 sufficiently large, ensuring convergence of (2.36). Employing the contour γ and branch cut Rθ as in Figure 1, and deforming γ so it eventually surrounds Rθ , one arrives at the following result. Theorem 2.9 Suppose S j , j = 1, 2, are self-adjoint operators in H satisfying Hypothesis 2.8 and that (cf. (2.17) and (2.26))   tr H (S2 − z IH )−1 − (S1 − z IH )−1

d (2.37) = − ln det H (S2 − z IH )1/2 (S1 − z IH )−1 (S2 − z IH )1/2 , dz z ∈ C\[inf(σ(S1 ) ∩ σ(S2 )), ∞). In addition, assume that for some ε > 0,    tr H (S2 − z IH )−1 [IH − P(0; S2 )] − (S1 − z IH )−1 [IH − P(0; S1 )]     O |z|−1−ε , as |z| → ∞, = O(1), as |z| → 0. Then, for Re(s) ∈ (−ε, 1), ζ(s; S1 , S2 ) = e

is(π−θ) −1

π

ˆ sin(πs)

∞

(2.38)

dt t −s

0

d iθ [m(0;S1 )−m(0;S2 )] ln (e t) dt  

× detH (S2 − eiθ t IH )1/2 (S1 − eiθ t IH )−1 (S2 − eiθ t IH )1/2 .

×

(2.39)

Proof Due to hypothesis (2.38) one can deform the contour γ so that it wraps around the branch cut Rθ , ζ(s; S1 , S2 ) = −(2πi)−1

= −(2πi)−1

‰ γ

‰ γ

 dz z −s tr H (S2 − z IH )−1 [IH − P(0; S2 )]

− (S1 − z IH )−1 [IH − P(0; S1 )]    dz z −s tr H (S2 − z IH )−1 − (S1 − z IH )−1



 + z −1 [m(0; S2 ) − m(0; S1 )]  ‰

d ln det H (S2 − z IH )1/2 (S1 − z IH )−1 (S2 − z IH )1/2 = (2πi)−1 dz z −s dz γ  − z −1 [m(0; S2 ) − m(0; S1 )]

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‰

d [m(0;S1 )−m(0;S2 )] ln z dz γ  1/2  −1  1/2

× det H S2 − z IH S1 − z IH S2 − z IH ˆ ∞

[m(0;S1 )−m(0;S2 )] d dt t −s ln teiθ = eis(π−θ) π −1 sin(πs) dt 0

1/2

−1

1/2

S1 − eiθ t IH S2 − eiθ t IH . × det H S2 − eiθ t IH (2.40) = (2πi)−1

dz z −s

Here we first applied tr H (P(0; S j )) = m(0; S j ) (cf. also (2.36)) and then (2.18). Carefully paying attention to the phases when shrinking the contour to the branch  cut Rθ , one obtains (2.39). Theorem 2.10 Suppose S j , j = 1, 2, are self-adjoint operators in H satisfying Hypothesis 2.8 and (2.38). Then   ζ (0; S1 , S2 ) = iπ(n 2 − n 1 ) − lim ln (eiθ t)[m(0;S1 )−m(0;S2 )] t↓0

   ×det H (S2 − eiθ t IH )1/2 (S1 − eiθ t IH )−1 (S2 − eiθ t IH )1/2  ,

(2.41)

where n j is the number of negative eigenvalues of S j , j = 1, 2. If n j = 0, j = 1, 2, then ζ (0; S1 , S2 ) = − lim ln (eiθ t)[m(0;S1 )−m(0;S2 )] t↓0  

× detH (S2 − eiθ t IH )1/2 (S1 − eiθ t IH )−1 (S2 − eiθ t IH )1/2 = −ln(C0 ),

(2.42)

where   detH (S2 − z IH )1/2 (S1 − z IH )−1 (S2 − z IH )1/2 = z [m(0;S2 )−m(0;S1 )] [C0 + O(z)], C0 > 0.

(2.43)

z→0

Proof First we note that (2.39) implies ˆ ∞ d   iθ [m(0;S1 )−m(0;S2 )] ln te ζ (0; S1 , S2 ) = dt dt 0  1/2  −1  1/2  . × det H S2 − eiθ t IH S1 − eiθ t IH S2 − eiθ t IH

(2.44)

In computing this quantity, one notes that for t ∈ [0, ∞), the graph of the function  [m(0;S1 )−m(0;S2 )] G(t) = teiθ  1/2  −1  1/2

S1 − eiθ t IH S2 − eiθ t IH × det H S2 − eiθ t IH

(2.45)

Computing Traces, Determinants, and ζ-Functions …

105

can cross the branch cut Rθ at several t-values. Therefore, the integral has to be split at these t-values and pursuant real and imaginary parts have to be summed. The real part between consecutive segments cancels except for the contributions from zero. This explains the real part of (2.41). The resulting imaginary part is found as follows. The sum defining the ζ-function can be split into contributions from negative and positive eigenvalues, namely, ζ(s; S j ) =

nj

λ−s − +

=1 λ− 0

For each negative eigenvalue, one computes  d  λ−s = −ln(λ− ) = iπ − ln(|λ− |). ds s=0 −

(2.47)

This yields the imaginary part in (2.41). Since 0 lies outside the essential spectra of S j , j = 1, 2, (2.43) follows, for instance, from [65, pp. 271–272]. Given the relation (2.43), the fact (2.42) follows from (2.44) and the Lebesgue dominated convergence theorem.  In the special case, where 0 ∈ ρ(S1 ) ∩ ρ(S2 ) (i.e., m(0; S1 ) = m(0; S2 ) = 0), one thus obtains  1/2

1/2  (2.48) e−ζ (0;S1 ,S2 ) = det H S2 S1−1 S2 and hence the ζ-function regularized relative determinant now equals the symmetrized (Fredholm) perturbation determinant for the pair (S1 , S2 ). Here any choice 1/2 of branch cut of the self-adjoint operator S2 (employing the spectral theorem) is permissible. For additional background and applications of (modified) Fredholm determinants to ordinary differential operators, we also refer to [5, 9, 13, 19, 21, 26, 38–46, 53], and the extensive literature cited therein.

3 Sturm–Liouville Operators on Bounded Intervals To illustrate the material of Section 2, we now apply it to the case of self-adjoint Sturm–Liouville operators on bounded intervals. We start by recalling a convenient parametrization of all self-adjoint extensions associated with a regular, symmetric, second-order differential expression as discussed in detail, for instance, in [64, Theorem 13.15] and [66, Theorem 10.4.3], and recorded in [9].

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Throughout this section, we make the following set of assumptions: Hypothesis 3.1 Suppose p, q, r satisfy the following conditions: (i) r > 0 a.e. on (a, b), r ∈ L 1 ((a, b); d x). (ii) p > 0 a.e. on (a, b), 1/ p ∈ L 1 ((a, b); d x). (iii) q ∈ L 1 ((a, b); d x), q is real-valued a.e. on (a, b). Given Hypothesis 3.1, we take τ to be the Sturm–Liouville-type differential expression defined by   1 d d − p(x) + q(x) for a.e. x ∈ (a, b), −∞ < a < b < ∞. r (x) dx dx (3.1) By definition, τ is called regular on [a, b] if p, q, r satisfy Hypothesis 3.1. In addition, the following convenient notation for the first quasi-derivative is introduced, τ=

y [1] (x) = p(x)y (x) for a.e. x ∈ (a, b), y ∈ AC([a, b]).

(3.2)

Here AC([a, b]) denotes the set of absolutely continuous functions on [a, b] and

= d/d x. For notational convenience, we will occasionally abbreviate L r2 ((a, b)) := L 2 ((a, b); r d x). Given that τ is regular on [a, b], the maximal operator Hmax in L 2 ((a, b); r d x) associated with τ is defined by Hmax f = τ f,

   f ∈ dom(Hmax ) = g ∈ L 2 ((a, b); r d x)  g, g [1] ∈ AC([a, b]); τ g ∈ L 2 ((a, b); r d x) ,

(3.3)

while the minimal operator Hmin in L 2 ((a, b); r d x) associated with τ is given by Hmin f = τ f,

  f ∈ dom(Hmin ) = g ∈ L 2 ((a, b); r d x)  g, g [1] ∈ AC([a, b]);

(3.4)

 g(a) = g [1] (a) = g(b) = g [1] (b) = 0; τ g ∈ L 2 ((a, b); r d x) .

One notes that the operator Hmin is symmetric and that ∗ = Hmax , Hmin

∗ Hmax = Hmin ,

(3.5)

(cf. Weidmann [63, Theorem 13.8]). Next, we summarize material found, for instance, in [64, Chap. 13] and [66, Sects. 10.3, 10.4] in which self-adjoint extensions of the minimal operator Hmin are characterized.

Computing Traces, Determinants, and ζ-Functions …

107

Theorem 3.2 (See, [62, Satz 2.6], [64, Theorem 13.14], [66, Theorem 10.4.2].)  is an extension of the minimal operator Assume Hypothesis 3.1 and suppose that H Hmin defined in (3.4). Then the following hold:  is a self-adjoint extension of Hmin if and only if there exist 2 × 2 matrices A (i) H and B with complex-valued entries satisfying1   0 −1 rank(A B) = 2, A J A = B J B , J = , 1 0 ∗

∗

(3.6)

with f = τ f H          g(a) g(b)   = B [1] . dom H = g ∈ dom(Hmax )  A [1] g (a) g (b)

(3.7)

 corresponding to the matrices A and B will Henceforth, the self-adjoint extension H be denoted by H A,B . (ii) For z ∈ ρ(H A,B ), the resolvent H A,B is of the form   (H A,B − z I L r2 ((a,b)) )−1 f (x) =

ˆ

b

r (x )d x G A,B (z, x, x ) f (x ),

a

(3.8)

f ∈ L ((a, b); r d x), 2

where the Green’s function G A,B (z, x, x ) is of the form given by 

m +j,k (x)u j (z, x)u k (z, x ), a ≤ x ≤ x ≤ b, −



j,k=1 m j,k (x)u j (z, x)u k (z, x ), a ≤ x < x ≤ b.

2

G A,B (z, x, x ) = 2j,k=1

(3.9)

Here {u 1 (z, · ), u 2 (z, · )} represents a fundamental set of solutions for (τ − z)u = 0 and m ±j,k (z), 1 ≤ j, k ≤ 2, are appropriate constants. (iii) H A,B has purely discrete spectrum with eigenvalues of multiplicity at most 2. Moreover, if σ(H A,B ) = {λ A,B, j } j∈N , then

|λ A,B, j |−1 < ∞.

(3.10)

j∈N λ A,B, j =0

In particular,   (H A,B − z I L r2 ((a,b)) )−1 ∈ B1 L 2 ((a, b); r d x) , z ∈ ρ(H A,B ).

1 The

expression (A B) represents a 2 × 4 rectangular matrix.

(3.11)

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F. Gesztesy and K. Kirsten

The characterization of self-adjoint extensions of Hmin in terms of pairs of matrices (A, B) ∈ C2×2 × C2×2 satisfying (3.6) is not unique in the sense that different pairs may lead to the same self-adjoint extension. The next result recalls a unique characterization for all self-adjoint extensions of Hmin and hence can be viewed as a refinement of Theorem 3.2. Theorem 3.3 (See, e.g., [64, Theorem 13.15], [66, Theorem 10.4.3]) Assume Hypothesis 3.1. Let Hmin be the minimal operator associated with τ and defined in (3.4) and H A,B a self-adjoint extension of the minimal operator as characterized in Theorem 3.2; then, the following hold: (i) H A,B is a self-adjoint extension of Hmin , with rank(A) = rank(B) = 1 if and only if A and B can be put in the form A=

    cos(α) sin(α) 0 0 , B= , 0 0 − cos(β) sin(β)

(3.12)

for a unique pair α, β ∈ [0, π). Hence, upon identifying H A,B with Hα,β , all selfadjoint extensions of Hmin with separated boundary conditions are of the form Hα,β f = τ f, α, β ∈ [0, π), f ∈ dom(Hα,β ) = {g ∈ dom(Hmax ) | sin(α)g [1] (a) + cos(α)g(a) = 0,

(3.13)

− sin(β)g [1] (b) + cos(β)g(b) = 0}. (ii) H A,B is a self-adjoint extension of Hmin with rank(A) = rank(B) = 2 if and only if A and B can be put in the form A = eiϕ R, B = I2 ,

(3.14)

for a unique ϕ ∈ [0, 2π), and unique R ∈ SL2 (R). Hence, upon identifying H A,B with Hϕ,R , all self-adjoint extensions of Hmin with coupled boundary conditions are of the form Hϕ,R f = τ f, ϕ ∈ [0, 2π), R ∈ SL2 (R),       g(b) g(a) iϕ  f ∈ dom(Hϕ,R ) = g ∈ dom(Hmax )  [1] = e R [1] . g (b) g (a)

(3.15)

(iii) All self-adjoint extensions of Hmin are either of type (i) (i.e., separated ) or of type (ii) (i.e., coupled ). Here SL2 (R) denotes the group of 2 × 2 matrices with real-valued entries and determinant one. For notational convenience, we will adhere to the notation Hα,β and Hϕ,R in the following. Next, we recall some results of [25]. For this purpose, we introduce the fundamental system of solutions θ(z, x, a), φ(z, x, a) of τ y = zy defined by

Computing Traces, Determinants, and ζ-Functions …

109

θ(z, a, a) = φ[1] (z, a, a) = 1, θ[1] (z, a, a) = φ(z, a, a) = 0,

(3.16)

such that W (θ(z, · , a), φ(z, · , a)) = 1,

(3.17)

where, for f, g (locally) absolutely continuous, W ( f, g)( · ) = f ( · )g [1] ( · ) − f [1] ( · )g( · ).

(3.18)

Furthermore, we introduce the boundary values for g, g [1] ∈ AC([a, b]), see [51, Sect. 1.2], [66, Sect. 3.2], Uα,β,1 (g) = sin(α)g [1] (a) + cos(α)g(a), Uα,β,2 (g) = − sin(β)g [1] (b) + cos(β)g(b),

(3.19)

in the case (i) of separated boundary conditions in Theorem 3.3, and Vϕ,R,1 (g) = g(b) − eiϕ R1,1 g(a) − eiϕ R1,2 g [1] (a), Vϕ,R,2 (g) = g [1] (b) − eiϕ R2,1 g(a) − eiϕ R2,2 g [1] (a),

(3.20)

in the case (ii) of coupled boundary conditions in Theorem 3.3. Moreover, we define Fα,β (z) = det

  Uα,β,1 (θ(z, · , a)) Uα,β,1 (φ(z, · , a)) , Uα,β,2 (θ(z, · , a)) Uα,β,2 (φ(z, · , a))

(3.21)

 Vϕ,R,1 (θ(z, · , a)) Vϕ,R,1 (φ(z, · , a)) , Vϕ,R,2 (θ(z, · , a)) Vϕ,R,2 (φ(z, · , a))

(3.22)

and  Fϕ,R (z) = det and note, in particular, that Fα,β (z) ⎧ ⎪ φ(z, b, a), ⎪ ⎪ ⎪ [1] ⎪ ⎪ ⎨− sin(β)φ (z, b, a) + cos(β)φ(z, b, a), = cos(α)φ(z, b, a) − sin(α)θ(z, b, a), ⎪ ⎪ ⎪ cos(α)[− sin(β)φ[1] (z, b, a) + cos(β)φ(z, b, a)] ⎪ ⎪ ⎪ ⎩− sin(α)[− sin(β)θ[1] (z, b, a) + cos(β)θ(z, b, a)],

(3.23) α = β = 0, α = 0, β ∈ (0, π), α ∈ (0, π), β = 0, α, β ∈ (0, π).

Given these preparations, we can state our first result concerning the computation of traces and determinants. Theorem 3.4 Assume Hypothesis 3.1 and denote by Hα,β and Hϕ,R the self-adjoint extensions of Hmin as described in cases (i) and (ii) of Theorem 3.3, respectively.

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(i) Suppose z 0 ∈ ρ(Hα,β ), then   det L r2 ((a,b)) I L r2 ((a,b)) − (z − z 0 )(Hα,β − z 0 I L r2 ((a,b)) )−1 = Fα,β (z)/Fα,β (z 0 ), z ∈ C.

(3.24)

In particular,   tr L r2 ((a,b)) (Hα,β − z I L r2 ((a,b)) )−1 = −(d/dz)ln(Fα,β (z)), z ∈ ρ(Hα,β ).

(3.25)

(ii) Suppose z 0 ∈ ρ(Hϕ,R ), then   det L r2 ((a,b)) I L r2 ((a,b)) − (z − z 0 )(Hϕ,R − z 0 I L r2 ((a,b)) )−1 = Fϕ,R (z)/Fϕ,R (z 0 ), z ∈ C.

(3.26)

In particular,   tr L r2 ((a,b)) (Hϕ,R − z I L r2 ((a,b)) )−1 = −(d/dz)ln(Fϕ,R (z)), z ∈ ρ(Hϕ,R ). (3.27) Proof In the special case p = r = 1 and given separated (but generally, nonselfadjoint) boundary conditions, the fact (3.24) was proved in [25] upon combining the eigenvalue results in [51, Sect. 1.2] and Lemma 2.2. The proof in [25] extends to the present situation with p, r satisfying Hypothesis 3.1 in the cases (i) and (ii) since actually the eigenvalues λ A,B, j of H A,B all have the universal leading Weyl-type asymptotics [33, Sect. VI.7] (see also [3, 66, Theorem 4.3.1]) lim j −2 λ A,B, j = π 2

j→∞

ˆ

b

d x [r (x)/ p(x)]1/2

−2

,

(3.28)

a

independently of the chosen boundary conditions, improving upon relation (3.10). More precisely, [33, Sect. VI.7] determines the leading asymptotic behavior of the eigenvalue counting function (independently of the underlying choice of boundary conditions) which is known to yield (3.28). Relations (3.25) and (3.27) are then clear from combining formula (2.3) with (3.24) and (3.26), respectively.  Remark 3.5 Considering traces of resolvent differences for various boundary conditions (separated and/or coupled) permits one to make a direct connection with the boundary data maps studied in [8, 9, 26]. More precisely, suppose the pairs A, B ∈ C2×2 and A , B ∈ C2×2 satisfy (3.6) and define H A,B and H A ,B according to (3.7). Then   tr L 2 ((a,b);r d x) (H A ,B − z I L r2 ((a,b)) )−1 − (H A,B − z I L r2 ((a,b)) )−1

d ,B (z) , z ∈ ρ(H A,B ) ∩ ρ(H A ,B ), = − ln det C2  AA,B dz

(3.29)

Computing Traces, Determinants, and ζ-Functions …

111



,B where the 2 × 2 matrix  AA,B (z) represents the boundary data map associated to the pair (H A,B , H A ,B ). A comparison of (3.29) with (3.25) and (3.27) yields



,B (z) = C0 FA ,B (z)/FA,B (z), z ∈ ρ(H A,B ) ∩ ρ(H A ,B ), det C2  AA,B

(3.30)

where, in obvious notation, FA ,B (z), FA,B (z) represent either Fα,β (z) or Fϕ,R (z), depending on which boundary conditions (separated or coupled) are represented by the pairs (A , B ), (A, B), and C0 = C0 (A , B , A, B) is a z-independent constant. Indeed, C0 = 1 for separated as well as coupled boundary conditions. The separated case was shown in [26], Lemma 3.4, using the explicit matrix representation of



 AABB (z) for that case, and the result for coupled boundary conditions follows along the same lines.  For the case of determinants of general higher order differential operators (with matrix-valued coefficients) and general boundary conditions on bounded intervals we refer to Burghelea, Friedlander, and Kappeler [5], Dreyfus and Dym [13], Falco, Fedorenko, and Gruzberg [15], and Forman [16, Sect. 3]. We also refer to [46] for a closed form of an infinite product of ratios of eigenvalues of Sturm–Liouville operators on bounded intervals. We briefly look at two prominent examples associated with coupled boundary conditions next: Example 3.6 (Floquet boundary conditions) Consider the family of operators Hϕ,I2 , familiar from Floquet theory, defined by taking R = B = I2 in (3.15). In this case

where

Fϕ,I2 (z) = −2eiϕ [(z) − cos(ϕ)], z ∈ C,

(3.31)

  (z) = θ(z, b, a) + φ[1] (z, b, a) 2, z ∈ C,

(3.32)

represents the well-known Floquet discriminant and hence   det L r2 ((a,b)) I L r2 ((a,b)) − (z − z 0 )(Hϕ,I2 − z 0 I L r2 ((a,b)) )−1 = [(z) − cos(ϕ)]/[(z 0 ) − cos(ϕ)], z ∈ C,   tr L r2 ((a,b)) (Hϕ,I2 − z I L r2 ((a,b)) )−1 = −(d/dz)ln([(z) − cos(ϕ)]), z ∈ ρ(Hϕ,I2 ).

(3.33) (3.34)

Example 3.7 (The Krein–von Neumann extension) Consider the case ϕ = 0, A = R K , with   θ(0, b, a) φ(0, b, a) . (3.35) R K = [1] θ (0, b, a) φ[1] (0, b, a) As shown in [9, Example 3.3], the resulting operator H0,R K represents the Krein–von Neumann extension of Hmin . Thus,

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F. Gesztesy and K. Kirsten

F0,R K (z) = −2[D K (z) − 1], z ∈ C,

(3.36)

where  D K (z) = φ[1] (0, b, a)θ(z, b, a) + θ(0, b, a)φ[1] (z, b, a) − φ(0, b, a)θ[1] (z, b, a)  − θ[1] (0, b, a)φ(z, b, a) 2, z ∈ C. (3.37) Hence,   det L r2 ((a,b)) I L r2 ((a,b)) − (z − z 0 )(H0,R K − z 0 I L r2 ((a,b)) )−1 = [D K (z) − 1]/[D K (z 0 ) − 1], z ∈ C,   tr L r2 ((a,b)) (H0,R K − z I L r2 ((a,b)) )−1 = −(d/dz)ln(D K (z) − 1), z ∈ ρ(H0,R K ).

(3.38) (3.39)

Because of the Wronskian relation (3.17), D K (0) = 1, furthermore D K (z) − 1 = z 2 [c + O(z)], z→0

(3.40)

where

.  . [1]  . [1] 1 . φ(0, b, a) θ (0, b, a) − φ (0, b, a) θ(0, b, a) 2

. 1 . = W φ(0, · , a), θ(0, · , a) (b), 2

c=

(3.41)

abbreviating . = d/dz. The small-z behavior (3.40) is clear from general results in [51, Sect. 1.2] or [66, Sect. 3.2] and is in accordance with a two-dimensional nullspace of the Krein–von Neumann extension on a bounded interval. To actually show that c = 0 (in fact,

.

.

c < 0) in (3.40) one recalls that φ(z, x, a) and θ(z, x, a) satisfy an inhomogeneous Sturm–Liouville equation and as a result one obtains ( for z ∈ C, x ∈ [a, b])

.

ˆ

x

θ(z, x, a) = θ(z, x, a)

r (x )d x φ(z, x , a)θ(z, x , a)

ˆ

a

x

− φ(z, x, a)  . [1] θ (z, x, a) = θ[1] (z, x, a)

r (x )d x θ(z, x , a)2 ,

ˆa x

(3.42) r (x )d x φ(z, x , a)θ(z, x , a)

ˆ

a

− φ[1] (z, x, a)

x a

r (x )d x θ(z, x , a)2 ,

Computing Traces, Determinants, and ζ-Functions …

ˆ

.

x

φ(z, x, a) = θ(z, x, a)

113

r (x )d x φ(z, x , a)2

ˆ

x

ˆ

x

a

− φ(z, x, a)  . [1] φ (z, x, a) = θ[1] (z, x, a)

r (x )d x φ(z, x , a)θ(z, x , a),

a

r (x )d x φ(z, x , a)2

ˆ

a [1]

x

− φ (z, x, a)

r (x )d x φ(z, x , a)θ(z, x , a).

a

Equations (3.42) and Cauchy’s inequality then imply ˆ x 2 . .  W φ(λ, · , a), θ(λ, · , a) (x) = r (x )d x φ(λ, x , a)θ(λ, x , a) (3.43) a  ˆ x  ˆ x − r (x )d x θ(λ, x , a)2 r (x )d x φ(λ, x , a)2 ≤ 0, λ ∈ R, x ∈ [a, b], a

a

using the fact that φ(λ, x, a) and θ(λ, x, a) are real-valued for λ ∈ R, x ∈ [a, b]. Equality in Cauchy’s inequality for x > a would imply that for some α, β ∈ [0, ∞), (α, β) = (0, 0), (3.44) α φ(λ, x , a) = β θ(λ, x , a), x ∈ (a, x],

. .  a contradiction. Thus, W φ(λ, · , a), θ(λ, · , a) (x) < 0 for λ ∈ R, x ∈ (a, b] and hence c < 0 in (3.40), (3.41). In the case of separated boundary conditions, that is, case (i) in Theorem 3.3, one can shed more light on Fα,β (z) in terms of appropriate Weyl solutions ψ− (z, · , a, α) and ψ+ (z, · , a, β) that satisfy the boundary conditions in dom(Hα,β ) in (3.13) at a and b, respectively. Up to normalizations, ψ± (z, · , a, α) are given by  ψ− (z, · , a, α) = c−

cos(α)φ(z, · , a) − sin(α)θ(z, · , a), α ∈ (0, π), φ(z, · , a), α = 0,

(3.45) ⎧  [1] − sin(β)θ (z, b, a) + cos(β)θ(z, b, a) φ(z, · , a) ⎪ ⎪ ⎪ ⎨+ sin(β)φ[1] (z, b, a) − cos(β)φ(z, b, a)θ(z, · , a), ψ+ (z, · , a, β) = c+ ⎪ β ∈ (0, π), ⎪ ⎪ ⎩ θ(z, b, a)φ(z, · , a) − φ(z, b, a)θ(z, · , a), β = 0, (3.46) and hence the Green’s function of Hα,β is of the semi-separable form, G α,β (z, x, x ) = (Hα,β − z I L r2 ((a,b)) )−1 (x, x )

(3.47)

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F. Gesztesy and K. Kirsten

1 W (ψ+ (z, · , a, β), ψ− (z, · , a, α))  ψ− (z, x, a, α)ψ+ (z, x , a, β), a ≤ x ≤ x ≤ b, × ψ− (z, x , a, α)ψ+ (z, x, a, β), a ≤ x ≤ x ≤ b.

=

A direct computation then reveals the following connection between Fα,β (z), ψ− (z, · , a, α), and ψ+ (z, · , a, β), 1 (3.48) W (ψ+ (z, · , a, β), ψ− (z, · , a, α)) ⎧ ψ− (z, b, a, 0)ψ+ (z, a, a, 0), α = β = 0, ⎪ ⎪   ⎪ [1] ⎪ ⎪ (z, b, a, 0) + cos(β)ψ− (z, b, a, 0) ψ+ (z, a, a, β), − sin(β)ψ− ⎪ ⎪ ⎪ ⎪ ⎪ α = 0, β ∈ (0, π), ⎨   [1] × ψ− (z, b, a, α) sin(α)ψ+ (z, a, a, 0) + cos(α)ψ+ (z, a, a, 0) , ⎪ ⎪ ⎪ α ∈ (0, π), β = 0, ⎪ ⎪   ⎪ [1] ⎪ ⎪ − sin(β)ψ− (z, b, a, α) + cos(β)ψ− (z, b, a, α) ⎪ ⎪  ⎩  [1] (z, a, a, β) + cos(α)ψ+ (z, a, a, β) , α, β ∈ (0, π). × sin(α)ψ+

Fα,β (z) =

Combining (3.48) and (3.25) thus yields   tr L r2 ((a,b)) (Hα,β − z I L r2 ((a,b)) )−1 ⎧   [1]  −(d/dz)ln ψ+ (z, a, a, 0) ψ+ (z, b, a, 0) ⎪ ⎪  [1]   ⎪ ⎪ ⎪ ⎪ ⎨ = −(d/dz)ln ψ− (z, b, a, 0) ψ− (z, a, a, 0) , = −(d/dz)ln(ψ+ (z, a, a, β)/ψ+ (z, b, a, β)), ⎪ ⎪ ⎪−(d/dz)ln(ψ− (z, b, a, α)/ψ− (z, a, a, α)), ⎪

⎪ ⎪ ⎩−(d/dz)ln W (ψ+ (z, · ,a,β),ψ− (z, · ,a,α)) , ψ+ (z,b,a,β)ψ− (z,a,a,α)

α = β = 0, α = 0, β ∈ (0, π), α ∈ (0, π), β = 0, α, β ∈ (0, π), z ∈ ρ(Hα,β ). (3.49)

Next, applying Theorem 3.4 in the context of Theorem 2.10 immediately yields results about ζ-regularized determinants (see also [18, 38, Chaps. 2, 3], [44]). Remark 3.8 In Example 3.19, we will consider a simple case with negative eigenvalues present. Otherwise, in the examples of this section, we will always assume that eigenvalues are nonnegative. If that is not the case, an appropriate imaginary part according to Theorem 2.10 has to be included.  To deal with ζ-regularized determinants we now strengthen Hypothesis 3.1 and introduce the following assumptions on p, q, r : Hypothesis 3.9 Suppose p, q, r satisfy the following conditions: (i) r > 0 a.e. on (a, b), r ∈ L 1 ((a, b); d x), 1/r ∈ L ∞ ((a, b); d x).

Computing Traces, Determinants, and ζ-Functions …

115

(ii) p > 0 a.e. on (a, b), 1/ p ∈ L 1 ((a, b); d x). (iii) q ∈ L 1 ((a, b); d x), q is real-valued a.e. on (a, b). (iv) p r and ( p r ) /r are absolutely continuous on [a, b]. The substitutions (cf. [47, p. 2]) 1 v(x) = c

ˆ

x

dt [r (t)/ p(t)]1/2 , u(x) = [r (x) p(x)]1/4 y(x),

(3.50)

a

where c is given by

ˆ c=

b

dt [r (t)/ p(t)]1/2 ,

(3.51)

a

such that v ∈ [0, 1], transforms the Sturm–Liouville problem (τ y)(x) = zy(x) into − u

(v) + V (v)u(v) = c2 zu(v),

(3.52)

with (μ(v) = [r (x) p(x)]1/4 ) q(x) μ

(v) + c2 μ(v) r (x)   1 (r (x) p(x))

c2 q(x) c2 1 d (r (x) p(x))

+ c2 =− + . 16 r (x) p(x) r (x) 4 r (x) d x r (x) r (x) (3.53)

V (v) =

Hypothesis 3.9 guarantees2 that V ∈ L 1 ((0, 1); d x), and as a consequence one has asymptotically for x ∈ (a, b], φ(z, x, a) = 2−1 z −1/2 [ p(x) p(a)r (x)r (a)]−1/4 |z|→∞   ˆ x    1/2 1/2 1 + O |z|−1/2 , × exp z dt [r (t)/ p(t)] a

φ[1] (z, x, a) = 2−1 [ p(a)r (a)]−1/4 [ p(x)r (x)]1/4 |z|→∞   ˆ x    × exp z 1/2 dt [r (t)/ p(t)]1/2 1 + O |z|−1/2 , a −1

θ(z, x, a) = 2 [r (a)/ p(a)]1/2 [ p(x) p(a)r (x)r (a)]−1/4 |z|→∞   ˆ x    × exp z 1/2 dt [r (t)/ p(t)]1/2 1 + O |z|−1/2 , a

2 We

are indebted to Bennewitz [4] for a very helpful discussion of this issue.

(3.54)

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F. Gesztesy and K. Kirsten

θ[1] (z, x, a) = 2−1 z 1/2 p(a)−1 [ p(x) p(a)r (x)r (a)]1/4 |z|→∞   ˆ x    1/2 1/2 1 + O |z|−1/2 . × exp z dt [r (t)/ p(t)] a

This asymptotic behavior is used to guarantee that assumption (2.38) is satisfied in several of the following examples. Theorem 3.10 Assume Hypothesis 3.9 and denote by Hα,β, j and Hϕ,R, j , j = 1, 2, the self-adjoint extensions of Hmin as described in Theorem 3.3 (i) and (ii), respectively. Here the index j refers to a potential q j in (3.1), j = 1, 2. Then the following items (i) and (ii) hold: (i)     Fα,β,2 (teiθ )   iθ [m(0,Hα,β,1 )−m(0,Hα,β,2 )]   (te ) ζ 0; Hα,β,1 , Hα,β,2 = − lim ln   . iθ t↓0 Fα,β,1 (te ) (3.55) (ii)

   Fϕ,R,2 (teiθ )    iθ [m(0,Hϕ,R,1 )−m(0,Hϕ,R,2 )]   ζ 0; Hϕ,R,1 , Hϕ,R,2 = − lim ln  (te )  . iθ t↓0 Fϕ,R,1 (te ) (3.56)

Proof This follows immediately from Theorems 2.9 and 2.10 applied to S j = Hα,β, j ,  respectively, S j = Hϕ,R, j , j = 1, 2. Remark 3.11 In the absence of zero eigenvalues of Hα,β, j , respectively, Hϕ,R, j , j = 1, 2, these results simplify to

respectively,

   Fα,β,1 (0)     , ζ 0; Hα,β,1 , Hα,β,2 = ln  Fα,β,2 (0) 

(3.57)

   Fϕ,R,1 (0)     .  ζ 0; Hϕ,R,1 , Hϕ,R,2 = ln  F (0) 

(3.58)



ϕ,R,2

In case there are zero eigenvalues, a suitable energy shift will again lead to this case.  Example 3.12 If one of the potentials vanishes, say q1 = 0, more explicit results can be obtained. We consider separated boundary conditions with no zero eigenvalues present for j = 1, 2. Then  Fα,β, j (0) = cos(α) − sin(β)φ[1] j (0, b, a) + cos(β)φ j (0, b, a)  − sin(α) − sin(β)θ[1] (0, b, a) + cos(β)θ (0, b, a) , j j

j = 1, 2. (3.59)

Computing Traces, Determinants, and ζ-Functions …

117

For q1 = 0, θ1 (0, x, a) and φ1 (0, x, a) satisfy the initial value problems   d d p(x) θ1 (0, x, a) = 0, − dx dx   d d φ1 (0, x, a) = 0, − p(x) dx dx

θ1 (0, a, a) = 1, θ1[1] (0, a, a) = 0, φ1 (0, a, a) = 0, φ[1] 1 (0, a, a) = 1, (3.60)

solutions of which are ˆ φ1 (0, x, a) =

x

dt p(t)−1 , θ1 (0, x, a) = 1.

(3.61)

a

Then one can show that ˆ Fα,β,1 (0) = − sin(α + β) + cos(α) cos(β)

b

dt p(t)−1 .

(3.62)

a

The relative ζ-regularized determinant in this case then reads  ´b    −1    cos(α) cos(β) a dt p(t) − sin(α + β)  ζ 0; Hα,β,1 , Hα,β,2 = ln   . (3.63)   Fα,β,2 (0)

Example 3.13 Restricting Example 3.12 to the case p(x) = r (x) = 1, the ζdeterminant ζ (0; Hα,β,2 ) can be computed explicitly. First, one notes that under Hypothesis 3.9, the ζ-function can be analytically continued to a neighborhood of s = 0 and ζ (0; Hα,β,2 ) is a well-defined quantity; similarly, ζ (0; Hα,β,1 ) is well-defined for q1 = 0. Computing ζ (0; Hα,β,1 ), employing (3.63), one obtains ζ (0; Hα,β,2 ). First we assume there are no zero eigenvalues for j = 1, which is the case if (b − a) cos(α) cos(β) − sin(α + β) = 0.

(3.64)

Next, one notes that     φ1 (z, x, a) = z −1/2 sin z 1/2 (x − a) , θ1 (z, x, a) = cos z 1/2 (x − a) ,

(3.65)

and thus from (3.23),      Fα,β,1 (z) = cos(α) − sin(β) cos z 1/2 (b − a) + z −1/2 cos(β) sin z 1/2 (b − a)      − sin(α) z 1/2 sin(β) sin z 1/2 (b − a) + cos(β) cos z 1/2 (b − a) . (3.66)

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The details of what follows depend on the boundary conditions imposed. In this example,  we consider α, β ∈ (0, π). Along the relevant contour, as |z| → ∞, one has Im z 1/2 > 0 and the asymptotics for the boundary conditions considered reads    asym Fα,β,1 (z) = Fα,β,1 (z) 1 + O z −1/2 , where

Fα,β,1 (z) = −(i/2)z 1/2 sin(α) sin(β)e−i z asym

1/2

(b−a)

(3.67) .

(3.68)

Adding and subtracting this asymptotics where applicable, we rewrite the ζ-function for Hα,β,1 in the form ˆ   sin(πs) 1 d  dt t −s ln Fα,β,1 teiθ π dt 0    ˆ ∞ Fα,β,1 teiθ sin(πs) d  dt t −s ln + eis(π−θ) asym  π dt Fα,β,1 teiθ 1

ζ(s; Hα,β,1 ) = eis(π−θ)

+ ζ asym (s; Hα,β,1 ),

(3.69)

where ζ asym (s; Hα,β,1 ) = eis(π−θ) This representation is valid for and yields

sin(πs) π

1 2

ˆ

∞

dt t −s

1

d  asym  iθ   ln Fα,β,1 te . dt

(3.70)

< Re(s) < 1. The term (3.70) is easily computed

ζ asym (s; Hα,β,1 ) = eis(π−θ)

  i sin(πs) 1 (b − a) − eiθ/2 , π 2s 2 s − (1/2)

(3.71)

yielding its analytic continuation to − 21 < Re(s) < 1. The ζ-determinant for Hα,β,1 then follows from 

 asym  ζ (0; Hα,β,1 ) = Re ln Fα,β,1 eiθ − ln(Fα,β,1 (0)) + ζ asym (0; Hα,β,1 )    2 [cos(α) cos(β) (b − a) − sin(α + β)]   . (3.72) = −ln   sin(α) sin(β) Using expression (3.72) in (3.63) yields    sin(α) sin(β)   .  ζ (0; Hα,β,2 ) = ln  2Fα,β,2 (0) 

If there is a zero eigenvalue for j = 1, namely, if

(3.73)

Computing Traces, Determinants, and ζ-Functions …

119

(b − a) cos(α) cos(β) − sin(α + β) = 0, the relevant formula for the relative ζ-determinant is   

  Fα,β,1 (0) 

ζ (0; Hα,β,1 , Hα,β,2 ) = ln   .  Fα,β,2 (0) 

(3.74)

(3.75)

Employing (3.66), this can be cast in the form    (b − a) [(b − a) sin(α + β) − 3 sin(α) sin β]   .  ζ (0; Hα,β,1 , Hα,β,2 ) = ln   3F (0)

α,β,2

(3.76) In order to compute ζ (0; Hα,β,2 ), the part ζ (0; Hα,β,1 ) can be computed as before, the only difference being that Fα,β,1 (z) is replaced by Fα,β,1 (z)/z. The final answer then reads      (b − a) sin(α + β)  (3.77) ζ (0; Hα,β,1 ) = −ln 2(b − a) 1 −  . 3 sin(α) sin β (This confirms the known result in the case of Neumann boundary conditions at x = a and x = b, that is, for α = β = π/2.) From (3.77) it is immediate that    sin(α) sin(β)   . (3.78) ζ (0; Hα,β,2 ) = ln  2Fα,β,2 (0)  Example 3.14 Next, consider the Dirichlet boundary condition at x = a and a Robin boundary condition at x = b, that is, α = 0 and β ∈ (0, π). In the absence of zero eigenvalues one then has    F0,β,1 (0)   . (3.79) ζ (0; H0,β,1 , H0,β,2 ) = ln  F0,β,2 (0)  In order to find the ζ-determinant ζ (0; H0,β,2 ) we consider as before q1 = 0. Then,     F0,β,1 (z) = − sin(β) cos z 1/2 (b − a) + z −1/2 cos β) sin(z 1/2 (b − a) . (3.80) The relevant asymptotic large-|z| behavior is F0,β,1 (z) = − sin(β)e−i z asym

1/2

(b−a)

/2,

(3.81)

and proceeding along the lines of previous computations, one finds     sin(β)   . ζ (0; H0,β,1 ) = ln  2 [sin(β) − (b − a) cos(β)] 

(3.82)

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F. Gesztesy and K. Kirsten

(Again, this confirms the known case where β = π/2.) From (3.82) it is immediate that    sin(β) 

 .  (3.83) ζ (0; H0,β,2 ) = ln  2F0,β,2 (0)  Example 3.15 For Dirichlet boundary conditions at both endpoints, that is, for α = β = 0, the relative zeta-determinant follows from    F0,0,1 (0) 

 .  ζ (0; H0,0,1 , H0,0,2 ) = ln  (3.84) F0,0,2 (0)  For q1 = 0, from

  F0,0,1 (z) = z −1/2 sin z 1/2 (b − a) ,

(3.85)

the relevant asymptotics is F0,0,1 (z) = (i/2)z −1/2 e−i z asym

Thus, one finds

and hence,

1/2

(b−a)

.

(3.86)

ζ (0; H0,0,1 ) = −ln(2(b − a)),

(3.87)

ζ (0; H0,0,2 ) = −ln(|2F0,0,2 (0)|).

(3.88)

Remark 3.16 Under the assumptions of Example 3.13 no additional computations are needed when considering certain relative determinants for different boundary conditions. Indeed, for α j , β j ∈ (0, π), j = 1, 2, Eq. (3.73) is valid replacing α, β → α1 , β1 and Fα,β,2 → Fα2 ,β2 ,2 , and Eq. (3.83) is valid replacing β → β1 , and F0,β,2 →  F0,β2 ,2 . Example 3.17 As an example for coupled boundary conditions we reconsider the Krein–von Neumann extension, Example 3.7. We first note that different potentials q1 = q2 lead to different Krein–von Neumann extensions R K 1 = R K 2 ; see, for instance, (3.35). Nevertheless, under the assumptions made, Theorem 3.10 (ii) remains valid and      F0,R K1 ,1 (z) 

(3.89) ζ (0; H0,R K1 ,1 , H0,R K2 ,2 ) = lim ln   , z→0  F0,R K2 ,2 (z)  and from (3.40) one finds ζ (0; H0,R K1 ,1 , H0,R K2 ,2 ) = ln(|c1 /c2 |),

(3.90)

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121

with cj =

  . . . [1] 1 . [1] θ j (0, b, a) φ j (0, b, a) − θ j (0, b, a)φ j (0, b, a) , 2

j = 1, 2. (3.91)

For the case of vanishing potential, q1 = 0, the constant c1 can be determined explicitly. To this end, we need the small-z expansion of the solutions of  −

 1 d d p(x) θ1 (z, x, a) = zθ1 (z, x, a), θ1 (z, a, a) = 1, θ1[1] (z, a, a) = 0, r (x) d x dx

(3.92)

and  −

 d 1 d p(x) φ1 (z, x, a) = zφ1 (z, x, a), φ1 (z, a, a) = 0, φ[1] 1 (z, a, a) = 1. r (x) d x dx

(3.93)

Expanding

.   θ1 (z, x, a) = θ1 (0, x, a) + z θ1 (0, x, a) + O z 2 ,

(3.94)

one compares O(z)-terms in (3.92) to find with (3.61) 

 d d . p(x) θ1 (0, x, a) = −r (x). dx dx

(3.95)

Integrating, this yields

. [1]

ˆ

. [1]

θ1 (0, x, a) − θ1 (0, a, a) = −

x

du r (u),

(3.96)

a

but given θ1[1] (z, a, a) = 0, one concludes that ˆ

. [1]

θ1 (0, x, a) = −

x

du r (u).

(3.97)

a

Similarly, integrating (3.97),

.

ˆ

θ1 (0, b, a) = −

b

dv p(v) a

−1

ˆ

v

du r (u).

(3.98)

a

Proceeding in the same way for φ1 (z, x, a), one first shows 

 ˆ x d d . φ1 (0, x, a) = −r (x) du p(u)−1 , p(x) dx dx a

(3.99)

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F. Gesztesy and K. Kirsten

and thus

ˆ

. [1]

φ1 (0, x, a) = −

ˆ

x

dv r (v)

a

v

du p(u)−1 .

(3.100)

a

Furthermore,

.

ˆ

φ1 (0, x, a) = −

x

dw p(w)

−1

ˆ

a

w

ˆ

v

dv r (v)

a

du p(u)−1 .

(3.101)

a

Altogether this yields ˆ b   ˆ b ˆ w ˆ v 1 r (v) c1 = dt r (t) dw dv du 2 p(w) p(u) a a a a  ˆ b  ˆ b ˆ v ˆ w r (u) r (w) . − dv du dw dt p(v) p(t) a a a a

(3.102)

Example 3.18 For the particular case r (x) = p(x) = 1, we now recompute the ζdeterminant ζ (0; H0,R K2 ,2 ) by choosing q1 = 0. First, from (3.102) one finds that c1 = −(b − a)4 /24.

(3.103)

From Example 3.7, one determines      F0,R K1 ,1 (z) = 2 1 − cos z 1/2 (b − a) − (b − a)z 1/2 sin z 1/2 (b − a) . (3.104) The relevant leading asymptotics reads asym

F0,R K

,1 (z) 1

= −i(b − a)z 1/2 e−i z

1/2

(b−a)



2.

(3.105)

The zeta-function for the Krein–von Neumann extension is therefore analyzed using   ˆ ∞ F0,R K1 ,1 (teiθ ) is(π−θ) sin(πs) −s d ln dt t ζ(s; H0,R K1 ,1 ) = e π dt t 2 e2iθ 0  ˆ 1 F0,R K1 ,1 (teiθ ) sin(πs) d = eis(π−θ) dt t −s ln π dt t 2 e2iθ 0  ˆ ∞ F0,R K1 ,1 (teiθ ) is(π−θ) sin(πs) −s d ln dt t +e asym π dt F0,R K ,1 (teiθ ) 1 1

+ ζ asym (s; H0,R K1 ,1 ),

(3.106)

where ζ

asym

(s; H0,R K1 ,1 ) = e

is(π−θ) sin(πs)

π

ˆ

∞

dt t 1

−s

d ln dt



asym

F0,R K

1

,1 (te

t 2 e2iθ

iθ

)

Computing Traces, Determinants, and ζ-Functions …

= eis(π−θ)

such that

123

  3 i sin(πs) 1 − − (b − a)eiθ/2 , π 2s 2 s − (1/2) (3.107)

ζ asym (0; H0,R K1 ,1 ) = i(b − a)eiθ/2 + [3i(θ − π)/2].

(3.108)

From (3.106) we then find   ζ (0; H0,R K1 ,1 ) = −ln (b − a)3 /6 ,

(3.109)

in agreement with [50]. Finally, this proves   b − a   . ζ (0; H0,R K2 ,2 ) = ζ (0; H0,R K1 ,1 ) + ln(|c1 /c2 |) = ln  4c 

(3.110)

2

Example 3.19 As our final example, we consider a case where negative eigenvalues are present. Let x ∈ (0, π) and τ j = −(d 2 /d x 2 ) − m 2j with m j ∈ (n j , n j + 1), n j ∈ N, j = 1, 2. Imposing Dirichlet boundary conditions at both endpoints, the ( j) eigenvalues are λ = 2 − m 2j ,  ∈ N, such that there are n j negative eigenvalues for H0,0, j . The ζ-function representation for each j = 1, 2, can be found following Examples 3.13 and 3.15. We note that   −1/2 1/2

sin z + m 2j π F0,0, j (z) = z + m 2j

 −1/2 2 1/2 2 1/2 = z + m 2j (i/2)e−i(z+m j ) π 1 − e2πi(z+m j )

2 1/2 asym =: F0,0, j (z) 1 − e2πi(z+m j ) ,

(3.111)

and hence, ζ(s; H0,0, j ) = e

is(π−θ) sin(πs)

ˆ

π

∞

dt t

−s

0

   F0,0, j teiθ d  + ζ asym (s; H0,0, j ), ln asym  dt F0,0, j teiθ (3.112)

where ζ asym (s; H0,0, j ) = eis(π−θ)

sin(πs) π

ˆ 0

∞

dt t −s

d  asym  iθ  ln F0,0, j te , dt

(3.113)

this representation being valid for (1/2) < Re(s) < 1. From [34, 3.193, 3.194] one infers   ˆ ∞ M (1/2)−s (1 − s) s − 21 t −s dt = , (3.114) (t + M)1/2 π 1/2 0

124

F. Gesztesy and K. Kirsten

ˆ

∞

dt 0

π t −s = s , t+M M sin(πs)

(3.115)

and hence one obtains ζ

asym

eisπ (s; H0,0, j ) = − 2s 2m j



 iπ 1/2 m j  (s − (1/2)) +1 , (s)

(3.116)

implying ζ asym (0; H0,0, j ) = −i(π/2) + im j π + ln(m j ).

(3.117)

It then follows that   ζ (0; H0,0, j ) = −ln 1 − e2πim j − i(π/2) + im j π + ln(m j ).

(3.118)

In order to obtain the final answer explicitly, showing the relation between the imaginary part and the number of negative eigenvalues, we first note that 1 − e2πim j = −2ieπim j sin(πm j ).

(3.119)

A careful analysis of the argument of 1 − e2πim j then shows that it equals π[m j − n j − (1/2)], such that    2 sin(πm j )   .  ζ (0; H0,0, j ) = iπn j − ln   mj

(3.120)

Considering instead ζ(s; H0,0,1 , H0,0,2 ) = e

is(π−θ) sin(πs)

π

ˆ

∞

dt t 0

−s

  F0,0,2 (teiθ ) d ln , dt F0,0,1 (teiθ )

(3.121)

in (3.120), one obtains    sin(πm 1 ) m 2   . ζ (0; H0,0,1 , H0,0,2 ) = iπ(n 2 − n 1 ) + ln  sin(πm 2 ) m 1 

(3.122)

The real part of this answer is readily reproduced from (2.41) in Theorem 2.10. However, even for this simple example, the behavior of F0,0,1 (teiθ )/F0,0,2 (teiθ ) along the integration range t ∈ [0, ∞) is quite intricate so that finding the correct imaginary part from (3.121), namely, from ζ (0; H0,0,1 , H0,0,2 ) =

ˆ

∞

dt 0

  F0,0,1 (teiθ ) d ln , dt F0,0,2 (teiθ )

(3.123)

is rather involved, and appears to be next to impossible for more general cases.

Computing Traces, Determinants, and ζ-Functions …

125

For the case of Schrödinger operators with strongly singular potentials at one or both endpoints of a bounded interval, see [41, 43, 45, 61].

4 Schrödinger Operators on a Half-Line: The Short-Range Case In this section, we illustrate some of the abstract notions in Section 2 with the help of self-adjoint Schrödinger operators on the half-line R+ = (0, ∞). We will focus on the case of short-range potentials q (cf. (4.1)) and hence the scattering theory situation which necessitates a comparison with the case q = 0 and thus illustrates the case of relative perturbation determinants, relative ζ-functions, and relative ζ-function regularized determinants. We assume that the potential coefficient q satisfies the following conditions. Hypothesis 4.1 Suppose q satisfies the short-range assumption q ∈ L 1 (R+ ; (1 + |x|)d x), q is real-valued a.e. on R+ .

(4.1)

Given Hypothesis 4.1, we take τ+ to be the Schrödinger differential expression τ+ = −

d2 + q(x) for a.e. x ∈ R+ , dx2

(4.2)

and note that τ+ is regular at 0 and in the limit point case at +∞. The maximal operator H+,max in L 2 (R+ ; d x) associated with τ+ is defined by H+,max f = τ+ f,

  f ∈ dom(H+,max ) = g ∈ L 2 (R+ ; d x)  g, g ∈ AC([0, b]) for all b > 0;

(4.3)  τ+ g ∈ L (R+ ; d x) , 2

while the minimal operator H+,min in L 2 (R+ ; d x) associated with τ+ is given by H+,min f = τ+ f,

  f ∈ dom(H+,min ) = g ∈ L 2 (R+ ; d x)  g, g ∈ AC([0, b]) for all b > 0;

(4.4)  g(0) = g (0) = 0; τ+ g ∈ L (R+ ; d x) .

2

Again, one notes that the operator H+,min is symmetric and that ∗ = H+,max , H+,min

∗ H+,max = H+,min .

(4.5)

Moreover, all self-adjoint extensions of H+,min are given by the one-parameter family H+,α in L 2 (R+ ; d x),

126

F. Gesztesy and K. Kirsten

H+,α f = τ+ f,

  f ∈ dom(H+,α ) = g ∈ L 2 (R+ ; d x)  g, g ∈ AC([0, b]) for all b > 0;



(4.6)

sin(α)g (0) + cos(α)g(0) = 0; τ+ g ∈ L (R+ ; d x) , α ∈ [0, π). 2

The corresponding comparison operator with vanishing potential coefficient q ≡ 0 (0) , α ∈ [0, π). will be denoted by H+,α Next, introducing the Jost solutions f + (z, x) = f +(0) (z, x) − f +(0) (z, x) = ei z

1/2

x

ˆ

∞

d x z −1/2 sin(z 1/2 (x − x ))q(x ) f + (z, x ),

(4.7)

x

  , z ∈ C, Im z 1/2 ≥ 0, x ≥ 0,

(4.8)

satisfying τ+ y = zy, z ∈ C, on R+ , and abbreviating I L 2 (R+ ;d x) = I+ , and v = |q|1/2 , u = v sign(q), such that q = uv = vu,

(4.9)

one infers the following facts:

 (0) −1 det L 2 (R+ ;d x) (H+,α − z I+ )1/2 H+,α − z I+ (H+,α − z I+ )1/2 −1

 (0) = det L 2 (R+ ;d x) I+ + u H+,α − z I+ v =

sin(α) f + (z, 0) + cos(α) f + (z, 0) , sin(α)i z 1/2 + cos(α)

(4.10)

(0) α ∈ [0, π), z ∈ ρ(H+,α ) ∩ ρ(H+,α ).

Here the first equality in (4.10) is shown as in the abstract context (2.18)–(2.22), and the second equality in (4.10) for the Dirichlet and Neumann cases α = 0, α = π/2 has been discussed in [21, 27, 28]; the general case α ∈ [0, π) is proved in [22, Theorem 2.6]. Since (4.11) σess (H+,α ) = [0, ∞), α ∈ [0, π), we now shift all operators H+,α by λ1 I+ , with a fixed λ1 > 0, and consider H+,α (λ1 ) = H+,α + λ1 I+ , α ∈ [0, π),

(4.12)

from this point on and hence obtain

 (0) −1 (λ1 ) − z I+ (H+,α (λ1 ) − z I+ )1/2 det L 2 (R+ ;d x) (H+,α (λ1 ) − z I+ )1/2 H+,α −1

 (0) = det L 2 (R+ ;d x) I+ + u H+,α (λ1 ) − z I+ v

Computing Traces, Determinants, and ζ-Functions …

=

127

sin(α) f + (z − λ1 , 0) + cos(α) f + (z − λ1 , 0) , sin(α)i(z − λ1 )1/2 + cos(α)   (0) (λ1 ) ∩ ρ(H+,α (λ1 )). α ∈ [0, π), z ∈ ρ H+,α

(4.13)

In this half-line context all discrete eigenvalues H+,α (λ1 ) (i.e., all eigenvalues of H+,α (λ1 ) below λ1 ) are simple and hence m(0; H+,α (λ1 )) ∈ {0, 1}, α ∈ [0, π).

(4.14)

In addition, it is known that under Hypothesis 4.1, the threshold of the essential spectrum of H+,α (λ1 ), λ1 , is never an eigenvalue of H+,α (λ1 ). Iterating the Volterra integral equation (4.7) for f + (z − λ1 , 0), and analogously for its x-derivative, yields uniform asymptotic expansions near z = 0 and as z → ∞ (in terms of powers of |z|−1/2 ). The same applies to their z-derivatives (cf., e.g., [7, Chap. I]) and explicit computations yield the following. For fixed 0 < ε0 sufficiently small, and using the abbreviation Cε0 = C\B(λ1 ; ε0 ), with B(z 0 ; r0 ) the open ball in C of radius r0 > 0 centered at z 0 ∈ C, one obtains f + (z − λ1 , 0) = 1 − |z|→∞ z∈Cε0

i 2z 1/2

f + (z − λ1 , 0) = i z 1/2 −

.

|z|→∞ z∈Cε0

ˆ

1 2

∞

ˆ

∞

    1/2 d x1 e2i(z−λ1 ) x1 − 1 q(x1 ) + O |z|−1 ,

∞

    1/2 d x1 e2i(z−λ1 ) x1 + 1 q(x1 ) + O |z|−1/2 ,

0

ˆ 0

f + (z − λ1 , 0) =

1 2z

.

i i − 1/2 2z 1/2 2z

|z|→∞ z∈Cε0

f + (z − λ1 , 0) =

|z|→∞ z∈Cε0

d x1 e2i(z−λ1 )

1/2

x1

  x1 q(x1 ) + O |z|−3/2 ,

0

ˆ

∞

d x1 e2i(z−λ1 )

1/2

x1

  x1 q(x1 ) + O |z|−1

0

(4.15) (abbreviating again . = d/dz). Given the asymptotic expansions (4.15) as |z| → ∞, and employing the fact that

.

.

( · − λ1 , 0) are all analytic with f + ( · − λ1 , 0), f + ( · − λ1 , 0), f + ( · − λ1 , 0), f + respect to z around z = 0, investigating the case distinctions α ∈ [0, π)\{0, π/2}, α = 0, α = π/2, f + (z − λ1 , 0) = 0, f + (z − λ1 , 0) = 0, f + (z − λ1 , 0) = 0, f + (z − λ1 , 0) = 0, etc., one verifies in each case that the logarithmic z-derivative of (4.10) satisfies hypotheses in Theorem 2.9, hence the latter applies with ε = 1/2 to the   the (0) (λ1 ), H+,α (λ1 ) , α ∈ [0, π). pairs H+,α More generally, we now replace the pair (0, q) by (q1 , q2 ), where q j , j = 1, 2, satisfy Hypothesis 4.1, and denote the corresponding Schrödinger operators in L 2 (R+ ; d x) with q (resp., u, v) replaced by q j (resp., u j , v j ) by H+,α, j and similarly, after the shift with λ1 , by H+,α, j (λ1 ), j = 1, 2. Analogously, we denote the

128

F. Gesztesy and K. Kirsten

corresponding Jost solutions by f +, j (z, · ), j = 1, 2. This then yields the following results: Theorem 4.2 Suppose q j , j = 1, 2, satisfy Hypothesis 4.1. Then,

 −1 det L 2 (R+ ;d x) (H+,α,2 (λ1 ) − z I+ )1/2 H+,α,1 (λ1 ) − z I+ (H+,α,2 (λ1 ) − z I+ )1/2

 −1 = det L 2 (R+ ;d x) I+ + u 1,2 H+,α,1 (λ1 ) − z I+ v1,2 =

sin(α) f +,2 (z − λ1 , 0) + cos(α) f +,2 (z − λ1 , 0) ,

sin(α) f +,1 (z − λ1 , 0) + cos(α) f +,1 (z − λ1 , 0)   α ∈ [0, π), z ∈ ρ H+,α,1 (λ1 ) ∩ ρ(H+,α,2 (λ1 )),

(4.16)

where v1,2 = |q2 − q1 |1/2 , u 1,2 = v1,2 sign(q2 − q1 ), such that q2 − q1 = u 1,2 v1,2 = v1,2 u 1,2 .

(4.17)

In addition, ζ(s; H+,α,1 (λ1 ), H+,α,2 (λ1 )) = eis(π−θ) π −1 sin(πs)  ˆ ∞ d dt t −s ln (eiθ t)[m(0;H+,α,1 (λ1 ))−m(0;H+,α,2 (λ1 ))] (4.18) × dt 0 

sin(α) f +,2 (eiθ t − λ1 , 0) + cos(α) f +,2 (eiθ t − λ1 , 0) , ×

sin(α) f +,1 (eiθ t − λ1 , 0) + cos(α) f +,1 (eiθ t − λ1 , 0) α ∈ [0, π), Re(s) ∈ (−1/2, 1). Proof Relations (4.16), (4.17) follow as summarized in (4.7)–(4.15) and the two paragraphs preceding Theorem 4.2. Thus, Theorem 2.9 applies to the pairs of selfadjoint operators (H+,α,1 (λ1 ), H+,α,2 (λ1 )), α ∈ [0, π).  The relative ζ-function regularized determinant now follows immediately from Theorem 2.10. Special cases of (4.16) (pertaining to the Dirichlet boundary conditions α j = 0, j = 1, 2) appeared in the celebrated work by Jost and Pais [36] and Buslaev and Faddeev [6] (see also [12, 21, 52, 53, 56]). Up to this point we kept the boundary condition, that is, α, fixed and varied the potential coefficient q. Next, we keep q fixed, but vary α. Returning to the operator H+,α , α ∈ [0, π), we turn to its underlying quadratic form Q H+,α next, ˆ

∞

  d x f (x)g (x) + q(x) f (x)g(x) − cot(α) f (0)g(0), 0   f, g ∈ dom(Q H+,α ) = dom |H+,α |1/2 = H 1 (R+ )

Q H+,α ( f, g) =

(4.19)

Computing Traces, Determinants, and ζ-Functions …

129

  = h ∈ L 2 (R+ ; d x) | h ∈ AC([0, b]) for all b > 0; h ∈ L 2 (R+ ; d x) , α ∈ (0, π),  Q H+,0 ( f, g) = d x f (x)g (x) + q(x) f (x)g(x) , 0   f, g ∈ dom(Q H+,0 ) = dom |H+,0 |1/2 = H01 (R+ ) (4.20)   2

2 = h ∈ L (R+ ; d x) | h∈ AC([0, b]) for all b > 0; h(0) = 0; h ∈ L (R+ ; d x) . ˆ

∞



Moreover, introducing the regular solution φα (z, · ) associated with H+,α satisfying τ+ y = zy, z ∈ C, on R+ , and sin(α)φ α (z, 0) + cos(α)φα (z, 0) = 0, α ∈ [0, π), z ∈ C,

(4.21)

one infers φα (z, x) =

φ(0) α (z, x)

ˆ

x

+

d x z −1/2 sin(z 1/2 (x − x ))q(x )φα (z, x ),

0

(4.22)

z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0, with −1/2 sin(z 1/2 x) − sin(α) cos(z 1/2 x), φ(0) α (z, x) = cos(α)z

z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0.

(4.23)

Given the solutions φα (z, · ), f + (z, · ) of τ+ y = z, z ∈ C, the resolvent of H+,α is given by   (H+,α − z I+ )−1 f (x) =

ˆ

∞

d x G +,α (z, x, x ) f (x ),

0

(4.24)

z ∈ ρ(H+,α ), x ≥ 0, f ∈ L (R+ ; d x), 2

with the Green’s function G +,α of H+,α expressed in terms of φα and f + by G +,α (z, x, x ) = (H+,α − z I+ )−1 (x, x )  1 φα (z, x) f + (z, x ), 0 ≤ x ≤ x < ∞, = W ( f + (z, ·), φα (z, ·)) φα (z, x ) f + (z, x), 0 ≤ x ≤ x < ∞,

(4.25)

z ∈ ρ(H+,α ). In the special case α = 0 one verifies W ( f + (z, ·), φ0 (z, ·)) = f + (z, 0), z ∈ ρ(H+,0 ), with f + (z, 0) the well-known Jost function.

(4.26)

130

F. Gesztesy and K. Kirsten

Next, we compare the half-line Green’s functions G +,α1 and G +,α2 , that is, we investigate the integral kernels associated with a special case of Krein’s formula for resolvents (cf. [2, & 106]): Assume α1 , α2 ∈ [0, π), with α1 = α2 . Then, ψ+,α1 (z, x)ψ+,α1 (z, x ) , cot(α2 − α1 ) + m +,α1 (z) z ∈ ρ(H+,α1 ) ∩ ρ(H+,α2 ), x, x ∈ [0, ∞),

G +,α2 (z, x, x ) = G +,α1 (z, x, x ) −

ˆ

and

∞

.

d x ψ+,α (z, x)2 = m +,α (z),

(4.27)

(4.28)

0

implying   tr L 2 (R+ ;d x) (H+,α2 (λ1 ) − z I+ )−1 − (H+,α1 (λ1 ) − z I+ )−1 d = − ln[cot(α2 − α1 ) + m +,α1 (z)], z ∈ ρ(H+,α1 ) ∩ ρ(H+,α2 ), dz

(4.29)

according to Lemma 2.2 and (A.44) in [24]. Here ψ+,α (z, · ) and m +,α (z) are the Weyl–Titchmarsh solution and m-function corresponding to H+,α . More precisely, ψ+,α (z, · ) = θα (z, · ) + m +,α (z)φα (z, · ), z ∈ ρ(H+,α ),

(4.30)

where θα (z, x) =

θα(0) (z, x)

ˆ +

x

d x z −1/2 sin(z 1/2 (x − x ))q(x )θα (z, x ),

0

(4.31)

z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0, with θα(0) (z, x) = cos(α) cos(z 1/2 x) + sin(α)z −1/2 sin(z 1/2 x), z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0.

(4.32)

Due to the limit point property of τ+ at +∞, one actually has ψ+,α (z, · ) = Cα (z) f + (z, · ), z ∈ ρ(H+,α ),

(4.33)

for some z-dependent complex-valued constant Cα (z). Actually, since ψ+,α (z, 0) = Cα f + (z, 0), one can show (using (A.18) in [24]) that Cα (z) =

1 cos(α) − sin(α)m +,α (z) = . f + (z, 0) cos(α) f + (z, 0) + sin(α) f + (z, 0)

(4.34)

Computing Traces, Determinants, and ζ-Functions …

131

Similarly,

(z, 0)/ψ+,0 (z, 0) = f + (z, 0)/ f + (z, 0), z ∈ ρ(H+,α ), m +,0 (z) = ψ+,0

(4.35)

and m +,α (z) =

cos(α) f + (z, 0) − sin(α) f + (z, 0) − sin(α) + cos(α)m +,0 (z) = , cos(α) + sin(α)m +,0 (z) sin(α) f + (z, 0) + cos(α) f + (z, 0) z ∈ ρ(H+,α ). (4.36)

Moreover, one verifies that cos(α2 − α1 ) + sin(α2 − α1 )m +,α1 (z) =

sin(α2 ) f + (z, 0) + cos(α2 ) f + (z, 0) . sin(α1 ) f + (z, 0) + cos(α1 ) f + (z, 0) (4.37)

Combining (4.29) and (4.37) yields   tr L 2 (R+ ;d x) (H+,α2 (λ1 ) − z I+ )−1 − (H+,α1 (λ1 ) − z I+ )−1   sin(α2 ) f + (z − λ1 , 0) + cos(α2 ) f + (z − λ1 , 0) d , = − ln dz sin(α1 ) f + (z − λ1 , 0) + cos(α1 ) f + (z − λ1 , 0)

(4.38)

z ∈ ρ(H+,α1 (λ1 )) ∩ ρ(H+,α2 (λ1 )), α1 , α2 ∈ [0, π).   In particular, (H+,α2 − z I+ )−1 − (H+,α1 − z I+ )−1 is rank-one and hence trace class. Moreover, since by (4.19)   f + (z, · ) ∈ dom |H+,α |1/2 = H 1 (R+ ), α ∈ (0, π),

(4.39)

Hypothesis 2.5 and hence relations (2.25), (2.26) are now satisfied for the pair (H+,α1 (λ1 ), H+,α2 (λ1 )) for α1 ∈ [0, π), α2 ∈ (0, π), implying the following result: Theorem 4.3 Suppose q j , j = 1, 2, satisfy Hypothesis 4.1. Then,   tr L 2 (R+ ;d x) (H+,α2 (λ1 ) − z I+ )−1 − (H+,α1 (λ1 ) − z I+ )−1   −1 d = − ln det L 2 (R+ ;d x) (H+,α2 (λ1 ) − z I+ )1/2 H+,α1 (λ1 ) − z I+ dz c

(4.40) × (H+,α2 (λ1 ) − z I+ )1/2  

sin(α2 ) f + (z − λ1 , 0) + cos(α2 ) f + (z − λ1 , 0) d = − ln , (4.41) dz sin(α1 ) f + (z − λ1 , 0) + cos(α1 ) f + (z − λ1 , 0) z ∈ ρ(H+,α1 (λ1 )) ∩ ρ(H+,α2 (λ1 )), α1 ∈ [0, π), α2 ∈ (0, π), α1 = α2 , temporarily abbreviating the operator closure symbol by {· · · }c due to lack of space in (4.40). In addition,

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F. Gesztesy and K. Kirsten

ζ(s; H+,α1 (λ1 ), H+,α2 (λ1 )) = eis(π−θ) π −1 sin(πs) ˆ ∞ d  × dt t −s ln (teiθ )[m(0,H+,α1 (λ1 ))−m(0,H+,α2 (λ1 ))] dt 0 ×

(4.42)

sin(α2 ) f + (teiθ − λ1 , 0) + cos(α2 ) f + (teiθ − λ1 , 0) , sin(α1 ) f + (teiθ − λ1 , 0) + cos(α1 ) f + (teiθ − λ1 , 0) α j ∈ (0, π), j = 1, 2, Re(s) ∈ (−1/2, 1).

Proof Relations (4.40), (4.41) summarize the discussion in (4.21)–(4.39). Applying  Theorem 2.9 in the case α1 , α2 ∈ (0, π) then yields (4.42). The relative ζ-function regularized determinant again follows immediately from Theorem 2.10,   [m(0,H+,α1 (λ1 ))−m(0,H+,α2 (λ1 ))] d

ζ (0; H+,α1 (λ1 ), H+,α2 (λ1 )) = − lim ln teiθ t↓0 dt  sin(α2 ) f + (teiθ − λ1 , 0) + cos(α2 ) f + (teiθ − λ1 , 0) . (4.43) × sin(α1 ) f + (teiθ − λ1 , 0) + cos(α1 ) f + (teiθ − λ1 , 0) Remark 4.4 The case α1 = 0, α2 ∈ (0, π), is more involved in that the representation (4.42) is only valid for Re(s) ∈ (0, 1). This is due to the fact that sin(α2 ) f + (z − λ1 , 0) + cos(α2 ) f + (z − λ1 , 0) f + (z − λ1 , 0)

=

|z|→∞

  O |z|1/2 ,

(4.44)

and hence assumption (2.38) is not satisfied. One then proceeds as follows. Let (0) (λ1 ) denote the case with vanishing potential q. We rewrite H+,α   tr L 2 (R+ ;d x) (H+,α2 (λ1 ) − z I+ )−1 − (H+,0 (λ1 ) − z I+ )−1   (0) (λ1 ) − z I+ )−1 = tr L 2 (R+ ;d x) (H+,α2 (λ1 ) − z I+ )−1 − (H+,α 2  (0)  (λ1 ) − z I+ )−1 − (H+,0 (λ1 ) − z I+ )−1 + tr L 2 (R+ ;d x) (H+,0  (0)  (0) + tr L 2 (R+ ;d x) (H+,α (λ1 ) − z I+ )−1 − (H+,0 (λ1 ) − z I+ )−1 . 2

(4.45)

For the first two of the three contributions on the right-hand side of (4.45) the relative ζ-determinants can be computed from Theorem 4.2. For the third contribution more care is needed as the subleading large-|z| behavior differs due to one boundary condition being Dirichlet and the other being Robin. The starting point is the representation

(0) (0) (λ1 ), H+,α (λ1 ) = eis(π−θ) π −1 sin(πs) ζ s; H+,0 2 ˆ ∞

 1/2 d × dt t −s ln sin(α2 )i teiθ − λ1 + cos(α2 ) . dt 0

(4.46)

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133

Along the lines of (3.69) one rewrites this as ˆ ∞

(0) (0) is(π−θ) −1 π sin(πs) dt t −s ζ s; H+,0 (λ1 ), H+,α2 (λ1 ) = e 0  1/2  sin(α2 )i teiθ − λ1 + cos(α2 ) d × ln + ζ asym (s), 1/2  iθ dt sin(α2 )i te − λ1

(4.47)

where, using (3.115), ζ

asym

(s) = e

is(π−θ) −1

π

ˆ sin(πs)

= λ−s 1 /2.

0

∞

dt t −s

1/2

 d ln i sin(α2 ) teiθ − λ1 dt (4.48)

The relative ζ-determinant for the last term in (4.45) then follows from    

 1/2  sin(α2 )i(−λ1 )1/2 + cos(α2 ) (0) (0) ζ 0; H+,0 (λ1 ), H+,α2 (λ1 ) = Re −ln − ln λ 1 sin(α2 )i(−λ1 )1/2   1/2 = −ln λ − cot(α2 ) . (4.49) 1

 One notes that formally, (4.41) extends to the trivial case α1 = α2 . For the case of a strongly singular potential on the half-line with x −2 -type singularity at x = 0 we refer to [41].

5 Schrödinger Operators on a Half-Line: The Case of Purely Discrete Spectra 2

We now illustrate (2.8) with the help of self-adjoint Schrödinger operators − ddx 2 + q on the half-line R+ = (0, ∞) in the particular case where the potential q diverges at ∞ and hence gives rise to a purely discrete spectrum (i.e, the absence of essential spectrum). To this end we introduce the following set of assumptions on q: Hypothesis 5.1 Suppose q satisfies 1 q ∈ L loc (R+ ; d x), q is real-valued a.e. on R+ ,

and for some ε0 > 0, C0 > 0, and sufficiently large x0 > 0,

(5.1)

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F. Gesztesy and K. Kirsten

q, q ∈ AC([x0 , R]) for all R > x0 ,

(5.2)

(2/3)+ε0

, x ∈ (x0 , ∞), q(x) ≥ C0 x  1/2 

q /q = o q , x→∞  −3/2 

q ∈ L 1 ((x0 , ∞); d x). q

(5.3) (5.4) (5.5)

Condition (5.3) guarantees that traces and modified determinants in this paper (see, e.g., the ones in Theorem 5.3) are well-defined. Conditions (5.4) and (5.5) are imposed so that [14, Corollary 2.2.1] is applicable, implying the asymptotic behavior (5.21). Given Hypothesis 5.1, we take τ+ to be the Schrödinger differential expression τ+ = −

d2 + q(x) for a.e. x ∈ R+ , dx2

(5.6)

and note that τ+ is regular at 0 and in the limit point case at +∞. The maximal operator H+,max in L 2 (R+ ; d x) associated with τ+ is defined by H+,max f = τ+ f,

  f ∈ dom(H+,max ) = g ∈ L 2 (R+ ; d x)  g, g ∈ AC([0, b]) for all b > 0;  τ+ g ∈ L 2 (R+ ; d x) ,

(5.7)

while the minimal operator H+,min in L 2 (R+ ; d x) associated with τ+ is given by H+,min f = τ+ f,

  f ∈ dom(H+,min ) = g ∈ L 2 (R+ ; d x)  g, g ∈ AC([0, b]) for all b > 0;  g(0) = g (0) = 0; τ+ g ∈ L 2 (R+ ; d x) .

(5.8)

One notes that the operator H+,min is symmetric and that ∗ = H+,max , H+,min

∗ H+,max = H+,min

(5.9)

(cf., eg., [64, Theorem 13.8]). Moreover, all self-adjoint extensions of H+,min are given by the one-parameter family in L 2 (R+ ; d x) H+,α f = τ+ f,

  f ∈ dom(H+,α ) = g ∈ L 2 (R+ ; d x)  g, g ∈ AC([0, b]) for all b > 0;

(5.10)  sin(α)g (0) + cos(α)g(0) = 0; τ+ g ∈ L (R+ ; d x) , α ∈ [0, π).

2

Next, we introduce the fundamental system of solutions φα (z, · ) and θα (z, · ), α ∈ [0, π), z ∈ C, associated with H+,α satisfying

Computing Traces, Determinants, and ζ-Functions …

135

(τ+ ψ(z, · ))(x) = zψ(z, x), z ∈ C, x ∈ R+ ,

(5.11)

and the initial conditions φα (z, 0) = − sin(α), φ α (z, 0) = cos(α), θα (z, 0) = cos(α), θα (z, 0) = sin(α).

(5.12)

Explicitly, one infers φα (z, x) =

φ(0) α (z, x)

ˆ +

x

dx

0

sin(z 1/2 (x − x )) q(x )φα (z, x ), z 1/2 z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0,

(5.13)

with φ(0) α (z, x) = cos(α)

sin(z 1/2 x) − sin(α) cos(z 1/2 x), z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0, z 1/2 (5.14)

and θα (z, x) =

θα(0) (z, x)

ˆ + 0

x

dx

sin(z 1/2 (x − x )) q(x )θα (z, x ), z 1/2 z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0,

(5.15)

with sin(z 1/2 x) , z ∈ C, Im(z 1/2 ) ≥ 0, x ≥ 0. z 1/2 (5.16) The Weyl–Titchmarsh solution, ψ+,α (z, · ), and Weyl–Titchmarsh m-function, m +,α (z), corresponding to H+,α , α ∈ [0, π), are then related via, θα(0) (z, x) = cos(α) cos(z 1/2 x) + sin(α)

ψ+,α (z, · ) = θα (z, · ) + m +,α (z)φα (z, · ), z ∈ ρ(H+,α ), α ∈ [0, π),

(5.17)

where ψ+,α (z, · ) ∈ L 2 (R+ ; d x), z ∈ ρ(H+,α ), α ∈ [0, π).

(5.18)

One then obtains for the Green’s function G +,α of H+,α expressed in terms of φα and ψ+,α , G +,α (z, x, x ) = (H+,α − z I+ )−1 (x, x )  φα (z, x) ψ+,α (z, x ), 0 ≤ x ≤ x < ∞, = φα (z, x ) ψ+,α (z, x), 0 ≤ x ≤ x < ∞,

z ∈ ρ(H+,α ), α ∈ [0, π), (5.19)

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F. Gesztesy and K. Kirsten

utilizing W (θα (z, ·), φα (z, ·)) = 1, z ∈ C, α ∈ [0, π),

(5.20)

implying W (ψ+,α (z, ·), φα (z, ·)) = 1, z ∈ ρ(H+,α ). By [14, Corollary 2.2.1], Hypothesis 5.1 implies the existence of two solutions f +, j (λ, · , x0 ), j = 1, 2, of τ+ ψ(λ, · ) = λψ(λ, · ), λ < 0 sufficiently negative (and below inf(σ(H+,α ))), satisfying   ˆ x −1/2 −1/4 j



1/2 [q(x) − λ] exp (−1) d x [q(x ) − λ] f +, j (λ, x, x0 ) = 2 x→∞

x0

× [1 + o(1)],

= f +, j (λ, x, x 0 ) x→∞

j −1/2

(−1) 2

[q(x) − λ]

1/4

 ˆ exp (−1) j

x







d x [q(x ) − λ]

1/2

x0

× [1 + o(1)],

j = 1, 2, (5.21)

with

  W f +,1 (λ, · , x0 ), f +,2 (λ, · , x0 ) = 1.

(5.22)

(Here we explicitly introduced the x0 dependence of f +, j , implied by the choice of normalization in (5.21), as keeping track of it later on will become a necessity.) In particular, f +,1 (λ, · , x0 ) now plays the analog of the Jost solution in the case of a short-range potential q (i.e., q ∈ L 1 (R+ ; (1 + x)d x), q real-valued a.e. on R+ ). By the limit point property of τ+ at +∞ and the asymptotic behavior of f +,1 in (5.21) one infers, in addition,  

ψ+,α (λ, · ) = f +,1 (λ, · , x0 ) sin(α) f +,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 ) , (5.23)  

φα (λ, · ) = cos(α) f +,1 (λ, 0, x0 ) + sin(α) f +,1 (λ, 0, x0 ) f +,2 (λ, · , x0 )  

− cos(α) f +,2 (λ, 0, x0 ) + sin(α) f +,2 (λ, 0, x0 ) f +,1 (λ, · , x0 ) (5.24) for λ < 0 sufficiently negative. Analytic continuation with respect to λ in (5.23) then yields the existence of a unique Jost-type solution f +,1 (z, · , x0 ) satisfying τ+ f +,1 (z, · , x0 ) = z f +,1 (z, · , x0 ), z ∈ C\R,

(5.25)

f +,1 (z, · , x0 ) ∈ L (R+ ; d x), z ∈ C\R,

(5.26)

2

coinciding with f +,1 (λ, · , x0 ) for z = λ < 0 sufficiently negative. In addition one has 

(z, 0, x0 ), W f +,1 (z, · , x0 ), φα (z, · , x0 )) = cos(α) f +,1 (z, 0, x0 ) + sin(α) f +,1 z ∈ ρ(H+,α ), (5.27)

Computing Traces, Determinants, and ζ-Functions …

137

which should be compared with the Jost function f + (z, 0) in the case where q represents a short-range potential and α = 0. In the following, we want to illustrate how Hypothesis 2.3 and (2.7) apply to H+,α in the case p = 2. For this purpose, we first recall the following standard convergence property for trace ideals in H: Lemma 5.2 Let q ∈ [1, ∞) and assume that R, Rn , T, Tn ∈ B(H), n ∈ N, satisfy s − limn→∞ Rn = R and s − limn→∞ Tn = T and that S, Sn ∈ Bq (H), n ∈ N, satisfy limn→∞ Sn − SBq (H) = 0. Then limn→∞ Rn Sn Tn∗ − RST ∗ Bq (H) = 0. (Here the strong limit of a sequence of bounded operators Bn , n ∈ N, as n → ∞, was abbreviated by s − limn→∞ Bn .) Lemma 5.2 follows, for instance, from [35, Theorem 1], [59, pp. 28–29], or [65, Lemma 6.1.3] with a minor additional effort (taking adjoints, etc.). Next, we recall a few facts that enable one to compute the trace of a nonnegative trace class integral operator in a straightforward manner: (i) Let 0 ≤ A ∈ B(H), {φm }m∈N an orthonormal basis in H (without loss of generality we assume dim(H) = ∞), then

(φm , Aφm )H ∈ [0, ∞) ∪ {∞}

(5.28)

m∈N

is independent of the orthonormal basis {φm }m∈N in H. Moreover,

(φm , Aφm )H < ∞ if and only if A ∈ B1 (H),

(5.29)

m∈N

and if A ∈ B1 (H), tr H (A) =

(φm , Aφm )H =

λ j (A) = AB1 (H) ,

(5.30)

j∈J

m∈N

where 0 ≤ λ j (A), j ∈ J , J ⊆ N, denote the eigenvalues of A counting multiplicity. (For details, see, e.g., 2.14 and 3.1].)   [59, Theorems (ii) Let 0 ≤ K ∈ B L 2 (; d n x) ,  ⊆ Rn , and suppose that K is an integral operator with continuous integral kernel K ( · , · ) on  × . Then, 0 ≤ K (x, x), x ∈ , |K (x, x)| ≤ |K (x, x)|

1/2

(5.31)



|K (x , x )|

1/2



, x, x ∈ ,

and for all orthonormal bases {em }m∈N in L 2 (; d n x), ˆ

(em , K em ) L 2 (;d n x) = d n x K (x, x). m∈N



(5.32)

(5.33)

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F. Gesztesy and K. Kirsten

In particular, the finiteness of either side in (5.33) implies that of the other. Hence,   K ∈ B1 L 2 (; d n x) if and only if K ( · , · ) ∈ L 1 (; d n x), and, in this case, ˆ tr L 2 (;d n x) (K ) =



d n x K (x, x) < ∞.

(5.34)

(For more details we refer, e.g., to [10, Proposition 5.6.9]. For more general measure spaces see, e.g., [54, pp. 65–66], [60, Sect. 3.11].) Next, we introduce the family of self-adjoint projections PR in L 2 (R+ ; d x) via (PR f )(x) = χ[0,R] (x) f (x),

f ∈ L 2 (R+ ; d x), R > 0,

(5.35)

with χ[0,R] ( · ) the characteristic function associated with the interval [0, R], R > 0. (PR will play the role of Rn , Tn in our application of Lemma 5.2 in the proof of Theorem 5.3 below.) One then obtains the following results. Theorem 5.3 Assume Hypothesis 5.1, z, z 0 ∈ ρ(H+,α ), and α ∈ [0, π). Then,     (H+,α − z I+ )−1 − (H+,α − z 0 I+ )−1 ∈ B1 L 2 (R+ ; d x) ,

(5.36)

and   tr L 2 (R+ ;d x) (H+,α − z I+ )−1 − (H+,α − z 0 I+ )−1   d  = − ln det 2,L 2 (R+ ;d x) I+ − (z − z 0 )(H+,α − z 0 I+ )−1 dz   d 

ln sin(α) f +,1 (z, 0, x0 ) + cos(α) f +,1 (z, 0, x0 )  = dz z=z 0  d 

− ln sin(α) f +,1 (z, 0, x0 ) + cos(α) f +,1 (z, 0, x0 ) dz 1 + I(z, z 0 , x0 ), 2

(5.37)

as well as,   det 2,L 2 (R+ ;d x) I+ − (z − z 0 )(H+,α − z 0 I+ )−1  

(z, 0, x0 ) + cos(α) f +,1 (z, 0, x0 ) sin(α) f +,1 =

sin(α) f +,1 (z 0 , 0, x0 ) + cos(α) f +,1 (z 0 , 0, x0 ) .

.   sin(α) f +,1 (z 0 , 0, x0 ) + cos(α) f +,1 (z 0 , 0, x0 ) × exp − (z − z 0 )

sin(α) f +,1 (z 0 , 0, x0 ) + cos(α) f +,1 (z 0 , 0, x0 )   ˆ z 1 × exp − dζ I(ζ, z 0 , x0 ) , 2 z0

(5.38)

Computing Traces, Determinants, and ζ-Functions …

139

where we abbreviated ˆ I(z, z 0 , x0 ) =

∞

  d x [q(x) − z]−1/2 − [q(x) − z 0 ]−1/2 .

(5.39)

x0

Proof Since the resolvents of H+,α , α ∈ (0, π), and H+,0 differ only by a rank-one operator, it suffices to choose α = 0 when establishing (5.36). We will first prove (5.36) for z = λ < 0, z 0 = λ0 < λ < 0, and employ monotonicity of resolvents with respect to λ < 0 sufficiently negative, implying   0 ≤ (H+,0 − λI+ )−1 − (H+,0 − λ0 I+ )−1 , λ0 < λ < 0,

(5.40)

with λ < 0 sufficiently negative (the latter will be assumed for most of the remainder of this proof). Subsequently, we will apply (5.34) to K given by the right-hand side of inequality (5.40). Equations (5.23) and (5.24) yield for α = 0, φ0 (λ, · )ψ+,0 (λ, · ) = f +,1 (λ, · , x0 ) f +,2 (λ, · , x0 ) − f +,1 (λ, 0, x0 )−1 f +,2 (λ, 0, x0 ) f +,1 (λ, · , x0 )2 ,

(5.41)

and since by (5.21) for j = 1 integrability properties of (5.41) over R+ depend on those of f +,1 (λ, · , x0 ) f +,2 (λ, · , x0 ), we now investigate the latter on [x0 , ∞). Employing (5.21) once more then yields 0 ≤ [φ0 (λ, x)ψ+,0 (λ, x) − φ0 (λ0 , x)ψ+,0 (λ0 , x)]   = 2−1 [q(x) − λ]−1/2 − [q(x) − λ0 ]−1/2 [1 + o(1)] x→∞

= 4−1 (λ − λ0 )q(x)−3/2 [1 + o(1)]

x→∞

= 4−1 (λ − λ0 ) C0 x −1−(3ε0 /2) [1 + o(1)],

x→∞

(5.42)

according to (5.3), proving integrability of [φ0 (λ, · )ψ+,0 (λ, · ) − φ0 (λ0 , · )ψ+,0 (λ0 , · )] near +∞. An application of (5.34) to K given by the right-hand side of inequality (5.40) then yields (5.36) with z = λ, z 0 = λ0 < λ < 0. Analytic continuation in λ and subsequently in λ0 proves (5.36). By (2.7) with p = 2 this proves the first equality in (5.37). To prove the second equality in (5.37), we now apply Lemma 5.2 in the trace class case q = 1 and combine it with (5.36) to arrive at   tr L 2 (R+ ;d x) (H+,α − λI+ )−1 − (H+,α − λ0 I+ )−1     = lim tr L 2 (R+ ;d x) PR (H+,α − λI+ )−1 − (H+,α − λ0 I+ )−1 PR R→∞ ˆ R d x [φα (λ, x)ψ+,α (λ, x) − φα (λ0 , x)ψ+,α (λ0 , x)] = lim R→∞

0

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F. Gesztesy and K. Kirsten

. .       = lim W φα (λ0 , · ), ψ +,α (λ0 , · ) (R) − W φα (λ, · ), ψ +,α (λ, · ) (R) R→∞

. .     + W φα (λ, · ), ψ +,α (λ, · ) (0) − W φα (λ0 , · ), ψ +,α (λ0 , · ) (0) . .       = lim W φα (λ0 , · ), ψ +,α (λ0 , · ) (R) − W φα (λ, · ), ψ +,α (λ, · ) (R) , R→∞

(5.43) since

.

. .   W φα (λ, ·), ψ +,α (λ, ·) (0) = − sin(α)ψ+,α (λ, 0) − cos(α)ψ +,α (λ, 0)  d 

sin(α)ψ+,α (λ, 0) + cos(α)ψ+,α (λ, 0) = 0. (5.44) =− dλ It remains to analyze the right-hand side of (5.43). To this end we note that

.

.

τ+ f +,1 (z, x, x0 ) = z f +,1 (z, x, x0 ) + f +,1 (z, x, x0 ),

(5.45)

and hence

.

f +,1 (z, x, x0 ) = c1 (z) f +,1 (z, x, x0 ) + c2 (z) f +,2 (z, x, x0 ) ˆ x + f +,1 (z, x, x0 ) d x f +,1 (z, x , x0 ) f +,2 (z, x , x0 ) 0 ˆ x − f +,2 (z, x, x0 ) d x f +,1 (z, x , x0 )2 ,

.

(5.46)

0

f +,1 (z, x, x0 )



= c1 (z) f +,1 (z, x, x0 ) + c2 (z) f +,2 (z, x, x0 ) ˆ x

+ f +,1 (z, x, x0 ) d x f +,1 (z, x , x0 ) f +,2 (z, x , x0 ) ˆ0 x

− f +,2 (z, x, x0 ) d x f +,1 (z, x , x0 )2 . 0

(5.47)

Next, we claim that ˆ c2 (z) =

∞

d x f +,1 (z, x , x0 )2 , z ∈ ρ(H+,α ),

(5.48)

0

and hence (5.46), (5.47) simplify to

.

f +,1 (z, x, x0 ) = c1 (z) f +,1 (z, x, x0 ) ˆ x + f +,1 (z, x, x0 ) d x f +,1 (z, x , x0 ) f +,2 (z, x , x0 ) 0 ˆ ∞ + f +,2 (z, x, x0 ) d x f +,1 (z, x , x0 )2 , x

(5.49)

Computing Traces, Determinants, and ζ-Functions …

141

.



f +,1 (z, x, x0 ) = c1 (z) f +,1 (z, x, x0 ) ˆ x

+ f +,1 (z, x, x0 ) d x f +,1 (z, x , x0 ) f +,2 (z, x , x0 ) ˆ0 ∞

+ f +,2 (z, x, x0 ) d x f +,1 (z, x , x0 )2 .

(5.50)

x

To infer the necessity of (5.48), one can argue by contradiction as follows: If (5.48)

.

does not hold, then integrating f +,1 (z, x) with respect to z from λ0 to λ along the negative real axis on the left-hand side of (5.46) yields ˆ

λ

λ0

.

dz f +,1 (z, x, x0 ) = f +,1 (λ, x, x0 ) − f +,1 (λ0 , x, x0 ) −→ 0 x→∞

(5.51)

by the first asymptotic relation in (5.21). However, with (5.48) violated, integrating the right-hand side of (5.46) with respect to z from λ0 to λ along the negative real axis now yields several contributions vanishing as x → ∞ (again invoking (5.21)), but there will also be one integral of the type ˆ

λ

λ0

dz f +,2 (z, x, x0 )A(z, x) −→  0

(5.52)

x→∞

where A(z, · ) is bounded with a finite nonzero limit, lim x→∞ A(z, x) = A(z, ∞) = 0. Relation (5.52) contradicts (5.51), proving (5.48). Investigating the asymptotics of the right-hand sides of (5.49), (5.50), invoking the leading asymptotic behavior (5.21), then shows that to obtain the leading asymptotic

.

.

behavior of f +,1 (λ, x, x0 ), f +,2 (λ, x, x0 ) one can formally differentiate relations (5.21) with respect to λ and hence obtains, ˆ x f +,1 (λ, x, x0 ) = 2−3/2 [q(x) − λ]−1/4 d x

[q(x

) − λ]−1/2 x→∞ x0   ˆ x



1/2 [1 + o(1)], × exp − d x [q(x ) − λ] x0 ˆ x .

f +,1 (λ, x, x0 ) = −2−3/2 [q(x) − λ]1/4 d x

[q(x

) − λ]−1/2 x→∞ x0   ˆ x



1/2 [1 + o(1)], × exp − d x [q(x ) − λ]

.

x0

for λ < 0 sufficiently negative according to our convention in this proof. Next, one utilizes (5.23) and (5.24) and computes

(5.53)

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.   W φα (λ, · ), ψ +,α (λ, · ) (R) .

=

R→∞

f +,2 (λ, R, x0 ) f +,1 (λ, R, x0 )

.

.

.

.

(λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 ) sin(α) f +,1

− f +,2 (λ, R, x0 ) f +,1 (λ, R, x0 )

sin(α) f +,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 )

.

(λ, R, x0 ) f +,1 (λ, R, x0 ) − f +,2

+

(λ, f +,2

.

=

R→∞

sin(α) f+,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 ) R, x0 ) f +,1 (λ, R, x0 )

sin(α) f +,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 )

.



f +,1 (λ, R, x0 ) f +,2 (λ, R, x0 ) − f +,1 (λ, R, x0 ) f +,2 (λ, R, x0 )

.

.

sin(α) f+,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 ) + ,

sin(α) f +,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 )

(5.54)

again for λ < 0 sufficiently negative. Insertion of (5.21) and (5.53) into (5.54) finally implies

.



.

.

sin(α) f+,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 )

R→∞ sin(α) f +,1 (λ, 0, x 0 ) + cos(α) f +,1 (λ, 0, x 0 ) ˆ R  −1 −1/2 [1 + o(1)]. −2 d x [q(x) − λ]

 W φα (λ, · ), ψ +,α (λ, · ) (R) =

x0

(5.55) Returning to (5.43) this yields   tr L 2 (R+ ;d x) (H+,α − λI+ )−1 − (H+,α − λ0 I+ )−1 . .       = lim W φα (λ0 , · ), ψ +,α (λ0 , · ) (R) − W φα (λ, · ), ψ +,α (λ, · ) (R) R→∞

=

R→∞

.

.

.

.

sin(α) f +,1 (λ0 , 0, x0 ) + cos(α) f +,1 (λ0 , 0, x0 )

sin(α) f +,1 (λ0 , 0, x0 ) + cos(α) f +,1 (λ0 , 0, x0 )

(λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 ) sin(α) f +,1 −

sin(α) f +,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 ) ˆ R    −1 −1/2 −1/2 +2 [1 + o(1)] d x [q(x) − λ] − [q(x) − λ0 ]

.

x0

.

sin(α) f +,1 (λ0 , 0, x0 ) + cos(α) f +,1 (λ0 , 0, x0 ) =

sin(α) f +,1 (λ0 , 0, x0 ) + cos(α) f +,1 (λ0 , 0, x0 )

.

.

(λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 ) sin(α) f +,1 −

sin(α) f +,1 (λ, 0, x0 ) + cos(α) f +,1 (λ, 0, x0 )

Computing Traces, Determinants, and ζ-Functions …

+ 2−1

ˆ

∞

143

   d x [q(x) − λ]−1/2 − [q(x) − λ0 ]−1/2 ,

(5.56)

x0

and hence (5.37) for z = λ < 0, z 0 = λ0 < 0, both sufficiently negative. In this context one observes that for x0 > 0 sufficiently large, 2−1

ˆ

R

   d x [q(x) − λ]−1/2 − [q(x) − λ0 ]−1/2

x0

=

R→∞

1 (λ − λ0 ) 4

ˆ

R

d x q(x)

−3/2

 [1 + o(1)]

(5.57)

x0

with q −3/2 ∈ L 1 ([x0 , ∞); d x) by Hypothesis (5.3). Analytic continuation in λ of both sides in (5.56) extends the latter to z ∈ ρ(H+,α ). Similarly, analytic continuation in λ0 of both sides in (5.56) extends the latter to z 0 ∈ ρ(H+,α ), completing the proof of (5.37). Relation (5.38) then follows from integrating (5.37) with respect to the energy  variable from z 0 to z. Remark 5.4 Employing the resolvent equation, (H+,0 − z I+ )−1 − (H+,0 − z 0 I+ )−1 = (z − z 0 )(H+,0 − z I+ )−1 (H+,0 − z 0 I+ )−1 , z, z 0 ∈ ρ(H+,0 ), (5.58) an alternative proof of relation (5.36) follows upon establishing   (H+,0 − z I+ )−1 ∈ B2 L 2 (R+ ; d x) , z ∈ ρ(H+,0 ).

(5.59)

To prove (5.59) in turn it suffices to establish the Hilbert–Schmidt property for some z = λ < 0 sufficiently negative, followed by analytic continuation with respect to z ∈ ρ(H+,0 ). Given the Green’s function of H+,0 in (5.19), it thus suffices to prove that ˆ ˆ d x d x |φ0 (λ, x) ψ+,0 (λ, x )|2 < ∞; (5.60) R+

R+

we omit further details at this point. Next, we apply Theorem 5.3 to the following explicitly solvable example concerning the linear potential and denote by Ai( · ), Bi( · ) the Airy functions as discussed, for instance, in [1, Sect. 10.4]. Example 5.5 Consider the special case q(x) = x, x ∈ R+ , and α = 0. Then, for x ∈ R+ , z, z 0 ∈ ρ(H+,0 ),

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f +,1 (z, x, x0 ) = (2π)1/2 e(2/3)(x0 −z) Ai(x − z),

(5.61)

f +,2 (z, x, x0 ) = (π/2)1/2 e−(2/3)(x0 −z) Bi(x − z),

(5.62)

W ( f +,1 (z, · , x0 ), f +,2 (z, · , x0 )) = 1, φ0 (z, x) = π[Ai(−z)Bi(x − z) − Bi(−z)Ai(x − z)],

(5.63) (5.64)

ψ+,0 (z, x) = Ai(x − z)/Ai(−z),

(5.65)

W (φ0 (z, · ), ψ +,0 (z, · ))(x)

(5.66)

3/2

3/2

.

= π[Ai (x − z)Bi (x − z) − (x − z)Ai(x − z)Bi(x − z)] − [Ai (−z)/Ai(−z)], ˆ ∞   I(z, z 0 , x0 ) = d x [x − z]−1/2 − [x − z 0 ]−1/2 x0   = 2 (x0 − z 0 )1/2 − (x0 − z)1/2 , (5.67)   −1 −1 tr L 2 (R+ ;d x) (H+,0 − z I+ ) − (H+,0 − z 0 I+ )



= ψ+,0 (z, 0) − ψ+,0 (z 0 , 0) = [Ai (−z)/Ai(−z)] − [Ai (−z 0 )/Ai(−z 0 )], (5.68)   −1 det 2,L 2 (R+ ;d x) I+ − (z − z 0 )(H+,0 − z 0 I+ )   = [Ai(−z)/Ai(−z 0 )] exp (z − z 0 )[Ai (−z 0 )/Ai(−z 0 )] . (5.69) We note that (5.69) was recently considered in [48], but the exponential factor in (5.69) was missed in [48]. Finally, we generalize Theorem 5.3 to the following setting. Theorem 5.6 Assume Hypothesis 5.1, z ∈ ρ(H+,α2 ), z 0 ∈ ρ(H+,α1 ), and α1 , α2 ∈ [0, π). Then,     (H+,α2 − z I+ )−1 − (H+,α1 − z 0 I+ )−1 ∈ B1 L 2 (R+ ; d x) ,

(5.70)

and (cf. (5.39))   tr L 2 (R+ ;d x) (H+,α2 − z I+ )−1 − (H+,α1 − z 0 I+ )−1  

(z, 0, x0 ) + cos(α2 ) f +,1 (z, 0, x0 ) sin(α2 ) f +,1 d , = − ln

dz sin(α1 ) f +,1 (z 0 , 0, x0 ) + cos(α1 ) f +,1 (z 0 , 0, x0 ) 1 + I(z, z 0 , x0 ). 2

(5.71)

Proof Equation (5.70) is established exactly as in the proof of Theorem 5.3. Furthermore, as argued there one has   tr L 2 (R+ ;d x) (H+,α2 − λI+ )−1 − (H+,α1 − λ0 I+ )−1 . .       = lim W φα1 (λ0 , · ), ψ +,α1 (λ0 , · ) (R) − W φα2 (λ, · ), ψ +,α2 (λ, · ) (R) . R→∞

(5.72)

Computing Traces, Determinants, and ζ-Functions …

Using Eq. (5.55) then immediately implies (5.71).

145



Setting z = z 0 , we obtain in particular   tr L 2 (R+ ;d x) (H+,α2 − z I+ )−1 − (H+,α1 − z I+ )−1  

sin(α2 ) f +,1 (z, 0, x0 ) + cos(α2 ) f +,1 (z, 0, x0 ) d . = − ln

dz sin(α1 ) f +,1 (z, 0, x0 ) + cos(α1 ) f +,1 (z, 0, x0 )

(5.73)

Remark 5.7 In order to prove Theorem 5.6, one could instead have proven the slightly simpler result (5.73) and then note that   tr L 2 (R+ ;d x) (H+,α2 − z I+ )−1 − (H+,α1 − z 0 I+ )−1   = tr L 2 (R+ ;d x) (H+,α2 − z I+ )−1 − (H+,α1 − z I+ )−1   + tr L 2 (R+ ;d x) (H+,α1 − z I+ )−1 − (H+,α1 − z 0 I+ )−1 ,

(5.74)

which, using (5.73) together with Theorem 5.3 implies Theorem 5.6.



Acknowledgements We are indebted to Christer Bennewitz for very helpful discussions.

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On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential Bernard Helffer and Jean Nourrigat

Abstract The aim of this paper is to review and compare the spectral properties of the Schrödinger operators −Δ + U (U ≥ 0) and −Δ + i V in L 2 (Rd ) for C ∞ real potentials U or V with polynomial behavior. The case with magnetic field will be also considered. We present the existing criteria for essential self-adjointness, maximal accretivity, compactness of the resolvent, and maximal inequalities. Motivated by recent works with X. Pan, Y. Almog, and D. Grebenkov, we actually improve the known results in the case with purely imaginary potential.

1 Introduction In this paper,1 , we will discuss the spectral theory for the Schrödinger operators −Δ + U (U ≥ 0) or −Δ + i V in L 2 (Rd ). More precisely, we present the state of the art (and sometime will go beyond) for the following properties: • essential self-adjointness or maximal accretivity, • compactness of the resolvent, • maximal inequalities, i.e., the existence of C > 0 such that, ∀u ∈ C0∞ (Rd ),   ||u||2H 2 (Rd ) + ||U u||2L 2 (Rd ) ≤ C ||(−Δ + U )u||2L 2 (Rd ) + ||u||2L 2 (Rd ) ,

(1)

or 1 The results of this paper have been presented by the first author at the Kato centennial confer-

ence in Tokyo in September 2017. B. Helffer (B) Laboratoire de Mathématiques Jean Leray, CNRS and Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex, France e-mail: [email protected] J. Nourrigat LMR EA 4535 and FR CNRS 3399, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039 51687 REIMS Cedex 2, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 149 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_8

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  ||u||2H 2 + ||V u||2 ≤ C ||(−Δ + i V )u||2 + ||u||2 .

(2)

We will also discuss the case with magnetic field where the operator reads: PA,W = −ΔA + W :=

d 

(Dx j − A j (x))2 + W (x) ,

j=1

where Dx j = −i ∂∂x j , A = (A1 , . . . , Ad ) is a C ∞ vector field on Rd and W is a complex-valued C ∞ potential, and the maximal regularity is expressed in terms of the magnetic Sobolev spaces:  ||(D − A)u||2L 2 (Rd ,Cd ) + j, ||(D j − A j )(D − A )u||2L 2 (Rd ) + || |W | u||2L 2 (Rd )   ≤ C ||PA,W u||2L 2 (Rd ) + ||u||2L 2 (Rd ) , (3) The question of analyzing −Δ + i V or more generally PA,i V := −ΔA + i V appears in many situations [1–3]. It seems therefore useful to present in a unified way, what is known on the subject in the self-adjoint case and try to go further in the accretive case, where much less is known. If we assume that the potential V is C ∞ , we know that the operator is essentially self-adjoint starting from C0∞ (Rd ) in the first situation and maximally accretive in the second case. Hence in the two cases, the closed operator in consideration is uniquely defined by its restriction to C0∞ . At least for the self-adjoint case, the subject has a long story, in which T. Kato and his school plays an important role. We refer to [36] for a rather complete presentation with an exhaustive list of reference. One should also mention the work of Avron–Herbst–Simon [5] which popularizes the basic questions on the subject and in particular the magnetic bottles. For the compactness of the resolvent, outside the easy case when U → +∞, the story starts around the 80s with the treatment of instructive examples (Simon [35], Robert [29]) and in the case with magnetic field [5] (the simplest example being for d = 2 and U = 0, when B(x) → +∞). In the polynomial case, many results are deduced as a byproduct of the analysis in Helffer–Nourrigat [15], at least in the case when V is a sum of square of polynomials. Using Kohn’s type inequality, Helffer and Morame (Mohamed) [14] obtain more general results which can be combined with the analysis of Iwatsuka [19]. Another family of results using the notion of capacity can be found in [21, 22] (see references therein). T. Kato proves, for example, the inequality ||Δu|| L 1 + ||U u|| L 1 ≤ 3 ||(−Δ + U )u|| L 1 , ∀u ∈ C0∞ (Rd ) ,

(4)

1 . under the condition that U ≥ 0 and U ∈ L loc p The generalization to the L ( p > 1) is only possible under stronger conditions 2 on U . We will mention some of these results but will focus on the  L estimates which are sometimes easier to obtain. In the case, when U (x) =  U (x)2 , the

On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential

151

maximal L 2 estimate is obtained as a byproduct of the analysis of the hypoellipticity (see Hörmander [17], Rothschild–Stein [30] and the book Helffer–Nourrigat [16] (including polynomial magnetic potentials)). This was then generalized to the case when V is a positive polynomial by J. Nourrigat in an unpublished paper [27] and used in the PhD of Guibourg [11, 12] defended in 1992, which considers the case when the electric potential U ≥ 0 and the magnetic potential A are polynomials (one chapter treats a more general situation). In his thesis, Zhong [37] proves the same result by showing that ∇ 2 (−Δ + U )−1 is a Calderon–Zygmund operator. Shen [31] generalizes the result to the case when U is in the reverse Hölder class R Hq (q ≥ d2 ), a class which contains the positive polynomials. Definition 1 A locally L q function ω and strongly positive almost everywhere belongs to R Hq if there exists a constant C > 0 such that for any cube Q in Rd 

1 |Q|

ωq d x

q1

 ≤C

1 |Q|



ω dx

.

One should also mention the unpublished thesis of Mba-Yébé [23] defended in 1995. Together with the techniques developed by Guibourg, some of his techniques are useful for the improvements presented in the last section. Z. Shen considers also the case with magnetic fields in 1996 [32]. Further progress is obtained in the thesis of Ben Ali [6], published in [4, 7, 8]. The methods applied by Shen and Auscher–Ben Ali include the Fefferman–Phong inequalities, the Calderón– Zygmund decompositions, and various techniques of interpolation. We will come back to one of these results in Sect. 2.3.

2 Kohn’s Approach This approach was mainly used for getting the compactness of the resolvent. Except in a few cases, these estimates do not lead to the maximal regularity but are sufficient for getting the compactness and we will see that surprisingly, they could also be a step for proving L 2 -maximal estimates.

2.1 Self-adjoint Case Here we mainly refer to [14] (see also [15, 24]). We analyze the problem for the family of operators: PA,U

p d   2 = (Dx j − A j (x)) + U (x)2 . j=1

=1

(5)

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B. Helffer and J. Nourrigat

Here, the magnetic potential A(x) = (A1 (x), A2 (x), · · · , An (x)) and the U j are assumed to be C ∞ . Under these conditions, the operator is essentially self-adjoint on C0∞ (Rd ). We note also that it can be written in the form PA,U =

d+ p 

X 2j =

j=1

d 

X 2j +

p 

Y2 ,

=1

j=1

with X j = (Dx j − A j (x)) , j = 1, . . . , d , Y = U ,  = 1, . . . , p . In particular, the magnetic field is recovered by observing that B jk =

1 [X j , X k ] = ∂ j Ak − ∂k A j , for j, k = 1, . . . , d . i

Of course, when U → +∞, it is well known that the operator has a compact resolvent. (see the argument below). On the opposite, when V = 0 and d = 2 if B(x) = B12 ≥ 0 , one immediately deduces from the trivial inequality: B(x)|u(x)|2 d x ≤ ||X 1 u||2 + ||X 2 u||2 = PA,U u | u .

(6)

that lim|x|→+∞ B(x) = +∞ implies that the operator has a compact resolvent. A typical example is A1 (x1 , x2 ) = −x2 x12 , A2 (x1 , x2 ) = +x1 x22 . In order to treat more general situations, we introduce the quantities: mˇ q (x) =

 

|∂xα U | +

|α|=q

 

|∂xα B jk (x)| .

(7)

j 0, C sup ω(x) ≤ ω(y) dy . (34) |B(x, r )| B(x,r ) y∈B(x,r ) Note that a nonnegative polynomial satisfies this condition. Then, other authors work on the subject with the aim of obtaining L p estimates [4, 6]. The case with magnetic fields is always considered (see [7, 8, 32, 33] and references therein).

2 There

are actually two  different proofs proposed by J. Nourrigat a rather direct one and another based to the analysis of j Xˇ 2j + i Xˇ 0 the difficulty (but this was sometimes treated in [15]) that this operator is no more hypoelliptic.

On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential

159

4 Maximal Estimates for the Complex Schrödinger Operator with Magnetic Potentials (Non Necessarily Polynomial Case) 4.1 Main Statement We consider as before W =



U2 + i V ,



and the associated complex Schrödinger operator PA,W . Theorem 5 If (A, W ) ∈ T (r, C0 ), there exists C > 0 such that, for all u ∈ C0∞ (Rd ):   |W |u2 ≤ C PA,W u2 + ||u||2 .

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4.2 Hörmander’s Metrics and Partition of Unity We introduce, for t ∈ [0, 1] and x ∈ Rd , Φ(x, t) =

  |α|≤r

t |α|+1 |∂xα U (x)| +

 



t |α|+2 |∂xα B jk (x)| +

j 1 such that, for all t ∈ (0, 1), we have the implication: |y − x| ≤ t =⇒ Φ(y, t) ≤ C2 Φ(x, t) + C2 t r +1 . Proof For all x et u in Rd such that |u| = 1, for all t and θ such that 0 < θ ≤ t ≤ 1, let us introduce:

Ψ (x, u, θ, t) = Φ(x + θ u, t) . Using Taylor’s formula with integral remainder, we can write, if θ ≤ t ,

Ψ (x, u, θ, t) ≤ C Φ(x, t) + CR(t) ,

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B. Helffer and J. Nourrigat

with R(t) := t r +1

  

... + Ct r +1

 j 0 such that θ Ψ (x, u, σ, t) dσ + C t r +1 . R(t) ≤ C 0

We now apply Gronwall’s Lemma and obtain the existence of C2 > 0 such that, for θ ≤ t ≤ 1,

Ψ (x, u, θ, t) ≤ C2 Φ(x, t) + C2 t r +1 . This achieves the proof of the proposition. Proposition 2 Let C2 > 1 the constant of Proposition 1. Then we have: |y − x| ≤

R(x, μ) 1 R(y, μ) =⇒ ≤ ≤ 2C2 . 2C2 2C2 R(x, μ)

Proof We apply Proposition 1 with t0 = R(x, μ) ≤ 1. If |y − x| ≤ R(x, μ), we have Φ(y, t0 ) ≤ C2 (Φ(x, t0 ) + t0r +1 ) ≤ 2C2 μ . We have indeed Φ(x, t0 ) ≤ μ and t0r +1 ≤ 1 ≤ μ . Consequently t1 = t0 /2C2 = R(x, μ)/(2C2 ) satisfies t1 ≤ 1 and Φ(y, t1 ) ≤ μ . Therefore we get t1 ≤ R(y, μ), and consequently the first inequality above. If now |y − x| ≤ R(x, μ)/(2C2 ), we deduce |y − x| ≤ R(y, μ), and, permuting the roles of x and y, we effectively get R(y, μ) ≤ 2C2 R(x, μ) . This achieves the proof of the proposition. This proposition shows that the metric defined on Rd by gx (t) = |t|2 /R(x, μ)2 (x ∈ Rd , t ∈ Rd ), is slowly varying in the sense of Definition 18.4.1 in [18]. Moreover, the constant in the definition can be chosen independently of μ. We deduce from Lemma 18.4.4 in [18] the following proposition. Proposition 3 For any μ ≥ 1, there exist a sequence of real valued functions (ϕm ) in C0∞ (Rd ), and a sequence (x m ) in Rd , such that:

On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential

•



ϕm (x)2 = 1 , ∀x ∈ Rd .

161

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m

•

supp ϕm ⊂ B(x m , R(x m , μ)) .

(38)

• For any multi-index α, there exists Cˆ α > 0, independent of μ, such that 

|∂ α ϕm (x)|2 ≤

m

Cˆ α . R(x, μ)2|α|

(39)

• There exists Cˆ > 0, independent of μ, such that, for k = 1, 2, for any u in C0∞ (Rd ),  ϕm (x)2 |u(x)|2 |u(x)|2 ˆ d x ≤ C dx . (40) 2k R(x m , μ)2k Rd R(x, μ) Rd j

4.3 Proof of Theorem 5 Just observing that: < PU +i V f, f >= ||(D − A) f ||2 +

U | f |2 d x ,

we obtain: Lemma 3 For all f ∈ C0∞ (Rd ), we have:  j

(Dx j − A j ) f 2 +

p 

U f 2 ≤ PA,W f   f  .

(41)

=1

Proposition 4 For any μ > 1, let (x m ) be the sequence of points in Rd constructed in Proposition 3 and let (A, W ) ∈ T (r, C0 ). Then, there exist μ0 > 1 and C3 (depending only on r and C0 ) such that, for any μ ≥ μ0 , for any m such that R(x m , μ) ≤ 1/2 , and for any f ∈ C0∞ (Rd ) supported in the ball Bm = B(x m , R(x m , μ)), μδ μδ/2 (D − A) f  ≤ C3 PA,W f   f  + R(x m , μ)2 R(x m , μ) where δ is the constant given by Theorem 3. Proof With Rm := R(x m , μ), we introduce

(42)

162

B. Helffer and J. Nourrigat

V (m,loc) (y) = Rm2 V (x m + Rm y) ,

U(m,loc) (y) = Rm U (x m + Rm y) ,

(y) = Rm Ak (x m + Rm y) , A(m,loc) k

B (m,loc) (y) = Rm2 B jk (x m + Rm y) . jk

If Rm ≤ 1, one verifies that, for any (A, W ) ∈ T (r, C0 ), the corresponding pair (A(m,loc) , W (m,loc) ) belongs to T (r, C0 ). If Rm ≤ 1/2, we have Φ(x m , R(x m , μ)) = μ. Applying Proposition 1 with t = Rm = R(x m , μ) ≤ 1, we have, if |y| ≤ 1 , Φ(x m , Rm ) ≤ C2 Φ(x m + Rm y, Rm ) + C2 ≤ C2 mˇ r,m,loc (y) , where mˇ r,m,loc (y) is the function associated, as in (17), with the localized operator PAm,loc ,W m,loc at the point x m . Consequently, we have μ ≤ C2 mˇ r,m,loc (y) for all y ∈ Rd such that |y| ≤ 1. By Theorem 3, for all g ∈ C0∞ (Rd ) with support in B(0, 1), we have: μδ (1/C2 )δ g ≤ C1 (PAm,loc ,W m,loc g + g) where C1 and C2 depend only on C0 . Then, one can find μ0 and C3 with the same properties such that, for μ ≥ μ0 , μδ g ≤ C3 PAm,loc ,W m,loc g .

(43)

If f is supported in B(x m , R(x m , μ)), we apply (43) to the function g(y) = f (x m + y R(x m , μ)) and obtain for μ ≥ μ0 μδ  f  ≤ C3 PA,W f  . R(x m , μ)2 Inequality (41) leads to (42).

4.4 End of the Proof of Theorem 5 Let u ∈ C0∞ (Rd ). For any μ ≥ 1, we apply (40) and get, distinguishing in the localization formula the x m such that R(x m , μ) > 21 from the terms such that R(x m , μ) ≤ 21 , 

Rd

 |u(x)|2 |(D − A)u(x)|2 d x ≤ C(u2 + (D − A)u2 ) + R , + R(x, μ)4 R(x, μ)2  ϕm u2 (D − A)(ϕm u)2 + . R := C R(x m , μ)4 R(x m , μ)2 R(x m ,μ)≤1/2

On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential

163

Here we have also used (39) for the control of commutators. If μ ≥ μ0 with μ0 large enough, and for any m such that R(x m , μ) ≤ 1/2, we apply Proposition 4 to the function f = ϕm u and obtain: R ≤ Cμ−2δ ≤



2 R(x m ,μ)≤ 21 PA,W (ϕm u)   Cμ−2δ PA,W u2 + Cμ−2δ m ∇ϕm

 · (∇ − iA)u2 + u(Δϕm )2 .

From (39), we deduce:

R ≤ Cμ−2δ PA,W u2 + Cμ−2δ



Rd

 |u(x)|2 |(D − A)u(x)|2 dx . + R(x, μ)4 R(x, μ)2

There exists a possibly larger new μ0 such that, for μ ≥ μ0 , 

Rd

 |u(x)|2 |(D − A)u(x)|2 d x ≤ C (u2 + (D − A)u2 ) + C μ−2δ PA,W u2 . + R(x, μ)4 R(x, μ)2

Using again (41), we get: 

Rd

 |u(x)|2 |(D − A)u(x)|2 d x ≤ C u2 + C (1 + μ−2δ )PA,W u2 . + R(x, μ)4 R(x, μ)2

Theorem 5 follows since Φ(x, R(x, μ)) ≤ μ and consequently R(x, μ)

 

|U (x)| + R(x, μ)2



|B jk (x)| + R(x, μ)2 |V (x)| ≤ μ .

j 0. However, if A and B are not commuting, then the question is rather complicated. Indeed, although we can prove the existence of the limit lim p→∞ (A p/2 B p A p/2 )1/ p , the description of the limit has a rather complicated combinatorial nature. Moreover, it is unknown so far whether the limit of (A p # B p )2/ p as p → ∞ exists or not. In the present paper, we consider a similar (but seemingly a bit simpler) question about what happens to the limits of (B A p B)1/ p and (A p # B)1/ p as p tends to ∞, the case where B is fixed without the p-power, in certain more general settings. The rest of the paper is organized as follows. In Section 2, we first prove the existence of the limit of (K A p K ∗ )1/ p as p → ∞ and give the description of the limit in terms of the diagonalization (eigenvalues and eigenvectors) data of A and the images of the eigenvectors by K . We then extend the result to the limit of (A p )1/ p as p → ∞ for a positive linear map  between matrix algebras. For instance, this limit is applied to the map (A ⊕ B) := (A + B)/2 to reformulate Kato’s limit theorem ((A p + B p )/2)1/ p → A ∨ B in [12]. Another application is given to find the limit formula as α  0 of the sandwiched α-Rényi divergence [14, 18], a new relative entropy relevant to quantum information theory. In Section 3, we discuss the limit behavior of (A p σ B)1/ p as p → ∞ for operator means σ. To do this, we may assume without loss of generality that B is an orthogonal projection E. Under a certain condition on σ, we prove that (A p σ E)1/ p is decreasing as 1 ≤ p → ∞, so that the limit as p → ∞ exists. Furthermore, when σ is the weighted geometric mean, we obtain an explicit description of the limit in terms of E and the spectral projections of A. It is worth noting that a limit formula in the same vein as those in [12] and this paper was formerly given in [2, 3] for the spectral shorting operation.

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

169

2 lim p→∞ ( A p )1/ p for Positive Linear Maps  For each n ∈ N, we write Mn for the n × n complex matrix algebra and M+ n for the set of positive semidefinite matrices in Mn . When A ∈ Mn is positive definite, we write A > 0. We denote by Tr the usual trace functional on Mn . For A ∈ M+ n, λ1 (A) ≥ · · · ≥ λn (A) are the eigenvalues of A in decreasing order with multiplicities, and ran A is the range of A. Let A ∈ M+ n be given, whose diagonalization is A = V diag(a1 , . . . , an )V ∗ =

n 

ai |vi vi |

(2.1)

i=1

with the eigenvalues a1 ≥ · · · ≥ an and a unitary matrix V = [v1 · · · vn ] so that Avi = ai vi for 1 ≤ i ≤ d. Let K ∈ Mn and assume that K = 0 (our problem below is trivial when K = 0). Consider the sequence of vectors K v1 , . . . , K vn in Cn . Let 1 ≤ l1 < l2 < · · · < lm be chosen so that if lk−1 < i < lk for 1 ≤ k ≤ m, then K vi is in span{K vl1 , . . . , K vlk−1 } (this means, in particular, K vi = 0 if i < l1 ). Then {K vl1 , . . . , K vlm } is a linearly independent subset of {K v1 , . . . , K vn }, so we perform the Gram–Schmidt orthogonalization to obtain an orthonormal vectors u 1 , . . . , u m from K vl1 , . . . , K vlm . In particular, u 1 = K vl1 /K vl1 . The next theorem is our first limit theorem. This implicitly says that the right-hand side of (2.2) is independent of the expression of (2.1) (note that vi ’s are not unique for degenerate eigenvalues ai ). Theorem 2.1 We have lim (K A p K ∗ )1/ p =

p→∞

m 

alk |u k u k |,

(2.2)

k=1

and in particular, ∗ 1/ p

lim λk ((K A K ) p

p→∞

 alk , 1 ≤ k ≤ m, )= 0, m < k ≤ n.

(2.3)

Proof Write Z p := (K A p K ∗ )1/ p and λi ( p) := λi (Z p ) for p > 0 and 1 ≤ i ≤ n. First we prove (2.3). Note that Z pp = K A p K ∗ = K V diag(a1 , . . . , anp )V ∗ K ∗  p  ∗ p = a1 K v1 a2 K v2 · · · anp K vn K v1 K v2 · · · K vn . p

Since

(2.4)

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F. Hiai

∗  p   λ1 ( p) p ≤ Tr Z pp = Tr K v1 · · · K vn a1 K v1 · · · anp K vn =

n 

p

p

ai K vi , K vi ≤ al1

i=1

n 

K vi 2 ,

i=1

we have lim sup λ1 ( p) ≤ al1 . p→∞

Moreover, since nλ1 ( p) p ≥ Tr Z pp =

n 

p

p

ai K vi , K vi ≥ al1 K vl1 2 ,

i=1

we have lim inf λ1 ( p) ≥ al1 . p→∞

Therefore, (2.3) holds for k = 1. To prove (2.3) for k > 1, we consider the antisymmetric tensor powers A∧k and ∧k K for each k = 1, . . . , n. Note that   ∧k ∧k p ∧k ∗ 1/ p Z ∧k p = (K )(A ) (K ) and

A∧k =



(2.5)

ai1 · · · aik |vi1 ∧ · · · ∧ vik vi1 ∧ · · · ∧ vik |.

1≤i 1 0 choose an orthonormal basis {u 1 ( p), . . . , u n ( p)} of Cn for which Z p u i ( p) = λi ( p)u i ( p) for 1 ≤ i ≤ n. To prove (2.2), write a˜ k := alk for 1 ≤ k ≤ m. If a˜ 1 = 0 then it is obvious that lim p→∞ Z p = 0. So assume that a˜ 1 > 0 and further-

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

171

more a˜ 1 > a˜ 2 , i.e., lim p→∞ λ1 ( p) > lim p→∞ λ2 ( p) at the moment. From (2.4) we have Z pp

=

n 

p

ai |K vi K vi |

i=1

=

l 2 −1

p ai |K vi K vi |

+

i=l1

=

l 2 −1

n 

p

ai |K vi K vi |

i=l2 p

ai K vi 2 |u 1 u 1 | +

i=l1

n 

p

ai |K vi K vi |

i=l2

so that

Zp a˜ 1

p =

l 2 −1

i=l1

ai a˜ 1

p K vi  |u 1 u 1 | + 2

 n

 ai p i=l2

a˜ 1

|K vi K vi | −→ α|u 1 u 1 |

as p → ∞ for some α > 0, since ai /a˜ 1 ≤ a˜ 2 /a˜ 1 < 1 for i ≥ l2 . Hence, for any p > 0 sufficiently large, the largest eigenvalue of (Z p /a˜ 1 ) p is simple and the corresponding eigen projection converges to |u 1 u 1 | as p → ∞. Since the eigen projection E 1 ( p) of Z p corresponding to the largest eigenvalue λ1 ( p) (simple for any large p > 0) is the same as that of (Z p /a˜ 1 ) p , we have E 1 ( p) = |u 1 ( p) u 1 ( p)| −→ |u 1 u 1 | as p → ∞. In the general situation, we assume that a˜ 1 ≥ · · · ≥ a˜ k > a˜ k+1 with 1 ≤ k ≤ m, where a˜ k+1 = 0 if k = m. From (2.3), note that ˜ 1 · · · a˜ k−1 a˜ k > a˜ 1 · · · a˜ k−1 a˜ k+1 = lim λ2 (Z ∧k lim λ1 (Z ∧k p )=a p ).

p→∞

p→∞

Hence, for any sufficiently large p > 0, the largest eigenvalue λ1 (Z ∧k p ) = λ1 ( p) · · · λk ( p) of Z ∧k is simple, and from the above case applied to (2.5) it follows that p |u 1 ( p) ∧ · · · ∧ u k ( p) u 1 ( p) ∧ · · · ∧ u k ( p)| −→ |u 1 ∧ · · · ∧ u k u 1 ∧ · · · ∧ u k | as p → ∞,

since the vector in the present situation corresponding to u 1 = K vl1 /K vl1  is K ∧k (vl1 ∧ · · · ∧ vlk ) K vl1 ∧ · · · ∧ K vlk = = u1 ∧ · · · ∧ uk . ∧k K (vl1 ∧ · · · ∧ vlk ) K vl1 ∧ · · · ∧ K vlk  (The last identity follows from the fact that, for linearly independent w1 , . . . , wk , if w1 , . . . , wk are the w1 ∧ · · · ∧ wk /w1 ∧ · · · ∧ wk  = w1 ∧ · · · ∧ wk Gram–Schmidt orthogonalization of w1 , . . . , wk .) By [6, Lemma 2.4], we see that the

172

F. Hiai

orthogonal projection E k ( p) onto span{u 1 ( p), . . . , u k ( p)} converges to the orthogonal projection E k of span{u 1 , . . . , u k }. Finally, let 0 = k0 < k1 < · · · < ks−1 < ks = m be such that a˜ 1 = · · · = a˜ k1 > a˜ k1 +1 = · · · = a˜ k2 > · · · > a˜ ks−1 +1 = · · · = a˜ ks . The above argument says that, for every r = 1, . . . , s − 1, the orthogonal projection E kr ( p) onto span{u 1 ( p), . . . , u kr ( p)} converges to the orthogonal projection E kr onto span{u 1 , . . . , u kr }. When a˜ ks > 0, this holds for r = s as well. Therefore, when a˜ ks > 0, we have Zp =

n 

λi ( p)|u i ( p) u i ( p)|

i=1

=

kr s  

λi ( p)|u i ( p) u i ( p)| +

r =1 i=kr −1 +1

=

kr s  

n 

(λi ( p) − a˜ i )|u i ( p) u i ( p)| +

r =1 i=kr −1 +1 n 

+

λi ( p)|u i ( p) u i ( p)|

i=ks +1 s 

a˜ kr (E kr ( p) − E kr −1 ( p))

r =1

λi ( p)|u i ( p) u i ( p)|

i=ks +1

−→

s 

a˜ kr (E kr − E kr −1 ) =

r =1

m 

a˜ i |u i u i |, where E 0 ( p) = E 0 = 0.

i=1

When a˜ ks = 0, we may modify the above estimate as Zp =

kr s−1  

λi ( p)|u i ( p) u i ( p)| +

r =1 i=kr −1 +1

−→

s−1  r =1

a˜ kr (E kr − E kr −1 ) =

n 

λi ( p)|u i ( p) u i ( p)|

i=ks−1 +1 ks−1  i=1

a˜ i |u i u i | =

m  i=1

a˜ i |u i u i |. 

The following corollary of Theorem 2.1 is an improvement of [7, Theorem 1.2]. Corollary 2.2 Let A ∈ Mn be positive definite. Then lim p→∞ λi ((K A p K ∗ )1/ p)=ai for all i = 1, . . . , n if and only if {K v1 , . . . , K vn } is linearly independent. Remark 2.3 Note that Theorem 2.1 can easily extend to the case where K is a  rectangle n  ×

n matrix. In fact, when n < n we may apply Theorem 2.1 to n × n   K matrices and A, and when n  > n we may apply to n  × n  matrices K O and O A ⊕ On  −n .

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

173

+ A linear map  : Mn → Mn  is said to be positive if (A) ∈ M+ n  for all A ∈ Mn , which is further said to be strictly positive if (In ) > 0, that is, (A) > 0 for all A ∈ Mn , A > 0. The following is an extended and refined version of Theorem 2.1.

Theorem Let  : Mn → Mn  be a positive linear map. Let A ∈ M+ n be given 2.4 n as A = i=1 ai |vi vi | with a1 ≥ · · · ≥ an and an orthonormal basis {v1 , . . . , vn } of Cn . Then lim p→∞ (A p )1/ p exists and lim (A p )1/ p =

p→∞

n 

ai PMi ,

i=1

where M1 := ran (|v1 v1 |), Mi :=

i 

ran (|v j v j |) 

j=1

i−1 

ran (|v j v j |),

2 ≤ i ≤ n,

j=1

and PMi is the orthogonal projection onto Mi for 1 ≤ i ≤ n. Proof Let C ∗ (I, A) be the commutative C ∗ -subalgebra of Mn generated by I, A. We can consider the composition of the conditional expectation from Mn onto C ∗ (I, A) with respect to Tr and |C ∗ (I,A) : C ∗ (I, A) → Mn  instead of , so we may assume that  is completely positive. By the Stinespring representation there are a ν ∈ N, a ∗ homomorphism π : Mn → Mnν and a linear map K : Cnν → Cn such that (X ) = K π(X )K ∗ for all X ∈ Mn . Moreover, since π : Mn → Mnν is represented, under a suitable change of an orthonormal basis of Cnν , as π(X ) = Iν ⊗ X for all X ∈ Mn under identification Mnν = Mν ⊗ Mn , we can assume that  is given (with a change of K ) as X ∈ Mn . (X ) = K (Iν ⊗ X )K ∗ , We then write   Iν ⊗ A = (Iν ⊗ V )diag a1 , . . . , a1 , a2 , . . . , a2 , . . . , an , . . . , an (Iν ⊗ V )∗          ν

=

n 

ν

ν

ai (|e1 ⊗ vi e1 ⊗ vi | + · · · + |eν ⊗ vi eν ⊗ vi |).

i=1 

Now, we consider the following sequence of nν vectors in Cn : K (e1 ⊗ v1 ), . . . , K (eν ⊗ v1 ), K (e1 ⊗ v2 ), . . . , K (eν ⊗ v2 ), . . . , K (e1 ⊗ vn ), . . . , K (eν ⊗ vn ),

and if K (e j ⊗ vi ) is a linear combination of the vectors in the sequence preceding it, then we remove it from the sequence. We write the resulting linearly independent

174

F. Hiai

subsequence as K (e j ⊗ vl1 ) ( j ∈ J1 ), K (e j ⊗ vl2 ) ( j ∈ J2 ), . . . , K (e j ⊗ vlm ) ( j ∈ Jm ), where 1 ≤ l1 < l2 < · · · < lm ≤ n and J1 , . . . , Jm ⊂ {1, . . . , ν}. Furthermore, by performing the Gram–Schmidt orthogonalization to this subsequence, we end up  making an orthonormal sequence of vectors in Cn as follows: u (lj 1 ) ( j ∈ J1 ), u (lj 2 ) ( j ∈ J2 ), . . . , u (lj m ) ( j ∈ Jm ). Since

(A p )1/ p = (K (Iν ⊗ A) p K ∗ )1/ p ,

Theorem 2.1 and Remark 2.3 imply that lim p→∞ (A p )1/ p exists and lim (A p )1/ p =

p→∞

m 

alk

k=1



  (lk )  (lk )  u  , u j j

j∈Jk

    where j∈Jk u (lj k ) u (lj k )  = 0 if Jk = ∅.     The next step of the proof is to find what is j∈Jk u (lj k ) u (lj k )  for 1 ≤ k ≤ m. For this we first note that ν 

|K (e j ⊗ vi ) K (e j ⊗ vi )| = K

j=1

 ν



|e j ⊗ vi e j ⊗ vi | K ∗

j=1

= K (Iν ⊗ |vi vi |)K ∗ = (|vi vi |). From Lemma 2.5 below this implies that Ri := ran (|vi vi |) = span{K (e j ⊗ vi ) : 1 ≤ j ≤ ν}. Through the procedure of the Gram–Schmidt diagonalization we see that Ri = 0,

1 ≤ i < l1 ,

Rl1 = span u (lj 1 ) : j ∈ J1 , 

Ri ⊂ Rl1 ,

l1 < i < l2 ,  (Rl1 ∨ Rl2 )  Rl1 = span u (lj 2 ) : j ∈ J2 , Ri ⊂ Rl1 ∨ Rl2 ,

l2 < i < l3 ,

 (Rl1 ∨ Rl2 ∨ Rl3 )  (Rl1 ∨ Rl2 ) = span u (lj 3 ) : j ∈ J3 , .. .

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

Ri ⊂ Rl1 ∨ · · · ∨ Rlm−1 ,

175

lm−1 < i < lm ,

 (Rl1 ∨ · · · ∨ Rlm )  (Rl1 ∨ · · · ∨ Rlm−1 ) = span u (lj m ) : j ∈ Jm , Ri ⊂ Rl1 ∨ · · · ∨ Rlm ,

lm < i ≤ n.

Now, let PMi be the orthogonal projections, respectively, onto the subspaces M1 := R1 , Mi := (R1 ∨ · · · ∨ Ri )  (R1 ∨ · · · ∨ Ri−1 ), 2 ≤ i ≤ n, / {l1 , . . . , lm } and PMlk is the orthogonal projection onto so that PMi = 0 if i ∈  span u (lj k ) : j ∈ Jk for 1 ≤ k ≤ m. Therefore, we have lim (A )

p 1/ p

p→∞

=

m 

alk



k=1

  m n   (lk )  (lk )  u  = u a P = ai PMi . l M k l j j k

j∈Jk

k=1

i=1

 

Lemma 2.5 For any finite set {w1 , . . . , wk } in Cn , span{w1 , . . . , wk } is equal to the range of |w1 w1 | + · · · + |wk wk |. More generally, for every B1 , . . . , Bk ∈ M+ n , k ran B is equal to the range of B + · · · + B . j 1 k j=1 Proof Let Q := |w1 w1 | + · · · + |wk wk |. Since Qx = w1 , x w1 + · · · + wk , x wk ∈ span{w1 , . . . , wk } 

for all x ∈ Cn , we have ran Q ⊂ span{w1 , . . . , wk }. Since |wi wi | ≤ Q, we have wi ∈ ran |wi wi | ⊂ ran Q, 1 ≤ i ≤ k. Hence, we have span{w1 , . . . , wk } ⊂ ran Q. The proof of the latter assertion is similar.  Thanks to the lemma we can restate Theorem 2.4 as follows: Theorem 2.6 Let  : Mn → Mn  be apositive linear map. Let A ∈ M+ n be given with the spectral decomposition A = m k=1 ak Pk , where a1 > a2 > · · · > am > 0. Define M1 := ran (P1 ), Mk := ran (P1 + · · · + Pk )  ran (P1 + · · · + Pk−1 ), 2 ≤ k ≤ m. Then lim (A p )1/ p =

p→∞

m  k=1

ak PMk .

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F. Hiai

Example 2.7 Consider a linear map  : M2n → Mn given by



X 11 X 12 X 21 X 22



X 11 + X 22 , 2

:=

X i j ∈ Mn .

Clearly,  is completely positive. For any A, B ∈ M+ n and p > 0 we have



A 0 0 B

p 1/ p

=

1/ p

Ap + B p 2

.

Thus, it is well-known [12] that

lim 

p→∞

A 0 0 B

p 1/ p

= lim

p→∞

1/ p

Ap + B p 2

= A ∨ B,

(2.7)

where A ∨ B is the supremum of A, B in the spectral order. Here, let us show (2.7) from Theorem 2.6. The spectral decompositions of A, B are given as A=

m 



ai Pi ,

B=

i=1

m 

bj Q j,

j=1

where a1 > · · · > am ≥ 0, b1 > · · · > bm  ≥ 0 and A⊕B =

l 

m i=1

Pi =

m  j=1

Q j = I . Then

ck Rk ,

k=1 

m where {ck }lk=1 = {ai }i=1 ∪ {b j }mj=1 with c1 > · · · > cl and

⎧ ⎪ ⎨ Pi ⊕ Q j if ai = b j = ck , Rk = Pi ⊕ 0 if ai = ck and b j = ck for all j, ⎪ ⎩ 0 ⊕ Q j if b j = ck and ai = ck for all i. Note that (R1 + · · · + Rk ) =

1 2



Pi +

i:ai ≥ck



 Qj

j:b j ≥ck

so that by Lemma 2.5 the support projection Fk (i.e., the orthogonal projection onto the range) of (R1 + · · · + Rk ) is Fk =

 i:ai ≥ck

 Pi

∨

 j:b j ≥ck

 Qj .

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

177

Theorem 2.6 implies that

lim 

p→∞

A 0 0 B

p 1/ p = C :=

l 

ck (Fk − Fk−1 ).

k=1

For every x ∈ R, we denote by E [x,∞) (A) the spectral projection of A corresponding to the interval [x, ∞), i.e.,  E [x,∞) (A) := Pi , i:ai ≥x

and similarly for E [x,∞) (B) and E [x,∞) (C). If ck ≥ x > ck+1 for some 1 ≤ k < l, then we have E [x,∞) (C) = Fk = E [x,∞) (A) ∨ E [x,∞) (B). This holds also when x > c1 and x ≤ cl . Indeed, when x > c1 , E [x,∞) (C) = 0 = E [x,∞) (A) ∨ E [x,∞) (B). When x ≤ cl , E [x,∞) (C) = I = E [x,∞) (A) ∨ E [x,∞) (B). This description of C is the same as A ∨ B in [12], so we have C = A ∨ B. Example 2.8 The example here is relevant to quantum information. For density matrices ρ, σ ∈ Mn (i.e., ρ, σ ∈ M+ n with Tr ρ = Tr σ = 1) and for a parameter α ∈ (0, ∞) \ {1}, the traditional Rényi relative entropy is 

  log Tr ρα σ 1−α if ρ0 ≤ σ 0 or 0 < α < 1, Dα (ρσ) := +∞ otherwise, 1 α−1

where ρ0 denotes the support projection of ρ. On the other hand, the new concept was recently introduced and called the sandwiched Rényi relative entropy [14, 18] is    1−α 1−α α  1 if ρ0 ≤ σ 0 or 0 < α < 1, log Tr σ 2α ρσ 2α α−1  Dα (ρσ) := +∞ otherwise. By taking the limit, we also consider D0 (ρσ) := lim Dα (ρσ) = − log Tr (ρ0 σ), α0

 1−α 1−α α   2α 2α . D0 (ρσ) := lim Dα (ρσ) = − log lim Tr σ ρσ α0

α0

α are interchanged from those in [8].) Here, (We remark that the notations Dα and D note that  1−α 1−α α = lim Tr (σ p/2 ρσ p/2 )1/ p = lim Tr (ρ0 σ p ρ0 )1/ p , lim Tr σ 2α ρσ 2α

α0

p→∞

p→∞

178

F. Hiai

where the existence of lim p→∞ Tr (ρ0 σ p ρ0 )1/ p follows from the Araki–Lieb– Thirring inequality [4] (also [1]), and the latter equality above follows since λρ0 ≤ ρ ≤ μρ0 for some λ, μ > 0 and λ1/ p Tr (σ p/2 ρ0 σ p/2 )1/ p ≤ Tr (σ p/2 ρσ p/2 )1/ p ≤ μ1/ p Tr (σ p/2 ρ0 σ p/2 )1/ p . It was proved in [8] that 0 (ρσ) ≤ D0 (ρσ) D

(2.8)

and equality holds in (2.8) if ρ0 = σ 0 . Let us here prove the following: 0 (ρσ) = − log Q 0 (ρσ), where (1) D  0 (ρσ) := max Tr (Pσ) : P an orthogonal projection, Q

[P, σ] = 0, (Pρ0 P)0 = P .

0 (ρσ) = D0 (ρσ) holds if and only if [ρ0 , σ] = 0. (Obviously, [ρ0 , σ] = 0 if (2) D ρ0 = σ 0 .) Indeed, to prove (1), first note that (Pρ0 P)0 = P means that the dimension of ran ρ0 P is equal to that of P, that is, ρ0 v1 , . . . , ρ0 vd are linearly independent when {v1 , . . . , vd } is an orthonormal basis of ran P. Choose 1 ≤ l1 < l2 < · · · < lm as in the first paragraph of this section (before Theorem 2.1) for A = σ and K = ρ0 . Let P0 be the orthogonal projection onto span{vl1 , . . . , vlm }. Then [P0 , σ] = 0, (P0 ρ0 P0 )0 = P0 , and Theorem 2.1 gives lim Tr (ρ0 σ p ρ0 )1/ p =

p→∞

m 

alk = Tr (P0 σ).

k=1

On the other hand, let P be an orthogonal projection [P, σ] = 0 and (Pρ0 P)0 = with d P. From [P, σ] = 0, we may assume that P = k=1 |vik vik | for some 1 ≤ i 1 < · · · < i d ≤ n (after, if necessary, changing vi for degenerate eigenvalues ai ). Since (Pρ0 P)0 = P implies that ρ0 vi1 , . . . , ρ0 vid are linearly independent, we have d ≤ m and d m   aik ≤ alk = Tr (P0 σ). Tr (Pσ) = k=1

k=1

Next, to prove (2), note that Tr (ρ0 σ p ρ0 )1/ p is increasing in p > 0 by the Araki– Lieb–Thirring inequality mentioned above, which shows that Tr (ρ0 σ) ≤ lim Tr (ρ0 σ p ρ0 )1/ p . p→∞

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

179

This means inequality (2.8), and equality holds in (2.8) if and only if Tr (ρ0 σ p ρ0 )1/ p is constant for p ≥ 1. By [10, Theorem 2.1] this is equivalent to the commutativity ρ0 σ = σρ0 . Finally, we consider the complementary convergence of (A p )1/ p as p → −∞, or (A− p )−1/ p as p → ∞. Here, the expression (A− p )−1/ p for p > 0 is defined in such a way that the (− p)-power of A is restricted to the support of A, i.e., defined in the sense of the generalized inverse, and the (−1/ p)-power of (A− p ) is also in this sense. The next theorem is the complementary counterpart of Theorem 2.6. Theorem 2.9 Let  : Mn → Mn  be apositive linear map. Let A ∈ M+ n be given with the spectral decomposition A = m k=1 ak Pk , where a1 > a2 > · · · > am > 0. Define k := ran (Pk + · · · + Pm )  ran (Pk+1 + · · · + Pm ), 1 ≤ i ≤ m − 1, M m := ran (Pm ). M Then lim (A− p )−1/ p =

p→∞

m 

ak PM k .

k=1

Proof The proof is just a simple adaptation of Theorem 2.6. We can write for any p>0  −1 (A− p )−1/ p = ((A−1 ) p )1/ p , where A−1 and {· · · }−1 are defined in the sense of the generalized inverse so that A

−1

=

m 

ak−1 Pk

=

k=1

m 

−1 am+1−k Pm+1−k

k=1

with am−1 > · · · > a1−1 > 0. By Theorem 2.6 we have lim ((A−1 ) p )1/ p =

p→∞

m 

−1 am+1−k PM m+1−k =

k=1

m 

ak−1 PM k ,

k=1

where m := ran (Pm+1−1 ) = ran (Pm ), M m+1−k := ran (Pm+1−1 + · · · + Pm+1−k )  ran (Pm+1−1 + · · · + Pm+1−(k−1) ) M = ran (Pm+1−k + · · · + Pm )  ran (Pm+2−k + · · · + Pm ), 2 ≤ k ≤ m.

According to the proofs of Theorems 2.1 and 2.4, we see that the ith eigenvalue λi ( p) of ((A−1 ) p )1/ p converges to a positive real as p → ∞, or otherwise, λi ( p) = 0

180

F. Hiai

for all p > 0. That is, λi ( p) → 0 as p → ∞ occurs only when λi ( p) = 0 for all p > 0. This implies that lim (A− p )−1/ p =



p→∞

lim ((A−1 ) p )1/ p

−1

p→∞

=

m 

ak PM k .

k=1

 Remark 2.10 Assume that  : Mn → Mn  is a unital positive linear map. Let A ∈ Mn be positive definite and 1 ≤ p < q. Since x p/q and x 1/ p are operator monotone on [0, ∞), we have (Aq ) p/q ≥ (A p ) and so (Aq )1/q ≥ (A p )1/ p . Hence (A p )1/ p increases as 1 ≤ p . Similarly, (A−q ) p/q ≥ (A− p ) and so (A−q )−1/q ≤ (A− p )−1/ p since x −1/ p is operator monotone decreasing on (0, ∞). Hence (A− p )−1/ p decreases as 1 ≤ p . Moreover, since x −1 is operator convex on (0, ∞), we have (A−1 )−1 ≤ (A). (See [5, Theorem 2.1] for more details.) Combining altogether, when A is positive definite, we have (A− p )−1/ p ≤ (Aq )1/q , and in particular,

p, q ≥ 1,

(2.9)

lim (A− p )−1/ p ≤ lim (A p )1/ p .

p→∞

p→∞

However, the latter inequality does not hold unless  is unital. For example, let

10 P1 := , 00

00 P2 := , 01



1/2 1/2 Q 1 := , 1/2 1/2



1/2 −1/2 Q 2 := , −1/2 1/2

and consider  : M2 → M2 given by 



a11 a12 := a11 P1 + a22 Q 1 , a21 a22

and A := a P1 + b P2 where a > b > 0. Since Pran (P1 +P2 ) = I , Pran (P1 ) = P1 and Pran (P2 ) = Q 1 , Theorems 2.6 and 2.9 give lim (A p )1/ p = a P1 + b(I − P1 ) = a P1 + b P2 ,

p→∞

lim (A− p )−1/ p = a(I − Q 1 ) + bQ 1 = a Q 2 + bQ 1 .

p→∞

We compute (a P1 + b P2 ) − (a Q 2 + bQ 1 ) = which is not positive semidefinite.

a−b

a−b 2 2 a−b b−a 2 2

,

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

181

Remark 2.11 We may always assume that  : Mn → Mn  is strictly positive. Indeed, we may consider  as  : Mn → Q 0 Mn  Q 0 ∼ = Mn  , where Q 0 is the support projection of (In ). Under this convention, another reasonable definition of (A− p )−1/ p for p ≥ 1 is (A− p )−1/ p := lim ((A + εIn )− p )−1/ p , ε0

which is well defined since ((A + εI )− p ) is increasing so that ((A + εI )− p )−1/ p is decreasing as ε  0. But this definition is different from the above definition of let  : M2 → M2 be given by (A) := K AK ∗ with an (A− p )−1/ p . For example,

ab 10 invertible K = , and let A = P = . Then A− p = P (in the generalized cd 00 inverse) so that

2 |a| ac −p ∗ KA K = ac |c|2 and so (K A

−p

∗ −1/ p

K )

=



1 1

(|a|2 + |c|2 )1− p

|a|2 ac . ac |c|2

(2.10)

On the other hand, lim (K (A + εI )− p K ∗ )−1/ p = lim (K ∗−1 (A + εI ) p K −1 )1/ p

ε0

ε0

= (K ∗−1 A p K −1 )1/ p = (K ∗−1 P K −1 )1/ p is equal to

1 1

|ad − bc|2/ p (|b|2 + |d|2 )1− p

|d|2 −bd . −bd |b|2

(2.11)

Hence we find that (2.10) and (2.11) are very different, even after taking the limits as p → ∞. Here is a simpler example. Let ϕ : M 2 → C = M1 be a state (hence, unital) with

1/2 1/2 10 density matrix , and let A = . For the first definition, we have 1/2 1/2 00 lim ϕ(A− p )−1/ p = lim 21/ p = 1.

p→∞

p→∞

For the second definition, lim ϕ((A + εI )− p )−1/ p = lim

ε0

ε0

(1 + ε)− p + ε− p 2

!−1/ p

=0

182

F. Hiai

for all p > 0. Moreover, since ϕ(A p )1/ p = 2−1/ p for p > 0, this example says also that (2.9) does not hold for general positive semidefinite A. Problem 2.12 It is also interesting to consider the limit of (A p B A p )1/ p as p → ∞ for A, B ∈ M+ n , a version different from the limit treated in Theorem 2.1. To consider lim p→∞ (A p B A p )1/ p , we may assume without loss of generality that B is an orthogonal projection E (see the argument around (3.2) below). Since (A p E A p )1/ p = (A p E 2 p A p )1/ p converges as p → ∞ by [6, Theorem 2.5], the existence of the limit lim p→∞ (A p B A p )1/ p follows. But it seems that the description of the limit is a combinatorial problem much more complicated than that in Theorem 2.1.

3 lim p→∞ ( A p σ B)1/ p for Operator Means σ In theory of operator means due to Kubo and Ando [13], a main result says that each operator mean σ is associated with a nonnegative operator monotone function f on [0, ∞) with f (1) = 1 in such a way that Aσ B = A1/2 f (A−1/2 B A−1/2 )A1/2 + for A, B ∈ M+ n with A > 0, which is further extended to general A, B ∈ Mn as

Aσ B = lim (A + εI )σ(B + εI ). ε0

We write σ f for the operator mean associated with f as above. For 0 ≤ α ≤ 1, the operator mean corresponding to the function x α (x ≥ 0) is the weighted geometric mean #α , i.e., A#α B = A1/2 (A−1/2 B A−1/2 )α A1/2 for A, B ∈ M+ n with A > 0. In particular, # = #1/2 is the so-called geometric mean first introduced by Pusz and Woronowicz [15]. The transpose of f above is given by  f (x) := x f (x −1 ),

x > 0,

which is again an operator monotone function on [0, ∞) (after extending to [0, ∞) by continuity) corresponding to the transposed operator mean of σ f , i.e., Aσ f B = Bσ f A. We also write  " f (x) :=

 f (x −1 ) = f (x)/x if x > 0, 0 if x = 0.

(3.1)

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

183

In the rest of the section, let f be such an operator monotone function as above and σ f be the corresponding operator mean. We are concerned with the existence and the description of the limit lim p→∞ (A p σ f B)1/ p , in particular, lim p→∞ (A p #α B)1/ p for A, B ∈ M+ n . For this, we may assume without loss of generality that B is an orthogonal projection. Indeed, let E be the support projection of B. Then we can choose λ, μ > 0 with λ < 1 < μ such that λE ≤ B ≤ μE. Thanks to monotonicity and positive homogeneity of σ f , we have λ(A p σ f E) = (λ A p )σ f (λE) ≤ A p σ f B ≤ (μA p )σ f (μE) = μ(A p σ f E). Hence, for every p ≥ 1, since x 1/ p (x ≥ 0) is operator monotone, λ1/ p (A p σ f E)1/ p ≤ (A p σ f B)1/ p ≤ μ1/ p (A p σ f E),

(3.2)

so that lim p→∞ (A p σ f B)1/ p exists if and only if lim p→∞ (A p σ f E)1/ p does, and in this case, both limits are equal. In particular, when B > 0, since (A p σ f I )1/ p =  f (A p )1/ p , we note that f (∞) (A) lim (A p σ f B)1/ p = 

p→∞

whenever  f (∞) (x) := lim p→∞  f (x p )1/ p exists for all x ≥ 0. For instance, • if f (x) = 1 − α + αx where 0 < α < 1, then σ f = α , the α-arithmetic mean f (∞) (x) = max{x, 1}, Aα B := (1 − α)A + αB, and  f (∞) (x) =  f (x) = x 1−α , • if f (x) = x α where 0 ≤ α ≤ 1, then σ f = #α and  • if f (x) = x/((1 − α)x + α) where 0 < α < 1, then σ f = !α , the α-harmonic f (∞) (x) = min{x, 1}. mean A !α B := (A−1 α B)−1 , and  But it is unknown to us that, for any operator monotone function f on [0, ∞), the limit lim p→∞ f (x p )1/ p exists for all x ≥ 0, while it seems so. When E is an orthogonal projection, the next proposition gives a nice expression for Aσ f E. This was shown in [11, Lemma 4.7], while the proof is given here for the convenience of the reader. Lemma 3.1 Assume that f (0) = 0. If A ∈ Mn is positive definite and E ∈ Mn is an orthogonal projection, then f (E A−1 E), Aσ f E = "

(3.3)

where " f is given in (3.1). Proof For every m = 1, 2, . . . we have A−1/2 (E A−1 E)m A−1/2 = (A−1/2 E A−1/2 )m+1 .

(3.4)

Note that the eigenvalues of E A−1 E and those of A−1/2 E A−1/2 are the same including multiplicities. Choose a δ > 0 such that the positive eigenvalues of E A−1 E and

184

F. Hiai

A−1/2 E A−1/2 are included in [δ, δ −1 ]. Then, since " f (x) is continuous on [δ, δ −1 ], one f (x) can choose a sequence of polynomials pk (x) with pk (0) = 0 such that pk (x) → " uniformly on [δ, δ −1 ] as k → ∞. By (3.4) we have A−1/2 pk (E A−1 E)A−1/2 = A−1/2 E A−1/2 pk (A−1/2 E A−1/2 ) for every k. Since " f (0) = 0 by definition, we have f (E A−1 E) pk (E A−1 E) −→ " and f (A−1/2 E A−1/2 ) A−1/2 E A−1/2 pk (A−1/2 E A−1/2 ) −→ A−1/2 E A−1/2 " as k → ∞. Since f (0) = 0 by assumption, we have f (x) = x " f (x) for all x ∈ [0, ∞). This implies that f (A−1/2 E A−1/2 ) = f (A−1/2 E A−1/2 ). A−1/2 E A−1/2 " Therefore,

f (E A−1 E)A−1/2 = f (A−1/2 E A−1/2 ) A−1/2 "

so that we have " f (E A−1 E) = A1/2 f (A−1/2 E A−1/2 )A1/2 = Aσ f E, as asserted.  Formula (3.3) can equivalently be written as Aσ f E =  f ((E A−1 E)−1 ),

(3.5)

where (E A−1 E)−1 is the inverse restricted to ran E (in the sense of the generalized inverse) and  f ((E A−1 E)−1 ) is also restricted to ran E. In particular, if f is symmetric (i.e., f =  f ) with f (0) = 0, then Aσ f E = f ((E A−1 E)−1 ). Example 3.2 Assume that 0 < α ≤ 1 and A, E are as in Lemma 3.1. (1) When f (x) = x α and σ f = #α , " f (x) = x α−1 for x > 0 so that A#α E = (E A−1 E)α−1 , where the (α − 1)-power in the right-hand side is defined with restriction to ran E. f (x) = (1 − α + αx)−1 for x > (2) When f (x) = x/((1 − α)x + α) and σ f = !α , " 0 so that A !α E = {(1 − α)E + αE A−1 E}−1 = {E((1 − α)I + α A−1 )E}−1 ,

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

185

where the inverse of E((1 − α)I + α A−1 )E in the right-hand side is restricted to ran E. f (x) = (1 − (3) When f (x) = (x − 1)/ log x and so σ f is the logarithmic mean, " x −1 )/ log x for x > 0 so that Aσ f E = (E − (E A−1 E)−1 )(log E A−1 E)−1 , where the right-hand side is defined with restriction to ran E. Theorem 3.3 Assume that f (0) = 0 and f (x r ) ≥ f (x)r for all x > 0 and all r ∈ (0, 1). Let A ∈ M+ n and E ∈ Mn be an orthogonal projection. Then (A p σ f E)1/ p ≥ (Aq σ f E)1/q if 1 ≤ p < q.

(3.6)

Proof First, note that  f (x r ) = x r f (x −r ) ≥ x r f (x −1 )r =  f (x)r for all x > 0, r ∈ (0, 1). By replacing A with A + εI and taking the limit as ε  0, we may assume that A is positive definite. Let 1 ≤ p < q and r := p/q ∈ (0, 1). By (3.5) we have f ((E A−q E)−1 )r ≤  f ((E A−q E)−r ). (Aq σ f E)r = 

(3.7)

Since x r is operator monotone on [0, ∞), we have by Hansen’s inequality [9] (E A−q E)r ≥ E A−qr E = E A− p E f (x) is operator monotone on [0, ∞), we so that (E A−q E)−r ≤ (E A− p E)−1 . Since  have  f ((E A− p E)−1 ) = A p σ f E. f ((E A−q E)−r ) ≤ 

(3.8)

Combining (3.7) and (3.8) gives (Aq σ f E)r ≤ A p σ f E. Since x 1/ p is operator monotone on [0, ∞), we finally have (Aq σ f E)1/q ≤ (A p σ f E)1/ p .  Corollary 3.4 Assume that f (0) = 0 and f (x r ) ≥ f (x)r for all x > 0, r ∈ (0, 1). Then for every A, B ∈ M+ n , the limit lim (A p σ f B)1/ p

p→∞

exists.

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F. Hiai

Proof From the argument around (3.2) we may assume that B is an orthogonal projection E. Then, Theorem 3.3 implies that (A p σ f E)1/ p converges as p → ∞.  Remark 3.5 Following [17], an operator monotone function f on [0, ∞) is said to be power monotone increasing (p.m.i. for short) if f (x r ) ≥ f (x)r for all x > 0, r > 1 (equivalently, f (x r ) ≤ f (x r ) for all x > 0, r ∈ (0, 1)), and power monotone decreasing (p.m.d.) if f (x r ) ≤ f (x)r for all x > 0, r > 1. These conditions play a role to characterize the operator means σ f satisfying Ando–Hiai’s inequality [1], see [17, Lemmas 2.1, 2.2]. For instance, the p.m.d. condition is satisfied for f in (1) and (2) of Example 3.2, while f in Example 3.2 (3) does the p.m.i. condition. Hence, for any α ∈ [0, 1], (A p #α E)1/ p and (A p !α E)1/ p converge decreasingly as 1 ≤ p  ∞. In fact, for the harmonic mean, we have the limit A ∧ B := lim p→∞ (A p ! B p )1/ p , the decreasing limit as 1 ≤ p  ∞ for any A, B ≥ 0, which is the infimum counterpart of A ∨ B in [12] (see also Example 2.7). The reader might be wondering if the opposite inequality to (3.6) holds (i.e., (A p σ f E)1/ p is increasing as 1 ≤ p  ∞) when f satisfies the p.m.i. condition. Although this is the case when σ = α the weighted arithmetic mean, it is not the case in general. In fact, if it were true, (A p #α E)1/ p must be constant for p ≥ 1 since x α satisfies both p.m.i. and p.m.d. conditions, that is impossible. Finally, for the weighted geometric mean #α we obtain the explicit description of lim p→∞ (A p #α E)1/ p for any A ∈ M+ n . For the trivial cases α = 0, 1 note that (A p #0 E)1/ p = A and (A p #1 E)1/ p = E for all p > 0. + Theorem 3.6 Assume  that 0 < α < 1. Let A ∈ Mn be given with the spectral a P where a > · · · > a decomposition A = m 1 m > 0, and E ∈ Mn be an orthogk=1 k k onal projection. Then

lim (A p #α E)1/ p =

p→∞

m 

ak1−α Q k ,

(3.9)

k=1

where Q 1 := P1 ∧ E, Q k := (P1 + · · · + Pk ) ∧ E − (P1 + · · · + Pk−1 ) ∧ E,

2 ≤ k ≤ m.

Proof First, assume that A is positive definite so that P1 + · · · + Pm = I . When f (x) = x α with 0 < α < 1, formula (3.5) is given as A#α E = (E A−1 E)−(1−α) . Since

Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

187

lim (A p #α E)1/ p = lim (E A− p E)−(1−α)/ p = lim (E(A1−α )− p E)−1/ p ,

p→∞

p→∞

p→∞

it follows from Theorem 2.9 that lim (A p #α E)1/ p =

p→∞

m 

ak1−α PM k ,

k=1

where k := ran E(Pk + · · · + Pm )E  ran E(Pk+1 + · · · + Pm )E, 1 ≤ k ≤ m − 1, M m := ran E Pm E. M From Lemma 3.7 below, we have 1 = ran E  ran E P1⊥ E = ran P1 ∧ E, M and for 2 ≤ k ≤ m, k = ran E(P1 + · · · + Pk−1 )⊥ E  ran E(P1 + · · · + Pk )⊥ E M     = ran E  ran E(P1 + . . . +Pk )⊥ E  ran E  ran E(P1 + · · · + Pk−1 )⊥ E = ran (P1 + · · · + Pk ) ∧ E  ran (P1 + · · · + Pk−1 ) ∧ E   = ran (P1 + · · · + Pk ) ∧ E − (P1 + · · · + Pk−1 ) ∧ E . Therefore, (3.9) is established when A is positive definite. Next, when A is not positive definite, let Pm+1 := (P1 + · · · + Pm )⊥ . For any ε ∈ (0, am ) define Aε := A + εPm+1 . Then the above case implies that lim (Aε #α E)

p→∞

1/ p

=

m 

ak1−α Q k + ε1−α Q m+1 ,

k=1

where Q m+1 := E − (P1 + · · · + Pm ) ∧ E. Assume that 0 < ε < ε < am . For every p ≥ 1, since Aεp ≤ Aε , we have Aεp #α E ≤ p p Aε #α E and hence (Aεp #α E)1/ p ≤ (Aε #α E)1/ p . Furthermore, since Aεp #α E → A p #α E as am > ε  0, we have p

(A p #α E)1/ p = lim (Aεp #α E)1/ p decreasingly. am >ε0

Now, we can perform a calculation of limits as follows:

(3.10)

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lim (A p #α E)1/ p =

lim

lim (Aεp #α E)1/ p

= lim

lim (Aεp #α E)1/ p

1≤ p→∞

1≤ p→∞ am >ε0

am >ε0 1≤ p→∞

= lim

 m

am >ε0

=

m 



ak1−α Q k + ε1−α Q m+1

k=1

ak1−α Q k .

k=1

In the above, the second equality (the exchange of two limits) is confirmed as follows. Let X p,ε := (Aεp #α E)1/ p for p ≥ 1 and 0 < ε < am . Then X p,ε is decreasing as 1 ≤ p  ∞ by Theorem 3.3 and also decreasing as am > ε  0 as seen in (3.10). Let X p := limε X p,ε (= (A p #α E)1/ p ), X ε := lim p X p,ε , and X := lim p X p . Since X p,ε ≥ X p , we have X ε ≥ X and hence limε X ε ≥ X . On the other hand, since X p,ε ≥ X ε , we have X p ≥ limε X ε and hence X ≥ limε X ε . Therefore, X = limε X ε , which gives the assertion.  In particular, when A = P is an orthogonal projection, we have (A p #α E)1/ p = P ∧ E for all p > 0 (see [13, Theorem 3.7]) so that both sides of (3.9) are certainly equal to P ∧ E. Lemma 3.7 For every orthogonal projections E and P, ran E P ⊥ E = ran (E − P ∧ E), or equivalently,

ran P ∧ E = ran E  ran E P ⊥ E.

Proof According to the well-known representation of two projections (see [16, pp. 306–308]), we write

I 0 E = I ⊕ I ⊕0⊕ ⊕ 0, 00

2 C CS ⊕ 0, P = I ⊕0⊕ I ⊕ C S S2 where 0 < C, S < I with C 2 + S 2 = I . We have P ∧ E = I ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0. Since P⊥ = 0 ⊕ I ⊕ 0 ⊕

S 2 −C S ⊕ I, −C S C 2



Matrix Limit Theorems of Kato Type Related to Positive Linear Maps …

we also have E P⊥ E = 0 ⊕ I ⊕ 0 ⊕



189

S2 0 ⊕ 0, 0 0

whose range is that of

I 0 0⊕ I ⊕0⊕ ⊕ 0 = E − P ∧ E, 00 which yields the conclusion.



Acknowledgements This work was supported by JSPS KAKENHI Grant Number JP17K05266.

References 1. Ando, T., Hiai, F.: Log majorization and complementary Golden-Thompson type inequalities. Linear Algebr. Appl. 197/198, 113–131 (1994) 2. Antezana, J., Corach, G., Stojanoff, D.: Spectral shorted matrices. Linear Algebr. Appl. 381, 197–217 (2004) 3. Antezana, J., Corach, G., Stojanoff, D.: Spectral shorted operators. Integr. Equ. Oper. Theory 55, 169–188 (2006) 4. Araki, H.: On an inequality of Lieb and Thirring. Lett. Math. Phys. 19, 167–170 (1990) 5. Audenaert, K.M.R., Hiai, F.: On matrix inequalities between the power means: counterexamples. Linear Algebr. Appl. 439, 1590–1604 (2013) 6. Audenaert, K.M.R., Hiai, F.: Reciprocal Lie-Trotter formula. Linear Multilinear Algebr. 64, 1220–1235 (2016) 7. Bourin, J.-C.: Convexity or concavity inequalities for Hermitian operators. Math. Inequalities Appl. 7, 607–620 (2004) 8. Datta, N., Leditzky, F.: A limit of the quantum Rényi divergence. J. Phys. A: Math. Theor. 47, 045304 (2014) 9. Hansen, F.: An operator inequality. Math. Ann. 246, 249–250 (1980) 10. Hiai, F.: Equality cases in matrix norm inequalities of Golden-Thompson type. Linear Multilinear Algebr. 36, 239–249 (1994) 11. Hiai, F.: A generalization of Araki’s log-majorization. Linear Algebr. Appl. 501, 1–16 (2016) 12. Kato, T.: Spectral order and a matrix limit theorem. Linear Multilinear Algebr. 8, 15–19 (1979) 13. Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980) 14. Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54, 122203 (2013) 15. Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975) 16. Takesaki, M.: Theory of Operator Algebras I, Encyclopaedia of Mathematical Sciences, vol. 124. Springer, Berlin (2002) 17. Wada, S.: Some ways of constructing Furuta-type inequalities. Linear Algebr. Appl. 457, 276– 286 (2014) 18. Wilde, M.M., Winter, A., Yang, D.: Strong converse for the classical capacity of entanglementbreaking and Hadamard channels via a sandwiched Rényi relative entropy. Commun. Math. Phys. 331, 593–622 (2014)

The Howland–Kato Commutator Problem Ira Herbst and Thomas L. Kriete

Dedicated to the memory of Tosio Kato

Abstract We investigate the following problem: For what bounded measurable real f and g is the commutator i[ f (P), g(Q)] positive? This problem originated in work of James Howland and was pursued by Tosio Kato who suggested what might be the answer. So far, there is no proof that Kato was correct but in this paper we discuss the problem and give some partial answers to the above question.

1 Introduction In a paper on spectral theory [1], J. Howland used the positive commutator of two bounded functions of the Heisenberg operators P and Q, i[ f (P), g(Q)], as a technical tool. Here, P = −id/d x and Q is multiplication by x in L 2 (R). The functions f and g were specifically f (t) = tan−1 (t/2), g(t) = tanh(t).

(1.1)

He sent his paper to T. Kato, his former thesis advisor, who got interested in the more general question: for what bounded real functions is the above commutator positive? Kato made much progress on this problem and in a beautiful paper, [2], he identified a very interesting class of pairs of such functions for which the commutator was I. Herbst (B) · T. L. Kriete Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA e-mail: [email protected]; [email protected] T. L. Kriete e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_10

191

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positive. In this paper, we will always assume that f and g are bounded measurable real functions. To state the main result of [2], for a > 0 define K a = { f : R → R| f is bounded and has an analytic continuation to the strip |Imz| < a with Im f (z)Imz  0}. Then Theorem 1.1 (Kato) If f ∈ K a , g ∈ K b with ab  π/2, then i[ f (P), g(Q)]  0. As we will see later, if f and g are two bounded real measurable functions for which i[ f (P), g(Q)] = C  0, then as noted by Kato, C is trace class. It is thus natural to look at the case where C is a nonzero, rank one operator. Kato does this in [2] and shows that in this case there exist a and b with ab = π/2 such that ˆ − ± f ∈ K a and ±g ∈ K b (the signs are correlated). In fact f (x) = c1 + d1 tanh(a(x ˆ − t2 )) where c j , d j , and t j are real, d1 d2 > 0, and t1 )) and g(x) = c2 + d2 tanh(b(x (following Kato) aˆ = π/2a and bˆ = π/2b, so that aˆ = b and bˆ = a. (Kato assumes that f and g are absolutely continuous with derivatives in L 1 (R) but this is not necessary as we will see later.) It is clear that from these functions, more pairs of functions with positive commutators can be constructed by convolution with a positive measure. In fact as Kato shows, the family of f of the form  f (x) =

R

tanh a(x ˆ − t)dν(t) + c,

(1.2)

where c is real and ν is a finite positive measure, exhausts all of K a . These results led Kato to state (in [2]): “In fact there is some reason to believe that these [ f ∈ K a , g ∈ K b with ab  π/2] are the only solutions to [i[ f (P), g(Q)]  0].” Note that since K a ⊃ K c whenever 0 < a < c, the meaning of Kato’s statement just quoted, as well as the meaning of the statement of Theorem 1.1, is unchanged if the inequality ab  π/2 is replaced by the equality ab = π/2. Kato also shows that if i[ f (P), g(Q)] = 0 and both f and g are absolutely continuous with L 1 (R) derivatives then at least one of them is constant. In much of this paper, we will relax this assumption and only assume that f and g are bounded measurable real functions. This opens up another interesting possibility. We will show that for f and g bounded, real, and measurable, Theorem 1.2 The commutator [ f (P), g(Q)] = 0 if and only if either f or g is almost everywhere constant or both have periodic versions with periods τ f and τg satisfying τ f τg = 2π. Actually, the theorem is still true without the assumption that f and g are real. There is a striking difference in the set of allowed f and g when C = i[ f (P), g(Q)]  0 and in addition we impose C = 0. We will show the following theorem.

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193

Theorem 1.3 Suppose C = i[ f (P), g(Q)]  0 and in addition C = 0. Then there are versions of f and g which are both monotone (either both increasing or both decreasing). For convenience, let us formulate a conjecture which we shall call K (after Kato): Conjecture 1.4 Suppose C = i[ f (P), g(Q)]  0 and in addition C = 0. Suppose both f and g are increasing. Then, there exist a and b with ab = π/2 such that f ∈ K a and g ∈ K b . Unfortunately, we are far from proving K. But in the rest of this paper, we will give several results which illuminate the properties of the set of f and g for which the commutator is positive. In addition to the theorems of this Introduction and their proofs, see in particular the section on 2 × 2 positivity which gives inequalities which f and g must satisfy under a mild assumption. Theorem 1.5 Suppose C = i[ f (P), g(Q)]  0 and in addition C = 0. Suppose both f and g are increasing. Then, both f and g have continuous versions which are strictly increasing. Taking f and g to be continuous, their inverse functions are absolutely continuous. A hint that K might be true is the following result: Theorem 1.6 Suppose g is nonconstant, lies in K b , and has the integral representation  ˆ − t)dμ(t) + d tanh b(x (1.3) g(x) = R

where d is real and μ is a finite positive measure such that 

ˆ

|t|e2bt dμ(t) < ∞ or



ˆ

|t|e−2bt dμ(t) < ∞.

If f ∈ L ∞ (R) and i[ f (P), g(Q)]  0 then f ∈ K bˆ . There is a reason that a condition such as this assumption of exponential decay is required. Just because we have assumed g ∈ K b does not mean that g is not in a smaller class K e with e > b. Suppose c is the largest number e such that g ∈ K e . ˆ We could not be able to prove that f ∈ K ˆ when only f ∈ K cˆ Then cˆ = π/2c < b. b is required. To see that our exponential decay assumption eliminates this possibility, note that if b and c are as above, then besides (1.3) there is another representation of g of the form  tanh c(x ˆ − t)dμ1 (t) + d1 g(x) = R

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where μ1 is a finite positive measure and d1 is a real constant. We have an explicit formula for the imaginary part of g on the line z = x + ib,  Img(x + ib) =

R

sin(bπ/c)(cosh(2c(x ˆ − t)) + cos(bπ/c))−1 dμ1 (t)  k(cosh 2cx) ˆ −1

for some positive k. If we note that (see [2]) dμ(t) = (2π)−1 Img(t + ib)dt, it follows  ˆ that |t|e±2bt dμ(t) = ∞. The analyticity requirements of the conjecture K remind us of a theorem of Loewner [3]. To state this theorem, let (a, b) be an open interval (finite or infinite). We say the continuous function f : (a, b) → R is matrix monotone if for all n and all self-adjoint n × n matrices A and B both with spectrum in (a, b) satisfying A  B we have f (A)  f (B). Theorem 1.7 (Loewner) f is matrix monotone if and only if f is the restriction to (a, b) of a function analytic in {Imz = 0} ∪ (a, b) satisfying Im f (z)Imz  0. It is not hard to see that if f is matrix monotone on (a, b) then for any two selfadjoint operators A and B on a Hilbert space with spectrum in (a, b) and satisfying A  B we have f (A)  f (B). We can formulate the positivity of the commutator i[ f (P), g(Q)] in a way that makes a connection with Loewner’s theorem in the following way, at least formally (since we have not shown that f is absolutely continuous, for example). Since   d/dt eit f (P) g(Q)e−it f (P) = eit f (P) i[ f (P), g(Q)]e−it f (P) , if the commutator is positive and nonzero we have g(eit f (P) Qe−it f (P) ) = g(Q + t f  (P))  g(Q) for positive t. As we have seen we can assume that f (and g) are increasing so that at least formally Q + t f  (P)  Q for positive t. Thus with A = Q + t f  (P) and B = Q we have A  B for t positive while g(A)  g(B) for all such t is the same as the positivity of the commutator. In the following sections, we prove Theorems 1.2, 1.3, 1.5, and 1.6 and add some further information about this fascinating problem. If C is an operator and C  0, we will say that C is positive, although perhaps nonnegative would be more accurate. Similarly, we sometimes call a nondecreasing function an increasing function. For the inner product of two vectors h and k in Hilbert space, we write (h, k), linear in k, and conjugate–linear in h. The Fourier transform of a function h on R is denoted by hˆ and includes the factor (2π)−1/2 , while hˇ denotes the inverse Fourier transform.

The Howland–Kato Commutator Problem

195

2 Finite Rank Commutators: i[tanh α P, tanh β Q] In this section, we consider the commutator i[ f (P), g(Q)] where f and g are the basic functions from which all functions in the Kato classes K a are constructed. We see that i[tanh αP, tanh β Q] = 4i[(1 + e2αP )−1 , (1 + e2β Q )−1 )] so that the positivity of the commutator with tanh is just a statement about the positivity of the commutator of resolvents of the exponential function. For that reason it is interesting to note the easily verified fact that for α and β real with (2α)(2β) = 2nπ and n ∈ Z e2iαP e2iβ Q e−2iαP = e2iβ(Q+2α) = e2iβ Q and thus

[(λ + e2iαP )−1 , (λ + e2iβ Q )−1 )] = 0,

for |λ| = 1. A formal calculation with unbounded operators would yield e2αP e2β Q e−2αP = e2β(Q−2iα) = e2β Q thus leading to

[(1 + e2αP )−1 , (1 + e2β Q )−1 )] = 0

if (2α)(2β) = 2nπ. But this is incorrect as we see from the next proposition. Proposition 2.1 The integral kernel of the commutator i[tanh αP, tanh β Q] = 4i[(1 + e2αP )−1 , (1 + e2β Q )−1 ], for α and β real and (2α)(2β) = 2πn > 0, is given by n−1  ψk (x)φk (y) (2.1) (β/nπ) k=0

where

and

ψk (x) = (cosh βx)−1 e(n−1−2k)βx/n φk (x) = (cosh βx)−1 e−(n−1−2k)βx/n

Thus i[tanh αP, tanh β Q] is a rank n operator if (2α)(2β) = 2πn. Note that for the rank one case (the one considered by Kato) where (2α)(2β) = 2π, the commutator is positive. We have assumed n > 0. Reversing the sign of n just amounts to a sign change in the commutator. If n > 1 this operator is not positive. This can be seen by looking at the matrix K i j := K (xi , x j ) in the 2 × 2 case. We have

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K i j = (β/nπ)(cosh βxi cosh βx j )−1

n−1 

f k (xi − x j )

(2.2)

k=0

where f k (x) = e(n−(2k+1))βx/n and thus in the 2 × 2 case 2 = det(K i j ) = K 11 K 22 − K 12

(2.3)

(β/nπ)2 (cosh βx1 cosh βx2 )−2 (n 2 − |

n−1 

f k (x1 − x2 )|2 )

(2.4)

k=0

Using the fact that for a  positive number a = 1 we have a + a −1 > 2, we see that if x1 − x2 = 0 and n > 1, n−1 k=0 f k (x 1 − x 2 ) > n and thus the determinant is negative. Remark 2.2 The noncommutativity of eαP and eβ Q for αβ = ±2π has been a source of counterexamples for the uniqueness of the representation of the canonical commutation relations [4] and the hypotheses under which the so-called virial theorem is true [5]. In the latter reference, one can also see in what sense these operators commute. Before proving Proposition 2.1 we need the following lemma. Lemma 2.3 If f and g are bounded, f is monotone increasing, and g is smooth with bounded derivatives then for ψ ∈ L 2 (R)  (i[ f (P), g(Q)]ψ)(x) = where

K (x, y)ψ(y)dy

(2.5)

1 g(x) − g(y)  K (x, y) = √ d f (y − x). x−y 2π

(2.6)

If g is smooth with bounded derivatives, and f is bounded then if ψ ∈ S(R) (the Schwartz space) we have  1 g(x) − g(y)  (i[ f (P), g(Q)]ψ)(x) = f (y − x)ψ(y)dy √ x−y 2π where f is considered a tempered distribution. Remarks 2.4 If f is monotone increasing, we take the version of f which is right continuous in the definition of d f . See Lemma 5.1 for a more complete result. Proof We first assume ψ ∈ S(R).   (i[ f (P), g(Q)]ψ)(x) = i

f (ξ)(g(y) − g(x))ψ(y)e

−iξ y

 dy eiξx dξ/2π.

The Howland–Kato Commutator Problem

Let h x (y) =

197

√ g(y + x) − g(x) ψ(y + x)/ 2π. y

Then  (i[ f (P), g(Q)]ψ)(x) = i

f (ξ)i( h x ) (ξ)dξ =



 h x (ξ)d f (ξ) =



h x (y)df (y)dy

where in the second equality we have used the integration by parts formula [6] in the case where we assume f is monotone. After a change of variable, this gives the first result (2.5) for ψ ∈ S(R). A limiting argument gives (2.5) for ψ ∈ L 2 (R). If we only assume f is bounded, the first equality in the last equation gives our result when we note that h x ∈ S(R) and remember the definition of the derivative and the Fourier transform of a tempered distribution.  Proof of Proposition 2.1 The kernel K (x, y) is given by (2π)−1/2

tanh βx − tanh β y ˆ f (y − x) x−y

with f (ξ) = tanh αξ. Using the known Fourier transform of (cosh x)−2 (see (6.11)), we obtain tanh βx − tanh β y K (x, y) = (β/nπ) . sinh(β(x − y)/n) We calculate tanh βx − tanh β y = sinh β(x − y)(cosh βx cosh β y)−1 . Using a n − bn = (a − b)(a n−1 + a n−2 b + · · · + bn−1 ) with a = eβx/n , b = e−βx/n , we obtain  sinh βx e(n−(2k+1))βx/n = sinh βx/n k=0 n−1

which gives the result.



3 Theorem 1.2—The Case [ f ( P), g( Q)] = 0 In this section, we consider the case where for real bounded measurable functions f and g, we have C = i[ f (P), g(Q)] = 0.

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Proof of Theorem 1.2 If f is periodic with period a = τ f then eia Q f (P)e−ia Q = f (P − a) = f (P), so eia Q commutes with f (P). If g is periodic with period τg = 2π/τ f , then g(Q) is a function of eia Q and thus commutes with f (P). For the converse first assume f and g are C ∞ with bounded derivatives and f (P) and g(Q) commute. Suppose f is not constant and suppose k1 and k2 are two points so that f (k1 ) = f (k2 ). Choose open neighborhoods N j of k j with N j compact such that f (N1 ) ∩ f (N2 ) = ∅. Choose a bounded continuous function F with F = 0 on f (N1 ) and F = 1 on f (N2 ). Note that (F ◦ f )(P) commutes with g(Q) so that if φ j ∈ C0∞ (N j ) 0 = (φ1 , [F ◦ f (P), g(Q)]φ2 ) = −(φ1 , g(Q)φ2 ). Considering g as a tempered distribution and denoting ψ1 = φˆ1 ∈ C0∞ (N1 ), ψ2 = φˆ2 ∈ C0∞ (N2 ), we have  √  0 = g(φ1 φ2 ) = gˆr (φ1 φ2 ) = gˆr (ψ2 (ξ + ·)ψ1 (ξ)dξ/ 2π where h r (x) = h(−x). Since this is true for all ψ1 ∈ C0∞ (N1 ), gˆr (ψ2 (· + ξ)) = 0 for all ξ ∈ N1 and thus gˆr (ψ) = 0 for all ψ ∈ C0∞ (N2 − ξ) for all ξ ∈ N1 or more ˆ ∩ (N1 − N2 ) = ∅. We concisely gˆr (ψ) = 0 for all ψ ∈ C0∞ (N2 − N1 ). Thus (supp g) have thus proved that / supp gˆ f (k1 ) = f (k2 ) implies k1 − k2 ∈ or k1 − k2 ∈ supp gˆ implies f (k1 ) = f (k2 ) or supp gˆ ⊂ P f := the set of periods of f

(3.1)

It follows that either supp gˆ = {0} in which case g is constant, or f is periodic. We have already assumed that f is not constant; now assume g is not constant. Then f has a smallest positive period, τ f , and supp gˆ ⊂ τ f Z. of the oriIf we choose φ ∈ C0∞ ((−τ f /2, τ f /2)) with φ = 1 in a neighborhood  } such that g ˆ = φ g ˆ where φ j (ξ) = gin, then there is a sequence of integers {n j j j  φ(ξ − n j τ f ). Thus gˆ = j c j δn j τ f with δk the Dirac delta at k. It follows that √  g = j c j ein j τ f x / 2π which implies g has period τg = 2π/τ f . This proves the result when f and g are smooth with bounded derivatives. In the general case con2 2 1 e−|x| /2σ . The resulting functions of volve f and g with the Gaussian δ0σ (x) = √2πσ P and Q still commute. Since

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σ ˆ supp (δ 0 ∗ f ) = supp f σ ˆ supp (δ 0 ∗ g) = supp g,

it follows that δ0σ ∗ f and f have the same periods as do δ0σ ∗ g and g. This completes the proof.  Remark 3.1 Actually, the result is true without the requirement that the functions f and g be real. The complex case follows from the result just proved and a result of Fuglede [7] (see also [8] for a very simple proof) which states that if a normal operator commutes with another operator then so does its adjoint. Thus the real and imaginary parts of f (P) and g(Q) commute with one another. With this information the proof is straightforward.

4 Theorem 1.3—Monotonicity If we assume that f and g are absolutely continuous with derivatives in L 1 (R), the proof of monotonicity is given by Kato in [2] and is straightforward. We give a sketch of the proof: The integral kernel of the positive operator i[ f (P), g(Q)] is √ g(x) − g(y)  f (y − x). K (x, y) = ( 2π)−1 x−y With [ f ] = f (∞) − f (−∞) the condition of positivity implies K (x, x) = g  (x) [ f ]/2π  0 and K (x, x)K (y, y)|K (x, y)|2 or g  (x)g  (y)[ f ]2 (2π)2 |K (x, y)|2 . Thus, unless K is identically zero, [ f ] = 0 which implies g is monotone. Using complex conjugation, C0 , and the Fourier transform, F, we note that F −1 C0 i[ f (P), g(Q)]C0−1 F = i[g(P), f (Q)]

(4.1)

which is therefore positive. Thus, the same argument gives that f is also monotone and it follows they are both either increasing or both decreasing. Proof of Theorem 1.3 We first assume that f and g are infinitely differentiable with bounded derivatives. In the proof, we use the distribution kernel of the operator C = i[ f (P), g(Q)] given by √ g(x) − g(y)  K (x, y) = ( 2π)−1 f (y − x) x−y We do not know that f  is integrable which accounts for the distribution nature of the kernel.

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Since the continuity of the kernel is not yet known, we use a smeared out version of the inequality |K (x1 , x2 )|2  K (x1 , x1 )K (x2 , x2 ). Put δaσ (x) = √ and note that

1 2πσ

e−|x−a|

√

2

/2σ 2

√

δaσ (x)δaσ (y) = δ0 2σ (x − y)δaσ/ 2 ((x + y)/2).

For φ ∈ C0∞ (R), let Iσ = (δ0σ ∗ φ, Cδ0σ ∗ φ) = and note



(δxσ1 , Cδxσ2 )φ(x1 )φ(x2 )d x1 d x2

(4.2)

|(δxσ1 , Cδxσ2 )|  (δxσ1 , Cδxσ1 )1/2 (δxσ2 , Cδxσ2 )1/2

so that  Iσ  (

(δxσ , Cδxσ )1/2 |φ(x)|d x)2

 =c



=c

 

√

√ σ/ 2

√

√ σ/ 2

δ0 2σ (x − y)K (x, y)δ0

δ0 2σ (x − y)K (x, y)δ0 √ =: c Jσ / 2π where c = write



 1d x supp φ

(δxσ , Cδxσ )|φ(x)|2 d x

((x + y)/2 − x1 )|φ(x1 )|2 d x1 d xd y ∗ |φ|2 ((x + y)/2)d xd y (4.3) √ σ/ 2

supp φ 1d x. We abbreviate ψσ = δ0

∗ |φ|2 and use Taylor’s theorem to

g(y + t) − g(y) = g  (y) + t G 1 (y, t) where t  1 G 1 (y, t) = g (2) (y + θt)(1 − θ)dθ.

(4.4) (4.5)

0

Thus, we obtain  Jσ = +



√

δ0 2σ (t)g  (y) f  (−t)ψσ (y + t/2)dtdy √

tδ0 2σ (t)G 1 (y, t) f  (−t)ψσ (y + t/2)dtdy

(4.6)

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and replacing ψσ (y + t/2) by ψσ (y) we pick up another error term so that  Jσ = +



√

δ0 2σ (t)g  (y) f  (−t)ψσ (y)dtdy √

tδ0 2σ (t)G σ2 (y, t) f  (−t)dtdy where

(4.7)

G σ2 (y, t) = g  (y)(ψσ (y + t/2) − ψσ (y))/t + G 1 (y, t)ψσ (y + t/2).

(4.8)

We now have Jσ = Jσ1 + Jσ2

 √    2σ 1   Jσ = δ0 (t) f (−t)dt g (y)ψσ (y)dy  I2σ (y)dy Jσ2 =   √ 2σ σ σ  I2 (y) = G 2 (y, t)tδ0 (t) f (−t)dt = Fyσ (ξ) f (ξ)dξ  √ √ Fyσ (ξ) = (−i/ 2π) G σ2 (y, t)t 2 δ0 2σ (t)eiξt dt. √ σ/ 2

We estimate D m ψσ = δ0

(4.9)

∗ D m |φ|2 to find for all m and n

|D m ψσ (x)|  cn,m (1 + |x|)−n uniformly for σ ∈ (0, 1). We use G σ2 (y, t) = (y + t/2) and obtain

 1/2 0

ψσ (y + θt)dθg  (y) + G σ1 (y, t)ψσ

|G σ2 (y, t)|  cn ((1 + |y + t/2|)−n + (1 + |y|)−n ). Thus for σ ∈ (0, 1) |Fyσ (ξ)|  cn



√

((1 + |y + t/2|)−n + (1 + |y|)−n )t 2 δ0 2σ (t)dt

 dn σ 2 (1 + |y|)−n .

(4.10)

We also need some decay of Fyσ (ξ) in ξ. Thus |(1 + ξ 2 )Fyσ (ξ)|  c



√

|(1 − d 2 /dt 2 )G σ2 (y, t)t 2 δ0 2σ (t)|dt.

(4.11)

We easily √see that |d m /dt m G σ2 (y, t)|  cm uniformly for σ ∈ (0, 1). Thus differentiating t 2 δ0 2σ (t) and integrating in (4.11), we find

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(1 + ξ 2 )|Fyσ (ξ)|  c .

(4.12)

Interpolating between (4.10) and (4.12), we obtain for θ ∈ (0, 1) |Fyσ (ξ)|  cn (1 + ξ 2 )−(1−θ) σ 2θ (1 + |y|)−nθ . It thus follows that for θ ∈ (0, 1/2) and any n |I2σ |  cn,θ σ 2θ (1 + |y|)−nθ which gives |J2σ |  cθ σ 2θ . We have thus shown  √ Jσ / 2π = (δxσ , Cδxσ )|φ(x)|2 d x =



√ 2σ

δ0

(t) f  (−t)dt

 

 g  (y)ψσ (y)dy + O(σ 2θ )

(4.13) for θ ∈ (0, 1/2). Notice that β(σ) :=

 √  √2σ 2 2π δ0 (t) f  (−t)dt = e−(σξ) f  (ξ)dξ

(4.14)

is independent of φ. Let us assume that g  is not of constant sign. Choose φ1 ∈ C0∞ (R) √   σ/ 2 ∗ |φ1 |2 , we have so that  g (y)|φ1 (y)|2 dy > 0. Then with ψσ1 = δ0  1  2 limσ→0 g (y)ψσ (y)dy = g (y)|φ1 (y)| dy > 0 and thus β(σ) = Jσ





g  (y)ψσ1 (y)dy

−1

+ O(σ 2θ )

(4.15)

which implies that for small σ, β(σ)  −c1 σ 2θ for some c1 > 0 since Jσ  0. Similarly choosing φ2 ∈ C0∞ (R) so that g  (y)|φ2 (y)|2 dy < 0, we find for small σ ∞ β(σ)  c2 σ 2θ for some c2 > 0. It follows that for small √ σ, for any φ ∈ C0 (R), σ σ 2θ Jσ = O(σ ). But since Iσ = (δ0 ∗ φ, Cδ0 ∗ φ)  c Jσ / 2π, we can take the limit σ → 0 and conclude (φ, Cφ) = 0 for all φ ∈ C0∞ (R) and thus C = 0. Hence, if C  0 and C = 0, g is a monotone function. By (4.1) the same holds for f . Combining (4.13) and (4.14), we see that √ Iσ = (δ0σ ∗ φ, Cδ0σ ∗ φ)  c Jσ / 2π

    √ −(σξ)2   f (ξ)dξ  c(1/ 2π) e g (y)ψσ (y)dy + O(σ 2θ ).

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Taking σ → 0, we obtain   √ f  (ξ)dξ g  (y)|φ(y)|2 dy (φ, Cφ)  c(1/ 2π)  (where again c = supp φ 1d x) so that g  (x) f  (ξ)  0. To deal with the general case where neither f nor g is known to be smooth note that if i[ f (P), g(Q)]  0 and nonzero, the same is true if we replace g with g ∗ ρ where ρ is a smooth nonnegative approximation to the identity. Similarly for f replaced with f ∗ ζ. It then follows that both g ∗ ρ and f ∗ ζ are monotone. By taking ρn a suitable sequence, g ∗ ρn → g on a set E of full Lebesgue measure and thus g is monotone on E. If we take G(x) = limu∈E,u↑x g(u), then G is monotone and equals g a.e. Clearly the same idea works for f . It is easy to see that G and F, the corresponding version of f , are either both increasing or both decreasing. 

5 Theorem 1.5—Continuity of g, Absolute Continuity of the Inverse of g In this section, we assume that the commutator i[ f (P), g(Q)] is nonzero and positive. Lemma 5.1 Suppose i[ f (P), g(Q)] = C where C is positive and f and g are 2 monotone  nondecreasing and bounded. Then for all ψ ∈ L (R), (i[ f (P), g(Q)]ψ) (x) = K (x, y)ψ(y)dy where 1 g(x) − g(y)  K (x, y) = √ d f (y − x). x−y 2π

(5.1)

The operator C is trace class with tr C = [ f ][g]/2π, and the kernel K is square integrable. Proof Let gt = φt ∗ g where φ is a nonnegative smooth function of compact support whose integral is 1 and φt (x) = t −1 φ(t −1 x). Then  i[ f (P), gt (Q)] =

R

C(a)φt (a)da

where C(a) = e−i Pa Cei Pa . The commutator i[ f (P), gt (Q)] is a positive operator with continuous kernel so we can calculate its trace which is easily seen to be [ f ][g]/2π. We have 

 tr R

C(a)φt (a)da =

(tr e−i Pa Cei Pa )φt (a)da =

 tr (C)φt (a)da = tr C

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which shows tr C = [ f ][g]/2π. Since i[ f (P), gt (Q)] has constant trace its Hilbert– Schmidt norm  is uniformly bounded. Thus if we abbreviate K t (x, y) = K (x − a, y − a)φt (a)da, we have 

 ∞ > lim inf t↓0

|K t (x, y)| d xd y  2

|K (x, y)|2 d xd y

by Fatou’s lemma. Here, we have used that K t → K a.e. It follows easily that K t → K in L 2 (R2 ). For an L 2 (R) function ψ, let Ft = K t (x, y)ψ(y)dy and F = K (x, y)ψ(y)dy. We have ||Ft − F||  ||K t − K || L 2 (R2 ) ||ψ||. But Ft → Cψ strongly which gives (5.1). This finishes the proof.  Lemma 5.2 Suppose i[ f (P), g(Q)] = C where C is nonzero and positive, while f and g are bounded and monotone increasing. Then g is continuous. Proof We use the square integrability of the kernel K . Suppose g has a jump of k  at 0. Then for all  > 0  |(g(y) − g(x))/(y − x)|2 |df (y − x)|2 d xd y < k x

for some k. Now |df (y − x)|  δ > 0 for y − x small, while g(y) − g(x)  k  > 0 in the range of integration. This contradicts the above inequality.  Lemma 5.3 Suppose i[ f (P), g(Q)] = C where C is positive and f and g are monotone nondecreasing and bounded. Then g is strictly increasing. Proof Suppose g is constant on I = (a, b). Then by the explicit form of the kernel K of the commutator C = i[ f (P), g(Q)], we see that for ψ ∈ C0∞ (I ), (ψ, Cψ) = 0. Since C  0, this means Cψ = 0 which in turn implies ( g(x)−g(y) )df (y − x) = 0 for x−y y ∈ I and x ∈ R. But df is continuous and nonzero at 0, so nonzero in an interval J = (−c, c), c > 0. Thus g is constant on the interval I + J . The result follows by induction.  We give a quick proof of Putnam’s theorem [9, 10] on positive commutators: Theorem 5.4 (Putnam) Suppose A and H are self-adjoint operators with A bounded. Suppose i[H, A] = C with C self-adjoint and positive. We assume this equality is true in the sense that for each ψ ∈ D(H ) i(H ψ, Aψ) − i(Aψ, H ψ) = ||C 1/2 ψ||2 . Then, the absolutely continuous subspace of H , Hac (H ) ⊃ Ran C. Remark 5.5 It will be clear in the proof that A need not be bounded but it rather suffices that D(A) contain all vectors in the range of E H (J ) for all finite intervals J . (Here E H (·) is the projection-valued spectral measure associated with the operator H .) This generalization may have limited applicability so we do not include a proof (although it should be obvious).

The Howland–Kato Commutator Problem

205

Proof For a finite interval I = (a, b), let λ = (a + b)/2, and E = E H (I ). We use the notation |J | for the Lebesgue measure of the Borel set J . Then note |(Eψ, i[H, A]Eψ)| = |(Eψ, i[H − λI, A]Eψ)| = |((H − λ)Eψ, AEψ) − (AEψ, (H − λ)Eψ)|  2(|I |/2)||A||||ψ||2 . Thus ||EC E||  |I |||A||. We also have ||EC E|| = ||(EC 1/2 )(EC 1/2 )∗ || = ||(EC 1/2 )∗ (EC 1/2 )|| = ||C 1/2 EC 1/2 || so that if ||ψ|| = 1,

(C 1/2 ψ, E H (I )C 1/2 ψ)  |I |||A||.

This inequality extends from finite open intervals, I , to arbitrary Borel sets and shows that Hac (H ) ⊃ Ran C 1/2 . But it is easily seen that Ran C 1/2 = Ran C which proves the theorem.  We use Putnam’s theorem to show absolute continuity of the inverse of the function g. Proposition 5.6 Suppose i[ f (P), g(Q)] = C, where f and g are real increasing bounded functions and C is positive and nonzero. Then the inverse function g −1 is absolutely continuous. Proof Let Mg be multiplication by g, that is, Mg = g(Q). From Putnam’s theorem, we have that Hac (Mg ) ⊃ Ran C and since for a ∈ R i[ f (P − a I ), g(Q)] = eia Q Ce−ia Q it follows that Hac (Mg ) ⊃ eia Q Ran C. Suppose (φ, eia Q Cψ) = 0 for all a ∈ R and all ψ ∈ L 2 (R). It follows that φ(x)(Cψ)(x) = 0 a.e. Choose a > 0 so that Re df (x) > 0 for x ∈ I = (−a, a). Choose a nonnegative ψ ∈ C0∞ ( 21 I ) with ψ(0) > 0. Then for  any x0 ∈ R, Re(Cψ(x)) = (2π)−1/2 g(x)−g(y) Re(df (y − x))ψ(y − x0 )dy > 0 for x−y 1 x ∈ 2 I + x0 . Thus for all x0 , φ(x) = 0 for a.e. x ∈ 21 I + x0 , or in other words φ is the zero vector in L 2 (R). It follows that Hac (Mg ) = L 2 (R). With H = Mg we have that E H (J ) is multiplication by 1{x:g(x)∈J } = 1g−1 (J ) . So Mg being absolutely continuous as an operator means that if the Borel set J has Lebesgue measure 0 then E Mg (J ) = 0 and thus g −1 (J ) has Lebesgue measure 0. Thus the strictly increasing function g −1 is absolutely continuous. (Note that even with the fact that g is strictly increasing, the absolute continuity of Mg as an operator alone does not imply that g is continuous.)  We remark that because of (4.1) all the results of this section apply to f as well as g.

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?

6 Theorem 1.6—(K b , L ∞ ) − → (K b , K bˆ ) Proof of Theorem 1.6 We assume g has the integral representation (1.3) where μ  ˆ is a nonzero finite positive measure with |t|e2bt dμ(t) < ∞. The proof where  ˆ |t|e−2bt dμ(t) < ∞ is similar. Assuming the commutator i[ f (P), g(Q)] is nonnegative, we are aiming to prove that f has the integral representation  f (x) =

tanh b(x − t)dν(t) + d

for some finite positive measure ν and with bbˆ = π/2. We first assume that f is as above but with dν(t) replaced with w(t)dt where w ∈ L 1 (R) ∩ L ∞ (R) and where w is real and continuous but not necessarily nonnegative. This will be justified at the end of the proof. Our first task is to prove that w  0. For the purpose of obtaining a contradiction, suppose that is not the case. Then the sets E 1 = {s : w(s) > 0} and E 2 = {s : w(s) < 0} are both non-empty. For j = 1, 2, put w j = 1 E j · |w|, where 1 F denotes the characteristic function of the set F. Clearly, w = w1 − w2 . A computation based on (2.5) shows that if ψ is in L 2 (R), then (ψ, i[ f (P), g(Q)]ψ) =

2    e−isx bˆ dμ(t) w(s)ds  0, ψ(x)d x ˆ − t) π cosh b(x (6.1)

or equivalently   

2 eisx ψ(x)d x w2 (s)ds dμ(t) ˆ − t) cosh b(x 2    eisx ≤ ψ(x)d x w1 (s)ds dμ(t), ˆ − t) cosh b(x

for all ψ ∈ L 2 (R) (we find it convenient to change signs in the complex exponential ¯ The last inequality can be written as in (6.1), and so have replaced ψ by ψ). A2 ψ ≤ A1 ψ,

(6.2)

with A j : L 2 (R) → L 2 (ν j × μ) defined by  (A j ψ)(x, t) =

∞

−∞

eisx ψ(x)d x ˆ − t) cosh b(x

for j = 1 or 2, where dν j (s) = w j (s)ds. These operators are in fact bounded, as one can see by applying (6.1) with ν = ν1 or ν = ν2 and with f correspondingly modified. We see (writing A for A1 or A2 as the case may be)

The Howland–Kato Commutator Problem

207

bˆ ¯ i[ f (P), g(Q)]ψ) ¯ Aψ2 = (ψ, π ≤ 2 f ∞ g∞ ψ2 . Let us turn to the inequality (6.2), which can be restated as A∗2 A2 ≤ A∗1 A1 . According to a lemma of Douglas [11], A2 = C A1 for some contraction operator C : L 2 (ν1 × μ) → L 2 (ν2 × μ), so A∗2 = A∗1 C ∗ and Ran A∗2 ⊂ Ran A∗1 . We will show that this range inclusion cannot occur, and to this end look closely at the operators A∗j . Note that if h ∈ L 2 (ν j × μ),  

(A∗j h)(x)

=

e−isx h(s, t)w j (s)ds dμ(t). ˆ − t) cosh b(x

It follows that ˆ

2−1 ebx (A∗j h)(x) =

 

ˆ

ˆ

ˆ

e−isx (1 − e2b(t−x) (1 + e2b(t−x) )−1 )h(s, t)w j (s)dsebt dμ(t).

Further ˆ

ˆ

ˆ

e2b(t−x) (1 + e2b(t−x) )−1 = 1(−∞,t) (x) − U (x − t), with |U (x)|  e−2b|x| . We see therefore that ˆ

2−1 ebx (A∗j h)(x) = I j (x) + I I j (x) + I I I j (x) where

 

ˆ

e−isx h(s, t)w j (s)dsebt dμ(t)   ˆ I I j (x) = − e−isx 1(−∞,t) (x)h(s, t)w j (s)dsebt dμ(t)   ˆ I I I j (x) = e−isx U (x − t)h(s, t)w j (s)dsebt dμ(t). I j (x) =

First consider I j . We define q j (s) =





 ˆ h(s, t)ebt dμ(t) w j (s).

Then q j ∈ L 2 (R) ∩ L 1 (R). Indeed 

  2 ˆ h(s, t)ebt dμ(t) w j (s)2 ds    ˆ 2bt  ||w||∞ e dμ(t) |h(s, t)|2 w j (s)dsdμ(t) < ∞

|q j (s)|2 ds =

(6.3) (6.4) (6.5)

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and  |q j (s)|ds   

ˆ

|h(s, t)|w j (s)1/2 ebt w j (s)1/2 dμ(t)ds 1/2  

  

|h(s, t)| w j (s)dμ(t)ds 2

 =

1/2 ||w j ||1

e

ˆ 2bt

e

ˆ 2bt

1/2 w j (s)dμ(t)ds

1/2   dμ(t)

1/2 |h(s, t)| w j (s)dsdμ(t) 2

< ∞.

Here, we have used the Schwarz inequality and the facts that w j ∈ L 1 (R) ∩ √  ˆ L ∞ (R), e2bt dμ(t) < ∞, and h ∈ L 2 (ν j × μ). We see that I j = 2π qˆj which implies I j ∈ L 2 (R) as well as in C0 (R), the space of continuous functions on R vanishing at infinity.  We turn to I I j . We first write g(t, x) = e−isx h(s, t)w j (s)ds and note that by the Plancherel theorem  (6.6) |g(t, x)|2 d xdμ(t)  2π||w j ||∞ ||h||2L 2 (ν j ×μ) We have  ˆ I I j (x) = − 1(−∞,t) (x)g(t, x)ebt dμ(t) (−∞,0)  ˆ 1(−∞,0) (x)g(t, x)ebt dμ(t) − [0,∞)  ˆ 1[0,t) (x)g(t, x)ebt dμ(t) − [0,∞)

Clearly from (6.6) and the Schwarz inequality, each of these three terms is in L 2 (R). Evidently, the first two terms vanish for x > 0 so that their sum can be written as a Fourier transform, vˆ j , with v j ∈ H−2 , the Hardy space of the lower half plane [12]. (Here, we follow the convention of identifying elements of H−2 , which are analytic in the open lower half plane, with their boundary value func ˆ tions which lie in L 2 (R).) Using the condition [0,∞) te2bt dμ(t) < ∞ and the Schwarz inequality, we see that the third piece of I I j is also in L 1 (R), thus of the form pˆ j with p j ∈ C0 (R) ∩ L 2 (R). In summary, I I j = vˆ j + pˆ j with v j ∈ H−2 and p j ∈ C0 (R) ∩ L 2 (R). Finally, I I I j ∈ L 2 (R) by essentially the same argument that shows each term of I I j ∈ L 2 (R). Moreover, I I I j ∈ L 1 (R) which can be seen as follows. On rewriting I I I j (x) in terms of g(t, x) defined above, we have

The Howland–Kato Commutator Problem

 |I I I j (x)|2  

209



ˆ

|U (x − t)|2 e2bt dμ(t)

|g(t, x)|2 dμ(t)

and thus

1/2  ˆ |I I I j (x)|d x  ||U ||2 2π||w j ||∞ ||h||2L 2 (ν j ×μ) e2bt dμ(t)

where we have used (6.6). Thus I I I j = uˆ j where u j ∈ C0 (R) ∩ L 2 (R). On putting the above together, we have ˆ

2−1 ebx (A∗j h)(x) = (qˆ j + vˆ j + pˆ j + uˆ j )(x).

(6.7)

To obtain a contradiction we now use the fact that RanA∗2 ⊂ RanA∗1 .

(6.8)

Choose h 2 ∈ L 2 (ν2 × μ) as h 2 (s, t) = 1 F (s), where F is a nondegenerate compact interval contained in the (assumed) non-empty open subset E 2 . By (6.8), there exists h 1 ∈ L 2 (ν1 × μ) with A∗2 h 2 = A∗1 h 1 .

(6.9)

For j = 1 or 2, associate q j , v j , p j , and u j to h j as they are associated to h in (6.7). By (6.9) we have qˆ2 + vˆ2 + pˆ 2 + uˆ 2 = qˆ1 + vˆ1 + pˆ 1 + uˆ 1 , or applying the inverse Fourier transform and rearranging q2 − q1 + p2 − p1 + u 2 − u 1 = v1 − v2 . Clearly,

 q2 (s) = 1 F (s)w2 (s)

(6.10)

ˆ

ebt dμ(t)

has nontrivial jump discontinuities at the end points of F since w2 is positive and continuous on E 2 . Let J ⊂ E 2 be a compact interval containing one of the endpoints of F, x0 , in its interior. Since q1 ≡ 0 on E 2 and p1 , p2 , u 1 , u 2 are continuous, v1 − v2 has a jump discontinuity at x0 . However, this will contradict a theorem of Lindelöf, since v1 − v2 ∈ H−2 . Indeed, the Poisson integral provides the analytic extension of v1 − v2 into the lower half plane. The left side of (6.10), and thus v1 − v2 are bounded in a neighborhood of J in R, so this analytic extension is bounded in the open half disk B in the lower half plane with the interior of J as its diameter. It follows that if φ is a conformal map of the open unit disk D onto B (such a φ must extend to a ¯ onto B) ¯ then (v1 − v2 ) ◦ φ is a bounded analytic function in D homeomorphism of D

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whose boundary value function has a jump discontinuity at φ−1 (x0 ). This contradicts Lindelöf’s theorem [13] (see also [14]) and shows that indeed w  0. Finally, we consider general f in L ∞ (R). Suppose our hypothesis i[ f (P), g(Q)] ≥ 0 is in effect. Let 1 u r (x) = rπ



sin r x x

2 ,

an approximate identity on R as r → ∞. Since u r  0 it follows that i[ f ∗ u r (P), g(Q)] ≥ 0. Let fr = f ∗ u r . Since fr is smooth, it also follows from Theorem 1.3 that ( fr ) ≥ 0. Since fr is bounded, we then see ( fr ) is in L 1 (R). By fr (taken in the sense of tempered direct computation u r has compact support, and so distributions) does as well. The same must be true for the Fourier transform of ( fr ) which is a continuous function. Now we let φ(x) = tanh bx and note

y 2  ˆ φ (y) = b . (6.11) ˆ π sinh(by)  on the support Let σ be a real even function in the Schwartz space agreeing with 1/φ     of fr . Then fr = φ · fr · σ, whence fr =

1  φ ∗ fr ∗ σ q. 2π

(6.12)

1  We put wr = 2π fr ∗ σ q, a convolution of real L 1 (R)-functions, and so in L 1 (R) and real valued. In particular,

 ∞  1  ( fr ) (x)d x  σ q1 wr 1 ≤ 2π −∞ 1 {( f ∗ u r )(∞) − ( f ∗ u r )(−∞)} q = σ 1 2π 1 ≤  f ∞ q σ 1 . π Further, wr is continuous and tends to zero at ±∞, hence wr ∈ L ∞ (R). On integrating equation (6.12) (that is fr = φ ∗ wr ), we have  fr (x) =

∞ −∞

tanh b(x − s)wr (s)ds + dr ,

(6.13)

where dr is a real constant. Since i[ f ∗ u r (P), g(Q)] ≥ 0, our previous results show that wr ≥ 0 a.e. On letting x → ±∞ in (6.13), we find by the dominated convergence theorem that

The Howland–Kato Commutator Problem

211

fr (∞) + fr (−∞) = 2dr , and fr (∞) − fr (−∞) = 2||wr ||1  so that |dr | ≤  f ∞ for r > 0. Since fr (±∞) = f (±∞) we have wr (s)ds = [ f ]/2. Now consider wr (s)ds as a measure on the two-point compactification [−∞, ∞] of R. Then there exists a finite positive measure ν on R and 1 , 2 ≥ 0, a real number d and a sequence rn → ∞ such that wrn (s)ds → ν + 1 δ−∞ + 2 δ+∞ , in the weak-∗ topology and drn → d as n → ∞; here, δ±∞ denotes the unit point mass at ±∞. Then taking a limit along rn in (6.13) yields f (x) = lim frn (x) n→∞  ∞ tanh b(x − s)dν(s) + 1 − 2 + d = −∞

for all x ∈ R. It follows that f ∈ K bˆ , and the proof is complete.



7 Finite Rank Positive Commutators Considering that the positive commutator is trace class, it is interesting to consider the case where the positive commutator has finite rank. The next lemma shows that, in particular, in considering the rank one case, Kato’s assumption that f and g are absolutely continuous with L 1 (R) derivatives can be proved. Lemma 7.1 Suppose the commutator C = i[ f (P), g(Q)] is positive, nonzero, and has finite rank. Then f and g can be taken differentiable with continuous derivatives. These derivatives are everywhere nonzero. Proof We can assume that f and g are increasing. The kernel of the commutator C = i[ f (P), g(Q)], given by 1 g(x) − g(y)  K (x, y) = √ d f (y − x), x−y 2π has the form K (x, y) =

N  j=1

φ j (x)φ j (y)

(7.1)

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where {φ1 , φ2 , . . . , φ N } is a linearly independent set of L 2 (R) functions. Let x0 ∈ R and I a small open interval centered at x0 . Denote the complement of I as J and the restriction of φi to J as φi,J . The N × N matrix with elements (φi,J , φk,J ) is continuous in |I | so we can choose a small such interval I such that the determinant of this matrix is nonzero, and thus the vectors φ1,J , φ2,J , . . . , φ N ,J are linearly independent. We can thus find N functions ψi ∈ C0∞ (J o ) so that the matrix with matrix elements (φi,J , ψk ) = (φi , ψk ) has nonzero determinant. If we define γ j (x) for all x ∈ R by  γ j (x) = (φi , ψ j )φi (x), i

the φi are thus linear combinations of the γ j and at least for x outside the support of the ψ j we have  γ j (x) =

K (x, y)ψ j (y)dy.

It follows from the explicit form of K that the φi can be taken continuous in a neighborhood of I . Since we can cover R with a countable number of such intervals, we  can assume all the φ √i are continuous functions. In addition, since d f (u) is continuous and df (0) = [ f ]/ 2π is nonzero, we can see from the explicit form of K that g is C 1 with  |φ j (x)|2 /[ f ]. g  (x) = 2π j

If φ j (x) = 0 for all j, this contradicts the fact that g is strictly increasing and that df (y − x) is nonzero for |y − x| small. Thus g  (x) is positive for all x. Since F f (P)F −1 = f (Q) and Fg(Q)F −1 = g(−P), i[−g(−P), f (Q)] has the integral kernel  j (ξ)φ j (η). φ K˜ (ξ, η) = j

It follows that what we have proved for g is also true for f . In particular, f ∈ C 1 (R) with  |φ j (ξ)|2 /[g]. (7.2) f  (ξ) = 2π j

 We can actually prove much more: Theorem 7.2 Suppose the commutator C = i[ f (P), g(Q)] is positive, nonzero, and finite rank. Then the functions f  and g  are in S(R).

The Howland–Kato Commutator Problem

213

Proof We write K (x, y) =

N 

φ j (x)φ j (y)

j=1

where the functions φ j are in L 2 (R) and linearly independent. We have shown that these functions can be assumed continuous. Let v(y) =< φ1 (y), · · · , φ N (y) > ∈ C N . The span of {v(y) : y ∈ R} is C N forotherwise there is a nonzero vector c ∈ C N perpendicular to this span. This implies c j φ j (y) = 0 contradicting the fact that C has rank N . Choose N points y j so that v(y1 ), · · · , v(y N) are linearly independent. Let ζ ∈ C0∞ (−1, 1) be a real nonnegative function with ζ(y)dy  = 1. Let ζt (y) = ζ (y − y). Then v := ζ ∗ v(y ) = ψ j (y)v(y)dy =< t−1 ζ(y/t) and ψ j (y) = j t j  t j ψ j (y)φ1 (y)dy, · · · , ψ j (y)φ N (y)dy >. We choose t > 0 small enough so that the v j span C N . Let  γ j (x) = φk (x)(φk , ψ j ) = (v j , v(x)). k

The φ j (x) are linear combinations of the γ j (x). In fact, a standard linear algebra calculation gives  (A−1 ) jl γl (x)v j , v(x) = j,l

where Ai j = (vi , v j ). We will show that ||D m φ j ||2 + ||D m φ j ||2 < ∞ for all m. We proceed to show m / I := ∪ j (y j − t, y j + t) we have D φ j ∈ L 2 (R) ∩ L ∞ (R) by induction. For x ∈  γ j (x) =

R

g(y) − g(x)  f (y − x)ψ j (y)dy y−x

or changing variables  γ j (x) =

R

g(x + u) − g(x)  f (u)ψ j (x + u)du. u

We note that for all i, ψi = 0 in a neighborhood of I c so that |γ j (x)|  c |ψ j (y)|/|x − y|dy and thus φ j (x) and thus g  (x) are bounded in a neighborhood of I c . Then γ j (x)

 =

R

[

g  (x + u) − g  (x) g(x + u) − g(x)  ψ j (x + u) + ψ j (x + u)] f  (u)du u u

or |γ j (x)|

 c

|g  (y) − g  (x)| |ψ j (y)|dy + c |y − x|



|g(y) − g(x)|  |ψ j (y)|dy |y − x|

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 C(|x| + 1)−1 for x in a neighborhood of I c . It follows that φj ∈ L 2 (I c ) ∩ L ∞ (I c ). We now choose a different set of y j , call them y˜ j , and replace I with J = ∪ j ( y˜ j − t, y˜ j + t). If, for example, 0 < | y˜ j − y j | is small enough, the v( y˜ j ) will be linearly independent and if t is small enough the v j will also be linearly independent and I ∩ J = ∅. Using the same technique, we find φj ∈ L 2 (J c ) ∩ L ∞ (J c ). Thus, we finally conclude the first step: The φ j are differentiable and φ j , φj ∈ L 2 (R) ∩ L ∞ (R). n 2 ∞ We now suppose  D φ j2 ∈ L (R) ∩ L (R) for 1  n  m and 1  j  N . Using  g (x) = 2π( j |φ j (x)| )/[ f ], c = 2π/[ f ], we calculate D

n+1

 n  [D k φ j (x)D n−k φ j (x)]. g(x) = c k j,k

This gives D n+1 g ∈ L 1 (R) ∩ L ∞ (R) for 1  n  m. Again using  γ j (x) =

R

g(x + u) − g(x)  f (u)ψ j (x + u)du, u

we see that γ j is m + 1 times continuously differentiable in a neighborhood of I c with

D

m+1

 m + 1  D k g(x + u) − D k g(x) γ j (x) = f  (u)D m+1−k ψ j (x + u)du k u k

which gives |D m+1 γ j (x)|  C

  |D k g(x + u) − D k g(x)| |D m+1−k ψ j (x + u)|du |u| k

  |D k g(y) − D k g(x)| |D m+1−k ψ j (y)|dy  C  (|x| + 1)−1 =C |y − x| k in a neighborhood of I c . So D m+1 γ j , and hence the D m+1 φk are in L 2 (I c ) ∩ L ∞ (I c ). Using J instead of I , we complete the induction step to learn D m+1 φ j ∈ L 2 (R) ∩ L ∞ (R). As before F f (P)F −1 = f (Q) and Fg(Q)F −1 = g(−P), so i[−g(−P), f (Q)] has the integral kernel  j (ξ)φ j (η). φ K˜ (ξ, η) = j

The Howland–Kato Commutator Problem

215

j . This completes the It follows that what we have proved for φ j is also true for φ proof.  In case the commutator C has finite rank with kernel as in (7.1), an important part of the kernel can be expressed entirely in terms of the φ j ’s. In particular, we have  φ j (x + u)φ j (y + u)du. f  (y − x) = (2π/[g]) j

There is another way of seeing this but let us calculate directly. Using the formula (7.2) for f  , we should calculate 

   √ e−i(y−x)ξ | φ(ξ)|2 dξ/ 2π = ei(x−y)ξ φ(x1 )e−iξx1 eiξx2 φ(x2 )d x1 d x2 dξ(2π)−3/2  √ = φ(x − y + x2 )φ(x2 )/ 2π  √ = φ(x + u)φ(y + u)du/ 2π.

8 Operator Monotone Functions The following proposition gives another connection between operator monotone functions and positive commutators. Proposition 8.1 Suppose f and g are bounded measurable functions with i[ f (P), g(Q)]  0. Suppose F is an operator monotone function on an open interval containing the range of f . Then i[F( f (P)), g(Q)]  0. Proof Looking at the derivative, we see that if t > 0, e−itg(Q) f (P)eitg(Q)  f (P). It follows that e−itg(Q) F( f (P))eitg(Q) = F(e−itg(Q) f (P)eitg(Q) )  F( f (P)). The result now follows from the fact that the derivative of the left side of the inequality at t = 0 is thus positive. 

9 Two Additional Representations of the Commutator The integral kernel of the commutator i[ f (P), g(Q)] given in (2.6) is clearly not very symmetric in the functions f and g although there is much symmetry in the operator (see (4.1) for example). In this section, we remedy this by giving two symmetrical representations. The integral kernel of our first symmetrical representation of i[ f (P), g(Q)] which is on L 2 (R2 ) appears in the next lemma.

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Lemma 9.1 Suppose i[ f (P), g(Q)] = C, where f and g are real measurable bounded functions and C is positive and nonzero. Then a representation of the commutator on L 2 (R2 ) has integral kernel √ √  2(y2 − x2 ))ei(x1 y2 −x2 y1 ) sin(y1 − x1 )(y2 − x2 ) /π K (x1 , x2 ; y1 , y2 ) = df ( 2(y1 − x1 ))dg( (y1 − x1 )(y2 − x2 )

Proof We define a different representation of the canonical commutation relations. We set p j = −i∂x j and √ √ p = ( p1 − x2 )/ 2, x = (x1 + p2 )/ 2 √ a = (x + i p)/ 2 √ √ p  = ( p2 − x1 )/ 2, x  = ( p1 + x2 )/ 2 √ a  = (x  + i p  )/ 2 We have [a, a ∗ ] = [a  , a ∗ ] = 1 [a, a  ] = [a, a ∗ ] = 0 For f, g ∈ L 1 (R) we have f ( p)ψ(x1 , x2 ) = π −1/2 g(x)ψ(x1 , x2 ) = π

−1/2

 

√ fˆ( 2(y1 − x1 ))ei x2 (x1 −y1 ) ψ(y1 , x2 )dy1 √ g( ˆ 2(y2 − x2 ))e−i x1 (x2 −y2 ) ψ(x1 , y2 )dy2 .

Thus

π −1 = π −1

f ( p)g(x)ψ(x1 , x2 ) =

 

√ √ fˆ( 2(y1 − x1 ))ˆg ( 2(y2 − x2 ))ei x2 (x1 −y1 )−i y1 (x2 −y2 ) ψ(y1 , y2 )dy1 dy2 √ √ fˆ( 2(y1 − x1 ))ˆg ( 2(y2 − x2 ))ei(x2 −y2 )(x1 −y1 )+i(x1 y2 −x2 y1 ) ψ(y1 , y2 )dy1 dy2

and

π −1 = π −1

g(x) f ( p)ψ(x1 , x2 ) =



√ fˆ( 2(y1 − x1 ))ˆg ( 2(y2 − x2 ))e−i x1 (x2 −y2 )+i y2 (x1 −y1 ) ψ(y1 , y2 )dy1 dy2



√ √ fˆ( 2(y1 − x1 ))ˆg ( 2(y2 − x2 ))e−i(x2 −y2 )(x1 −y1 )+i(x1 y2 −x2 y1 ) ψ(y1 , y2 )dy1 dy2 .

√

The Howland–Kato Commutator Problem

217

It follows that  i[ f ( p), g(x)] =

K (x1 , x2 ; y1 , y2 )ψ(y1 , y2 )dy1 dy2

where K (x1 , x2 ; y1 , y2 ) √ √ = −2π −1 fˆ( 2(y1 − x1 ))g( ˆ 2(y2 − x2 )) sin(x1 − y1 )(x2 − y2 )ei(x1 y2 −x2 y1 ) √ √  2(y2 − x2 )) sin(x1 − y1 )(x2 − y2 ) ei(x1 y2 −x2 y1 ) = π −1 df ( 2(y1 − x1 ))dg( (x1 − y1 )(x2 − y2 ) If f, g are increasing and bounded, it is not hard to make sense of these manipulations using distributions. This proves the result.  Remark 9.2 Note that if  is the vacuum vector (a = a   = 0), then L 2 (R2 )  F1 ⊗ F2 , where F j are the Fock spaces which are the spans of {(a ∗ )n  : n = 0, 1, 2, ...} and {(a ∗ )n  : n = 0, 1, 2, ...} and i[ f (P), g(Q)] = k ⊗ I for some operator k acting in F1 . Another representation on L 2 (R): Lemma 9.3 i[ f (P), g(Q)] = (2π)−1



 sin(uξ/2) dudξ. ei(ξ Q+u P) df (u)dg(ξ) uξ/2

Proof We calculate 

 ∞ dg(t + x) = lim e−t dg(t + x) ↓0 0 0  ∞  = lim(2π)−1/2 e−t ei(t+x)ξ dg(ξ)dξdt

g(∞) − g(x) =

∞

↓0

= lim(2π)−1/2 ↓0

0



ei xξ

 dg(ξ) dξ  − iξ

and [ f (P), ei Qξ ] = ei Qξ ( f (P + ξ) − f (P))  1  1 f  (P + sξ)ξds = (2π)−1/2 ξ ei Qξ ei(P+sξ)u df (u)duds = ei xξ 0

0

and thus   1 ξ  ei Qξ ei(P+sξ)u df (u)dg(ξ)dudsdξ i[ f (P), g(Q)] = lim −i(2π)−1 ↓0  − iξ 0   1  ei Qξ ei(P+sξ)u df (u)dg(ξ)dudsdξ = (2π)−1 0

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1



 = (2π) ei(Qξ+Pu) ei(s−1/2)ξu df (u)dg(ξ)dudsdξ 0   sin(ξu/2)   ei(Qξ+Pu) = π −1 d f (u)dg(ξ)dudξ. ξu −1

Since we do not know a priori that f is differentiable, we should first mollify our bounded f and then take a limit at the end of the calculation.  As an application of this representation, we calculate expectations of the com√ −x 2 /2 mutator in coherent states: With a = (Q + i P)/ 2 and  = π −1/4 e√ we have ∗ a = 0. Let ψ(z) = e za . Then aψ(z) = zψ(z). Let ζ = (ξ + iu)/ 2. We calculate ∗ ¯ 2 ∗ ¯ ei(ξ Q+u P) = ei(ζa +ζa) = e|ζ| /2 ei ζa eiζa ¯ (ψ(w), ψ(z)) = ewz .

Thus 

2 sin(ξu/2)   e|ζ| /2 e(w−iζ)(z+iζ) d f (u)dg(ξ)dudξ ξu  2 ¯ sin(ξu/2)  ¯ ¯ ζz)  e−|ζ| /2+i(ζ w+ d f (u)dg(ξ)dudξ = π −1 ewz ξu

(ψ(w), i[ f (P), g(Q)]ψ(z)) = π −1

(9.1) Putting z = w we obtain 

e−(ξ

2

+u 2 )/4 i(ξx+uy) sin(ξu/2) 

e

ξu

 d f (u)dg(ξ)dudξ 0

(9.2)

√ for all z = (x + i y)/ 2.

10 Some Results that Follow from 2 × 2 Positivity We assume that i[F(P), G(Q)] = C  0, C = 0. It then follows that F and G can be taken strictly increasing, continuous, with inverses which are absolutely continuous. We now look more carefully at the condition that the kernel Hx y =

G(x) − G(y)  d F(y − x) x−y

gives a positive semidefinite 2 by 2 matrix for pair (x, y). In this section only, √  any −i xξ  2π and use d F(x) = e d F(ξ). Let us restrict to F with we drop the factor of  d F = 1. An important parameter will make its appearance, namely,

The Howland–Kato Commutator Problem

 σ 2 = inf{

219

 ξ 2 d f (ξ) − (

 ξd f (ξ))2 : i[ f (P), G(Q)] = C  0, C  = 0,

d f = 1}

which we believe must be related √ to the Kato class of G (see some discussion below). We assume σ 2 < ∞. Let aˆ = 3σ. Theorem 10.1 Suppose i[F(P), G(Q)] = C  0, C = 0 where F and G are real bounded measurable functions. Then, both are monotone increasing or monotone decreasing and continuous. Suppose there exists an increasing f with i[ f (P),  G(Q)]  0 and with ξ 2 d f (ξ) < ∞, d f = 1. Then in fact G is C 1 with a Lipschitz derivative and thus G  is absolutely continuous. G satisfies the following estimates: ˆ ˆ 0 )| 0 )|  G  (x)  G  (x0 )e2a|(x−x G  (x0 )e−2a|(x−x   |G (x)|  2aG ˆ (x), a.e. and both

(10.1) (10.2)

G  (x)  2a|G(±∞) ˆ − G(x)|.

(10.3)

If G is odd then G  (x) 

G  (0) . (cosh(ax)) ˆ 2

(10.4)

This inequality is an equality in the rank one case [2] and if K is true the rank, one case is the case of minimal variance (see Eq. 10.12). The proof will require some preliminary work. Let gt = φt ∗ G where φ is a nonnegative smooth function of compact support whose integral is 1and φt (x) = t −1 φ(t −1 x). We  take f to satisfy i[ f (P), G(Q)] = C  0, C = 0 and d f = 1 and additionally ξ 2 d f (ξ) < ∞. The idea is to get uniform bounds on the derivatives of gt and then take limits to learn about G. In the following, we drop the subscript t. Lemma 10.2 The function g  is positive. The function ψ = (g  )−1/2 satisfies − ψ  + aˆ 2 ψ  0

(10.5)

Proof Let y = x + h with h small and nonzero. Then g(x) − g(y) = (g(x + h) − g(x))/ h = g  (x) + hg  (x)/2 + h 2 g  (x)/6 + o(h 2 ) x−y (10.6)   df (y − x) = e−i hξ d f (ξ) = e−i hξ e−i h(ξ−ξ) d f (ξ) = e−i hξ (1 −

h2 2 (ξ  − ξ2 ) + o(h 2 )) 2

(10.7)

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where we have written



ξ n d f (ξ) = ξ n . We have

Hx x Hyy − |Hx y |2 = g(x) − g(y) 2  g  (x)g  (y) − ( ) |d f (y − x)|2 x−y = g  (x)(g  (x) + hg  (x) + h 2 g  (x)/2) − (g  (x) + hg  (x)/2 + h 2 g  (x)/6)2 + (g  (x) + hg  (x)/2 + h 2 g  (x)/6)2 (1 − |df (y − x)|2 ) + o(h 2 ) = h 2 g  g  /6 − h 2 (g  )2 /4 + (g  )2 (1 − [1 − h 2 (ξ 2  − ξ2 )]) + o(h 2 ) =

h 2   [g g − 3(g  )2 /2 + 6(g  )2 (ξ 2  − ξ2 )] + o(h 2 ) 6

(10.8)

We thus find g  (x)g  (x) − 3(g  (x))2 /2 + 6g  (x)2 (ξ 2  − ξ2 )  0

(10.9)

  Since this is true for all monotone f with d f = 1, ξ 2 d f < ∞ and i[ f (P), G(Q)] positive semidefinite, we take the infimum of the variance of d f over all such f to find (10.10) g  (x)g  (x) − 3(g  (x))2 /2 + 6σ 2 g  (x)2  0 Since g  (x) > 0 (see the formula for the determinant above and use the fact that g is strictly increasing) we can introduce ψ = (g  )−1/2 . Then recalling aˆ 2 = 3σ 2 , (10.10) becomes (10.11) − ψ  + aˆ 2 ψ  0  To see why we have used the abbreviation aˆ 2 = 3σ 2 , suppose we assume K and take g in its smallest Kato class, K a . Let us find σ 2 . We must have f ∈ K aˆ , viz., f (ξ) = tanh aξ ∗ μ/2 + c forsome positive measure μ and where aˆ = π/2a. Then if μ is a probability measure, f  (ξ)dξ = 1. We compute  2   ξ cosh−2 (ξ)dξ 2 ] + t dμ − ( tdμ)2 ξ 2  − ξ2 = a −2 [  cosh−2 (ξ)dξ

(10.12)

This is minimized by taking μ a point measure. We compute with the help of Grad∞ shteyn and Ryzhik 3527.3, −∞ x 2 cosh−2 (x)d x = π 2 /6 and thus σ 2 = π 2 /12a 2 = aˆ 2 /3

(10.13)

Continuing with the proof of Theorem 10.1 we set u = ψ  /ψ and obtain the Ricattitype inequality u   aˆ 2 − u 2 .

The Howland–Kato Commutator Problem

221

Lemma 10.3 |ψ  (x)/ψ(x)| < a. ˆ ˆ aˆ 2 − u(x)2  −(u(x) − a) ˆ 2 so that if Proof If u(x0 ) > aˆ then as long as u(x)  a,  2 w = u − a, ˆ w  −w . Thus w tends to +∞ for some x < x0 , a contradiction. If u(x0 ) < −aˆ then as long as u(x)  −a, ˆ u   −(u + b)2 . This leads to u(x) tendˆ We know ing to +∞ for some x > x0 , another contradiction. Suppose u(x0 ) = a. that |u(x)|  aˆ for all x. If  > 0 let aˆ  = aˆ + . We know that −ψ  + aˆ 2 ψ  0.   e−2aˆ x is monotone decreasing. Thus if x < x0 , A computation shows that aaˆˆ  +u −u    aˆ +u(x) −2aˆ x aˆ −2ax e  aˆ + e ˆ 0 . If we take  → 0 we find u(x) = a. ˆ Integrating we find aˆ  −u(x)  ax ˆ  −2ax ˆ 2 /c . This contradicts the boundedness of ψ(x) = ce for x < x0 or g (x) = e ˆ u(x) = −aˆ for all x > x0 and this also contradicts the g. Similarly if u(x0 ) = −a,  boundedness of g. So we have shown −aˆ < ψ  /ψ < aˆ for all x. Let m(t) =

a+u ˆ , a−u ˆ

Lemma 10.4

p(t) =

1 2

log m(t). Then for any x0 ∈ R,

g  (x) = g  (x0 )e

−2aˆ

x x0

tanh( p(t))dt

; p  (t)  a. ˆ

(10.14)

ˆ ˆ 0 )| 0 )| g  (x0 )e−2a|(x−x  g  (x)  g  (x0 )e2a|(x−x

|g  (x)|  2ag ˆ  (x), and both g  (x)  2a|g(±∞) ˆ − g(x)|. (10.15) Proof The fact that p  (t)  aˆ is a computation which uses u   aˆ 2 − u 2 . We comaˆ = − 21 (log g  ) . Integration gives (10.14). The first two lines of pute u = m−1 m+1 (10.15) follow directly from (10.14). To prove the last inequalities first note that lim x→∞ g  (x) = 0. To see this suppose the contrary. Then there is a sequence xn with xn+1 > xn + 1 so that g  (xn )  δ > 0. But then from the first inequality of the lemma g  (x)  δe−2aˆ for x ∈ [xn , xn + 1]. This contradicts the integrability of g  . Similarly lim x→−∞ g  (x) = 0. From the second line of (10.15), ˆ  (x) or − g  (x)  2ag  (g  (x) + 2ag(x)) ˆ  0.

Integrating from x to infinity gives one of the last inequalities. The other follows in a similar way.  To make a connection with the conjecture K, consider the case where g  is even. Corollary 10.5 g  (x) 

g  (0) (cosh ax) ˆ 2

(10.16)

222

I. Herbst and T. L. Kriete

Proof This follows directly from p  (t)  aˆ and the formula in (10.14). This inequality is an equality in the rank one case [2] and if K is true the rank one case is the case of minimal variance (see Eq. 10.12).  Proof of Theorem 10.1 We have g = gt = φt ∗ G satisfying 0 < gt (x)  2a[G], ˆ ˆ 2 [G]2 . By the Arzela–Ascoli theorem there is a sequence gn = |gt (x)|  (2a) gtn , tn ↓ 0, such that gn → G 1 and gn →  yG 2 uniformly on compact subsetsofy R. Clearly, G 1 = G and gn (x) − gn (y) = x gn (t)dt so that G(x) − G(y) = x G 2 1  (t)dt. Since G 2 is continuous it follows that G  uniformly on  yG is C and gn →   2 ˆ [G]2 |x − y| giving compacts. We also have |gn (x) − gn (y)| = | x gn (t)dt|  (2a)   2 2  ˆ [G] |x − y|. Thus G is absolutely continuous. We have |G (x) − G (y)|  (2a)  x+h   x+h   g (x + h) − gn (x) = x gn (u)du  x 2ag ˆ n (u)du. Thus G  (x+h)−G  (x)  nx+h   2aG ˆ (u)du. It follows that G (x)  2aG ˆ  (x). Similarly, −G  (x)  2aG ˆ  (x). x We have used that gn (±∞) = G(±∞). We take φ to be even and then g = gt is odd. Thus, the last inequality follows simply from Corollary 10.5.  If we now saturate (10.5) by√ making the inequality an equality, we obtain g −1/2 = jx − jx where j = 3σ. Taking b and c positive to insure that g  does not ψ = be + ce blow up at ±∞, we can write ψ = aˆ cosh( j (x − t)) for some aˆ > 0 and t ∈ R. This gives g(x) = aˆ −2 tanh( j (x − t)) + c for some constant c. We can check that indeed j = aˆ so that g ∈ K a . This is to be expected from Theorem 1.6 which in particular states that if i[ f (P), g(Q)]  0 for all f ∈ K aˆ , then g ∈ K a . But it does show that the inequality (10.5) is in some sense sharp, so that cases of equality occur if the Kato conjecture is correct.

11 An Interesting Formula Here is an interesting formula: Suppose g is a bounded real function with a bounded analytic continuation to a strip of width > 2λ. Consider i[tanh λP, g(Q)] = −2i[(1 + e2λP )−1 , g(Q)] = 2i(1 + e2λP )−1 [e2λP , g(Q)](1 + e2λP )−1   = 2i(1 + e2λP )−1 eλP (eλP g(Q)e−λP )eλP − eλP (e−λP g(Q)eλP )eλP (1 + e2λP )−1   = (i/2)(cosh λP)−1 eλP g(Q)e−λP − e−λP g(Q)eλP (cosh λP)−1 = (i/2)(cosh λP)−1 (g(Q − iλ) − g(Q + iλ))(cosh λP)−1 = (cosh λP)−1 Img(Q + iλ)(cosh λP)−1

(11.1)

Given the problems which occurred with manipulating unbounded functions of P in Section 2, we assure the reader that the computations above can be made rigorous. Perhaps (11.1) makes Theorem 1.6 intuitive. Acknowledgements We are grateful to Brian Hall for many useful conversations about this problem.

The Howland–Kato Commutator Problem

223

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14.

Howland, J.: Perturbation theory of dense point spectrum. J. Funct. Anal. 94, 52–80 (1987) Kato, T.: Positive commutators i[ f (P), g(Q)]. J. Funct. Anal. 96, 117–129 (1991) Löwner, K.: Über monotone Matrixfunktionen. Math. Z. 38, 177–216 (1934) Fuglede, B.: On the relation P Q − Q P = −i I . Math. Scand. 20, 79–88 (1967) Georgescu, V., Gerard, C.: On the virial theorem in quantum mechanics. Commun. Math. Phys. 208, 275–281 (1999) Folland, G.: Real Analysis, p. 107. Wiley, New York (1999) Fuglede, B.: A commutativity theorem for normal operators. Proc. Natl. Acad. Sci. 36, 35–40 (1950) Rosenblum, M.: On a theorem of Fuglede and Putnam. J. Lond. Math. Soc. 33, 376–377 (1958) Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, p. 61. Springer, Berlin (1987) Putnam, C.: Commutation Properties of Hilbert Space Operators and Related Topics, p. 20. Springer, Berlin (1967) Douglas, R.: On majorization, factorization and range inclusion of operators in Hilbert space. Proc. Am. Math. Soc. 17, 413–415 (1966) Duren, P.: Theory of H p Spaces, pp. 196–197. Academic, New York (1970) Lindelöf, E.: Sur un principe gènèral de l’Analyse. Acta Societatis Scientiarum Finnicae 66, 1–35 (1915) Garnett, J.: Bounded Analytic Functions, p. 92. Academic, New York (1981). (Problem 7)

Pointwise Exponential Decay of Bound States of the Nelson Model With Kato-Class Potentials Fumio Hiroshima

Abstract Pointwise exponential decay of bound states of the so-called Nelson model with Kato-class potentials is shown by constructing a martingale derived from the semigroup generated by the Nelson Hamiltonian.

1 Introduction To show the existence of the ground state of a model in quantum field theory has been a crucial issue. In particular, the so-called infrared regular condition is a critical condition for a scalar model to have the ground state. In this article, we are concerned with the so-called Nelson model [16, 17] describing an interaction between nonrelativistic nucleons and spinless scalar mesons. The time evolution of the nonrelativistic matters studied in this article is given by a Schrödinger operator. Then the model can be regarded as Schrödinger operator coupled to a quantum field. The existence of the ground state of this kind of model has been shown under some general conditions so far. Next interesting issue concerning the ground state is to make properties of the ground state clear, which includes to estimate the number of bosons in the ground state and the decay properties on both field variable φ and matter variable x. In this article, we treat Kato-class potentials V , which were introduced and studied by Aizenman and Simon [1] and the definition of Kato-class potentials is based on a condition considered by Kato in [11]. The Nelson Hamiltonian H with Kato-class potential is defined via functional integrations. The main purpose of this article is to show pointwise exponential decay of bound states Φ of the Nelson Hamiltonian: (1) Φ(x)F ≤ Ce−c|x| , a.e.x ∈ Rd . The strategy is an extension of Carmona, Masters and Simon [4]. Let H Φ = EΦ. Then Φ = e+t E e−t H Φ. By the functional integration of e−t H , we can express Φ as F. Hiroshima (B) Faculty of Mathematics, Kyushu University, Fukuoka, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_11

225

226

F. Hiroshima

Φ(x) = et E Ex [ξ Φ(Bt )] for each x ∈ Rd , where ξ = e−

t 0

V (Bs )ds ∗ −φ(K ) I0 e It

(2)

is the integral kernel and E denotes the expectation with respect to Brownian motion (Bt )t≥0 . See (18) below. We estimate Ex [ξ Φ(Bt )] to getthe bound (1). Statement (1) is stronger rather than localization: Rd d xeC|x| Φ(x)2F < ∞, which can be shown by IMS localization [5, Theorem 3.2] in, e.g., [2, 6, 8]. The pointwise exponential decay is also shown for the so-called semi-relativistic Pauli– Fierz model in quantum electrodynamics by the author oneself [9]. Here, the integral kernel of the semigroup generated by the semi-relativistic Pauli–Fierz Hamiltonian is of the form t ξPF = e− 0 V (X s )ds J0∗ e−i A(K P F ) Jt , where (X t )t≥0 is a Lévy process and Jt the family of isometries and e−i A(K P F ) a unitary operator. Hence, J0∗ e−i A(K P F ) Jt is contractive, i.e., J0∗ e−i A(K P F ) Jt  ≤ 1. However, the integral kernel of the semigroup generated by the Nelson Hamiltonian is of the form (2), where It is also a family of isometries but e−φ(K ) is unbounded. To see the bound I0∗ e−φ(K ) It , one can apply the hypercontractivity [19] of I0∗ It for massive cases, but it is nontrivial to have a bound for massless cases. In this article, we derive the operator bound ˆ I0∗ e−φ(K ) It  ≤ et E(ϕ) for massless cases in Corollary 4.6 but under infrared-regular condition, and consequently show pointwise exponential decay (1).

2 Basic Facts on Fock Space 2.1 Boson Fock Space Let H be a separable Hilbert space over C. Define F (n) = ⊗ns H , where ⊗ns H denotes the n-fold symmetric tensor product with ⊗0s H = C. The space (n) (H ) F = ⊕∞ n=0 F

is called boson Fock space over H . The Fock space F can be identified with the space of 2 -sequences (Ψ (n) )n∈N such that Ψ (n) ∈ F (n) and Ψ 2F =

∞  n=0

Ψ (n) 2F (n) < ∞.

(3)

Pointwise Exponential Decay of Bound States of the Nelson Model …

227

 (n) Set (Ψ , Φ)F = ∞ , Φ (n) )F (n) . The vector Ω = (1, 0, 0, . . . ) is called the n=0 (Ψ Fock vacuum. The creation operator denoted by a ∗ ( f ) and the annihilation operator by a( f ) are defined by (a ∗ ( f )Ψ )(n) =

√

nSn ( f ⊗ Ψ (n−1) ), n ≥ 1,

(a ∗ ( f )Ψ )(0) = 0     (n−1) 2 with domain D(a ∗ ( f )) = (Ψ (n) )n≥0 ∈ F  ∞ )F (n) < ∞ n=1 nSn ( f ⊗ Ψ and a( f ) = (a ∗ ( f¯))∗ . Furthermore, since both operators are closable and we denote their closed extensions by the same symbols. The space Ffin = {(Ψ (n) )n≥0 ∈ F | Ψ (m) = 0 for all m ≥ M with some M} is called finite particle subspace. Operators a, a ∗ leave Ffin invariant and satisfy the canonical commutation relations on Ffin : [a( f ), a ∗ (g)] = ( f¯, g), [a( f ), a(g)] = 0, [a ∗ ( f ), a ∗ (g)] = 0. Given a contraction operator T on H , the second quantization of T is defined by n Γ (T ) = ⊕∞ n=0 (⊗ T ).

Here ⊗0 T = 1. For a self-adjoint operator h on H , {Γ (eith ) : t ∈ R} is a strongly continuous one-parameter unitary group on F . Then by Stone’s theorem, there exists a unique self-adjoint operator dΓ (h) on F such that Γ (eith ) = eitdΓ (h) . Let N = dΓ (1). To obtain the commutation relations between a ( f ) and dΓ (h), suppose that f ∈ D(h). Then [dΓ (h), a ∗ ( f )] = a ∗ (h f ), [dΓ (h), a( f )] = −a(h f¯),

(4)

for Ψ ∈ D(dΓ (h)3/2 ) ∩ Ffin . The Segal field Φ( f ) on the boson Fock space F (H ) is defined by 1 Φ( f ) = √ (a ∗ ( f ) + a( f¯)), f ∈ H . 2 Here f¯ denotes the complex conjugate of f . Field operator Φ( f ), f ∈ H , is a self∗ adjoint operator, but a ∗ ( f ) and a( f ) are not. Nevertheless, we can define ea ( f ) and ea( f ) by a geometric series. Let f ∈ H and we define the exponential of creation operators F f by ∞  1 ∗ a ( f )n Ff = n! n=0

228

F. Hiroshima

  ∞ ∗ n  and D(F f ) = Φ ∈ ∩∞ n=1 D(a ( f ) ) n=0 Thus we have F f Φ ≤ Φ +

∞  n=1

1 a ∗ ( n!

 f )n Φ < ∞ . Let Φ ∈ F (m) .

√ √ m + n − 1··· m  f n Φ < ∞. n!

Then Ffin ⊂ D(F f ) follows. We also define the exponential of annihilation operators by ∞  1 Gf = a( f )n n! n=0   ∞ n  with D(G f ) = Φ ∈ ∩∞ D(a( f ) )  n=0 n=1

1 a( n!

 f )n Φ < ∞ . We simply write

∗

¯

∗

F f = ea ( f ) and G f = ea( f ) . Then we can see that (ea ( f ) )∗ ⊃ ea( f ) and this implies ∗ ∗ that ea ( f ) is closable. The closure of ea ( f ) is denoted by the same symbol. Similarly, the closure of ea( f ) is denoted by the same symbol. We can represent eΦ( f ) in terms ∗ of both ea ( f ) and ea( f ) . Let Db = L.H.{C(g), Φ| g ∈ H , Φ ∈ Ffin }. Proposition 2.1 (Baker–Campbell–Hausdorff formula) Let f ∈ H and α ∈ C. Then it holds on Db that eαΦ( f ) = eαa

∗

√ √ ( f )/ 2 αa( f¯)/ 2

e

e2α 1

2

 f 2

.

(5)

Proof We shall show (5) on C(g). The proof of (5) on Ffin is similar. We have eαa

∗

( f ) αa( f¯)

e

C(g) = eα( f,g) C(α f + g).

(6)

Let ψ( f ) = a ∗ ( f ) + a( f¯). Then ψ( f ) is self-adjoint and it holds that eαψ( f ) =

∞  α n ψ( f )n n=0

on Ffin . Let Cm (g) = as e

m

αψ( f )

n=0

a ∗ (g)n Ω. n!

Cm (g) =

By using the expansion (7), we can compute

m  (a ∗ (g) + α( f, g))n

n!

n=0

Together with eαψ( f ) Ω = e 2 α 1

eαψ( f ) Cm (g) =

2

 f 2 αa ∗ ( f )

e

eαψ( f ) Ω.

Ω, we see that

m  (a ∗ (g) + α( f, g))n n=0

(7)

n!

n!

e2α 1

2

 f 2 αa ∗ ( f )

e

Ω.

Pointwise Exponential Decay of Bound States of the Nelson Model …

229

Hence we have eψ( f ) C(g) = eα( f,g) e 2 α 1

2

 f 2

C( f + g).

(8) 

By (6) and (8) the proposition follows.

2.2 Bounds In this section, we show several bounds concerning the exponential of annihilation operators and the creation operators. We learned all these bounds from the papers by Guneysu, Matte and Møller [7] and Matte and Møller [15]. We consider the case where H = L 2 (Rd ). In this case, for n ∈ N, the space F (n) can be identified with the set of symmetric functions on L 2 (Rdn ). The creation and annihilation operators act as  √ (n) (a( f )Ψ ) (k1 , . . . , kn ) = n + 1 f (k)Ψ (n+1) (k, k1 , . . . , kn )dk, n ≥ 0, 1 (a ∗ ( f )Ψ )(n) (k1 , . . . , kn ) = √ n

n 

Rd

f (k j )Ψ (n−1) (k1 , . . . , kˆ j , . . . , kn ), , n ≥ 1,

j=1

with (a ∗ ( f )Ψ )(0) = 0. Let Hf = dΓ (ω) with ω = ω(k) =



|k|2 + ν 2 . We have

⎛ ⎞ n  ω(k j )⎠ Ψ (n) (k1 , . . . , kn ). (Hf Ψ )(n) (k1 , . . . , kn ) = ⎝ j=1

We can see the lemma below: Lemma 2.2 Let h : Rd → C be measurable, and  g j ∈ D(h) for j = 1, ...., m. Then for every Ψ ∈ D(dΓ (|h|2 )m/2 ), we have Ψ ∈ D( mj=1 a(hg j )), and it follows that   ⎛ ⎞ m  m     a(hg j )Ψ  ≤ ⎝ g j ⎠ dΓ (|h|2 )m/2 Ψ .    j=1  j=1 In particular   ⎛ ⎞  n  n    √  a(g j )Φ  ≤ ⎝ g j / ω⎠ H n/2 Φ, Φ ∈ D(H n/2 ). f f    j=1  j=1

(9)

230

F. Hiroshima

Proof Let Ψ ∈ D(dΓ (|h|2 )m ). First note that (Ψ

(n)

(n)



, (dΓ (|h| ) Ψ ) ) = 2 m

Rnd

|Ψ

(n)

(k1 , · · · , kn )|

2

 n 

m |h(ki )|

2

dk1 · · · dkn .

i=1

 n   2 m By the symmetry, we can replace with C(n, m) mj=1 |h(k j )|2 . Here i=1 |h(ki )| C(n, m) = n(n − 1) · · · (n − m + 1). Then we have  m  (Ψ (n) , (dΓ (|h|2 )m Ψ )(n) ) = C(n, m) |Ψ (n) (k1 , · · · , kn )|2 |h(k j )|2 dk1 · · · dkn . Rnd

j=1

On the other hand by the definition of annihilation operators, we have  2    m  ( a(hg j )Ψ )(n−m)     j=1  ⎛ ⎞  m  2⎠ ⎝ ≤ C(n, m) g j  ⎛ =⎝

j=1 m 

⎞

⎛ ⎝ Rnd

m 

⎞ |h(k j )|2 ⎠ |Ψ (n) (k1 , · · · , kn )|2 dk1 · · · dkn

j=1

g j 2 ⎠ (Ψ (n) , (dΓ (|h|2 )m Ψ )(n) )

j=1

and  summation over n gives (9). By the closedness of both operators dΓ (|h|2 )m/2 and mj=1 a(hg j ), we can extend to Ψ ∈ D(dΓ (|h|2 )m/2 ).      Next we estimate  nj=1 a ∗ ( f j )Φ . √ n/2 Lemma 2.3 Let f i , g j ∈ D(1/ ω) for i, j = 1, ..., n and Φ ∈ D(Hf ). Then ⎛ ⎞  n  n  n  n    1   ∗  m/2 ∗ n ⎝ a (g j )Φ, Hf Φ2 , a ( f j )Φ ⎠ ≤ n!2  fl ω gl ω  m!  j=1  m=0 j=1 l=1 √ where  f ω =  f  +  f / ω. Proof Let Φ ∈ Ffin and f i , g j ∈ H for i, j = 1, ..., m. Then n  j=1

a(g¯ j )

n  j=1

a ∗ ( f j )Φ

Pointwise Exponential Decay of Bound States of the Nelson Model …

=

n   



m=0 Cm A Cn−m B



σ :Ac →B bijection



⎛ (gl , f σ (l) ) ⎝

l∈Ac



231

⎞⎛ a ∗ ( f p )⎠⎝

p∈B c

⎞



a(g¯ q )⎠ Φ. (10)

q∈A

 Here Ck = {A ⊂ {1, ..., n}|# A = k}, C0 = ∅, and σ :Ac →B is understood to take bijection c summation over all bijections from A to B. In particular ⎡ ⎤ n n   ⎣ a(g¯ j ), a ∗ ( f j )⎦ j=1

=

j=1

n−1    m=0 Cm A Cn−m B





σ :Ac →B bijection



⎞ ⎛   (gl , f σ (l) ) ⎝ a ∗ ( f p )⎠ a(g¯ q ).

l∈Ac

p∈B c

q∈A

By this formula we have ⎛ ⎞ n n   ⎝ a ∗ (g j )Φ, a ∗ ( f j )Φ ⎠ j=1

=

j=1

n    m=0 Cm A Cn−m B

 σ :Ac →B bijection

⎛ ⎝Φ,

 l∈Ac

(gl , f σ (l) )

 p∈B c

a∗( f p )



⎞ a(g¯ q )Φ ⎠ .

(11)

q∈A

  n/2 By  nj=1 a(h j )Φ ≤ nj=1 h j Hf Φ and # B c = m = # A, the right-hand side of (11) can be estimated as ⎛ ⎞ ⎛ ⎞⎛ ⎞         √ √ m/2 ∗ ⎝Φ, a ( f p) a(g¯ q )Φ ⎠ ≤ ⎝  f p / ω⎠ ⎝ gq / ω⎠ Hf Φ2 .    p∈B c q∈A p∈B c q∈A

√ Since  f  ≤  f ω and  f / ω ≤  f ω , we have ⎛ ⎞     n       m/2 ∗ ⎝Φ, (gl , f σ (l) ) a ( f p) a(g¯ q )Φ ⎠ ≤ gl ω  fl ω Hf Φ2 .    l∈Ac p∈B c q∈A l=1 (12) Hence by (11) and (12) ⎛ ⎞   n   n n n       1 m/2 ∗ n ⎝Φ, a(g¯ j ) a ( f j )Φ⎠ ≤ n!2 gl ω  fl ω Hf Φ2 .   m!   m=0 j=1 j=1 l=1 Then the lemma follows.



232

F. Hiroshima

By Lemma 2.3, we have bounds for products of annihilation operators√and creation operators. We summarize them as follows. Suppose that f j ∈ D(1/ ω) for j = 1, ..., n. By introducing a scaling parameter 0 < s < 1, we also have   ⎛ ⎞   n n    √  a( f j )Φ  ≤ s −n/2 ⎝  f j / ω⎠ (s Hf )n/2 Φ, (13)     j=1 j=1    n  n 1/2   n   1  √  ∗ m/2 2  a ( f j )Φ  ≤ n!2n/2 s −n/2  fl ω . (14) (s Hf ) Φ   m!   j=1 m=0 l=1 ∗

Although exponential operator ea ( f ) is unbounded, it can be seen in the proposition t ∗ below that ea ( f ) e− 2 Hf is bounded for any t > 0. √ t t ∗ Proposition 2.4 Let t > 0 and f ∈ D(1/ ω). Then both ea ( f ) e− 2 Hf and e− 2 Hf ea( f ) are bounded. n Proof Let Ψ ∈ ∩∞ n=1 D(Hf ). Suppose that t < 1. By (14) for any s < t we have

 m   n 1/2 m  1    1 1 t t   a ∗ ( f )n e− 2 Hf Φ  ≤ (s Hf )k/2 e− 2 Hf Φ2 . √ 2n/2 s −n/2  f nω    n! k! n! n=0

n=0

k=0

 1 ∗ n − 2t Hf We can see that sequence { m Φ}∞ m=0 is a Cauchy sequence in F . n=0 n! a ( f ) e t ∗ − 2 Hf a (f) Φ ∈ D(e ) and as m → ∞ on both sides above, we have Hence e ea where A( f, s) =

∞ n=0

− 21 (t−s)Hf

∗

( f ) − 2t Hf

e

Φ ≤ A( f, s)e− 2 (t−s)Hf Φ, 1

√1 2n/2 s −n/2  f n . Choosing ω n! a ∗ ( f ) − 2t Hf

s such that s < t, we can see

Φ ≤ Φ and e e for t < 1 is bounded. Suppose 1 ≤ t. that e Choosing s = 1 in the above discussion, we have ea

∗

( f ) − 2t Hf

e

Φ ≤ A( f, 1)e− 2 (t−1)Hf Φ ≤ A( f, 1)Φ. 1

∗  t t t ∗ ∗ ¯ Thus ea ( f ) e− 2 Hf for t ≥ 1 is bounded. Finally since e− 2 Hf ea( f ) ⊃ ea ( f ) e− 2 Hf , the second statement follows. Then the lemma follows.  √ Corollary 2.5 Let f ∈ D(1/ ω). Then √ t 1 ∗ 2 ea ( f ) e− 2 Hf  ≤ 2e(2/s) f ω e− 2 (t−s)Hf , 0 < s < t < 1, √ t 1 ∗ 2 ea ( f ) e− 2 Hf  ≤ 2e2 f ω e− 2 (t−1)Hf , 1 ≤ t. In particular, we have

Pointwise Exponential Decay of Bound States of the Nelson Model … ∗

¯

233

ea ( f ) e−t Hf ea( f )  ≤ 2e(4/s) f ω , 0 < s < t < 1, ∗ 2 ¯ ea ( f ) e−t Hf ea( f )  ≤ 2e4 f ω , 1 ≤ t. 2

Proof We can estimate A( f, s) as ∞ 1/2  ∞ 1/2  1  √ 2 n 2n −n −n 2 −n 1 ·2 ≤ 2e(2/s) f ω . A( f, s) ≤ 8  f ω s 2 n! n=0 n=0 

Then (1) follows from Proposition 2.4.

3 Definition of the Nelson model Let S  (Rd ) be the tempered distribution on Rd . Let s ∈ R and H s (Rd ) be the inhomogeneous Sobolev space, i.e., H s (Rd ) = {u ∈ S  (Rd )|uˆ ∈ L 1loc (Rd ), (1 + |k|2 )s/2 uˆ ∈ L 2 (Rd )}. Here uˆ describes the Fourier transform on S  (Rd ). Homogeneous Sobolev space H˙ s (Rd ) is defined by H˙ s (Rd ) = {u ∈ S  (Rd )|uˆ ∈ L 1loc (Rd ), |k|s uˆ ∈ L 2 (Rd )}. The scalar product on H˙ s (Rd ) is defined by ( f, g) H s (Rd ) = (|k|s fˆ, |k|s g) ˆ L 2 (Rd ) . It is known that H˙ s (Rd ) is a Hilbert space if and only if s < d/2. Now we modify H s (Rd ) to apply quantum field theory. We define Hνs (Rd ) by H s (Rd ) with (1 + |k|2 )1/2 replaced by ω. Hence H s (Rd ) and Hνs (Rd ) are equivalent for ν > 0, and H˙ s (Rd ) = Hνs (Rd ) for ν = 0. We set  Hs (R ) = d

Hνs (Rd ), ν > 0, H˙ s (Rd ), ν = 0.

We set HM = H−1/2 (Rd ) and HE = H−1 (Rd+1 ). We define the Fourier transform (in the sense of tempered distribution) of HM and HE by HˆM and Hˆ E , respectively. Although HM , HˆM , HE and Hˆ E depend on the space dimension and ν ≥ 0, we do not write the dependence explicitly. We also define real Hilbert spaces below: (1) M = { f ∈ HM | f is real valued}, (2) E = { f ∈ HE | f is real valued}. Both M and E are Hilbert spaces over R, and note that MC = HM and EC = HE . Hilbert space L 2 (Rd ) describes the state space of the nonrelativistic matter, and FN = F (HˆM ) that of the scalar bose field. The joint state space is described by the tensor product

234

F. Hiroshima

H = L 2 (Rd ) ⊗ FN . The free particle Hamiltonian is described by the Schrödinger operator 1 Hp = − Δ + V 2 acting in L 2 (Rd ). We introduce a class R of potentials. Let V be relatively bounded with respect to −(1/2)Δ with a relative bound strictly smaller than one, i.e., D(V ) ⊂ D(−(1/2)Δ) and V f  ≤ a − (1/2)Δf  + b f  for f ∈ D(V ) with some a < 1 and b ≥ 0. Then we say V ∈ R. We introduce Assumption 3.1. Assumption 3.1 The following conditions hold: √ ˆ = ϕ(−k) ˆ and ϕ/ω, ˆ ϕ/ ˆ ω ∈ L 2 (Rd ). (1) ϕ ∈ S  (Rd ), ϕ(k) (2) V ∈ R. The free field Hamiltonian Hf = dΓ (ω) on FN accounts for the energy carried by the field configuration. The matter -field interaction Hamiltonian HI acting on the Hilbert space H describes then the interaction energy between the bose field and the matter . To give a definition of this operator we identify H as the space of FN -valued L 2 -functions on Rd :     ⊕  FN d x = F : Rd → FN  F(x)2F N d x < ∞ . H ∼ = Rd

Rd

For each x ∈ Rd , HI (x) is defined by  1  ˜ˆ ikx ) . ˆ −ikx ) + aM (ϕe HI (x) = √ aM∗ (ϕe 2 Here aM and aM∗ denote the creation operator and the annihilation operator in the boson Fock space F (HˆM ), respectively. Since ϕ(k) ˆ = ϕ(−k), ˆ HI (x) is symmetric, and it can be shown by using Nelson’s analytic vector theorem that HI (x) is essentially self-adjoint on Ffin (HˆM ) of FN . We denote the self-adjoint extension of HI (x) by HI (x). The interaction HI is then defined by the self-adjoint operator  HI =

⊕ Rd

HI (x)d x.

Under the conditions of Assumption 3.1 the operator H = Hp ⊗ 1l + 1l ⊗ Hf + HI

(15)

Pointwise Exponential Decay of Bound States of the Nelson Model …

235

acting in H is called the Nelson Hamiltonian. Let H0 = Hp ⊗ 1l + 1l ⊗ Hf ,

D(H0 ) = D(Hp ⊗ 1l) ∩ D(1l ⊗ Hf ).

Then H0 is self-adjoint on D(H0 ) and bounded below. Suppose Assumption 3.1. Then H is also self-adjoint on D(H0 ) and bounded below, furthermore, it is essentially self-adjoint on any core of H0 . This follows from Kato–Rellich theorem [10].

4 Pointwise Exponential Decay 4.1 Integral Kernels We review a family of Gaussian random variables indexed by a real vector space M . We say that (φ( f ), f ∈ M ) is a family of Gaussian random variables on a probability space (Q, Σ, μ) indexed by a real inner product space M whenever (1) φ : M  f → φ( f ) is a map from M to a Gaussian random variable on (Q, Σ, μ) with Eμ [φ( f )] = 0 and covariance Eμ [φ( f )φ(g)] = 21 ( f, g)M , (2) φ(α f + βg) = αφ( f ) + βφ(g), α, β ∈ R, (3) Σ is the completion of the minimal σ -field generated by {φ( f )| f ∈ M }. Also let (φE ( f ), f ∈ E ) be a family of Gaussian random variables on a probability space (QE , ΣE , μE ) indexed by a real inner product space E . Let O ⊂ R and put E (O) = { f ∈ E |supp f ⊂ O × Rd } and the projection E → E (O) is denoted by eO . Let ΣO = σ ({φE ( f )| f ∈ E (O)}) . Define EO = {Φ ∈ L 2 (QE )|Φ is ΣO -measurable}. Let et = τt τt∗ , t ∈ R. Then {et }t∈R is a family of projections from E to Ran(τt ). Let Σt , t ∈ R, be the minimal σ -field generated by {φE ( f )| f ∈ Ran(et )}. Define Et = {Φ ∈ L 2 (QE )|Φ is Σt -measurable}, t ∈ R. We will see below that F ∈ E[a,b] can be characterized by suppF ⊂ [a, b] × Rd . Lemma 4.1 (1) E ({t}) = Ran(et ) and any f ∈ Ran(et ) can be expressed as f = δt ⊗ g for some g ∈ M . In particular, e{t} = et . ·−1 (2) E ([a, b]) = L.H. { f ∈ E | f ∈ Ran(et ), a ≤ t ≤ b} holds.

236

F. Hiroshima



Proof Refer to see [18].

We will define a family of transformations It from L 2 (Q) to L 2 (QE ) through the second quantization of a specific transformation τt from M to E . Define τt : M → E by τt : f → δt ⊗ f . Here δt (x) = δ(x − t) is the delta function with mass at t. Note that δt ⊗ f = δt ⊗ f , which implies that τt preserves realness. It follows that τs∗ τt = e−|s−t|ωˆ , s, t ∈ R. In particular, τt is isometry between M and E for each t ∈ R. Let It = Γ (τt ) : L 2 (Q) → L 2 (QE ), t ∈ R, be the family of isometries: It 1lM = 1lE , It :φ( f 1 ) · · · φ( f n ): = :φE (δt ⊗ f 1 ) · · · φE (δt ⊗ f n ):. Let ωˆ = ω(−i∇) =

√

−Δ + ν 2 . The self-adjoint operator ˆ Hˆ f = dΓ (ω)

−1 . From is called the free field Hamiltonian in L 2 (Q). It follows that Hˆ f = θW Hf θW ∗ −|s−t|ωˆ , it follows that the identity τs τt = e ˆ

It∗ Is = e−|s−t| Hf , s, t ∈ R.

(16)

On L 2 (Rd ) ⊗ L 2 (Q), we define the Nelson Hamiltonian by Hˆ = Hp ⊗ 1l + 1l ⊗ Hˆ f + g Hˆ I , ⊕ ˜ − x)). Since H and Hˆ are unitarily where Hˆ I = Rd Hˆ I (x)d x with Hˆ I (x) = φ(ϕ(· equivalent, we denote H for Hˆ for simplicity in what follows. Let (Bt )t≥0 be the brownian motion on the probability space (X , B(X ), W ). Proposition 4.2 Suppose Assumption 3.1. Then for t ≥ 0 and F, G ∈ H ,  (F, e−t H G)H =

Rd

 t  t d xEx e− 0 V (Bs )ds (F(B0 ), I0∗ e−φE ( 0 δs ⊗ϕ(·−Bs )ds) It G(Bt )) L 2 (Q ) .

(17)

Here F, G ∈ H are regarded as L 2 (Q)-valued L 2 -functions on Rd . 

Proof See Appendix and see [14]. We call

t

I[0,t] = I0∗ e−φE (

0

δs ⊗ϕ(·−Bs )ds)

It

the integral kernel of the semigroup generated by H . Thus (F, e

−t H

G)H

  t  = d xEx e− 0 V (Bs )ds (F(B0 ), I[0,t] G(Bt )) L 2 (Q ) Rd

Pointwise Exponential Decay of Bound States of the Nelson Model …

237

 t  e−t H G(x) = Ex e− 0 V (Bs )ds I[0,t] G(Bt ) .

and we have

Moreover if H G = E G we have

 t  G(x) = et E Ex e− 0 V (Bs )ds I[0,t] G(Bt ) .

(18)

We discuss the boundedness of the norm of Ia∗ eφE ( f ) Ib for not only massive case but also massless case. We see some intertwining properties of It and τt . We can identify φE ( f ) (resp.φ( f )) with √1 (a ∗E ( fˆ) + a E ( f˜ˆ)) (rep. √1 (a ∗M ( fˆ) + a M ( f˜ˆ))). 2

2

Under this identification It can be recognized as a map from F (HˆM ) to F (Hˆ E ), i.e., n n   It a ∗M ( fˆj )Ω = a ∗E (τˆt f j )Ω j=1

j=1

and It∗ as that from F (Hˆ E ) to F (HˆM ). It follows that on the finite particle subspace, ˆ  It a ∗M ( fˆ) = a ∗E (τ t f )It , It a M ( f ) = a E (τt f )It , It∗ a ∗E (

fˆ) =

a ∗M ( τt∗ f )It∗ ,

It∗ a E (e t f)

=

a M ( τt∗ f )It∗ ,

(19) (20)

where et = τt τt∗ . In particular ∗  ˆ  τt∗ f ) = a ∗E (e It a ∗M ( t f )It , It a M (τt f ) = a E (et f )It = a E ( f )It ,

It∗ a ∗E (τ t f)

=

a ∗M (

fˆ)It∗ ,

It∗ a E (τ t f)

= aM (

fˆ)It∗ .

(21) (22)

√ Theorem 4.3 Suppose that fˆ ∈ Hˆ E and  τt∗ f / ω ∈ HˆM for t = a, b with a = b. Then Ia∗ eφE ( f ) Ib is bounded and Ia∗ eφE ( f ) Ib 

! ! 1 ˆ 1 2 2 ∗ ∗    f Hˆ E + 1 ∨ (τa f ω + τb f ω ) . ≤ 2 exp 4 |a − b|

Here x ∨ y = max{x, y} and √ τa∗ f / ω L 2 (Rd ) +  τa∗ f /ω L 2 (Rd ) .  τa∗ f ω =  Proof By Baker–Campbell–Hausdorff formula, we have eφE ( f ) = e

√1 a ∗ ( 2 E

The intertwining property yields that

fˆ)

e

√1 a E ( 2

f˜ˆ)

1

e4

 fˆ2

Hˆ E

.

238

F. Hiroshima

Ia∗ eφE ( f ) Ib = e √1

a ∗ ( τ∗ f )

√1 a ∗ ( τ∗ f ) 2 M a

|a−b|

Since e 2 M a e− 2 Hf and e− ator bounds are given by

|a−b| 2

Hf

e

e−|a−b|Hf e  √1 a M ( τb∗ f ) 2

 √1 a M ( τb∗ f ) 2

1

e4

 fˆ2

Hˆ E

.

(23)

are bounded operators which oper-

!  √ 2  1 2 ∗ ∗   τa f Hˆ + τa f / ωHˆ , e 1∨ e M M |a − b| !      √ √ |a−b| 1 √1 a ( τ∗ f )  τb∗ f 2Hˆ +  . e− 2 Hf e 2 M b  ≤ 2 exp 1 ∨ τb∗ f / ω2Hˆ M M |a − b| √1 a ∗ ( τ∗ f ) 2 M a

− |a−b| 2 Hf

√  ≤ 2 exp



Hence together with them, we have Ia∗ eφE ( f ) Ib  ≤ 2 exp

! ! 1 ˆ 2 1  f Hˆ + 1 ∨  τa∗ f 2ω +  τb∗ f 2ω . E 4 |a − b| 

Then the proof is complete. Corollary 4.4 (Integral kernel) The integral kernel is given by I[0,t] = e where Ut = and

√1 a ∗ (Ut ) 2 M

e−t Hf e

˜t ) √1 a M (U 2

1

e4W,

t

t √ √ ω(k), Ut = 0 dse−sω(k) eik Bs ϕ(k)/ ω(k) dse−sω(k) e−ik Bs ϕ(k)/ ˆ ˆ  t  t  2 |ϕ(k)| ˆ e−|s−r |ω(k) e−ik(Bs −Br ) . dr ds dk W = ω(k) 0 0 Rd 0

Proof This follows from (23) and the definition of I[0,t] .



We consider special cases of Theorem 4.3.

√ Corollary 4.5 Let T ≥ 0. Let f ∈ HM , i.e., fˆ/ ω ∈ L 2 (Rd ). We set 

T

Φ X = φE

! (τs f )(· − Bs )ds .

0

Then (1) and (2) follow. √ (1) Suppose that fˆ/ω, fˆ/ ω3 ∈ L 2 (Rd ). Then I0∗ eΦ X IT 

! √ T ˆ 2 2 2 3 ˆ ˆ  f /ω + 2(T ∨ 1)( f /ω +  f / ω  ) . ≤ 2 exp 2

(24)

(2) Suppose that fˆ/ω ∈ L 2 (Rd ). Then I0∗ eΦ X IT  ≤ 2 exp

! √ T ˆ  f /ω2 + 2T (T ∨ 1)( fˆ/ ω2 +  fˆ/ω2 ) . 2

(25)

Pointwise Exponential Decay of Bound States of the Nelson Model …

239

Proof We see that    

0

T

2  τs f ds  

Hˆ E





T

=

T

ds 0

0

dt (e−|s−t|ω fˆ, fˆ)Hˆ M ≤ 2T  fˆ/ω2 .

We can also see that   T 2  ∗  τ τs f ds  ≤ T  fˆ/ω2 ,  T  0 HM   T 2  ∗  τ τs f ds  ≤ T  fˆ/ω2 ,  0  HM

0

  T 2 √  ∗ √  τ τs f ds/ ω ≤ T  fˆ/ ω3 2 ,  T  0 HM   T 2 √  ∗  √ τ τs f ds/ ω ≤ T  fˆ/ ω3 2 .  0  HM

0

Then (1) follows from Theorem 4.3. For (2) we can estimate as   T 2  ∗  √ τ  τ f ds ≤ T 2  fˆ/ ω2 , s  T  0 HM   T 2  ∗  √ τ τs f ds  ≤ T 2  fˆ/ ω2 ,  0  HM

0

  T 2  ∗ √  τ  τ f ds/ ω ≤ T 2  fˆ/ω2 , s  T  0 HM   T 2  ∗ √  τ τs f ds/ ω ≤ T 2  fˆ/ω2 .  0  0

HM



Then (2) follows. We can plug (24) and (25). Let E( fˆ) = max

1

√   fˆ/ω2 + 2( fˆ/ω2 +  fˆ/ ω3 2 ) √ . fˆ/ω2 + 2( fˆ/ ω2 +  fˆ/ω2 )

2 1  2

(26)

Definition 4.1 (Infrared regular condition) The condition  Rd

2 |ϕ(k)| ˆ dk < ∞ 3 ω(k)

is called the infrared regular condition. Corollary 4.6 (Integral kernel√under infrared regular condition) Let Φ X be in 2 ˆ /ω(k)3 dk < ∞. Corollary 4.5. Suppose that fˆ/ ω, fˆ/ω ∈ L 2 (Rd ) and Rd |ϕ(k)| Then it follows that ˆ , T ≥ 0. I0∗ eΦ X IT  ≤ 2e T E(ϕ)

Proof We have I0∗ eΦ X IT 

! √ T ˆ 2 2 2 3 ˆ ˆ ≤ 2 exp  f /ω + 2T ( f /ω +  f / ω  ) 2

(27)

240

F. Hiroshima

for T ≥ 1, and √ T ˆ  f /ω2 + 2T ( fˆ/ ω2 +  fˆ/ω2 ) 2

I0∗ eΦ X IT  ≤ 2 exp

!

for T ≤ 1. Then (27) is shown.



4.2 Kato-class Potentials Definition 4.2 (Kato-class potentials [1, 11]) (1) V : Rd → R is called a Kato-class potential whenever  lim sup

r →0 x∈Rd

Br (x)

|g(x − y)V (y)| dy = 0

holds, where Br (x) is the closed ball of radius r centered at x, and ⎧ ⎪ d = 1, ⎨|x|, g(x) = − log |x|, d = 2, ⎪ ⎩ 2−d |x| , d ≥ 3. We denote this linear space by K (Rd ). (2) V is Kato-decomposable whenever V = V+ − V− with V+ ∈ L 1loc (Rd ) and V− ∈ K (Rd ). Lemma 4.7 ([3]) Let 0 ≤ V ∈ K (Rd ). Then there exist β, γ > 0 such that sup Ex [e

t 0

V (Bs )ds

] < γ eβt .

(28)

x∈Rd

Furthermore, if V ∈ L p (Rd ) with p > d/2 and 1 ≤ p < ∞, then β ≤ c( p)1/ε Γ (ε)1/ε V 1/ε p , where ε = 1 −

d 2p

and  c( p) =

with

1 p

+

1 q

(29)

p = 1, (2π )−d/2 (2π )−d/2 p q d/(2q) p > 1

= 1. In particular L p (Rd ) ⊂ K (Rd ) for p > d/2 and 1 ≤ p < ∞.

Pointwise Exponential Decay of Bound States of the Nelson Model …

241

t Proof There exists t ∗ > 0 such that αt = supx∈Rd Ex [ 0 V (Bs )ds] < 1, for all t ≤ t ∗ , and αt → 0 as t → 0. By Khasminskii’s lemma we have sup Ex [e

t 0

V (Bs )ds

]
0, where [z] = max{w ∈ Z|w ≤ z}. Setting γ = ( 1−α t∗   1 1/t ∗ . This proves (28). Next we prove (29). Suppose V ∈ L p (Rd ) with log ( 1−αt ∗ ) p > d/2 and 1 ≤ p < ∞. We let p > 1. By Schwarz inequality, we have

E [V (Bt )] ≤ (2π t) x

−d/2

 e

−|x−y|2 q/(2t)

!1/q

Rd

V  p = (2π t)−d/(2 p) q d/(2q) V  p .

In particular, we have Ex [V (Bt )]∞ ≤ c( p)t −d/(2 p) V  p ,

p > 1.

 1 k We introduce a Mittag-Leffler function which is defined by m b (x) = ∞ k=0 Γ (1+kb) x , where Γ denotes the Gamma function, x ∈ R and b > 0. It is known that Mittag1/b Leffler function m b (x) satisfies that lim x→∞ (m b (x) − b1 e x ) = 0 and there exists 1/b kb > 0 such that m b (x) ≤ kb e x for all x > 0. Let 0 ≤ s1 ≤ s2 ≤ · · · ≤ sk . By the Markov property of Brownian motion, we have −d/(2 p)

Ex [V (Bs1 ) · · · V (Bsk )] ≤ c( p)k V kp s1

(s2 − s1 )−d/(2 p) · · · (sk − sk−1 )−d/(2 p) .

Then & E

x

1 k!

'

≤ c( p)

t

(k ) V (Bs )ds

0 k

V kp



t

ds1 · · · 0



t

−d/(2 p)

dsk s1

sk−1

(s2 − s1 )−d/(2 p) · · · (sk − sk−1 )−d/(2 p)

242

F. Hiroshima

=

(c( p)V  p t ε Γ (ε))k , Γ (1 + kε)

where ε = 1 −

d 2p

sup Ex [e

t 0

> 0. Then it can be derived that V (Bs )ds

x∈Rd

  1/ε 1/ε 1/ε ] ≤ m ε c( p)V  p Γ (ε)t ε ≤ kε ec( p) V  p Γ (ε) t .

Then (29) is proven. Let p = 1. Then d = 1 and it follows directly that Ex [V (Bt )] ds ≤ (2π t)−1/2 V 1 = c(1)t −1/2 V 1 . Hence (29) is proven in a similar way to the case of p > 1.



We introduce Assumption 4.1. Assumption 4.1 The following conditions hold: √ (1) ϕ ∈ S  (Rd ), ϕ(k) ˆ = ϕ(−k) ˆ and ϕ/ω, ˆ ϕ/ ˆ ω ∈ L 2 (Rd ). (2) V is Kato-decomposable. Suppose Assumption 4.1 and define the family of operators (Tt F) (x) = Ex [e−

t 0

V (Br )dr

I[0,t] F(Bt )].

Lemma 4.8 Tt is bounded. Proof Let F ∈ HN . Since I[0,t] F(Bt ) ≤ 2e E(t) F(Bt ), where E(t) is given by E(t) =

√ √ t 2 ϕ/ ˆ ω2 + 2t (1 ∨ t)(ϕ/ ˆ ω2 + ϕ/ω ˆ ), 2 

from Tt F2H N

=

Rd

d xEx [e−

t 0

V (Br )dr

I[0,t] F(Bt )]2L 2 (Q )

it follows that  Tt F2H N ≤ 2

d xE[e−2 Rd

t 0

V (Br +x)dr

]E[F(Bt + x)2 ]e2E(t) . t

Since V is Kato-decomposable, we have supx∈Rd E[e−2 0 V (Br +x)dr ] = C < ∞, and  thus Tt F2H N ≤ 4Ce2E(t) F2H N follows.  2 We note that if infrared regular condition Rd |ϕ(k)| ˆ /ω(k)3 dk < ∞ holds, then E(t) can be replaced with I[0,t]  ≤ 2e ˆ . I[0,t]  ≤ 2et E(ϕ)

(32)

Pointwise Exponential Decay of Bound States of the Nelson Model …

243

In what follows, we show that {Tt : t ≥ 0} is a symmetric C0 -semigroup. To do that, we introduce the time-shift operator u t on L 2 (Rd ) by u t f (x) = f (x0 − t, x), x = (x0 , x) ∈ R × Rd . It is straightforward that u ∗t = u −t and u ∗t u t = 1. We denote the second quantization of u t by Ut = ΓE (u t ) which acts on L 2 (QE ) and is a unitary map. We can see that Ut Is = Is+t . We set  t

Kt =

δs ⊗ ϕ(· − Bs )ds.

0

Lemma 4.9 Ts Tt = Ts+t holds for s, t ≥ 0. Proof By the definition of Tt we have Ts Tt F = Ex [e−

s 0

V (Br )dr

I[0,s] E Bs [e−

t 0

V (Br )dr

I[0,t] F(Bt )]].

(33)

∗ ∗ = Es U−s and It = U−s It+s , Eq. (33) is equal to By the formulae Is I0∗ = Is Is∗ U−s

Ex [e−

s 0

V (Br )dr ∗ −φE (K s ) I0 e Es E Bs [e−

t 0

V (Br )dr

∗ −φE (K t ) U−s e U−s It+s F(Bt )]].

(34)

∗

∗ −φE (K t ) e U−s = e−φE (u −s K t ) as an operator. The test Since Us is unitary, we have U−s ∗ function of the exponent u −s K t is given by

u ∗−s K t =



t

δr +s ⊗ ϕ(· ˜ − Br )dr.

0

Moreover by the Markov property of Et , t ∈ R, we may neglect Es in (34), and by the Markov property of (Bt )t≥0 we have Ts Tt F = Ex [e− = E [e x

where K ss+t = (Bt )t≥0 .

s

−

 s+t

 s+t s

V (Br )dr ∗ −φE (K s ) x − I0 e E [e

0

0

 s+t s

V (Br )dr −φE (K ss+t )

e

V (Br )dr ∗ −φE (K s+t ) I0 e Is+t F(Bs+t )]

Is+t F(Bs+t )|Fs ]]

= Ts+t F,

δr ⊗ ϕ(· ˜ − Br )dr and (Ft )t≥0 denotes the natural filtration of 

The strong continuity of the map t → Tt on HN can be checked, while T0 = 1 is trivial. Theorem 4.10 Semigroup {Tt : t ≥ 0} is a symmetric C0 -semigroup. Proof Since it was shown that {Tt : t ≥ 0} is a C0 -semigroup, it is enough to show that Tt is symmetric, i.e., (F, Tt G) = (Tt F, G). Let R = ΓE (r ) be the second quantization of the reflection r , and Ut = ΓE (u t ). Then we have

244

F. Hiroshima

 (F, Tt G) =

Rd

 t  d xEx e− 0 V (Bs )ds (It F(B0 ), e−φE (u t r K t ) I0 G(Bt )) ,

t d where u t r K t = 0 δt−s ⊗ ϕ(· ˜ − Bs )ds. Noticing that B˙ s = Bt−s − Bt = Bs for 0 ≤ s ≤ t. Thus we can replace Bs with B˙ s . Thus  (F, Tt G) =

d xE[e−

t 0

V ( B˙ s +x)ds

Rd

˙ (It F( B˙ 0 + x), e−φE (u t r K t ) I0 G( B˙ t + x))].

t   ˜ − B˙ s )ds. Exchanging Rd d x and X dW 0 , and changing Here K˙ t = 0 δs ⊗ ϕ(· variable −Bt + x to y, we have  (F, Tt G) =

dyE[e−

t 0

V (Bt−s +y)ds

Rd

˜

(It F(Bt + y), e−φE ( K t ) I0 G(B0 + y))].

t ˜ − Bt−s − y)ds. Thus we conclude that (F, Tt G) = Here K˜ t = 0 δt−s ⊗ ϕ(·  (Tt F, G) and the theorem follows. By Theorem 4.10, there exists a self-adjoint operator HKato such that Tt = e−t HKato for t ≥ 0. Definition 4.3 (Nelson Hamiltonian with Kato-class potential) Let V be a Kato decomposable potential. Then we call the self-adjoint operator HKato Nelson Hamiltonian with Kato-class potential V . In what follows notational simplicity, we also write H for HKato . That is, the Nelson Hamiltonian is written by H but with Kato-decomposable potential is understood as HKato .

4.3 Martingales Let us consider the Schrödinger operator Hp = − 21 Δ + V . Let f be an eigenvector of Hp ; Hp f = E p f . Then we define the random process h t (x) = et Ep e−

t 0

V (Br +x)dr

f (Bt + x), t ≥ 0.

Note that E[h t (x)] = f (x) for all t ≥ 0. We see that the random process (h t (x))t≥0 is martingale with respect to (Ft )t≥0 . We extend this to the Nelson Hamiltonian. Suppose that E is an eigenvalue associated with a bound state Φ; H Φ = EΦ. Define

Pointwise Exponential Decay of Bound States of the Nelson Model …

Ht (x) = et E e−

t 0

t

V (Br +x)dr −φE (

e

0

δr ⊗ϕ(·−x−Br )dr )

245

It Φ(Bt + x).

(35)

Then (Ht (x))t≥0 is a random process on (X × QE , B(X ) × ΣE , W × μE ) for each x ∈ Rd . By the functional integral representation we have (Ψ , Φ) = (Ψ , e−t (H −E) Φ) =

 Rd

¯ (x)E[I0∗ Ht (x)]]. d xEμE [Ψ

Hence it follows that Φ(x) = (e−t (H −E) Φ)(x) = E[I0∗ Ht (x)]. Define the filtration (Mt )t≥0 by Mt = Ft × Σ(−∞,t] , t ≥ 0. Theorem 4.11 Suppose Assumption 4.1. Then (Ht (x))t≥0 is martingale with respect to (Mt )t≥0 . Proof Let us set K (0, s) = e−

s 0

s

V (Br +x)dr −φE (

e

0

δr ⊗ϕ(·−x−Br )dr )

.

We have EμE E [Ht (x)|Ms ] = et E K (0, s)EμE E [K (s, t)Φ(Bt + x)|Ms ] We compute the conditional expectation of the right-hand side above. From Markov property of (Bt )t≥0 , it follows that EμE E [K (s, t)It Φ(Bt + x)|Ms ]    t−s  t = EμE E Bs e− 0 V (Br +x)dr e−φE ( s δr ⊗ϕ(·−x−Br −s )dr ) It Φ(Bt−s + x)|Σ(−∞,s] . From the Markov property of projection Es = Is Is∗ it furthermore follows that    t−s  t = EμE E Bs e− 0 V (Br +x)dr e−φE ( s δr ⊗ϕ(·−x−Br −s )dr ) It Φ(Bt−s + x)|Σs   t−s  t = Es E Bs e− 0 V (Br +x)dr e−φE ( s δr ⊗ϕ(·−x−Br −s )dr ) It Φ(Bt−s + x) . Let Us = Γ (u s ), where u s : E → E denotes the time-shift operator defined by u s f (t, x) = f (t + s, x). Since the shift operator and the projection are related to E s = Is I0∗ U−s , we have   t−s  t = Is I0∗ U−s E Bs e− 0 V (Br +x)dr e−φE ( s δr ⊗ϕ(·−x−Br −s )dr ) It Φ(Bt−s + x)   t−s  t = Is I0∗ E Bs e− 0 V (Br +x)dr e−φE ( s δr −s ⊗ϕ(·−x−Br −s )dr ) It−s Φ(Bt−s + x) * + = Is I0∗ E Bs K (0, t − s)It−s Φ(Bt−s + x) .

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Hence we conclude that EμE E [Ht (x)|Ms ] = es E K (0, s)e−(t−s)(H −E) Φ(Bs + x) = Hs (x). Then (Ht (x))t≥0 is martingale.



Let τ be a stopping time with respect to Mt . Then Ht∧τ (x) is also martingale. In particular EμE E[Ht (x)] = EμE E[Ht∧τ (x)].

4.4 Main Theorem In this section we assume that Φ is a bound state of H : H Φ = EΦ. Lemma 4.12 Suppose Assumption 4.1. Then Φ(·) L 2 (Q ) ∈ L ∞ (Rd ). Proof It follows that Φ(x) = EμE E[I0∗ Ht (x)] for an arbitrary t > 0. Since we can see that I[0,t] is bounded and I[0,t]  ≤ 2e E(t) , we obtain  1/2  t 1/2 Φ(x) ≤ 2et E e E(t) Ex [e−2 0 V (Br )dr ] Ex [Φ(Bt )2 ] . Since supx∈Rd Ex [e−2 follows.

t 0

V (Br )dr

] < ∞ and Ex [Φ(Bt )2 ] ≤ CΦ, the lemma 

Now we state the main theorem in this article. Theorem 4.13 (Pointwise exponential decay) Suppose Assumption 4.1 and infrared  2 ˆ /ω(k)3 dk < ∞. Assume either (1) or (2): regular condition Rd |ϕ(k)| (1) lim|x|→∞ V (x) = ∞, ˆ < 0. (2) lim|x|→∞ V− (x) + E + E(ϕ) Then there exist constants c and C such that Φ(x) L 2 (Q ) ≤ Ce−c|x| . Proof Suppose (1). Let τ R = inf{t||Bt | > R}. Then τ R is a stopping time with respect to the natural filtration of the Brownian motion (Bt )t≥0 . Let W R (x) = inf{V (y)||x − y| < R}. Note that W R (x) ≤ V (x + y) for |y| < R, and we see that ˆ → ∞ (|x| → ∞). X R (x) = W R (x) − E − E(ϕ) Let Ψ ∈ L 2 (Q). Then I0 Ψ · Ht (x) is also martingale. Actually we can see that

Pointwise Exponential Decay of Bound States of the Nelson Model …

247

EμE E[I0 Ψ · Ht (x)|Ms ] = I0 Ψ · EμE E[Ht (x)|Ms ] = I0 Ψ · Hs (x). Hence, it follows that (I0 Ψ , Φ(x)) L 2 (Q) = EμE E[I0 Ψ · H0 (x)] = EμE E[I0 Ψ · Ht∧τ R (x)]. ˆ is derived from the infrared regular condition and the expoBoud I[0,t]  ≤ 2et E(ϕ) nent is linear in t. Then

Φ(x) =

sup Ψ ∈L 2 (Q ),Ψ =1

|(I0 Ψ , Φ(x))| = |EμE E[I0 Ψ · Ht∧τ R (x)]|

≤ E[I0∗ Ht∧τ R (x)] ≤ 2E[e− Then it is enough to estimate E[e− E[e−

 t∧τ R 0

(V (Br +x)−E−E(ϕ))dr ˆ

 t∧τ R 0

 t∧τ R 0

(V (Br +x)−E−E(ϕ))dr ˆ

] sup Ψg (x). x∈Rd

(V (Br +x)−E−E(ϕ))dr ˆ

]. Let

] ≤ E[1l{τ R 0 such that for all |x| > R it holds that ˆ < −ε < 0. By the definition of stopping time |Br + x| ≥ R for V− (x) + E + E(ϕ)  t∧τ ˆ is less than r ≤ t ∧ τ R (x). Then the integrand in 0 R (V− (Br + x) + E + E(ϕ))dr −ε. Hence E[e

 t∧τ R (x) 0

(V− (Br +x)+E+E(ϕ))dr ˆ

] ≤ E[e−(t∧τ R (x))ε ]

= E[1l{t 0. We can also see that e−t H → e−t H strongly as ε ↓ 0. We shall conε struct a functional integral representation of (F, e−t H G) for ε > 0 and by a limiting argument we construct that of (F, e−t H G) for ε = 0. Assume that V ∈ C0∞ (Rd ). Let ˆ h = (−1/2)Δ. By the Trotter–Kato product formula [12, 13] and e−|s−t| Hf = Is∗ It , we have ⎛ ⎞ ⎛ ⎞ n−1  ε t ˆ t t ε (F, e−t H G) = lim ⎝I0 F, ⎝ I jt e− n HI e− n h e− n V I∗jt ⎠ It G ⎠ . n→∞

n

n

j=0

 ˆε ˆε Here nj=1 t j = t1 · · · tn . Using the identity Is e− HI Is = E s e− HI (s) E s for s ∈ R, where  ⊗ ε ρε (φE (δs ⊗ ϕ(· − x))) d x Hˆ I (s) = Rd

and E s = Is Is∗ is a projection, we can see that ⎛

⎛

ε (F, e−t H G) = lim ⎝I0 F, ⎝

n→∞

n−1 

⎞

⎞ E jt e

− nt

n

ε Hˆ I ( jtn

)

e

− nt h

e

− nt

V

E jt ⎠ It G ⎠ . n

j=0

By the Markov property of E s ’s, we can neglect E s ’s. Then ⎛

⎛

(F, e−t H G) = lim ⎝I0 F, ⎝ ε

n→∞

n−1 

⎞ e

− nt

ε Hˆ I ( jtn ) − nt h − nt V ⎠

e

e

⎞ It G ⎠ .

j=0

The right-hand side above can be represented in terms of the Wiener measure as (F, e

−t H ε

 G) = lim

' n−1  ( n−1 t ε jt − j=0 V (B jt ) ˆI ( ) − H j=0 n n I G(B ) n I0 F(B0 )e d xE e . t t

n→∞ Rd

x

Pointwise Exponential Decay of Bound States of the Nelson Model …

249

Note that s → δs ⊗ ϕ(· − Bs ) is strongly continuous as a map R → E , almost surely. Hence, s → φE (δs ⊗ ϕ(· − Bs )) is also strongly continuous as a map R → L 2 (Q E ). Then we can compute the limit and the result is ε

(F, e−t H G) =

Here Q t =

t 0

 Rd

 t   t d xEx e− 0 V (Bs )ds I0 F(B0 ), e−ε Q t −φE ( 0 δs ⊗ϕ(·−Bs )ds) It G(Bt ) .

φE (δs ⊗ ϕ(· − Bs ))2 ds. Take ε ↓ 0 on both sides above we have

(F(B0 ), e−t H G) =

 Rd

 t   t d xEx e− 0 V (Bs )ds I0 F(B0 ), e−φE ( 0 δs ⊗ϕ(·−Bs )ds) 8It G(Bt ) .

Then the theorem follows for V ∈ C0∞ (Rd ). By a simple limiting argument, we can prove (17) for V ∈ R.  Acknowledgements The author thanks Oliver Matte for useful discussions and comments for bounds in Section 2.2, and also thanks organizers: Kenji Yajima, Arne Jensen, Hisashi Okamoto, Yoshio Tsutsumi, Shu Nakamura, Keiichi Kato, Norikazu Saito, and Fumihiko Nakano for inviting him to “Tosio Kato Centennial Conference” held in the University of Tokyo at September 4–8, 2017. Finally, this work is financially supported by JSPS KAKENHI 16H03942, CREST JPMJCR14D6 and JSPS open partnership joint research with Denmark 1007867.

References 1. Aizenman, M., Simon, B.: Brownian motion and Harnack’s inequality for Schrödinger operators. Commun. Pure Appl. Math. 35, 209–271 (1982) 2. Bach, V., Fröhlich, J., Sigal, I.M.: Renormalization group analysis of spectral problems in quantum field theory. Adv. Math. 137, 205–298 (1998) 3. Carmona, R.: Pointwise bounds for Schrödinger eigenstates. Commun. Math. Phys. 62, 97–106 (1978) 4. Carmona, R., Masters, W.C., Simon, B.: Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91, 117–142 (1990) 5. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators With Application to Quantum Mechanics and Global Geometry. Springer, Berlin (1987) 6. Griesemer, M., Lieb, E., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145, 557–595 (2001) 7. Guneysu, B., Matte, O., Møller, J.: Stochastic differential equations for models of nonrelativistic matter interacting with quantized radiation fields. Probab. Theory Relat. Fields 167, 817–915 (2017) 8. Hirokawa, M., Hiroshima, F., Spohn, H.: Ground state for point particles interacting through a massless scalar bose field. Adv. Math. 191, 339–392 (2005) 9. Hiroshima, F.: Functional integral approach to semi-relativistic Pauli-Fierz models. Adv. Math. 259, 784–840 (2014) 10. Kato, T.: Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Am. Math. Soc. 70, 195–211 (1951) 11. Kato, T.: Schrödinger operators with singular potentials. Isr. J. Math. 13, 135–148 (1972) 12. Kato, T.: On the Trotter-Lie product formula. Proc. Jpn. Acad. 50, 694–698 (1974) 13. Kato, T., Masuda, K.: Trotter’s product formula for nonlinear semigroups generated by the subdifferentiables of convex functionals. J. Math. Soc. Jpn. 30, 169–178 (1978)

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14. L˝orinczi, J., Hiroshima, F., Betz, V.: Feynman-Kac Type Theorems and Its Applications. De Gruyter (2011) 15. Matte, O., Møller, J.: Feynman-Kac Formulas for the Ultra-violet Renormalized Nelson Model (2017). arXiv:1701.02600 16. Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1990–1997 (1964) 17. Nelson, E.: Schrödinger particles interacting with a quantized scalar field. In: Martin, W.T., Segal, I. (eds.) Proceedings of a conference on analysis in function space, p. 87. MIT Press (1964) 18. Simon, B.: The P(φ)2 Euclidean (Quantum) Field Theory. Princeton University Press (1974) 19. Simon, B., Høegh-Krohn, R.: Hypercontractive semigroup and two dimensional self-coupled bose fields. J. Funct. Anal. 9, 121–180 (1972)

Regular KMS States of Weakly Coupled Anharmonic Crystals and the Resolvent CCR Algebra Tomohiro Kanda and Taku Matsui

Abstract We consider equilibrium states of weakly coupled anharmonic quantum oscillators(= anharmonic crystal) on an integer lattice Z. We employed standard functional analytic methods for Schrödinger operators and we show existence of the infinite volume limit of equilibrium states, and uniqueness of the regular KMS (Kubo–Martin–Schwinger) states in the frame of Resolvent CCR Algebra introduced by D. Buchholz and H. Grundling.

1 Introduction In this article, we consider equilibrium states of weakly coupled anharmonic quantum oscillators on Z. We first consider finite quantum systems of particles located in sites  L := {−L + 1, −L + 2, · · · L − 1, L} ⊂ Z. Let HL be the Schrödinger operator (defined the domain of the quantum harmonic oscillator) on L 2 (R2L ) defined by HL =

L  

L−1   pk2 + ω2 xk2 + V (xk ) + ϕ(xk − xk+1 )

k=−L+1

(1)

k=−L+1

where the potential V (x) and ϕ(x) are rapidly decreasing smooth functions defined on R and pk is the quantum mechanical momentum operator pk = −i ∂∂xk as usual. Bose particles are supposed to be fixed at sites in Z, and we consider the nearestneighbor interaction. Due to the presence of the term V (x) in (1), the case of one-site double-well potentials is within reach of our study while ϕ(x) gives rise to the (weak)

T. Kanda · T. Matsui (B) Faculty of Mathematics, Kyushu University, Fukuoka, Japan e-mail: [email protected] T. Kanda e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_12

251

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interaction between Bose particles. By equilibrium states for finite quantum systems, we mean the state ψ L ,β is defined by ψ L ,β (Q) =

tr (e−βHL Q) tr (e−βHL )

where Q is any element in an algebra R L of bounded operators we introduce below. Our interest is to examine the infinite volume limit lim L→∞ ψ L ,β and uniqueness of KMS states. We will not discuss the decay of correlation nor analyticity of correlation functions as these topics were already investigated before. To this end, we proceed with two steps. (a-1) The first step is to find the physically reasonable choice of an algebra R L of operators. More precisely, we require that the multiplication operators f (xk ) of the variable xk and the function g( pk ) of momentum operators pk (−L + 1 ≤ k ≤ L) are contained in R L where f (x) and g(x) are rapidly decreasing smooth functions. (a-2) We require that αtL (Q) = eit HL Qe−it HL ,

Q ∈ RL

(2)

gives rise to a one-parameter group of automorphisms on R L , hence Q ∈ R L should imply αtL (Q) ∈ R L . (b) R L can be embedded into R L+1 so that the infinite volume limit lim L→∞ is well defined and we can consider the uniqueness of the infinite volume states (the states obtained by the infinite volume limit) satisfying the KMS conditions. At a first look, these questions are mathematical problems of operator algebras, however, real mathematical contents are functional analysis related to Schrödinger operators as we see below. The conditions (a-1) and (a-2) are satisfied if we employ the resolvent CCR algebra introduced by D. Buchholz and H. Grundling in their investigation of supersymmetric quantum field theories. (c.f. [7–9] and D. Buchholz considered KMS states of lattice models recently in [10]). To show the uniqueness of the infinite volume states, we first show the existence of a state satisfying regularity and KMS conditions using the Trotter–Kato formula for the heat kernel associated with HL and next, we apply known results on relative entropy due to H. Araki, F. Hiai and A. Uhlmann in [26]. To state our results more precisely, we define the resolvent CCR algebra now. Let R L be the unital C∗ -algebra on L 2 (R2L ) generated by the unit and operators g(s−L+1 x−L+1 + · · · + s L x L + t−L+1 p−L+1 + · · · + t L p L ) where g(x) is a continuous function (with a single variable) vanishing at infinity, sk , tl (k, l ∈  L ) are real constants. The sub-algebra generated by g(s−L+2 x−L+2 + · · · + s L−1 x L−1 + t−L+2 p−L+2 + · · · + t L−1 p L−1 )

Regular KMS States of Weakly Coupled Anharmonic …

253

can be identified with R L−1 , in fact, the tensor product structure of L 2 (R2M ) = L 2 (R2(M−L) ) ⊗ L 2 (R2L ) induces the natural inclusion R L ⊂ R M if L < M. The inductive limit C∗ -algebra of ∪ L R L is denoted by R. Both R L and R are referred to as the resolvent CCR algebras. D. Buchholz and H. Grundling named R the resolvent algebra because they defined R as the universal C∗ -algebra generated by resolvents 1 (s−L+1 x−L+1 + · · · + s L x L + t−L+1 p−L+1 + · · · + t L p L ) + iλ where λ is an arbitrary real number. The definition of the resolvent algebra due to D. Buchholz and H. Grundling coincides with the one we introduced here. The quantum mechanical Hamiltonian of the harmonic oscillators is denoted by HLhar HLhar =

L  

 pk2 + ω2 xk2 ,

αthar,L (Q) = eit HL Qe−it HL , har

har

Q ∈ RL .

k=−L+1

By definition, αthar,L (Q) ∈ R L if Q ∈ R L . Theorem 1 Let HL be the Schrödinger operator defined in (1) where the potential V and ϕ are rapidly decreasing smooth functions. Then, αtL (Q) = eit HL Qe−it HL , Q ∈ R L gives rise to a one-parameter group of automorphisms on R L . The above result was shown by D. Buchholz and H. Grundling in [8] for the case of ω = 0. The key point of their argument is based on the fact that eith e−ith 0 − 1 is a compact operator for Schrödinger operators h and h 0 defined by h = p 2 + V (x), h 0 = p 2 . We can show Theorem 1 in the same manner for the case of ω = 0. Combined with Lieb-Robinson bound techniques, (c.f. [13, 15, 17–19, 23]) existence of the limit lim L→∞ αtL (Q) can be proved. Application of the techniques to anharmonic crystals was investigated by B. Nachtergaele and R. Sims in [21, 23], by B. Nachtergaele, B. Schlein, H. Raz, R. Sims, S. Starr in [20] and by B. Nachtergaele, B. Schlein, R. Sims, S. Starr and V. Zagrebnov in [22]. Even though they did not consider the case of resolvent CCR algebras, there is no essential difference between our case and that of papers cited above, hence we omit the detailed proof of the existence of the infinite volume limit of the Heisenberg time evolution. Theorem 2 The infinite volume limit αt (Q) = lim αtL (Q) L→∞

(3)

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exists in the norm topology of R. Let H free be the Hamiltoninan of decoupled oscillators ∞   2  H free = pk + ω2 xk2 + V (xk ) (4) k=−∞ free free and set αtfree (Q) = eit H Qe−it H . αt ◦ α−t (Q) and α−t ◦ αt (Q) are continuous in t for the norm topology of R. free

free

Thus, we obtain the Heisenberg time evolution for weakly coupled anharmonic oscillators on the infinite lattice Z. Now we state our main result. Uniqueness of the KMS states for one-dimensional quantum spin systems is a well-known fact, however, in our case, the time evolution αt is not norm continuous in t. αt is continuous in weak topology on the GNS representations which is locally quasi-equivalent to the standard Fock representation. We will see that there exists an infinite volume limit of ψ L ,β that the restriction ψ to each R L is obtained by a density matrix on L 2 (R2L ). In another word, the restriction ψ to R L is normal to the Fock representation and ψ(Qαt (R)) is continuous in t. By regularity of states, we mean states locally normal to the standard Fock state. Definition 1 Suppose that β > 0. Let αt , t ∈ R, be the one-parameter group of ∗-automorphism above on R. The state ψ is an (αt , β)-KMS state, if ψ is αt invariant state, ψ(αt (Q)) = ψ(Q), Q ∈ R, and ψ(Qαt (R)) is a continuous function in t ∈ R for any Q, R ∈ R satisfying the KMS boundary condition, namely there exists a holomorphic function FQ,R (t) in the strip Iβ = {z ∈ C , | 0 < I mz < β} of the complex plain, which is bounded continuous in the closure of Iβ such that FQ,R (t) = ψ(Qαt (R)),

FQ,R (t + iβ) = ψ(αt (R)Q)

(5)

for any Q, R ∈ R. By a direct computation, we can verify that ψ L ,β satisfies the KMS condition for αtL and for Q, R ∈ R L . Theorem 3 The unique regular KMS state, ψβ associated with the Hamiltonian (1) exists, and ψβ is obtained by the infinite volume limit lim L→∞ ψ L ,β = ψβ of ψ L ,β Statistical mechanics of anharmonic crystals with the Hamiltonian (6) defined below has been extensively studied by several people. (c.f. [1, 16] and the references therein)   { p 2j + V (x j )} + |xi − x j |2 (6) H= j∈Zd

j,i∈Zd ,||i− j||=1

where V is a polynomial giving rise to a double well potential. Results obtained so far are based mainly on perturbation theories. If β is small, it is possible to show exponential decay and and analyticity of correlation functions by using cluster expansion due to R. Minlos, A. Verbeure and V. Zagrebnov in

Regular KMS States of Weakly Coupled Anharmonic …

255

[16], however, at the low-temperature regime, it seems that different techniques are required. In [16], the confining potential has the form ax 4 + bx 2 and decay property of eigenfunctions of the anharmonic oscillators of [16] is different from our perturbed harmonic potential. In fact, R. Minlos, A. Verbeure and V. Zagrebnov used the fact of (each decoupled single) anharmonic oscillator that the ratio ψn (x)/ψ0 (x) of the eigenfunction ψn (x) for the nth eigenvalue and the ground state eigenfunction ψ0 (x) is bounded as a function of x. Obviously, this property does not hold for our case even though our weakly coupled model looks mathematically simpler to handle. With suitable modification, the methods of [16] may be applied at low temperatures regime for our model but in this article, we concentrate on (non-perturbative) methods of operator theories. We obtained proof of existence of the Heisenberg time evolution for the same Hamiltonian (6), with the potential V (x) = ω2 x 2 + v(x) and v(x) is smooth, rapidly decreasing, and we consider the application elsewhere.

2 Weyl and Resolvent CCR Algebras For our proof of results, we use the Weyl CCR algebra generated by unitaries ei(s−L+1 x−L+1 +···+sL x L +t−L+1 p−L+1 +···+tL pL ) as well. Let f = (sk , tm ) be a pair of real valued functions sk , tm with finite supports defined on Z. As both sk and tm have a finite support, the following sum is finite. ( f ) =

∞  k=−∞

sk x k +

∞ 

tl pl

l=−∞

We regard the domain of ( f ) is the set of rapidly decreasing smooth functions and by abuse of notation, we denote its selfadjoint extension by the same symbol ( f ). The Weyl CCR algebra is the C∗ -algebra generated by unitaries W ( f ) where W ( f ) is written as W ( f ) = ei( f ) (For a precise definition of the Weyl CCR algebra, see e.g., [6, Theorem 5.2.8.].) The Weyl CCR algebra is denoted by W and W L is defined as the unital sub-algebra of W generated by W ( f ) where f has its support in  L . We also set 1 R(λ, f ) = iλ + ( f )

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For any positive integer L, we set  L = { j ∈ Z | −L < j ≤ L}. For simplicity, we set R L as the unital sub-algebra of R generated by R(λ, f ) where f has its support in  L and λ is any real number. Next, we consider regular states of Weyl and Resolvent CCR algebras. Let ψ be a state of a C∗ -algebra A and by {πψ (·), ψ , Hψ }, we denote the GNS representation associated with ψ where πψ (·) is the representation on the GNS Hilbert space  Hψ and ψ is the GNS cyclic vector in Hψ such that ψ(Q) =  ψ , πψ (Q)ψ , Q ∈ A. Any state ψ of R L is regular if the GNS representation {πψ (·), ψ , Hψ } is quasiequivalent to the Fock representation. A state ψ of R is referred to as a regular state if the restriction of ψ to R L is regular for any L. In the same manner, a state ψ of W L regular if the GNS representation {πψ (·), ψ , Hψ } is quasi-equivalent to the Fock representation. A state ψ of R is referred to as a regular state if the restriction of ψ to W L is regular for any L. D. Buchholz and H. Grundling proved that the regularity of a state ψ of R L is equivalent to the condition if and only if ker(πψ (R(λ, f ))) = {0} for any f and λ ∈ R\{0}. In another word, πψ (R(λ, f )) is the resolvent of a closed operator if ψ is regular. Note that, there is a one-to-one correspondence between a regular state of R and that of W . And by abuse of notations, we employ the same notation, ψ or ϕ, etc., for the regular states of R and W . (See Definition 4.3 and Corollary 4.4 of [8].) Lemma 1 We identify R L with operators on L 2 (R2L ) and let K L be the algebra of compact operators on L 2 (R2L ) which we regard as a sub-algebra of R L . If ψ is a regular state on R L ( L ∈ N), πψ (K L ) is weekly dense in πψ (R L ) . Proposition 1 Let ψ be a regular state on R L , L ∈ N. Then, ψ(αtL (Q)R) is continuous on t ∈ R for any Q, R ∈ R L where αtL is defined in (2). For our resolvent CCR algebra, there exists the trivial (= of no interest) state ψtrivial defined by (7) ψtrivial (R(λ1 , f 1 )R(λ2 , f 2 ) · · · R(λk , f k )) = 0 for any λi ∈ C and f j . For any finite system, we have a canonical decomposition of the KMS state into the regular part and the singular part. Lemma 2 As before, we identify R L with operators in Fock representation on L 2 (R2L ) and let K L be the algebra of compact operators on L 2 (R2L ). Let H be a positive selfadjoint operator on L 2 (R2L ) satisfying the following conditions: (a) eit H Qe−it H (Q ∈ R L ) gives rise to a one-parameter group of automorphisms of R L denoted by γt (Q). γt (Q) = eit H Qe−it H . (b) e−βHL ,ψ is a trace class operator on L 2 (R2L ). Suppose a β-KMS state ψβ for γt is given. There exist β-KMS states ψs and ψr satisfying the following properties. (i) The kernel of the GNS representation for ψs contains the compact operator algebra K L on L 2 (R2L ).

Regular KMS States of Weakly Coupled Anharmonic …

257

(ii) ψr is the regular KMS state defined by ψr (Q) =

tr L 2 (R2L ) (e−βH Q) , Q ∈ RL . tr L 2 (R2L ) (e−βH )

(8)

(iii) ψβ is a convex combination of ψs and ψr , ψβ = λψr + (1 − λ)ψs for some a positive real number λ 0 ≤ λ ≤ 1. Proof Let p j ( j = 0, 1, 2, · · · ) be the mutually orthogonal rank one projections in R L associated with an eigenvector for an eigenvalue ε j of H , hence the spectral decomposition of H is written as follows: H=



ε j πL ( p j ).

j

 Set Pm = mj=1 p j and P = w − lim m→∞ πψβ (Pm ) on the GNS representation associated with ψβ . We claim that the projection P is in the center of the von Neumann algebra πψβ (R L ) . In fact, by definition P commutes with any elements in ¨ representaπψβ (K L ) and for any Q ∈ R L Q Pm is of finite rank in the Schrodinger tion, πψβ (Q Pm ) and its weak limit commutes with πψβ (R L ). Set λ = limm→∞ ψ(Pm ) and ψr (Q) = lim ψβ (Q Pm ), ψs (Q) = lim ψβ (Q(1 − Pm )) m→∞

m→∞

for any Q ∈ R L . As P is in the center of πψβ (R L ) , ψr and ψs are β−KMS states. For any Q ∈ K L , ψs (Q ∗ Q) = limm→∞ ψβ (Q ∗ Q(1 − Pm )) = 0 as {Pm } is an approximate unit for Q ∈ K L . As the GNS vector of the KMS state ψs is separating for πψs (R L ), the kernel of πψs contains K L , πψs (K L ) = 0. Now suppose that λ = limm→∞ ψ(Pm ) = 0. Then, ψ( p j ) = ψr ( p j ) = 0 for any j. As λ does not vanish, there is at least j satisfying ψr ( p j ) = 0. On the other hand, for a matrix unit system pi j of K L satisfying pi j p ji = pi , the KMS condition implies ψr ( pi ) = ψr ( pi j p ji ) = eε j −εi ψr ( p j ), ψr ( pkl ) = 0(k = l) which shows that ψr ( p j ) = 0 for any j. These equations tell us that (8) for Q = A Pm (A ∈ R L ). By taking the limit  m → ∞, we obtain (8) holds for any Q ∈ R L . In the case of a single harmonic oscillator, the singular part of any KMS state is ψtrivial . We denote the time evolution of a single harmonic oscillator by αthar and the resolvent CCR algebra for one degree of freedom by R.

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Lemma 3 For any β > 0, if ψ is a (αthar , β)-KMS state of a single harmonic oscillator, ψs = ψtrivial where ψtrivial is defined in (7). Proof Let ϕ be a KMS state for the single harmonic oscillator αthar such that the kernel of the GNS representation for ϕ contains the compact operator algebra. Let {πϕ (·), , H} be the GNS triple associated with ϕ. ˜ = R/K1 is an abelian algebra. Note that the quotient R Assuming πϕ (Q) = 0, Q ∈ K L , we show πϕ (R(λ, f )) = 0. If πϕ (Q) = 0, Q ∈ ˜ = R/K1 for the time K L , ϕ gives rise to the KMS state ϕ˜ of the quotient algebra R evolution α˜ thar induced by αthar . ˜ Due to the KMS boundary condition Let Q and R be entire analytic elements in R. ˜ and commutativity of R, ϕ(Qσ ˜ t (R)) is bounded on the whole complex plane and is entire, so ϕ(Qσ ˜ t (R)) is a constant, ˜ R). ϕ(Qσ ˜ t (R)) = ϕ(Q We set Q = R = πϕ ( f (x)) where f is a real continuous function with one variable vanishing at infinity. (x is the position operator.) As αthar ( f (x)) = f (cos ωt · x + sin ωt · p), for t = π/(2ω) f (x)αthar ( f (x)) is a compact operator. Thus, ϕ( f (x)2 ) = 0. ˜ , πϕ ( f (x)) = 0. It turns out As  is separating for πϕ ( R) πϕ (αthar,L ( f (x))) = πϕ ( f (cos ωt · x + sin ωt · p)) which shows that ϕ = ψtrivial .



We are not certain that ψs = ψtrivial holds for more general finite quantum systems, though, the physical meaning of singular KMS states is not clear and we shall consider only regular KMS states.

3 Existence of Regular KMS States In this section, we establish the existence of regular KMS states for infinite systems. Lemma 4 For any positive integers L < L and any positive function F ∈ ⊗k∈L L ∞ (R), the following estimates are valid: e−2β ϕ ∞ Tr L \L (e−βHL \L )Tr L (e−βHL M F ) ≤ Tr L (e−βHL M F ) Tr L (e−βHL M F ) ≤ e2β ϕ ∞ Tr L \L (e−βHL \L )Tr L (e−βHL M F ) where we set  L = { j ∈ Z | −L < j ≤ L } ⊂ Z and

(9) (10)

Regular KMS States of Weakly Coupled Anharmonic …



HL \L =

( pk2 + ω2 xk2 + V (xk )) +

k∈ L \ L

259



ϕ(xk − xk+1 )

k,k+1∈ L \ L

and M F is the multiplication operator of F on L 2 (R2L ) and ϕ ∞ is the supremum norm of ϕ. −βHL Proof Note that for β > 0, e−βHL is a positive trace is an integral  operator. e  class operator with its kernel e−βHL (x, y) satisfying R|L | R|L | e−βHL (x, y)2 d xd y < ∞. For L < L , we have   

() = V (xk ) + ϕ(xk − xk+1 ) ϒL = ⊂ L

≥



k∈ L

V (xk ) +

k∈ L

=



k,k+1∈ L





ϕ(xk − xk+1 ) +



() +

⊂ L

ϕ(xk − xk+1 ) − 2 ϕ ∞

k,k+1∈ L \ L

k,k+1∈ L

() − 2 ϕ ∞

⊂ L \ L

= ϒ L + ϒ L \L − 2 ϕ ∞ and ϒ L ≤ ϒ L + ϒ L \L + 2 ϕ ∞ .

e−βHL is the Mehler kernel kβh (x, y) ∈ S (R2L ), h

kβh (x, y) =

ω 2π sinh(2ωβ)

n

2



exp −

ω(xk2 + yk2 )coth(2ωβ) − 2cosech(2ωβ)xk yk 2

k∈ L



for x = (x−L +1 , · · · , x−1 , x0 , x1 , · · · , x L ), y = (y−L +1 , · · · , y L ) ∈ R2L . For any positive functions f, g ∈ S (R2L ), we have  

f, (e−

βH har L n

e− 

βϒ L n

 )n g L2

 βϒ L βϒ L h h f (w) kβ/n (w, z 1 )e− n (z 1 ) kβ/n (z 1 , z 2 )e− n (z 2 ) 2L R2L R2L  R βϒ L h ×··· × kβ/n (z n−1 , z n )e− n (z n )g(z n )dz n · · · dz 2 dz 1 dw R2L   ϒ L +ϒ L \L h ≥ e−2β ϕ ∞ f (w) kβ/n (w, z 1 )e−β n (z 1 ) 2L R2L R ϒ L +ϒ L \L h ×··· × kβ/n (z n−1 , z n )e−β n (z n )g(z n )dz n · · · dz 2 dz 1 dw 2L R  ϒ L +ϒ L \L βH har −2β ϕ ∞ − nL −β n n f, (e =e e ) g .

=

L2

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As a consequence, 

f, e

−βHL

g



 L2

f, (e

= lim

n→∞

≥ lim e

Then,

βH har L n

−2β ϕ ∞

n→∞

=e

−

 −2β ϕ ∞

e

−

βϒ L n

 ) g n

L2

 f, (e

βH har − nL

−

βϒ L +βϒ L \L

e  f, e−βHL −βHL \L g L 2 .

n

 ) g n

L2



e−βHL (x, y) ≥ e−2β ϕ ∞ e−βHL −βHL \L (x, y), x, y ∈ R2L .

 Since e−βHL is a trace class operator, R 2L e−βHL (x,x)d x < ∞. Thus, we obtain the following estimates for any positive function F ∈ k∈L L ∞ (R): Tr L (e−βHL M F ) =



e−βHL (x, x)F(x)d x  −2β ϕ ∞ ≥e e−β(HL \L +HL ) (x, x)F(x)d x R2L

R2L

−2β ϕ ∞

Tr L (e−βHL \L e−βHL M F ) =e = e−2β ϕ ∞ Tr L \L (e−βHL \L )Tr L (e−βHL M F ) and Tr L (e−βHL M F ) ≤ e2β ϕ ∞ Tr L (e−βHL \L e−βHL M F ) = e2β ϕ ∞ Tr L \L (e−βHL \L )Tr L (e−βHL M F ). We obtain (9) and (10).



Proposition 2 For any positive integers L ≤ L and any F ∈ R L such that π0 (F) is a positive multiplication operator on HL , the following estimate hold: e−4β ϕ ∞ ψ L (F) ≤ ψ L (F) ≤ e4β ϕ ∞ ψ L (F),

(11)

where ψ L and ψ L are KMS states of finite systems R L and R L . Proof By (9) and (10), we obtain the following inequalities: e−2β ϕ ∞ Tr L\L (e−βHL\L )Tr L (e−βHL M F ) ≤ Tr L (e−βHL M F ), Tr L (e−βHL M F ) ≤ e2β ϕ ∞ Tr L\L (e−βHL\L )Tr L (e−βHL M F ), e−2β ϕ ∞ Tr L\L (e−βHL\L )Tr L (e−βHL ) ) ≤ Tr L (e−βHL ), Tr L (e−βHL ) ≤ e2β ϕ ∞ Tr L\L (e−βHL\L )Tr L (e−βHL ). Thus, we obtain (11).



Regular KMS States of Weakly Coupled Anharmonic …

261

Note that ψ L can be regarded as a state on W L , i.e., ψ L (W ) =

Tr L (e−βHL π0 (W )) , W ∈ WL . Tr L (e−βHL )

(12)

Also, for the Weyl CCR algebra, the following statement follows. Proposition 3 For any positive integers L ≤ L and any F ∈ W L such that π0 (F) is a positive multiplication operator on HL , the following estimate hold: e−4β ϕ ∞ ψ L (F) ≤ ψ L (F) ≤ e4β ϕ ∞ ψ L (F)

(13)

where ψ L and ψ L are states on W L and W L defined in (12), respectively. Since e−β( p +ω x ) is a trace class operator on L 2 (R, d x) and by [6, Proposition 2 2 2 5.2.27], e−βd( p +ω x ) is a trace class on F+ (L 2 (R, d x)), where d( p 2 + ω2 x 2 ) is the second quantization of p 2 + ω2 x 2 and F+ (L 2 (R, d x)) is the Boson Fock space of L 2 (R, d x) (see also [6, Section 5.2.1]). Set 2

2 2

Tr F + (L 2 (R,d x)) (e−βH (Z\L ) π0 (·)) , Tr F + (L 2 (R,d x)) (e−βH har (Z\L ) ) har

 L := ψ L ⊗ ψ

 L |W L = ψ L . Thus, the regular state ψ L on W L  L is a regular state on W and ψ then ψ can extend to a regular state on W . Since W is a unital C∗ -algebra, the family of  L } L∈N has at least one cluster point ψ. Next, we show that ψ is a regular states {ψ state. Theorem 4 The state ψ defined in the above is a regular state on W . Proof To show regularity of ψ, we show that for t in |t| ≤ δ  L (QW0 (t)) ψ is equicontinuous with respect to L where Q = Q(x) is an arbitrary essentially bounded function on R2L and (W0 (t)) =eit p0 . (We can show the continuity of L tk pk  L (QW0 (t)) for the general W = ei k=−L+1 lim L ψ in the same way.) For simplicity of presentation, we consider the case ω = 1 here. The case ω = 1 can be shown analogously. Note that  L (QW0 (t)) = ψ   1 e−β/2HL (x + t (0) , y) −β/2HL −β/2HL e (x, y)Q(x) (x, y)d xd y e 2 e−β/2HL (x, y) Z βL

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2 where Z βL is the normalization constant

  2 = Z βL

(e−β/2HL (x, y))2 d xd y =



e−βHL (x, x)d x.

and x + t (0) is the addition of t to x at the component corresponding to the origin of the integer lattice Z. For x = (· · · , x−1 , x0 , x1 , · · · ) and y = (· · · , y−1 , y0 , y1 , · · · ), we claim that e−c(t)

kβh (x0 + t, y0 ) kβh (x0 , y0 )

≤

k h (x + t, y0 ) e−β/2HL (x + t (0) , y) c(t) β 0 ≤ e e−β/2HL (x, y) kβh (x0 , y0 )

(14)

where c(t) = sup |V (x0 + t) − V (x0 )| + sup |ϕ(x0 − x1 + t) − ϕ(x0 − x1 )| x0 ,x1

x0

+ sup |ϕ(x−1 − x0 + t) − ϕ(x−1 − x0 )|. x0 ,x−1

limt→0 c(t) = 0 due to uniform continuity of V and ϕ and this bound implies regularity. We now show (14). Note the following tautological equalities holds. For any n ∈ N,   h h h kβh (x0 + t, y0 ) = · · · kβ/n (x0 + t, z 1 )kβ/n (z 1 , z 2 ) · · · kβ/n (z n , y0 )dz 1 · · · dz n   h h h = · · · kβ/n (x0 + t, z 1 + s1 )kβ/n (z 1 + s1 , z 2 + s2 ) · · · kβ/n (z n + sn , y0 )dz 1 · · · dz n

(15) nt , for any constants sk . Then, up to a multipliticave factor, C h h h (x0 + t, z 1 + s1 )kβ/n (z 1 + s1 , z 2 + s2 ) · · · kβ/n (z n + sn , y0 ) = kβ/n

nt × exp[− C

 1 {cosh(2β/n)((z k + sk )2 + (z k+1 + sk+1 )2 ) 2 sinh(2β/n) k=0 n

−2(z k + sk )(z k+1 + sk+1 )}]

(16)

where we set z 0 = x, z n+1 = y. In the exponent, we can write n  {cosh(2β/n)((z k + sk )2 + (z k+1 + sk+1 )2 ) − 2(z k + sk )(z k+1 + sk+1 )} k=0 n  2 = {cosh(2β/n)(z k2 + z k+1 ) − 2z k z k+1 } k=0

Regular KMS States of Weakly Coupled Anharmonic …

+

n+1 

263

An,k (s)z k + n (t, x0 , y0 )

(17)

k=0

where An,k (s)(linear in sk ) and n (t, x0 , y0 )(quadratic in sk ) are terms independent of z k . Now, we choose the constants sk satisfying the condition An,k (s) = 0 s0 = t sn+1 = 0. We do not need the exact form of An,k (s) and n (s, t) here but what we  independent of n. need are bounds |sk | ≤ A|t| Thus, we obtain   n (t, x0 , y0 ) h × kβ (x0 + t, y0 ) = exp − 2 sinh(2β/n)   h h h · · · kβ/n (x0 , z 1 )kβ/n (z 1 , z 2 ) · · · kβ/n (z n , y0 )dz 1 · · · dz n (18) and

 kβh (x0 + t, y0 ) n (t, x0 , y0 ) = lim exp − n→∞ 2 sinh(2β/n) kβh (x0 , y0 ) 

(19)

To show (14), we apply the Trotter–Kato formula again to β

e−βHL (x + t (0) , y) = lim (e− n HL e− har

n→∞

We consider now (e− βH har − nL

βH Lhar n

e−

βϒ L n

har

βϒ L n

)n (x + t (0) , y)

(20)

)n (x + t (0) , y) at each n in (20). The integral kernel of

βϒ − nL

(e e )n is iteration of integral in which change of the variable x + t (0) affects only to the integral associated to the particle at the origin and its nearest neighbor. In that integral, we denote the variable at the site −1 at the lattice by z k(−1) and that at the site 1 at the lattice by z k(1) . Then, the contribution to the iterated integral from the origin in (e− 

βH Lhar n

e−

βϒ L n

)n (x, y) is

  β h (x0 , z 1 ) exp − (V (z 1 ) + ϕ(z 1 − z 1(−1) ) + ϕ(z 1(1) − z 1 )) kβ/n n   β h (z 1 , z 2 ) exp − (V (z 2 ) + ϕ(z 2 − z 2(−1) ) + ϕ(z 2(1) − z 2 )) ×kβ/n n   β h · · · kβ/n (z n , y0 ) exp − (V (y0 ) + ϕ(y0 − y−1 ) + ϕ(y1 − y0 )) dz 1 · · · dz n n (21) 

···

After change of variable z k → z k + sk as in (15), the corresponding integral for (e−

βH Lh n

e−

βϒ L n

)n (x + t (0) , y) is

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T. Kanda and T. Matsui

  β h (x0 , z 1 ) exp − (V (z 1 ) + ϕ(z 1 − s1 − z 1(−1) ) + ϕ(z 1(1) − z 1 + s1 )) kβ/n n   β h (z 1 , z 2 ) exp − (V (z 2 − s2 ) + ϕ(z 2 − s2 − z 2(−1) ) + ϕ(z 2(1) − z 2 + s2 )) ×kβ/n n   n (t, x0 , y0 ) · · · dz 1 · · · dz n × exp − 2 sinh(2β/n) (22) 

···

Then,   n (t, x0 , y0 )  (21) × e−c( At) exp − 2 sinh(2β/n)   n (t, x0 , y0 )  ≤ (22) ≤ (21) × ec( At) exp − 2 sinh(2β/n)

(23)

By taking the limit n → ∞ we obtain the bound (14). Finally, we can show the regularity of ψ by using (14), Proposition 2 and the Lebesgue dominated convergence theorem. 

4 Relative Entropy First, let us recall the definition of quasi-containment. Let A be a C∗ -algebra and let (H1 , π1 ) and (H2 , π2 ) be nondegenerate representations of A . The representations π1 and π2 is quasi-equivalent, if there exists an isomorphism γ : π1 (A ) → π2 (A ) such that γ (π1 (A)) = π2 (A) for all A ∈ A (see also [6, Definition 2.4.25] and [6, Theorem 2.4.26]). If a subrepresentation of π1 is quasi-equivalent to π2 , then π1 is quasi-contain π2 . The next lemma is essentially due to [2, Lemma 1]. Lemma 5 Let ψ1 and ψ2 be regular states on R and (H1 , π1 , ξ1 ) and (H2 , π2 , ξ2 ) be the GNS representations associated with ψ1 and ψ2 , respectively.If π1 does not quasi-contain π2 , then there exists a sequence of projections em ∈ L∈N R L such that lim ψ1 (em ) = 0,

(24)

lim ψ2 (em ) = a > 0.

(25)

m

m

Proof Put H = H1 ⊕ H2 , π = π1 ⊕ π2 ,  ξ1 = ξ1 ⊕ 0,  ξ2 = 0 ⊕ ξ2 and M = π(R) . Note that π is a regular representation of R. Let E 1 and E 2 be the projections from H onto H1 and H2 , respectively. By the assumption, there exists a central projection E ∈ M ∩ M such that E E 1 = 0, E ≤ E 2 . By Kaplansky’s density theorem, there exists selfadjoint elements an ∈ R L(n) such that st- lim π(an ) = E, n

Regular KMS States of Weakly Coupled Anharmonic …

265

where the st-lim is the strong limit in H. By the regularity of π, Lemma 1 and Kaplansky’s density theorem, there exists selfadjoint elements bm(n) ∈ K L(n) such that st- lim π(bm(n) ) = π(an ). m

Note that the spectral projections of bm(n) are also contained in R L(n) . Let en be the spectral projection of bm(n) for an interval [1 − δ, 1 + δ], where δ ∈ (0, 1) is fixed. Then, en ∈ K L(n) ⊂ R L(n) and st- limn π(en ) = E.  Thus, limn ψ1 (en ) = 0, limn ψ2 (en ) = a > 0. Next, let us recall basic properties of the relative entropy of normal states of von Neumann algebras and ∗-algebras. The relative entropy for positive normal linear functionals of a von Neumann algebra was introduced by H. Araki in [3, 4]. Let ψ1 and ψ2 be positive normal linear functionals over a von Neumann algebra M. In a standard form of our von Neumann algebra M acting as a Hilbert space H, let ξ1 (respectively ξ2 ) be the vector in the positive cone H satisfying ψ1 (a) = (ξ1 , aξ1 ) (resp. ψ2 (a) = (ξ2 , aξ2 )) for a ∈ M. As usual, Sξ2 ,ξ1 is the densely defined closed conjugate linear operator Sξ2 ,ξ1 satisfying a ∈ M, Sξ2 ,ξ1 aξ1 = a ∗ ξ2 , and the relative modular operator ξ2 ,ξ1 is, by definition, the square root of the positive part of in the polar decomposition of Sξ2 ,ξ1 . ξ2 ,ξ1 = Sξ∗2 ,ξ1 Sξ2 ,ξ1 ,

(26)

We denote the orthogonal projection to the subspace M ξ1 (respectively M ξ2 on H) by s(ψ1 ) (respectively s(ψ2 )). The relative entropy S A (ψ1 , ψ2 ) is defined by  S A (ψ1 , ψ2 ) =

∞ − 0 log(λ)d(ξ1 , E(λ)ξ1 ), if (s(ψ1 ) ≤ s(ψ2 )) . ∞ otherwise

(27)

The relative entropy we use later is neither that of finite matrix algebras nor that of a von Neuman algebra but that of the resolvent algebra so that a caution is in order. The one we use there is due to A. Uhlmann who extended the notion of relative entropy to positive linear functionals on a (not necessarily normed) ∗-algebra. (c.f. [26].) We denote the relative entropy defined by A. Uhlmann by SU . SU coincides with that of H. Araki when the ∗-algebra we concern is a von Neumann algebra as we explain below. (See [14].) Lemma 6 ([14, Lemma 3.1]) Let A be a unital C∗ -algebra and π be a nondegenerate representation of A on a Hilbert space. Let ψ1 and ψ2 be positive linear 1 and ψ 2 . Then, functionals of A and we denote normal extensions to π(A ) by ψ 1 , ψ 2 ). SU (ψ1 , ψ2 ) = S A (ψ

(28)

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T. Kanda and T. Matsui

In view of this lemma, by the relative entropy of states ψ1 and ψ2 on a unital 1 and ψ 2 if ψ1 C∗ -algebra, we mean the relative entropy of normal extension of ψ and ψ2 are quasi-equivalent. If ψ1 and ψ2 are not quasi-equivalent, the relative entropy is infinite. Lemma 7 ([24, Corollary 5.12 (iii)]) Let N ⊂ M be von Neumann algebras and ψ1 and ψ2 be normal states on M. Assume there exists a norm one projection from M to N. Then (29) 0 ≤ S(ψ1 N , ψ2 N ) ≤ S(ψ1 , ψ2 ). By the same argument as that in [2], we have the following. Lemma 8 Let ψ1 and ψ2 be regular states on R. If sup S(ψ1 RL , ψ2 RL ) ≡ μ < ∞,

(30)

L∈N

then π2 quasi contains π1 where π j is the GNS representation of R associated with ψ j , j = 1, 2. Proof Assume that π2 does not quasi-contain π1 . By Lemma 5, there exists a sequence of projections en ∈ R L(n) such that lim ψ1 (en ) = a > 0, n

lim ψ2 (en ) = 0. n

Then, −ψ1 (en ) log ψ2 (en ) → ∞. Consider the C∗ -sub-algebra Bn of R L(n) generated by en and 1 − en . Due to Lemma 7, 1 π0 (RL(n) ) , ψ 2 π0 (RL(n) ) ) S(ψ1 RL , ψ2 RL ) = S(ψ 1 π0 (B n ) , ψ 2 π0 (B n ) ) ≥ S(ψ ψ1 (en ) ψ1 (1 − en ) + ψ1 (1 − en ) log . = ψ1 (en ) log ψ2 (en ) ψ2 (1 − en ) The above estimate is contradictory to our assumption.



We will use the continuity of the relative entropy. Lemma 9 ([24, Corollary 5.12 (i)]) Let ψi , ψ, φi and φ be normal states on a von Neumann algebra M. If ψi and φi converge to ψ and φ in σ (M∗ , M) topology, respectively, then (31) S(ψ, φ) ≤ lim inf S(ψi , φi ). i

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5 Proof of Theorem 3 For regular states, the KMS condition is equivalent to the Arak–Gibbs condition. We consider the perturbation of a regular state and the automorphism α. The perturbation of an automorphism and a state on a C∗ -algebra or a von Neumann algebra is defined in [6, Proposition 5.4.1] and [6, Theorem 5.4.4]. Proposition 4 Let φ be a regular (α, β)-KMS state on R. −1 Put W (L) := π−1 0 ( ({L , L + 1})) + π0 ( ({−L , −L + 1})), L ∈ N. Then φ satisfies the following condition: φ φβW (L) = ψ L ⊗ 

(32)

for all L ∈ N, where  φ is a state over R L c , φβW (L) is a perturbed state of φ by βW (L).

Proof For positive integers L < L , let γtL ,L be the perturbed automorphism of αtL by βW (L). Since HL − π0 (W (L)) = HL \L + HL and HL and HL \L are commute, L \L L \L γtL ,L = αtL ⊗ αt . The automorphism αt converges strongly to an automorc phism αtL on R L c when L → ∞. Note that the perturbed state φβW (L) is a (γ , β)c KMS state by construction, where γt = αtL ⊗ αtL . For 0 < R ∈ R L c , we define the βW (L) on R L by state φ R βW (L)

φR

(Q) =

φβW (L) (Q R) , φβW (L) (R)



Q ∈ RL .

βW (L)

c

βW (L)

Note that φ R is a regular state by construction and by γt = αtL ⊗ αtL , φ R is L an (α , β)-KMS state. Regular states of our Weyl algebra those of resolvent CCR βW (L) = ψ L . For all Q ∈ R L and 0 < R ∈ R L c algebras φ R φβW (L) (Q R) = ψ L (Q)φβW (L) (R).

(33)

For any selfadjoint element R ∈ R L c and any ε > 0, R + ( R + ε)1 is a strictly positive operator. Then we obtain φβW (L) (Q(R + ( R + ε)1)) = ψ L (Q)φβW (L) (R + ( R + ε)1). Since for any element R ∈ R L c can decompose two selfadjoint elements R1 and R2 such that R = R1 + i R2 . By the linearity of φβW (L) , the equation (33) holds for any elements Q ∈ R L and R ∈ R L c . Thus, φβW (L) = ψ L ⊗ φβW (L) |RL c . Put  φ = φβW (L) |RL c , then we get the claim.

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Remark 1 For Q ∈ R, it may not be a linear combination of the form of A ⊗ B for Theorem 4.2 (v)], for a regular A ∈ R L and B ∈ R L c . However, by Lemma 1 and [8, state φ on R and any positive integer L, πφ (R L ) πφ (R L c ) is a weakly dense sub-algebra in πφ (R). For Q ∈ R, there exists a positive integer {L(n)}n∈N such  that L ≤ L(n) for any n ∈ N and a sequence i ai(n) Ri(n) ⊗ K i(n) such that ai(n) ∈ C, Ri(n) ∈ R L , K i(n) ∈ K( L(n) \ L ) and πφ (Q) = w-lim n



ai(n) πφ (Ai(n) ) ⊗ πφ (K i(n) ).

i

and we can define the product state ψ L ⊗ φβW (L) |RL c for any Q ∈ R. Finally, we show uniqueness of (α, β)-KMS state for β > 0 in Theorem 3. Due to Theorem 4, ψ gives rise to a regular state on W and hence a regular state of R. Proof of Theorem 3  First, we show ψ(Qαt (R)) is continuous in t ∈ R for any Q, R ∈ R. Since L∈N R L is norm dense in R and R L ⊂ R L whenever L ≤ L , we show ψ(Qαt (R)) is continuous in t ∈ R for any Q, R ∈ R L . For any positive integer L and any Q, R ∈ R L , ψ(Qαt (R)) is continuous in t ∈ R. In fact, for any ε > 0, there exists a positive integer L such that ||αt (R) − αtL (R)|| < ε , |ψ(Q R) − ψ L (Q R)| < 4ε and |ψ(QαtL (R)) − ψ L (Qαt (R))| < 4ε and a δ > 0 4 such that |ψ L (QαtL (R) − Q R)| < 4ε for |t| < δ. Then, for |t| < δ |ψ(Qαt (R) − Q R)| ≤ |ψ(Qαt (R) − QαtL (R))| + |ψ(QαtL (R)) − ψ L (QαtL (R))| +|ψ L (QαtL (R)) − ψ L (Q R)| + |ψ L (Q R) − ψ(Q R)| < ε. Next, we show that ψ is an (α, β)-KMS state as specified in Definition 1. Note that the following inequality are valid for any Q, R ∈ R: |ψ(Qαt (R)) − ψ L (QαtL (R))| ≤ |ψ(Qαt (R)) − ψ L (Qαt (R))| +|ψ L (Qαt (R)) − ψ L (QαtL (R))|.

(34)

Using the integral representation of an analytic function in a strip Iβ = {z ∈ C | 0 < Imz < β} (c.f. the proof of [25, Theorem 2.2]), ψ is an (α, β)-KMS state. Finally, we show the uniqueness of (α, β)-KMS state. Let φ be an arbitrary extremal (α, β)-KMS regular state at β. Let (Hψ , πψ , ψ ) (resp. (Hφ , πφ , φ )) be  and  the GNS representation associated with ψ(rest. φ). By ψ φ, we denote the normal extension to the von Neumann algebras πψ (R) and πφ (R) . φβW (N ) , N ∈ N, be the perturbed state of  φ by βW (N ), where W (N ) is Let  φN =  defined in Proposition 4. Put M = πφ (R) and N L = πφ (R L ) , L ∈ N. By Lemma 7, for L ≤ N we obtain

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269

      0 ≤ S(φ N |N L , φ|N L ) ≤ S(φ N , φ) = φ N (βW (N )) − log φ N (1)   ≤ φ N (βW (N )) − φ(βW ((N )) ≤ 4|β| · |ϕ|∞ . This follows from Peierls–Bogoliubov inequality: 

φ(βW (N ))  = φ(βW (N )). log φ N (1) ≥ log e

Due to Lemmas 4 and 6, for L ≤ N   S(φ N |N L , φ|N L ) = S(φ N |R L , φ|R L ) = S(ψ N |R L , φ|R L ) N |π0 (RL ) ,  = S(ψ φ|π0 (RL ) ) ≤ 4|β| · |ϕ|∞ .  RL in σ (B(HL )∗ , B(HL )) topology. By Lemma N |RL converge to ψ| Note that ψ 9, it follows that  π0 (RL ) ,  φ|π0 (RL ) ) S(ψ|RL , φ|RL ) = S(ψ|  ≤ lim inf S(ψ N |π0 (RL ) ,  φ|π0 (RL ) ) ≤ 4|β| · |ϕ|∞ . N

By Lemma 5 and [2, Lemma 3], we complete our proof.



References 1. Albeverio, S., Kondratiev, Y., Kozitsky, Y., Rockner, M.: The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach. Ems Tracts in Mathematics. European Mathematical Society, Zürich (2009) 2. Araki, H.: On uniqueness of KMS states of one-dimensional quantum lattice systems. Commun. Math. Phys. 44(1), 1–7 (1975) 3. Araki, H.: Relative entropy of states of von Neumann algebras. Publ. RIMS, Kyoto Univ. 13, 173–192 (1977) 4. Araki, H.: Relative entropy for states of von Neumann algebras II. Publ. RIMS, Kyoto Univ. 11, 809–833 (1976) 5. Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics I, 2nd edn. Springer, Berlin (1987) 6. Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics II, 2nd edn. Springer, Berlin (1997) 7. Buchholz, D., Grundling, H.: Algebraic supersymmetry: a case study. Commun. Math. Phys. 272, 699–750 (2007) 8. Buchholz, D., Grundling, H.: The resolvent algebra: a new approach to canonical quantum systems. J. Funct. Anal. 254, 2725–2779 (2008) 9. Buchholz, D., Grundling, H.: Lie algebras of derivations and resolvent algebras. Commun. Math. Phys. 320, 455–467 (2013) 10. Buchholz, D.: The resolvent algebra for oscillating lattice systems: dynamics, ground and equilibrium states. Commun. Math. Phys. 353, 949–981 (2018) 11. Buchholz, D.: The resolvent algebra of non-relativistic Bose fields: observables, dynamics and states. Commun. Math. Phys. 362, 691–716 (2018)

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12. Fannes, M., Verbeure, A.: On the time evolution automorphisms of the CCR-algebra for quantum mechanics. Commun. Math. Phys. 35, 257–264 (1974) 13. Hastings, M.B., Koma, T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006) 14. Hiai, F., Ohya, M., Tsukada, M.: Sufficiency and relative entropy in ∗-algebras with applications in quantum systems. Pac. J. Math. 107(1), 117–140 (1983) 15. Lieb, E.H., Robinson, D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972) 16. Minlos, R., Verbeure, A., Zagrebnov, V.: A quantum crystal model in the light-mass limit: Gibbs states. Rev. Math. Phys. 12(7), 981–1032 (2000) 17. Nachtergaele, B., Sims, R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006) 18. Nachtergaele, B., Sims, R.: Recent progress in quantum spin systems. Markov Process. Relat. Fields 13, 315–329 (2007) 19. Nachtergaele, B., Sims, R.: Locality estimates for quantum spin systems. arXiv:math-ph/0712.3318v1 20. Nachtergaele, B., Raz, H., Schlein, B., Sims, R.: Lieb-Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286, 1073–1098 (2009) 21. Nachtergaele, B., Sims, R.: Lieb-Robinson Bounds in Quantum Many-body Physics. Entropy and the Quantum. Contemporary Mathematics, vol. 529, pp. 141–176. American Mathematical Society, Providence (2010) 22. Nachtergaele, B., Sims, R., Schlein, B., Starr, S., Zagrebnov, V.: On the existence of the dynamics for anharmonic quantum oscillator systems. Rev. Math. Phys. 22, 207–231 (2010) 23. Nachtergaele, B., Sims, R.: On the dynamics of lattice systems with unbounded on-site terms in the Hamiltonian. arXiv:math-ph/1410.8174 24. Ohya, M., Petz, D.: Quantum Entropy and its Use, Corrected Second Printing. Texts and Monographs in Physics. Springer, Berlin (2004) 25. Powers, R.T., Sakai, S.: Existence of ground states and KMS states for approximately inner dynamics. Commun. Math. Phys. 39, 273–288 (1975) 26. Uhlmann, A.: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Commun. Math. Phys. 54(1), 21–32 (1977)

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces Hagen Neidhardt, Artur Stephan and Valentin A. Zagrebnov

Abstract The paper is devoted to evolution equations of the form ∂ u(t) = −(A + B(t))u(t), t ∈ I = [0, T ], ∂t on separable Hilbert spaces where A is a non-negative self-adjoint operator and B(·) is family of non-negative self-adjoint operators such that dom(Aα ) ⊆ dom(B(t)) for some α ∈ [0, 1) and the map A−α B(·)A−α is Hölder continuous with the Hölder exponent β ∈ (0, 1). It is shown that the solution operator U (t, s) of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by A and B(t) provided the condition β > 2α − 1 is satisfied. The convergence rate for the approximation is given by the Hölder exponent β. The result is proved using the evolution semigroup approach.

1 Introduction A closer look to Kato’s work shows that abstract evolution equations and Trotter product formula were topics of high interest for Kato. Already at the beginning of his scientific career, Kato was interested in evolution equations [16, 17]. This interest On the occasion of the 100th birthday of Tosio Kato H. Neidhardt · A. Stephan Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany e-mail: [email protected] A. Stephan e-mail: [email protected] V. A. Zagrebnov (B) Institut de Mathématiques de Marseille (UMR 7373), Université d’Aix-Marseille, CMI - Technopôle Château-Gombert, 39 rue F. Joliot Curie, 13453 Marseille, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_13

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has lasted a lifetime [18–21, 26, 27, 29]. Another topic of great interest for him was the so-called Trotter product formula [22–24, 28]. Even the paper [23] has inspired further developments in this field [15]. The topic of the present paper is to link evolution equations with the Trotter product formula. To this end, we consider an abstract evolution equation of type ∂u(t) = − C(t)u(t), u(s) = xs , s ∈ [0, T ), ∂t C(t) =A + B(t),

t ∈ I := [0, T ],

(1.1)

on the separable Hilbert space H. Evolution equations of that type on Hilbert or Banach spaces are widely investigated, cf. [1–4, 7, 29, 31, 32, 43–48, 51–53, 55, 56] or the books [5, 49, 54]. We consider the equation (1.1) under the following assumptions. Assumption 1.1 (S1) The operator A is self-adjoint in the Hilbert space H such that A ≥ I . Let {B(t)}t∈I be a family of non-negative self-adjoint operators in H such that the function (I + B(·))−1 : I −→ L(H) is strongly measurable. (S2) There is an α ∈ [0, 1) such that for a.e. t ∈ I the inclusion dom(Aα ) ⊆ dom(B(t)) holds. Moreover, the function B(·)A−α : I −→ L(H) is strongly measurable and essentially bounded, i.e., Cα := ess sup B(·)A−α  < ∞.

(1.2)

t∈I

(S3) The map A−α B(·)A−α : I −→ L(H) is Hölder continuous, i.e, for some β ∈ (0, 1) there is a constant L α,β > 0 such that the estimate A−α (B(t) − B(s))A−α  ≤ L α,β |t − s|β , (t, s) ∈ I × I, holds.

(1.3)

Notice that under the assumption (S2), the operator C(t) is also an invertible non-negative self-adjoint operator for each t ∈ I. Assumptions of that type were made in [13, 14, 34, 35, 56]. One checks that the assumptions (S1)–(S3) and the additional assumption β > α imply the assumptions (I), (VI), and (VII) of [56] for the family {C(t)}∈I . Hence, Proposition 3.1 and Theorem 3.2 of [56] yield the existence of a so-called solution (or evolution) operator for the evolution equation (1.1), i.e., a strongly continuous, uniformly bounded family of bounded operators {U (t, s)}(t,s)∈Δ , Δ := {(t, s) ∈ I × I : 0 ≤ s ≤ t ≤ T }, such that the conditions U (t, t) =I, for t ∈ I, U (t, r )U (r, s) =U (t, s), for t, r, s ∈ I with s ≤ r ≤ t,

(1.4)

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are satisfied and u(t) = U (t, 0)x is for every x ∈ H a strict solution of (1.1), see Definition 1.1 of [56]. Because the involved operators are self-adjoint and nonnegative one checks that the solution operator consists of contractions. The aim of the present paper is to analyze the convergence of the following approximation to the solution operator {U (t, s)}(t,s)∈Δ . Let , s =: t0 < t1 < . . . < tn−1 < tn := t, t j := s + j t−s n

(1.5)

j = {0, 1, 2, . . . , n}, n ∈ N, be a partition of the interval [s, t]. Let G j (t, s; n) :=e−

t−s t−s A − n e n B(t j ) ,

j = 0, 1, 2, . . . , n,

Vn (t, s) :=G n−1 (t, s; n)G n−1 (t, s; n) × · · · × G 2 (t, s; n)G 0 (t, s; n),

(1.6)

n ∈ N. The main result in the paper is the following. If the assumptions (S1)–(S3) are satisfied and in addition the condition β > α holds, then the solution operator {U (t, s)}(t,s)∈Δ of [56] admits the approximation ess sup Vn (t, s) − U (t, s) ≤ (t,s)∈Δ

Rβ , n ∈ N, nβ

(1.7)

with some constant Rβ > 0. The result shows that the convergence of the approximation {Vn (t, s)}(t,s)∈Δ is determined by the smoothness of the perturbation B(·). If the map A−α B(·)A−α : I −→ L(H) is Lipschitz continuous, then the map is of course Hölder continuous with any exponent γ ∈ (α, 1). Hence from (1.7), it immediately follows that for any γ ∈ (α, 1), there is a constant Rγ such that ess sup Vn (t, s) − U (t, s) ≤ (t,s)∈Δ

Rγ , n ∈ N. nγ

(1.8)

In particular, for any γ close to one the estimate (1.8) holds. In [14], the Lipschitz case was considered. It was shown that there is a constant Υ0 > 0 such that the estimate ess sup Vn (t, 0) − U (t, 0) ≤ Υ0 t∈I

log(n) , n = 2, 3, . . . . n

(1.9)

holds. It is obvious that the estimate (1.9) is stronger than ess sup Vn (t, 0) − U (t, 0) ≤ t∈I

Rγ , n ∈ N. nγ

(which follows from (1.8)) for any γ independent of how close it is to one. To prove (1.7), we use the so-called evolution semigroup approach which allows not only to verify the estimate (1.7) but also to generalise it. The approach is quite different from the technique used in [14, 56]. We have successfully applied this

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approach already in [34, 35]. The key idea is to forget about the evolution equation (1.1) and to consider instead of it the operators K0 and K on K = L 2 (I, H). The operator K0 is the generator of the contraction semigroup {U0 (τ )}τ ∈R+ , (U0 (τ ) f )(t) := e−τ A χI (t − τ ) f (t − τ ),

f ∈ L 2 (I, H),

(1.10)

and K is given by K = K0 + B, dom(K) = dom(K0 ) ∩ dom(B), where B is the multiplication operator induced by the family {B(t)}∈I in L 2 (I, H) which is self-adjoint and non-negative, for more details see Section 2. It turns out that under the assumptions (S1) and (S2), the operator K is the generator of a contraction semigroup {U(τ )}τ ∈R+ on L 2 (I, H). For the pair {K0 , B}, we consider the Lie–Trotter product formula. From the original paper of Trotter [50], one gets that  τ τ n s- lim e− n K0 e− n B = e−τ K , τ ∈ R+ := [0, ∞), n→∞

(1.11)

holds uniformly in τ on any bounded interval of R+ . Since e−τ K0 = 0 and e−τ K = 0 for τ ≥ T one gets even uniformly in τ ∈ R+ . Previously, it was shown that under certain assumptions, the strong convergence can be improved to operator-norm convergence on Hilbert spaces, see [9, 10, 15, 38, 42] as well as on Banach spaces, see [11]. For an overview, the reader is referred to [37]. To consider the Trotter product formula for evolution, equations is relatively new and was first realized in [34, 35] for Banach spaces. In the following, we improve the convergence (1.11) to operator-norm convergence. We show that under the assumptions (S1)–(S3) and β > 2α − 1, there is a constant Rβ > 0 such that  τ  τ n Rβ   sup  e− n K0 e− n B − e−τ K  ≤ β , n ∈ N, n τ ∈R+

(1.12)

holds. It turns out that K is the generator of an evolution semigroup. This means, there is a propagator {U (t, s)}(t,s)∈Δ0 , Δ0 := {(t, s) ∈ I0 × I0 : s ≤ t}, I0 = (0, T ], such that the contraction semigroup {U(τ ) = e−τ K }τ ∈R+ admits the representation (U(τ ) f )(t) = U (t, t − τ )χI (t − τ ) f (t − τ ),

f ∈ L 2 (I, H).

(1.13)

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275

We recall that a strongly continuous, uniformly bounded family of bounded operators {U (t, s)}(t,s)∈Δ0 is called a propagator if (1.4) is satisfied for I0 and Δ0 instead of I and Δ, respectively. Roughly speaking, a propagator is a solution operator restricted to Δ0 where the assumption that U (t, 0)x should be a strict solution is dropped. Obviously, the notion of a propagator is weaker then that one of a solution operator. For its existence, one needs only the assumptions (S1) and (S2), see Theorem 4.4 and 4.5 in [34] or Theorem 3.3 [35]. Of course, the propagator coincides with the solution operator of [56] if the assumptions (S1)–(S3) are satisfied and β > α. By Proposition 3.8 of [37] and (1.12) we immediately get that under the assumptions (S1)–(S3) and β > 2α − 1 the estimate ess sup Vn (t, s) − U (t, s) ≤ (t,s)∈Δ0

Rβ , n ∈ N, nβ

(1.14)

holds, where the constant Rβ is that one of (1.12). Notice that the condition β > 2α − 1 is weaker than β > α, i.e., if β > α, then β > 2α − 1 holds. If α satisfies > α > β, then the assumptions (I), (VI), and (VII) of [56] for the condition 1+β 2 the family {C(t)}∈I are not valid but nevertheless, we get an approximation of the corresponding propagator {U (t, s)}(t,s)∈Δ0 . The results are stronger than those in [34, 35] for Banach spaces. In [34], a convergence rate O(n −(β−α) ) was found, whereas in [35], the Lipschitz case has been considered and the rate O(n −(1−α) ) for α ∈ ( 21 , 1) was proved. It turns out that the result (1.7) can be hardly improved. Indeed in [36], the simple case H := C and A = 1 was considered. In that case, the family {B(t)}t∈I reduces to a non-negative bounded measurable function: b(·) : I −→ R which has to be Hölder continuous with exponent β ∈ (0, 1). For that case, it was found in [36] that the convergence rate is O(n −β ) which coincides with (1.7). For the Lipschitz case, it was found O(n −1 ) which suggests that (1.8) and (1.9) might be not optimal. The paper is organised as follows. In Section 2, we give a short introduction into evolution semigroups. For more details, the reader referred to [33, 34, 39, 40]. The results are proven in Section 3. In Section 3.1, we prove auxiliary results which are necessary to prove the main results of Section 3.2. Notation: Spaces, in particular, Hilbert are denoted by Gothic capital letters like H, K, etc. Operators are denoted by Latin or italic capital letters. The Banach space of bounded operators on space is denoted by L(·), like L(H). We set R+ := [0, ∞). If a function is called measurable, then it means Lebesgue measurable. The notation “a.e.” means that a statement or relation fails at most for a set of Lebesgue measure zero. In the following, we use the notation ess sup(t,s)∈Δ or ess sup(t,s)∈Δ0 . In that case, the Lebesgue measure of R2 is meant. We point out that we call operator K to be generator of a semigroup {e−τ K }τ ∈R+ , see e.g., [41], although in [12, 25] it is the operator −K, which is called the generator.

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2 Evolution Semigroups Below, we consider the Hilbert space K = L 2 (I, H) consisting of all measurable functions f (·) : I −→ H such that the norm function  f (·) : I −→ R+ is square integrable. Further, let D0 be the generator of the right-hand sift semigroup on L 2 (I, H), i.e., (e−τ D0 f )(t) = χI (t − τ ) f (t − τ ), t ∈ I, τ ∈ R+ ,

f ∈ L 2 (I, H).

Notice that e−τ D0 = 0 for τ ≥ T . The generator D0 is given by ∂ f (t), ∂t f ∈ dom(D0 ) :=W01,2 (I, H) = { f ∈ W 1,2 (I, H) : f (0) = 0}. (D0 f )(t) :=

(2.1)

We remark that D0 is a closure of the maximal symmetric operator and its semigroup is contractive. Further, we consider the multiplication operator A in L 2 (I, H), (A f )(t) :=A f (t),   f (t) ∈ dom(A) for a.e. t ∈ I 2 f ∈ dom(A) := f ∈ L (I, H) : A f (t) ∈ L 2 (I, H). If (S1) is satisfied, then A is self-adjoint and A ≥ I L 2 (I,H) . For the resolvent, one has the representation ((A − z)−1 f )(t) = (A − z)−1 f (t), t ∈ I0 , f ∈ L 2 (I, H), z ∈ ρ(A) = ρ(A), and the corresponding semigroup {e−τ A }τ ∈R+ is given by (e−τ A f )(t) = e−τ A f (t), t ∈ I, f ∈ L 2 (I, H), τ ∈ R+ .

(2.2)

Notice that the operators e−τ D0 and e−τ A commute. Let us consider the contraction semigroup (2.3) U0 (τ ) := e−τ D0 e−τ A , τ ∈ R+ . Obviously, the semigroup {U0 (τ )}τ ∈R+ admits the representation (1.13). Due to the maximal L 2 -regularity of A, cf. [6], its generator K0 is given by K0 := D0 + A, dom(K0 ) := dom(D0 ) ∩ dom(A). Further, we consider the multiplication operator B, defined as

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

(B f )(t) :=B(t) f (t)   f (t) ∈ dom(B(t)) for a.e. t ∈ I . f ∈ dom(B) := f ∈ L 2 (I, H) : B(t) f (t) ∈ L 2 (I, H)

277

(2.4)

If (S1) is satisfied, then B is self-adjoint and non-negative. For the resolvent, we have the representation ((B − z)−1 f )(t) = (B(t) − z)−1 f (t),

f ∈ L 2 (I, H), z ∈ C−− ,

for a.e. t ∈ I. The semigroup {e−τ B }τ ∈R+ , admits the representation (e−τ B f )(t) = e−τ B(t) f (t),

f ∈ L 2 (I, H),

for a.e. t ∈ I. By [34, Proposition 4.4], we get that under the assumptions (S1) and (S2) the operator K := K0 + B, dom(K) := dom(K0 ) ∩ dom(B), is a generator of a contraction semigroup on L 2 (I, H). From [34, Proposition 4.5], we obtain that K is the generator of an evolution semigroup. Because K is a generator of a contraction semigroup it turns out that the corresponding propagator consists of contractions. If {U (t, s)}(t,s)∈Δ0 is a propagator, then by virtue of (1.13) it defines a semigroup, which by definition is an evolution semigroup. It turns out that there is a one-to-one correspondence between the set of evolution semigroups on L 2 (I, H) and propagators. It is interesting to note that evolution generators can be characterised quite independent from a propagator, see [33, Theorem 2.8] or [34, Theorem 3.3].

3 Results We start with a general observation concerning the conditions (S1)–(S3). Remark 3.1 If the conditions (S1)–(S3) are satisfied for some α ∈ [0, 1), then they are also satisfied for each α  ∈ (α, 1]. Indeed, the condition (S1) is obviously satisfied.  To show (S2), we note that dom(Aα ) ⊆ dom(Aα ) ⊆ dom(B(t)) for a.e. t ∈ I. Using the representation   (3.1) B(t)A−α = B(t)A−α A−(α −α) 

for a.e. t ∈ I we get that the map B(·)A−α : I −→ L(H) is strongly measurable. Further, from (3.1) 

Cα := ess sup B(t)A−α  ≤ ess sup B(t)A−α  = Cα < ∞. t∈I

t∈I

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Moreover we have 



A−α (B(t) − B(s))A−α  ≤ A−α (B(t) − B(s))A−α  ≤ L α,β |t − s|β , (t, s) ∈ I × I, which shows that there is a constant L α ,β ≤ L α,β such that 



A−α (B(t) − B(s))A−α  ≤ L α ,β |t − s|β (t, s) ∈ I × I. holds for (t, s) ∈ I × I.

γ −τ A

γ

 ≤ 1/τ for any τ ∈ Since A is self-adjoint and non-negative, one has A e R+ and γ ∈ [0, 1]. Then by virtue of (2.2) and of (1.10), (2.3), one gets the estimates Aγ e−τ A  = e−τ A Aγ  ≤

1 1 and Aγ e−τ K0  = e−τ K0 Aγ  ≤ γ . τγ τ

(3.2)

3.1 Auxiliary Estimates In this section, we prove a series of estimates necessary to establish (1.12). The following lemma can be partially derived from [34, Lemma 7.4]. Lemma 3.2 Let the assumptions (S1) and (S2) be satisfied. Then for any γ ∈ [α, 1), there is a constant Λγ ≥ 1 such that Aγ e−τ K  ≤

Λγ τγ

and e−τ K Aγ  ≤

Λγ , τγ

τ > 0,

(3.3)

holds. Proof The proof of the first estimate follows from Lemma 7.4 of [34] and Remark 3.1. The second estimate can be proved similarly as the first one. One has only to modify the proof of Lemma 7.4 of [34] in a suitable manner and to apply again Remark 3.1.  Remark 3.3 Lemma 2.1 of [14] claims that for the Lipschitz case, the solution operator {U (t, s)}(t,s)∈Δ of (1.1) admits the estimates sup (t − s)γ Aγ U (t, s) < ∞ and

(t,s)∈Δ

sup (t − s)γ U (t, s)Aγ  < ∞

(t,s)∈Δ

for γ ∈ [0, 1]. Proposition 2.1 of [36] immediately yields that the corresponding evolution semigroup {U(τ ) = e−τ K }τ ∈R+ satisfies the estimates (3.3) for γ = 1. Now we set

T (τ ) = e−τ K0 e−τ B , τ ∈ R+ .

Notice that T (τ ) = 0 for τ ≥ T .

(3.4)

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279

Lemma 3.4 Let the assumptions (S1) and (S2) be satisfied. Then for any γ ∈ [α, 1), the estimates A−γ (T (τ ) − U(τ )) ≤ 2Cγ τ and (T (τ ) − U(τ ))A−γ  ≤ 2Cγ τ, hold for τ ≥ 0, where

Cγ := ess sup B(t)A−γ .

(3.5)

(3.6)

t∈I

Proof The proof of the first estimate follows from Lemma 7.6 of [34] and Remark 3.1. The specific constant 2Cγ is obtained following carefully by the proof of Lemma 7.6 of [34]. The second estimate can be proved by modifying the proof of the first estimate in an obvious manner.  Lemma 3.5 Let the assumptions (S1)–(S3) be satisfied. Then for any γ ∈ [α, 1) and β ∈ (0, 1), there is a constant Z γ ,β > 0 such that A−γ (T (τ ) − U(τ ))A−γ  ≤ Z γ ,β τ 1+κ , τ ∈ R+ ,

(3.7)

holds where κ := min{γ , β}. Proof We use the representation   d −(τ −σ )K −σ K0 −σ B e e e =e−(τ −σ )K Ke−σ K0 − e−σ K0 K0 − e−σ K0 B e−σ B dσ   =e−(τ −σ )K Be−σ K0 − e−σ K0 B e−σ B which yields   e−(τ −σ )K Be−σ K0 − e−σ K0 B e−σ B

=(e−(τ −σ )K − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B (e−σ B − I )+

e−(τ −σ )K0 Be−σ K0 − e−σ K0 B (e−σ B − I )+

(e−(τ −σ )K − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B +

e−(τ −σ )K0 Be−σ K0 − e−σ K0 B . Hence, we obtain the identity   A−γ e−(τ −σ )K Be−σ K0 − e−σ K0 B e−σ B A−γ

=A−γ (e−(τ −σ )K − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B (e−σ B − I )A−γ +

e−(τ −σ )K0 A−γ Be−σ K0 − e−σ K0 B (e−σ B − I )A−γ +

A−γ (e−(τ −σ )K − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B A−γ +

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e−(τ −σ )K0 A−γ Be−σ K0 − e−σ K0 B A−γ which leads to the estimate      −γ −(τ −σ )K  −σ K0 Be − e−σ K0 B e−σ B A−γ  ≤ A e  

 −γ −(τ −σ )K  − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B (e−σ B − I )A−γ + (3.8) A (e  

 −(τ −σ )K0 −γ  (3.9) Be−σ K0 − e−σ K0 B (e−σ B − I )A−γ + A e  

 −γ −(τ −σ )K  (3.10) − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B A−γ + A (e  

 −(τ −σ )K0 −γ  (3.11) Be−σ K0 − e−σ K0 B A−γ . A e Note that by (3.2) and (3.6), one gets e−σ K0 B =e−σ K0 Aγ A−γ B ≤ σ −γ A−γ B = Cγ σ −γ , Be−σ K0  =BA−γ Aγ e−σ K0  ≤ σ −γ BA−γ  = Cγ σ −γ ,

(3.12)

for σ > 0. Due to (3.12), one estimates (3.8) as  

 −γ −(τ −σ )K  − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B (e−σ B − I )A−γ  A (e       ≤2 Cγ σ −γ A−γ (e−(τ −σ )K − e−(τ −σ )K0 )(e−σ B − I )A−γ . Since the fundamental properties of semigroups and (3.6) yield    −σ B  − I )A−γ  ≤ BA−γ  σ ≤ Cγ σ, σ ∈ R+ , (e and

(3.13)

    −γ −(τ −σ )K − e−(τ −σ )K0 ) ≤ Cγ (τ − σ ), σ ∈ R+ , A (e

we get for (3.8) the estimate  

  −γ −(τ −σ )K − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B (e−σ B − I )A−γ  A (e ≤2 Cγ3 σ 1−γ (τ − σ ), 0 ≤ σ ≤ τ.

(3.14)

To estimate (3.9), we recall that A and K0 commute. Then by (3.6), one gets  

  −(τ −σ )K0 −γ Be−σ K0 − e−σ K0 B (e−σ B − I )A−γ  A e ≤2Cγ (e−σ B − I )A−γ  ≤ 2 Cγ2 σ, 0 ≤ σ ≤ τ , where (3.13) was used for the last inequality.

(3.15)

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

To estimate (3.10), we have  

 −γ −(τ −σ )K  − e−(τ −σ )K0 ) Be−σ K0 − e−σ K0 B A−γ  A (e ≤2 Cγ A−γ (e−(τ −σ )K − e−(τ −σ )K0 ) ≤ 2 Cγ2 (τ − σ ), 0 ≤ σ ≤ τ.

281

(3.16)

To estimate (3.11), we use the representation

A−γ Be−σ K0 − e−σ K0 B A−γ =A−γ B(e−σ A − I )A−γ e−σ D0 − e−σ D0 A−γ (e−σ A − I )BA−γ +

A−γ Be−σ D0 − e−σ D0 B A−γ , that yields  

  −γ Be−σ K0 − e−σ K0 B A−γ  A         ≤A−γ B(e−σ A − I )A−γ  + A−γ (e−σ A − I )BA−γ +  

 −γ  Be−σ D0 − e−σ D0 B A−γ  . A Then by (3.6) and by semigroup properties one gets   C γ γ  −γ  σ , σ ∈ R+ , A B(e−σ A − I )A−γ  ≤ γ and

 C  γ γ   −γ −σ A σ , σ ∈ R+ . − I )BA−γ  ≤ A (e γ

The last term is obtained by using (S3) (for α substituted by γ ) and the definitions (2.1), (2.4):  

  −γ Be−σ D0 − e−σ D0 B A−γ  ≤ σ β L γ ,β , σ ∈ R+ . A Summing up one finds that  2C 

γ γ   −γ σ + L γ ,β σ β , σ ∈ R+ . Be−σ K0 − e−σ K0 B A−γ  ≤ A γ Using the estimates (3.14)–(3.17), we get

(3.17)

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    −γ −(τ −σ )K  −σ K0  Be − e−σ K0 B e−σ B A−γ  A e ≤2Cγ3 σ 1−γ (τ − σ ) + 2Cγ2 σ + 2Cγ2 (τ − σ ) + =2Cγ3 σ 1−γ (τ − σ ) + 2Cγ2 τ +

2Cγ γ σ + L γ ,β σ β γ

2Cγ γ σ + L γ ,β σ β , γ

or returning back to its derivative    −γ d −(τ −σ )K −σ K0 −σ B −γ  e e A  e A dσ ≤2Cγ3 σ 1−γ (τ − σ ) + 2Cγ2 τ +

2Cγ γ σ + L γ ,β σ β , 0 ≤ σ ≤ τ. γ

Since A

−γ

(e

−τ K0 −τ B

e

−e

−τ K

)A

−γ

 =

τ

A−γ

0

d −(τ −σ )K −σ K0 −σ B −γ e e e A dσ, dσ

we find the estimate     −γ −τ K0 −τ B −τ K −γ  e −e )A  ≤ A (e

τ 0

   −γ d −(τ −σ )K −σ K0 −σ B −γ  e e A dσ, e A dσ

which yields the estimate    −γ −τ K0 −τ B  e − e−τ K )A−γ  A (e  τ ≤2Cγ3 σ 1−γ (τ − σ )dσ + 2Cγ2 τ 2 + 0

2Cγ L γ ,β 1+β τ 1+γ + τ (1 + γ )γ 1+β

or after integration     −γ −τ K0 −τ B e − e−τ K )A−γ  A (e ≤

2Cγ3 (2 − γ )(3 − γ )

τ 3−γ + 2Cγ2 τ 2 +

2Cγ L γ ,β 1+β τ 1+γ + τ , τ ∈ R+ . (1 + γ )γ 1+β

If γ ∈ [α, 1) and γ ≤ β < 1, then one gets    −γ −τ K0 −τ B  e − e−τ K )A−γ  A (e ≤



2Cγ3 (2 − γ )(3 − γ )

τ 2−2γ + 2Cγ2 τ 1−γ +

τ ∈ R+ , which immediately yields (3.7).

(3.18) L γ ,β β−γ  1+γ 2Cγ + τ τ , (1 + γ )γ 1+β

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

283

If γ ∈ [α, 1) and 0 < β < γ , then one can rewrite it as    −γ −τ K0 −τ B  (3.19) e − e−τ K )A−γ  A (e

2Cγ3 2Cγ L γ ,β ≤ τ 2−γ −β + 2Cγ2 τ 1−β + τ γ −β + τ 1+β , (2 − γ )(3 − γ ) (1 + γ )γ 1+β τ ∈ R+ , which shows (3.7) for this choice of γ and β.



Remark 3.6 For γ ∈ [α, 1) and γ ≤ β < 1, we find from (3.18) that Z γ ,β :=

2Cγ3 (2 − γ )(3 − γ )

T 2−2γ + 2Cγ2 T 1−γ +

L γ ,β β−γ 2Cγ + T . (3.20) (1 + γ )γ 1+β

For γ ∈ [α, 1) and 0 < β < γ , we get from (3.19) that Z γ ,β :=

2Cγ3 (2 − γ )(3 − γ )

T 2−γ −β + 2Cγ2 T 1−β +

2Cγ L γ ,β T γ −β + . (1 + γ )γ 1+β

Here, Cγ := ess supt∈I BA−γ , see (3.3), and L γ ,β is the Hölder constant of the function A−γ B(·)A−γ : I −→ L(H), see (S3). Lemma 3.7 Let the assumptions (S1) and (S2) be satisfied. Then Aγ (U(τ ) − T (τ ))A−γ  ≤



 + 1 Cγ τ 1−γ , τ ∈ R+ ,

Λγ 1−γ

(3.21)

for γ ∈ [α, 1). Proof We use the representation U(τ ) − T (τ ) = e−τ K − e−τ K0 + e−τ K0 (I − e−τ B ) which yields Aγ (U(τ ) − T (τ ))A−γ = Aγ (e−τ K − e−τ K0 )A−γ + Aγ e−τ K0 (I − e−τ B )A−γ Using the semigroup property we obtain for the first term the representation: Aγ (e−τ K − e−τ K0 )A−γ = −



τ 0

Hence, by (3.3) and (3.6) one gets

Aγ e−(τ −x)K BA−γ e−xK0 d x .

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 τ Aγ (e−τ K − e−τ K0 )A−γ  ≤ Aγ e−(τ −x)K BA−γ d x ≤ 0  τ Λγ Cγ 1−γ 1 dx = . Λγ Cγ τ γ 1−γ 0 (τ − x)

(3.22)

To estimate the second term we use the inequality Aγ e−τ K0 (I − e−τ B )A−γ  ≤ Aγ e−τ K0 (I − e−τ B )A−γ . Using (3.2) and (3.13) we estimate the second term as Aγ e−τ K0 (I − e−τ B )A−γ  ≤ Cγ τ 1−γ .

(3.23) 

Now the estimates (3.22) and (3.23) yield (3.21).

Lemma 3.8 Let the assumption (S1) be satisfied. If for each γ ∈ [α, 1) there is a constant Mγ > 0 such that Aγ T (τ )m  ≤

Mγ , m ∈ N, τ ∈ R+ , (mτ )γ

(3.24)

holds for T (τ ) defined in (3.4), then Aσ T (τ )m  ≤

Mγδ (mτ )σ

, m ∈ N,

holds for σ ∈ [0, γ ] and δ := σ/γ . Proof If (3.24) is satisfied, then (T (τ )∗ )m Aγ  ≤

Mγ , m ∈ N, (mτ )γ

holds, which is equivalent to Aγ T (τ )m (T (τ )∗ )m Aγ ≤ or T (τ )m (T (τ )∗ )m ≤

Mγ2 (mτ )2γ

Mγ2 (mτ )2γ

, m ∈ N,

A−2γ , m ∈ N.

Let δ ∈ (0, 1). Using the Heinz inequality [8, Theorem X.4.2] we get  δ T (τ )m (T (τ )∗ )m ≤

Mγ2δ (mτ )2δγ

A−2δγ , m ∈ N.

(3.25)

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285

Since T (τ )m (T (τ )∗ )m is a self-adjoint contraction we get δ  T (τ )m (T (τ )∗ )m ≤ T (τ )m (T (τ )∗ )m , m ∈ N, which yields T (τ )m (T (τ )∗ )m ≤

Mγ2δ (mτ )2δγ

A−2δγ , m ∈ N,

or Aδγ T (τ )m (T (τ )∗ )m Aδγ ≤

Mγ2δ (mτ )2δγ

, m ∈ N.

Therefore, one gets     (T (τ )∗ )m Aδγ  ≤ or

   δγ  A T (τ )m  ≤

Mγδ (mτ )δγ Mγδ

(mτ )δγ

, m ∈ N,

, m ∈ N.

Setting δ = σ/γ we obtain the proof of (3.25).



Lemma 3.9 Let the assumptions (S1) and (S2) be satisfied and let γ ∈ (α, 1). Then there is a constant Mγ > 0 such that Aγ T (τ )m  ≤ holds for any T > 0

if

Mγ , m = 1, 2, . . . n, (mτ )γ

(3.26) Λ

τ ∈ (0, Tn ) and n ≥ n 0 where n 0 := (2( 1−γγ + 1)

1

Cγ ) 1−γ T  + 1 and x denotes the largest integer smaller than x. Proof Let Mγ > 0 be a constant which satisfies the inequality  5Λγ + 2

 α Λγ 1 γ + 1 Cγ Mγ T 1−γ 1−γ + 4Λγ Cγ Mγ B(1 − α, 1 − γ ) T 1−α ≤ Mγ 1−γ n

(3.27)

for n ≥ n 0 . Here, constants Λγ and Cγ are defined by Lemmas 3.2 and 3.4, respectively, while B(·, ·) denotes the Euler Beta-function. (Note that such Mγ > 0 always exists, see Remark 3.10 below.) Let m = 1. Then by (3.2) and (3.4) we get Aγ T (τ ) ≤ Aγ e−τ K0  ≤

Λγ Mγ 1 ≤ γ ≤ γ , γ τ τ τ

for τ > 0 and, in particular, for τ ∈ (0, T /n). Hence (3.26) holds for m = 1.

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Let us assume that (3.26) holds for l = 1, 2, . . . , m − 1, with m ≤ n, i.e., Aγ T (τ )l  ≤

Mγ , l = 1, 2, . . . m − 1, (lτ )γ

(3.28)

for τ ∈ (0, T /n). We are going to show that (3.28) holds for l = m. To this aim, we use the representation U(τ )m − T (τ )m =

m−1 

U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k , m = 2, 3, . . . ,

k=0

which implies T (τ )m = U(τ )m −

m−1 

U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k , m = 2, 3, . . . .

k=0

Hence Aγ T (τ )m = Aγ U(τ )m −

m−1 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k

k=0

or Aγ T (τ )m = Aγ U(τ )m − Aγ U(τ )m−1 (U(τ ) − T (τ )) − Aγ (U(τ ) − T (τ ))T (τ )m−1 −

m−2 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k .

k=1

for m = 3, 4, . . . . This yields the inequality Aγ T (τ )m  ≤ Aγ U(τ )m  + Aγ U(τ )m−1 (U(τ ) − T (τ ))+ Aγ (U(τ ) − T (τ ))T (τ )m−1  +

m−2 

(3.29)

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k 

k=1

for m = 3, 4, . . . . From Lemma 3.2, we get the estimates Aγ U(τ )m  ≤

Λγ , m = 2, 3, . . . , (mτ )γ

and consequently Aγ U(τ )m−1 (U(τ ) − T (τ )) ≤

2Λγ 4Λγ ≤ , m = 2, 3, . . . . γ ((m − 1)τ ) (mτ )γ

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

287

Then, summing up estimates for the first two terms in the right-hand side of (3.29), we obtain Aγ U(τ )m  + Aγ U(τ )m−1 (U(τ ) − T (τ )) ≤

5Λγ , m = 2, 3, . . . . (3.30) (mτ )γ

Next, we get for the third term in the right-hand side of (3.29), the estimate Aγ (U(τ ) − T (τ ))T (τ )m−1  ≤ Aγ (U(τ ) − T (τ ))A−γ Aγ T (τ )m−1 , m = 2, 3, . . . . Then using Lemma 3.7 we find that Aγ (U (τ ) − T (τ ))T (τ )m−1  ≤



Λγ +1 1−γ



Cγ τ 1−γ Aγ T (τ )m−1 , m = 2, 3, . . . .

By assumption (3.28) this yields Aγ (U (τ ) − T (τ ))T (τ )m−1  ≤



Λγ +1 1−γ

 Mγ C γ

1 τ 1−γ , m = 2, 3, . . . , ((m − 1)τ )γ

Mγ C γ

2 τ 1−γ , m = 2, 3, . . . . (mτ )γ

for τ ∈ (0, T /n), which leads to Aγ (U (τ ) − T (τ ))T (τ )m−1  ≤



Λγ +1 1−γ



(3.31)

Finally, one gets for the sum in (3.29) m−2 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k 

k=1

≤

m−2 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))A−α Aα T (τ )k , m = 2, 3, . . . .

k=1

Then by Lemma 3.2 this implies m−2 

Aγ U (τ )m−1−k (U (τ ) − T (τ ))T (τ )k 

k=1

≤ Λγ

m−2  k=1

1 (U (τ ) − T (τ ))A−α Aα T (τ )k , m = 2, 3, . . . . ((m − 1 − k)τ )γ

Taking into account Lemma 3.4 we get

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H. Neidhardt et al. m−2 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k 

k=1

≤ 2Λγ Cα

m−2  k=1

τ Aα T (τ )k , m = 2, 3, . . . . ((m − 1 − k)τ )γ

Finally, using assumption (3.28) and Lemma 3.8 one obtains m−2 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k 

k=1 α m−2  γ

≤ 2Λγ Cα Mγ

k=1

or

m−2 

τ 1 , m = 2, 3, . . . , ((m − 1 − k)τ )γ (kτ )α

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k 

k=1

≤

α γ 2Λγ Cγ Mγ

m−2  k=1

1 1 γ (m − 1 − k) k α

τ 1−γ −α , m = 2, 3, . . . ,

for τ ∈ (0, T /n). Since Lemma 3.11 below yields m−2  k=1

1 1 ≤ B(1 − α, 1 − γ )(m − 1)1−γ −α , m = 2, 3, . . . , (3.32) (m − 1 − k)γ k α

where B(·, ·) is the Euler Beta-function, we get m−2 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k 

k=1 α γ

≤2Λγ Cγ Mγ B(1 − α, 1 − γ )τ 1−γ −α (m − 1)1−γ −α , m = 2, 3, . . . , which in turn leads to m−2 

Aγ U(τ )m−1−k (U(τ ) − T (τ ))T (τ )k 

k=1

≤

α γ 4Λγ Cγ Mγ

B(1 − α, 1 − γ ) 1−α 1−α τ m , (mτ )γ

for m = 2, 3, . . . and any τ ∈ (0, T /n).

(3.33)

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289

Now we take into account (3.29)–(3.31) and (3.33) to conclude that Aγ T (τ )m  ≤   α

1 Λγ γ , 5Λγ + 2 + 1 Mγ Cγ τ 1−γ + 4Λγ Cγ Mγ B(1 − α, 1 − γ )τ 1−α m 1−α 1−γ (mτ )γ

for m = 2, 3, . . . and τ ∈ (0, T /n). Then Aγ T (τ )m  ≤   α

1 Λγ 1 γ + 1 Mγ Cγ T 1−γ 1−γ + 4Λγ Cγ Mγ B(1 − α, 1 − γ ) T 1−α 5Λγ + 2 . 1−γ (mτ )γ n

From assumption (3.27) we get  5Λγ + 2

 α Λγ 1 γ + 1 Mγ Cγ T 1−γ 1−γ + 4Λγ Cγ Mγ B(1 − α, 1 − γ ) T 1−α ≤ Mγ 1−γ n

for n ≥ n 0 , which shows that (3.28) holds for l = 1, 2, 3, . . . , n and n ≥ n 0 which proves (3.26).  Remark 3.10 One checks that condition (3.27) is always satisfied for sufficiently large M = Mγ and n ≥ n 0 . Indeed, after setting 

c0 := 5Λγ , c1 := 2

 Λγ + 1 Cγ T 1−γ , c2 := 4Λγ Cγ B(1 − α, 1 − γ ) T 1−α 1−γ

we get the condition c0 +

α

c1 n 1−γ

which yields

M + c2 M γ ≤ M

α

c0 + c2 M γ ≤ (1 − or

c0 + M

c2 M

1−

α γ

c1 1−γ n

≤1−

)M

c1 1−γ n

1 1−γ

Since n > c1 we have 1 − c1 /n 1−γ > 0. The left-hand side tends to zero if M → ∞. Hence, choosing M sufficiently large we guarantee the existence of Mγ such that condition (3.27) is satisfied for any n ≥ n 0 . It remains only to verify the following statement. Lemma 3.11 Let α ∈ [0, 1) and γ ∈ [α, 1). Then n−1  k=1

1 1 ≤ B(1 − α, 1 − γ )n 1−γ −α , n ∈ 2, 3, . . . . (n − k)γ k α

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the estimate holds where B(·, ·) is the Euler Beta-function. 

1

B(1 − α, 1 − γ ) := 0

x α (1

1 dx − x)γ

Proof If x ∈ (k − 1, k], then 1 1 1 1 ≤ α and ≤ α γ k x (n − k) (n − 1 − x)γ for k = 1, 2, . . . , n − 1. Hence 1 1 ≤ , x ∈ (k − 1, k]. γ α (n − k) k (n − 1 − x)γ x α Therefore 1 = (n − k)γ k α



k

k−1

1 dx ≤ (n − k)γ k α



k k−1

1 d x, x ∈ (k − 1, k], (n − 1 − x)γ x α

or n−1  k=1

n−1  k n−1  k   1 1 1 = d x ≤ dx γ α γ α (n − k) k (n − k) k (n − 1 − x)γ x α k=1 k−1 k=1 k−1  n−1 1 = d x = B(1 − α, 1 − γ )n 1−α−γ . (n − 1 − x)γ x α 0



3.2 Main Results In this section, we collect our main results and their proofs. They are based on preliminaries established in Section 3.1. Theorem 3.12 Let the assumptions (S1)–(S3) be satisfied and let β > 2α − 1. Then there is a constant Rβ > 0 such that sup U(τ ) − T (τ/n)n  ≤

τ ∈R+

holds for n ∈ N and τ ∈ R+ . Proof Taking into account the representation

Rβ nβ

(3.34)

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

U(τ/n)n − T (τ/n)n =

n−1 

291

U(τ/n)n−m−1 (U(τ/n) − T (τ/n))T (τ/n)m , n ∈ N,

m=0

or, identically, U(τ/n)n − T (τ/n)n =U(τ/n)n−1 (U(τ/n) − T (τ/n)) + (U(τ/n) − T (τ/n))T (τ/n)n−1 + n−2 

U(τ/n)n−m−1 (U(τ/n) − T (τ/n))T (τ/n)m , n = 3, 4, . . . ,

m=1

we obtain the estimate U(τ/n)n − T (τ/n)n  ≤ U(τ/n)n−1 Aγ A−γ (U(τ/n) − T (τ/n)) + (U(τ/n) − T (τ/n))A−γ Aγ T (τ/n)n−1  +

n−2 

(3.35)

U(τ/n)n−m−1 Aγ A−γ (U(τ/n) − T (τ/n))A−γ Aγ T (τ/n)m ,

m=1

n = 3, 4, . . . . Note that using Lemmas 3.2 and 3.4 one gets U(τ/n)n−1 Aγ A−γ (U(τ/n) − T (τ/n)) ≤ 2

Λγ Cγ τ , γ (τ (n − 1)/n) n

which yields 1 U(τ/n)n−1 Aγ A−γ (U(τ/n) − T (τ/n)) ≤ 21+γ Λγ Cγ T 1−γ . n

(3.36)

for n = 3, 4, . . . and τ ∈ [0, T ]. Now using Lemmas 3.4 and 3.9 for m = n − 1 we find (U(τ/n) − T (τ/n))A−γ Aγ T (τ/n)n−1  ≤ 2 Cγ

Mγ τ , n (τ (n − 1)/n)γ

for n ≥ n 0 , where n 0 is defined in Lemma 3.9 and τ ∈ [0, T ]. Hence, (U(τ/n) − T (τ/n))A−γ Aγ T (τ/n)n−1  ≤ 21+γ Cγ Mγ T 1−γ

1 . n

(3.37)

Taking into account Lemmas 3.2, 3.5, and 3.9 (for κ = min{γ , β}) one gets

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H. Neidhardt et al. n−2 

U(τ/n)n−m−1 Aγ A−γ (U(τ/n) − T (τ/n))A−γ Aγ T (τ/n)m 

m=1

≤

 τ 1+κ Λγ Z γ ,β Mγ γ ((n − m − 1) τ /n) n (m τ/n)γ m=1

=

n−2 Λγ Z γ ,β Mγ τ 1+κ−2γ  1 1 , 1+κ−2γ γ n (n − m − 1) m γ m=1

n−2 

for n > max {2, n 0 } and τ ∈ [0, T ]. Then by (3.32) we obtain n−2 

U(τ/n)n−m−1 Aγ A−γ (U(τ/n) − T (τ/n))A−γ Aγ T (τ/n)m 

m=1

≤

Λγ Z γ ,β Mγ τ 1+κ−2γ B(1 − γ , 1 − γ ) n 1−2γ , n 1+κ−2γ

or n−2 

U(τ/n)n−m−1 Aγ A−γ (U(τ/n) − T (τ/n))A−γ Aγ T (τ/n)m  (3.38)

m=1

≤ Λγ Z γ ,β Mγ B(1 − γ , 1 − γ ) T

1+κ−2γ

1 . nκ

Therefore, by virtue of (3.35)–(3.38) we get for n > max {2, n 0 } and τ ∈ [0, T ] the estimate U(τ ) − T (τ/n)n  = U (τ/n)n − T (τ/n)n  1 1 ≤21+γ Λγ Cγ T 1−γ + 21+γ Cγ Mγ T 1−γ + Λγ Z γ ,β Mγ B(1 − γ , 1 − γ )T 1+κ−2γ n n

≤ 21+γ Λγ Cγ T 1−γ + 21+γ Cγ Mγ T 1−γ + Λγ Z γ ,β Mγ B(1 − γ , 1 − γ )T 1+κ−2γ

1 nκ 1 . nκ

If α < β < 1, then we choose γ = β, i.e., κ = β and 1 + κ − 2γ = 1 − β ≥ 0. Setting Rβ := 21+β Λβ Cβ T 1−β + 21+β Cβ Mβ T 1−β + Λβ Z β,β Mβ B(1 − β, 1 − β)T 1−β one obtains the estimate U(τ )n − T (τ/n)n  ≤ for n > max {2, n 0 } and τ ∈ [0, T ] .

Rβ nβ

,

(3.39)

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293

Now let 0 < β ≤ α. Since 1 + β − 2α > 0, there exists γ ∈ (α, 1) such that 1 + β − 2γ ≥ 0. Indeed, there is a ε > 0 verifying 1 + β − 2α > 2ε. Setting γ = α + ε we get 1 + β − 2γ > 0. Notice that κ = β. Then setting Rβ := 21+γ Λγ Cγ T 1−γ + 21+γ Cγ Mγ T 1−γ + Λγ Z γ ,β Mγ B(1 − γ , 1 − γ )T 1+β−2γ ,

we obtain (3.39) for n > max {2, n 0 }. Both results immediately imply that there is a constant Rγ such that (3.34) holds for τ ∈ [0, T ] and n ∈ N. Finally, using U(τ ) = 0 and T (τ/n)n = 0 for τ ≥ T we  obtain (3.28) for any τ ∈ R+ . Now we set

T(τ ) := e−τ B e−τ K0 , τ ∈ R+ .

Corollary 3.13 Let the assumptions (S1)–(S3) be satisfied and β > 2α − 1. Then β > 0 such that estimate there exists R (τ/n)n  ≤ sup U (τ ) − T

τ ∈R+

β R

(3.40)

nβ

holds for n ∈ N and τ ∈ R+ . Proof Notice that T(τ/n)n+1 = e−τ B/n T (τ/n)n e−τ K0 /n , τ ∈ R+ , n ∈ N. Hence U((n + 1)τ/n) − T(τ/n)n+1 = e−(n+1)τ K/n − e−τ B/n T (τ/n)n e−τ K0 /n =e−(n+1)τ K/n − e−τ B/n e−τ K e−τ K0 /n + e−τ B/n (U(τ ) − T (τ/n)n )e−τ K0 /n =(I − e−τ B/n )e−τ K e−τ K0 /n + e−τ K (e−τ K/n − e−τ K0 /n )+ e−τ B/n (U(τ ) − T (τ/n)n )e−τ K0 /n , τ ∈ R+ , n ∈ N, which yields the estimate U((n + 1) τn ) − T( τn )n+1  τ

τ

τ

≤(I − e− n B )e−τ K  + e−τ K (e− n K − e− n K0 )+ U(τ ) − T ( τn )n , τ ∈ R+ , n ∈ N.

(3.41)

Obviously, one has τ

τ

(I − e− n B )e−τ K  ≤ (I − e− n B )A−α Aα e−τ K , τ ∈ R+ , n ∈ N.

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Using τ

(I − e− n B )A−α =



τ n

e−σ B BA−α dσ, τ ∈ R+ , n ∈ N,

0

we get the estimate τ τ (I − e− n B )A−α  ≤ Cα , τ ∈ R+ , n ∈ N. n

Taking into account condition (S2) and Lemma 3.2 we find τ

(I − e− n B )e−τ K  ≤ Cα Λα

τ 1−α 1 ≤ Cα Λα T 1−α , τ ∈ R+ , n ∈ N, n n

(3.42)

where we have used that e−τ K = 0 for τ ≥ T . Further, we have τ

τ

τ

τ

e−τ K (e− n K − e− n K0 ) ≤ e−τ K Aα  A−α (e− n K − e− n K0 ), τ ∈ R+ , n ∈ N. Then using A

−α

τ (e− n K

−

τ e− n K0 )



τ n

=−

e−σ K0 A−α Be−(τ −σ )K dσ,

0

τ ∈ R+ , n ∈ N, we find the estimate τ

τ

A−α (e− n K − e− n K0 ) ≤ Cα

τ , τ ∈ R+ , n ∈ N. n

Applying again Lemma 3.2 one gets τ

τ

e−τ K (e− n K − e− n K0 ) ≤ Cα Λα T 1−α

1 , τ ∈ R+ , n ∈ N. n

(3.43)

The insertion of (3.42) and (3.43) into (3.41) yields U((n + 1) τn ) − T( τn )n+1  ≤ 2Cα Λα

1 + U(τ ) − T ( τn )n ), τ ∈ R+ , n ∈ N. n

Then by Theorem 3.12 we obtain U((n + 1) τn ) − T( τn )n+1  ≤ 2Cα Λα

1 1 + Rγ γ , τ ∈ R+ , n ∈ N. n n

Therefore, by setting Rγ := 2Cα Λα + Rγ we obtain

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

U((n + 1) τn ) − T( τn )n+1  ≤

Rγ nγ

295

, τ ∈ R+ , n ∈ N.

which yields sup U((n + 1) τn ) − T( τn )n+1  ≤

τ ∈R+

Rγ nγ

, τ ∈ R+ , n ∈ N.

Let τ = τ  n/(n + 1) for τ  ∈ R+ . Then τ )n+1  ≤ sup U((n + 1) τn ) − T( τn )n+1  = sup U(τ  ) − T( n+1 

τ  ∈R+

τ ∈R+

or τ  n+1 sup U(τ  ) − T( n+1 )  ≤ 2γ

τ  ∈R+

Rγ (n + 1)γ

Rγ nγ

,

,

γ := max{2, 2γ Rγ } we prove (3.40). τ ∈ R+ , n ∈ N. Setting R



These results can be immediately extended to propagators. To this end we set t−s

t−s

 j (t, s; n) :=e− n B(t j ) e− n A , j = 0, 1, 2, . . . , n, G n (t, s; n)G n−1 (t, s; n) × · · · × G 2 (t, s; n)G 1 (t, s; n), n (t, s) :=G V

(3.44)

, j = 0, 1, 2, . . . , n, in analogy to (1.6). t j := s + j t−s n Theorem 3.14 Let the assumptions (S1)–(S3) be satisfied. Further, let {U (t, s)}(t,s)∈Δ0 be the propagator corresponding to the evolution generator K and let {Vn (t, s)}(t,s)∈Δ0 n (t, s)}(t,s)∈Δ0 be defined by (1.6) and (3.44), respectively. If β > 2α − 1, then and {V the estimates ess sup U (t, s) − Vn (t, s) ≤

(t,s)∈Δ0

Rβ

nβ

and

n (t, s) ≤ ess sup U (t, s) − V

(t,s)∈Δ0

β R

nβ

(3.45)

γ are those of Theorem 3.12 and hold for n ∈ N, where the constants Rγ and R Corollary 3.13. Proof Note that Proposition 2.1 of [36] yields sup U(τ ) − T ( τn )n  = ess sup U (t, s) − Vn (t, s), n ∈ N.

τ ∈R+

(t,s)∈Δ0

Then applying Theorem 3.12 we prove (3.45).

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To proof the second estimate, we use Proposition 3.8 of [37] where the relation n (t, s), n ∈ N. sup U(τ ) − T( τn )n  = ess sup U (t, s) − V

τ ∈R+

(t,s)∈Δ0

was shown. Applying Corollary 3.13 we complete the proof.



4 Example As an example, we consider the diffusion equation perturbed by a time-dependent scalar potential. For this aim, let H = L 2 (Ω), where Ω ⊂ R3 is a bounded domain with sufficiently smooth boundary. Domains in higher dimension can be treated analogously. The equation reads as u(t) ˙ = Δu(t) − B(t)u(t), u(s) = u s ∈ H, t, s ∈ [0, T ] ,

(4.1)

where Δ denotes the Laplace operator in L 2 (Ω) with Dirichlet boundary conditions, i.e., Δ : dom(Δ) = H 2 (Ω) ∩ H01 (Ω) → L 2 (Ω) and H01 (Ω) denotes the subset of functions that vanish at the boundary. Then operator −Δ is self-adjoint on H and positive. For any α ∈ (0, 1) the fractional power of operator −Δ is defined on the domain dom((−Δ)α ), i.e., (−Δ)α : dom((−Δ)α ) → L 2 (Ω). The domain is given by a fractional Sobolev space and for α > 1/2, we have dom((−Δ)α ) = H02α (Ω) ⊂ H 2α (Ω) (see [30] for more information). Moreover, let B(t) denote a time-dependent scalar-valued multiplication operator given by (B(t) f )(x) =V (t, x) f (x), (4.2) dom(B(t)) ={ f ∈ L 2 (I, H) : V (·, x) f (x) ∈ L 2 (I, H)} where V : I × Ω → R is measurable. We assume that the potential V (·, ·) is real and non-negative. Then B(t) is obviously self-adjoint and non-negative on H. Theorem 4.1 Let A be the Laplacian operator −Δ with Dirichlet boundary conditions in L 2 (Ω), see above. Further, let {B(t)}i∈I be the family of multiplication operators defined by (4.2). If V (·, ·) : I × Ω −→ R is measurable, real, non-negative with regularity V ∈ L ∞ (I, L 2+ε (Ω)) ∩ C β (I, L 1+ε (Ω)) for β ∈ (0, 1) and some ε > 0, then the assumptions (S1)–(S3) are satisfied with α ∈ [3/4, 1). Moreover, if β > 2α − 1 then the converging rates of Theorem 3.12, Corollary 3.13 and Theorem 3.14 hold. Proof Since Ω is bounded there one has inf σ (A) > 0 which does not satisfy A ≥ I in general and, hence, assumption (S1) is not satisfied. Nevertheless, inf σ (A) > 0 is sufficient to prove the converging results. So we can believe that (S1) is satisfied. Let α ≥ 3/4. Using the Sobolev space embeddings, we get that H 2α (Ω) ⊂ L γ (Ω) for any γ ∈ [2, ∞[. Hence, if V ∈ L ∞ (I, L 2+ε (Ω)), we conclude that the function

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297

[0, T ]  t → B(t)(−Δ)−α is essentially operator-norm bounded in t ∈ I and thus, (S2) is satisfied. Now, we consider F(t) := (−Δ)−α B(t)(−Δ)−α : L 2 (Ω) → H 2α (Ω) ⊂ L 2 (Ω). The function F(·) : I → L(H) is bounded for fixed t ∈ [0, T ] if for any f, g ∈ H 2α (Ω) the function  f, B(t)g is bounded. This holds since V (t, ·) ∈ L 1+ε (Ω) and H 2α (Ω) ⊂ L γ (Ω) for any γ ∈ [2, ∞[. Hence we conclude that (S3) is satisfied and the claim is proved.  Theorem 4.1 provides a convergence rate of an approximation of the solution of (4.1) by the time-ordered product n (t, s) = V

n 

e−

t−s n

j)s V ( jt+(n− ,·) n

e

t−s n Δ

(4.3)

j=1

This looks elaborate, but is indeed simple. There are strategies to compute the semigroup of the Laplace operator for bounded domains and there are also explicit formulas on special domains like disks, etc. The factors e−τ V (t j ) , j = 1, 2, . . . , n are scalar valued and can be easily computed. Acknowledgements We thank Takashi Ichinose and Hideo Tamura for the explanation of details of the proof of Theorem 1.1 of [14], which makes possible to prove Lemmas 3.8 and 3.9.

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Exact Solutions to Problems with Perturbed Differential and Boundary Operators I. N. Parasidis and E. Providas

On the occasion of the 100th birthday of Tosio Kato

Abstract We elaborate on a perturbation technique for examining the existence and uniqueness and delivering the solution in closed form of a composite boundary value problem involving the sum of a linear differential and an integral or loaded operator with nonlocal or integral boundary conditions, assuming that the exact solution for the differential operator with conventional boundary conditions is known. We apply this perturbation method to solve partial integro-differential, or loaded differential, equations with nonlocal, or integral, boundary conditions.

1 Introduction Kato has completed and advanced the perturbation theory for linear operators originated by Rayleigh and Schrödinger and developed further by Rellich, Nagy and others [2]. In the introduction of his book [3] it is stated that: ‘Perturbations theories are based on the idea of studying a system deviating slightly from a simple ideal system for which the complete solution of the problem under consideration is known; but the problems they treat and the tools they use are quite different’. Recently, an increasing number of papers have appeared using perturbation techniques to study the existence of solutions and other properties of composite problems, termed ’perturbed’ I. N. Parasidis Department of Electrical Engineering, TEI of Thessaly, 41110 Larissa, Greece e-mail: [email protected] E. Providas (B) Department of Mechanical Engineering, TEI of Thessaly, 41110 Larissa, Greece e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_14

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problems, by presupposing that the corresponding characteristics for a simpler problem, named ‘unperturbed’ problem, are known. For example, Dhage [1] investigates the solvability of ordinary nonlinear differential or integral or integro-differential equations involving linear or quadratic perturbations related to whether they can be formulated as the sum or the product of two operators, respectively. Sadybekov and Imanbaev [8] study the spectral properties of boundary value problems encompassing an operator with integral perturbations of its boundary conditions assuming that the corresponding spectral properties of the unperturbed operator are known. In [6, 7] closed form solutions are obtained for the compound problem Bu = Au + Qu = f, D(B) = {u : u ∈ D(A), Γ u = 0},

(1)

for all f ∈ Y , where X, Y, Z are Banach spaces, A : X → Y is a linear maximal on Γ : X A → Z is a linear bounded boundary operator, X A =  differential operator, D(A), || · || X A , || · || X A denotes a graph norm and Q is a linear or nonlinear integral operator, presuming that the unique solution to the unperturbed problem  = f, Au  ={u : u ∈ D(A), Γ u = 0}, D( A)

(2)

 is a correct restriction of A, exists and can be derived in closed form. where A In this article, we deal with problems of the type (1) but over and above that with integral perturbations of its boundary conditions. More specifically, we examine solvability conditions and find the exact solution for every f ∈ Y of the following two boundary value problems: Bu = Au − g F(Au) = f, D(B) = {u : u ∈ D(A), Γ u = vΨ (u)},

(3)

where F, Ψ are vectors of linear bounded functionals on Y and X A , respectively, and the vectors g ∈ Y n , v ∈ Z m , and B1 u = Au − gG(u) = f, D(B1 ) ={u : u ∈ D(A), Γ u = vΨ (u) + wF(Au)},

(4)

where now G, Ψ and F are vectors of linear bounded functionals on X A and Y , respectively, and the vectors g ∈ Y n , v ∈ Z m , w ∈ Z l . The proposed method is easily programmable to any Computer Algebra System and can be handled by scientists and practitioners from different disciplines. We apply this perturbation method to solve partial integro-differential, or loaded differential, equations with nonlocal, or integral, boundary conditions.

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The rest of the paper is organized as follows: In Section 2 we give some definitions and preliminary results. In Section 3 we present the perturbation technique for constructing closed form solutions to perturbed boundary value problems. Finally, in Section 4, two applications are included to illustrate the implementation and efficiency of the abstract theory explicated in Section 3.

2 Preliminaries Let X, Y be complex Banach spaces, A : X → Y an operator and D(A) and R(A) the domain and the range of A, respectively. The following definitions are useful in the sequel. Definition 1: An operator A2 is said to be an extension of an operator A1 , or A1 is said to be a restriction of A2 , in symbol A1 ⊂ A2 , if D(A2 ) ⊇ D(A1 ) and A1 x = A2 x for all x ∈ D(A1 ). Definition 2: An operator A : X → Y is called closed if for every sequence xn in D(A) converging to x0 with Axn → f 0 , it follows that x0 ∈ D(A) and Ax0 = f 0 . Definition 3: An operator A is called maximal if R(A) = Y and ker A = {0}.  : X → Y is called correct if R( A)  = Y and the inverse Definition 4: An operator A −1 exists and is bounded on Y . A  is called a correct restriction of the maximal operator Definition 5: An operator A  ⊂ A. A if it is a correct operator and A  : X → Y is called injective and everywhere solvable Definition 6: An operator A  = Y and the inverse A −1 exists. if R( A) Denote by X ∗ the adjoint space of X , i.e. the set of all complex-valued linear and bounded functionals on X . Consider the functionals Ψi ∈ X ∗ , i = 1, . . . , m and the vector Ψ =col(Ψ1 , . . . , Ψm ) with Ψ (x) =col(Ψ1 (x), . . . , Ψm (x)). Let g = (g1 , . . . , gn ) be a vector of X n . Then Ψ (g) is the m × n matrix whose i, jth entry Ψi (g j ) is the value of the functional Ψi on element g j . Note that Ψ (gC) = Ψ (g)C, where C is a n × l constant matrix. We designate by 0ln the l × n zero matrix, In the n × n identity matrix, c a constant column vector and 0 the zero column vector.

3 Main Results Let now X, Y, Z be complex Banach spaces, A : X → Y a linear maximal closed  a correct restriction of A defined by operator and A  = Au, Au on

 = {u : u ∈ D(A), Γ u = 0} = ker Γ, D( A)

where Γ : X A → Z is a linear bounded boundary operator and

(5)

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  X A = D(A), || · || X A , ||x|| X A = ||x|| X + ||Ax||Y , ∀x ∈ D(A).

(6)

Note that the space X A is a Banach space if A is a closed operator. Further, let Γ be a restriction of Γ defined by Γu = Γ u,

D(Γ) = ker A,

(7)

where we mention that ker A is a Banach space in the induced topology of X . We prove the next lemma initiated in [5]. on Lemma 1 Let the operator Γ be defined by (7). Then Γ : ker A → Z is bounded on and injective and has a bounded inverse Γ−1 : Z → ker A, i.e. Γ is correct.

Proof By assumption the operator Γ is bounded from X A onto Z , i.e. there exists a constant c > 0 not depending on x ∈ X A such that ||Γ x|| Z ≤ c||x|| X A = c(||x|| X + ||Ax||Y ), ∀x ∈ D(A).

(8)

For every x0 ∈ ker A, we have ||Γ x0 || Z = ||Γx0 || Z ≤ c||x0 || X . Hence, the operator Γ is bounded on ker A. Moreover, Γ is closed because ker A is closed. From the  and the decomposition [4], condition ker Γ = D( A)  ⊕ ker A, D(A) = D( A)

(9)

it follows that ker Γ ∩ ker A = {0} and the operator Γ is injective. From (9) it is implied that Z = R(Γ ) = Γ D(A) = Γ ker A = Γ ker A, thus Z = R(Γ) and the domain of Γ−1 is the whole of Z . Since Γ−1 is closed, because Γ is closed, D(Γ−1 ) = Z and Z is a Banach space, then by the Closed-Graph Theorem the operator Γ−1 is bounded. The lemma has been proved. We first study perturbed boundary value problems as in (3) and prove the following theorem for examining the existence and uniqueness of their solutions and obtaining them in closed form. Theorem 1 Let X, Y, Z be complex Banach spaces, A : X → Y a linear maximal  a correct restriction of A defined by (5). Let X A as in (6), closed operator and A on Γ : X A → Z be a bounded boundary operator and Γ a restriction of Γ defined by (7). Consider the functionals Fi ∈ Y ∗ , i = 1, . . . , n, Ψ j ∈ X ∗A , j = 1, . . . , m, and the vectors F = col(F1 , . . . , Fn ), Ψ = col(Ψ1 , . . . , Ψm ) and suppose that the components of vectors g = (g1 , . . . , gn ) ∈ Y n and v = (v1 , . . . , vm ) ∈ Z m are linearly independent. Then: (i) The operator B : X → Y defined by Bu = Au − g F(Au) = f,

f ∈ Y,

D(B) = {u : u ∈ D(A), Γ u = vΨ (u)},

(10)

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is injective if and only if det W = det [In − F(g)] = 0, and   det V = det Im − Ψ (Γ−1 v) = 0,

(11)

where In , Im designate respectively the n × n and m × m identity matrices. (ii) If (i) is true then B is correct and for all f ∈ Y the unique solution of (10) is obtained as follows  −1  −1 f + A  g + Γ−1 vV −1 Ψ ( A −1 g) W −1 F( f ) u = B −1 f = A −1 f ). + Γ−1 vV −1 Ψ ( A

(12)

Proof (i) Let det W = 0, det V = 0 and u ∈ ker B. Then from (10) we have in succession Bu = Au − g F(Au) = 0, F(Au − g F(Au)) = F(0), [In − F(g)] F(Au) = 0.

(13)

Since det W = det [In − F(g)] = 0, it follows from (13) that F(Au) = 0 and therefore Bu = Au = 0 and consequently u ∈ ker A. Then by means of (7) and (10), we have Γ u = Γu = vΨ (u),   Γ u − Γ−1 vΨ (u) = 0.

(14)

By Lemma (1) and acting with Ψ , we get u − Γ−1 vΨ (u) = 0,   Ψ u − Γ−1 vΨ (u) = Ψ (0),   Im − Ψ (Γ−1 v) Ψ (u) = 0.

(15)

Because det V = det [Im − Ψ (Γ−1 v)] = 0, it follows from (15) that Ψ (u) = 0 and therefore u = 0. That is ker B = {0} and hence B is an injective operator. Conversely: Let det V = 0. Then there exists a vector c = col(c1 , . . . , cm ) = 0 such that V c = 0. Take the element u 0 = Γ−1 vc and note that u 0 ∈ ker A ⊂ D(A) and u 0 = 0, otherwise c = 0 since v1 , . . . , vm are linearly independent. Moreover, u 0 ∈ D(B) since Γ (u 0 ) = vc, Ψ (u 0 ) = Ψ (Γ−1 v)c and Γ (u 0 ) − vΨ (u 0 ) = vc − vΨ (Γ−1 v)c = v[Im − Ψ (Γ−1 v)]c = vV c = 0.

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Then from (10) it is implied that u 0 ∈ ker B and hence ker B = {0} meaning that B is not injective. Let now det V = 0, but det W = 0. Then there exists a vector of constants c = col(c1 , . . . , cn ) = 0 such that W c = 0. Note that gc = 0, because the vector components g1 , . . . , gn are linearly independent, and that the element −1 g + Γ−1 vV −1 Ψ ( A −1 g))c = 0, otherwise g = 0. Moreover, u 0 ∈ D(B) u0 = ( A since −1 g)c − vΨ ( A −1 g)c − vΨ (Γ−1 v)V −1 Ψ ( A −1 g)c Γ (u 0 ) − vΨ (u 0 ) = vV −1 Ψ ( A −1 g)c − vΨ ( A −1 g)c = v[Im − Ψ (Γ−1 v)]V −1 Ψ ( A   −1 g) − vΨ ( A −1 g) c = 0, = vΨ ( A (16) and Bu 0 = Au 0 − g F(Au 0 ) = gc − g F(g)c = g[In − F(g)]c = gW c = 0.

(17)

Thus u 0 ∈ ker B, ker B = {0} and B is not injective. The statement (i) holds. (ii) Let det W = 0 and det V = 0. By applying the vector F on (10), we obtain [In − F(g)] F(Au) = F( f ), F(Au) = W −1 F( f ).

(18)

By discerning that for every v ∈ Z m the element Γ−1 vΨ (u) ∈ ker A and Γ ⊂ Γ , the problem (10) may be written as   Bu = A u − Γ−1 vΨ (u) − g F(Au) = f, f ∈ Y, D(B) = {u : u ∈ D(A), Γ (u − Γ−1 vΨ (u)) = 0}.

(19)

 and  ⊂ A, we have u − Γ−1 vΨ (u) ∈ D( A) Since (5) and A    u − Γ−1 vΨ (u) − g F(Au) = f, u ∈ D(B), Bu = A

f ∈ Y.

(20)

 and thus applying the inverse A −1 on both sides of (20), Utilizing the correctness of A we get −1 g F(Au) + A −1 f. (21) u − Γ−1 vΨ (u) = A Then acting by the vector Ψ and using (18), we obtain

Exact Solutions to Problems with Perturbed Differential and Boundary Operators



−1 g)F(Au) + Ψ ( A −1 f ), Ψ (u) − Ψ (Γ−1 v)Ψ (u) = Ψ ( A  −1 g)W −1 F( f ) + Ψ ( A −1 f ), Im − Ψ (Γ−1 v) Ψ (u) = Ψ ( A   −1 g)W −1 F( f ) + Ψ ( A −1 f ) . Ψ (u) = V −1 Ψ ( A

307

(22)

Substituting into (21), we have −1 f + A −1 gW −1 F( f ) u = B −1 f = A   −1 g)W −1 F( f ) + Ψ ( A −1 f ) , +Γ−1 vV −1 Ψ ( A

(23)

and hence the unique solution (12) to the problem (10). Because f in (12) is arbitrary, −1 , Γ−1 and the it is implied that R(B) = Y . Moreover, because the operators A functionals F1 , . . . , Fn , Ψ1 , . . . , Ψm are bounded, it is derived that B −1 is bounded. That is, the operator B is correct if and only if (11) holds and the unique solution of (10) is given by (12). The theorem is proved.  Remark 1 In the case where Γ u = col(Γ1 u, . . . , Γk u), with the boundary operators on Γi : X A → Z , i = 1, . . . , k, the elements of v = (v1 , . . . , vm ) are column vectors v j = col(v1 j , . . . , vk j ), j = 1, . . . , m, and in matrix form we have ⎛

⎞ ⎛ Γ1 u v11 · · · ⎜ .. ⎟ ⎜ .. ⎝ . ⎠ = ⎝ . ··· vk1 · · · Γk u

⎞⎛ ⎞ v1m Ψ1 (u) .. ⎟ ⎜ .. ⎟ . . ⎠⎝ . ⎠ vkm

(24)

Ψm (u)

Boundary value problems of the type in (4) contain drastic perturbations of their boundary conditions and therefore they are more difficult to solve. The theorem that follows elaborates in the construction of their exact unique solutions. Theorem 2 Let X, Y, Z be complex Banach spaces, A : X → Y be a linear max a correct restriction of A defined by (5). Further, let imal closed operator and A on X A as in (6), Γ : X A → Z be a linear bounded boundary operator and Γ a restriction of Γ defined by (7). Consider the functionals G i ∈ X ∗A , i = 1, . . . , n, Ψ j ∈ X ∗A , j = 1, . . . , m, Fk ∈ Y ∗ , k = 1, . . . , l, and the vectors G = col(G 1 , . . . , G n ), Ψ = col(Ψ1 , . . . , Ψm ), F = col(F1 , . . . , Fl ). Suppose that the components of the vectors g = (g1 , . . . , gn ) ∈ Y n and v = (v1 , . . . , vm ) ∈ Z m are linearly independent and that w = (w1 , . . . , wl ) ∈ Z l . Then: (i) The operator B1 : X → Y defined by B1 u = Au − gG(u) = f, D(B1 ) ={u : u ∈ D(A), Γ u = vΨ (u) + wF(Au)}, is injective if and only if

(25)

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⎛

⎞ −1 g) Ψ (Γ−1 w) Ψ (Γ−1 v) − Im Ψ ( A −1 g) − In G(Γ−1 w) ⎠ = 0. det L 1 = det ⎝ G(Γ−1 v) G( A F(g) −Il 0lm

(26)

(ii) If (i) holds then B is correct and for every f ∈ Y the unique solution of (25) is given by ⎛ ⎞ −1 f ) Ψ(A   −1 f − Γ−1 v, A −1 g, Γ−1 w L −1 ⎝ G( A −1 f ) ⎠ . u = B1−1 f = A 1 F( f )

(27)

Proof (i) Let det L 1 = 0 and u ∈ ker B1 . Then B1 u = Au − gG(u) = 0,

(28)

Γ u = vΨ (u) + wF(Au),   Γ u − Γ−1 vΨ (u) − Γ−1 wF(Au) = 0,

(29)

and sequentially

by Lemma 1 and observing that Γ−1 vΨ (u), Γ−1 wF(Au) ∈ ker A. From (5) and (29)  Hence, follows that u − Γ−1 vΨ (u) − Γ−1 wF(Au) ∈ D( A). B1 u = Au − gG(u)   = A u − Γ−1 vΨ (u) − Γ−1 wF(Au) − gG(u)   −1 gG(u) = 0,  u − Γ−1 vΨ (u) − Γ−1 wF(Au) − A =A and as a result

−1 gG(u). u = Γ−1 vΨ (u) + Γ−1 wF(Au) + A

(30)

(31)

Acting by the vectors Ψ , G on (31) and by the vector F on (28), we obtain [Im − Ψ (Γ−1 v)]Ψ (u) − Ψ (A−1 g)G(u) − Ψ (Γ−1 w)F(Au) = 0, −1 g)]G(u) − G(Γ−1 w)F(Au) = 0, − G(Γ−1 v)Ψ (u) + [In − G( A F(g)G(u) − F(Au) = 0, or in matrix form ⎛ ⎞⎛ ⎛ ⎞ ⎞ −1 g) Ψ (Γ−1 w) Ψ (Γ−1 v) − Im Ψ ( A Ψ (u) Ψ (u) ⎝ G(Γ−1 v) G( A −1 g) − In G(Γ−1 w) ⎠ ⎝ G(u) ⎠ = L 1 ⎝ G(u) ⎠ = 0. F(Au) F(Au) F(g) −Il 0lm (32)

Exact Solutions to Problems with Perturbed Differential and Boundary Operators

309

Since by assumption det L 1 = 0, it is implied that Ψ (u) = 0, G(u) = 0 and F(Au) = 0, and as a consequence from (28) and (29) follows that B1 u = Au = 0  and u = 0. Therefore ker B1 = {0} and and Γ u = 0, respectively. Thus u ∈ D( A) B1 is an injective operator. Conversely: Let ⎛

⎞ −1 g) Ψ (Γ−1 w) Ψ (Γ−1 v) − Im Ψ ( A −1 g) − In G(Γ−1 w) ⎠ = 0. det L 1 = det ⎝ G(Γ−1 v) G( A F(g) −Il 0lm

(33)

Multiplying the third column of the matrix L 1 from the right by F(g) and then adding to the second column, we get ⎛

⎞ −1 g) + Ψ (Γ−1 w)F(g) Ψ (Γ−1 w) Ψ (Γ−1 v) − Im Ψ ( A −1 g) − In + G(Γ−1 w)F(g) G(Γ−1 w) ⎠ det L 1 = det ⎝ G(Γ−1 v) G( A 0ln −Il 0lm 

−1 −1 −1  Ψ (Γ v) − Im Ψ ( A g) + Ψ (Γ w)F(g) l = (−1) det −1 g) − In + G(Γ−1 w)F(g) G(Γ−1 v) G( A = (−1)l det L = 0.

(34)

Then det L = 0 and hence there exists a nonzero vector c = col(c1 , c2 ), with c1 = col(c11 , . . . , c1m ) and c2 = col(c21 , . . . , c2n ), such that Lc = 0 or

−1 g) + Ψ (Γ−1 w)F(g) Ψ (Γ−1 v) − Im Ψ ( A −1 −1  G(Γ v) G( A g) − In + G(Γ−1 w)F(g)



c1 c2

 = 0.

(35)

−1 g + Γ−1 wF(g))c2 and notice that u 0 = 0; Take the element u 0 = Γ−1 vc1 + ( A −1 −1  Γ v, Γ−1 wF(g) ∈ ker A and D( A)  ∩ ker A =  g ∈ D( A), otherwise since A −1  gc2 = 0 and Γ−1 vc1 + Γ−1 wF(g)c2 = 0 from where it is implied {0}, we have A that c2 = 0, because the components of the vector g are linearly independent, and as a consequence Γ−1 vc1 = 0 meaning that c1 = 0, since v1 , ..., vm are linearly independent, and therefore c = col(c1 , c2 ) = 0, which contradicts the hypothesis that  c = 0. By substituting u 0 into (25), taking into account that Γ ⊂ Γ , ker Γ = D( A),  ⊂ A and using (35), we obtain A Γ u 0 − vΨ (u 0 ) − wF(Au 0 ) = vc1 + wF(g)c2 − vΨ (Γ−1 v)c1  −1   g + Γ−1 wF(g) c2 − wF(g)c2 − vΨ A   = −v Ψ (Γ−1 v) − Im c1   −1 g) + Ψ (Γ−1 w)F(g) c2 − v Ψ(A = 0. Thus u 0 ∈ D(B1 ). Analogously, we have

(36)

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B1 u 0 = Au 0 − gG(u 0 )  −1   g + Γ−1 wF(g) c2 = gc2 − gG(Γ−1 v)c1 − gG A     −1 g) − In + G(Γ−1 w)F(g) c2 = −g G(Γ−1 v) c1 − g G( A = 0.

(37)

Hence u 0 ∈ ker B1 . That is ker B1 = {0} and B1 is not injective. The statement (i) holds true. (ii) Let det L 1 = 0. From (25), working in like manner as above, we get B1 u = Au − gG(u)  − Γ−1 vΨ (u) − Γ−1 wF(Au) − A −1 gG(u)] = f. = A[u

(38)

−1 and then solving for u, we have Multiplying by A −1 gG(u) + A −1 f. u = Γ−1 vΨ (u) + Γ−1 wF(Au) + A

(39)

Acting by the vectors Ψ , G on (39) and by the vector F on (25), we acquire  Im − Ψ (Γ−1 v) Ψ (u) − Ψ (A−1 g)G(u) − Ψ (Γ−1 w)F(Au) = Ψ (A−1 f ),   −1 g) G(u) − G(Γ−1 w)F(Au) = G(A−1 f ), − G(Γ−1 v)Ψ (u) + In − G( A



− F(g)G(u) + F(Au) = F( f ). ⎛

⎞ ⎞ ⎛ −1 f ) Ψ (u) Ψ(A −1 f ) ⎠ , L 1 ⎝ G(u) ⎠ = − ⎝ G( A F(Au) F( f )

or in matrix form

(40)

which, since det L 1 = 0, yields ⎛

⎛ ⎞ ⎞ −1 f ) Ψ (u) Ψ(A ⎝ G(u) ⎠ = −L −1 ⎝ G( A −1 f ) ⎠ . 1 F(Au) F( f )

(41)

Writing (39) in the matrix form −1

u=A

⎛ ⎞  −1  Ψ (u) −1 −1  g, Γ w ⎝ G(u) ⎠ , f + Γ v, A F(Au)

(42)

and substituting (41) into (42), we obtain (27). Notice that (27) holds for every f ∈ Y −1 , Γ−1 and the functionals and therefore R(B1 ) = Y. Moreover, the operators A G 1 , . . . , G n , Ψ1 , . . . , Ψm and F1 , . . . , Fl are bounded and therefore B1−1 is bounded

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311

too. Hence, the operator B1 is correct if and only if (26) holds and the unique solution to the problem (25) is provided by (27). The proof is completed.  on

Remark 2 When Γ u = col(Γ1 u, . . . , Γk u), where Γi : X A → Z , i = 1, . . . , k, the elements of vectors v = (v1 , . . . , vm ) and w = (w1 , . . . , wl ) are column vectors v j = col(v1 j , . . . , vk j ), j = 1, . . . , m and w j = col(w1 j , . . . , wk j ), j = 1, . . . , l, respectively, and in matrix form we write ⎞ ⎛ v11 · · · Γ1 u ⎜ .. ⎟ ⎜ .. = ⎝ . ⎠ ⎝ . ··· vk1 · · · Γk u ⎛

⎞⎛ ⎞ ⎛ v1m ψ1 (u) w11 · · · .. ⎟ ⎜ .. ⎟ + ⎜ .. . ⎠⎝ . ⎠ ⎝ . ··· vkm wk1 · · · ψm (u)

⎞⎛ ⎞ w1l F1 (Au) ⎟ .. ⎟ ⎜ .. ⎠. . ⎠⎝ . wkl

(43)

Fl (Au)

The analysis above and the results presented so far demand the operator A be closed and the operator Γ be linear and bounded. These two requirements are necessary for the correctness of the operator Γ and hence the operators B and B1 . However,  Γ to be injective and everywhere solvable. Thus, for most applications it suffices A,  we can allow A : X → Y be linear maximal, generally not closed operator, and A an injective and everywhere solvable restriction of A defined by  = Au, Au

 = {u : u ∈ D(A), Γ u = 0}, D( A)

(44)

where the linear boundary operator on Γ : X¯ A → Z ,

  X¯ A = D(A), || · || X¯ A ,

(45)

and X¯ A is a Banach subspace of X . Note that for any maximal closed operator A, we can take X¯ A = X A with the graph norm ||u|| X A = ||u|| X + ||Au||Y . In the special case wherein A = A2 A1 , A1 : X → S , A2 : S → Y , where S is a Banach space, we can take the norm ||u|| X¯ A = ||u|| X + ||A1 u|| S + ||A2 A1 u||Y , which is useful if the functionals contain derivatives of the unknown function. Accordingly, Lemma 1 and Theorems 1, 2 can be made less formal by stating the following corresponding Lemma and two Theorems. Lemma 2 Let the operator Γ be defined by (45) and Γ be a restriction of Γ to on on ker A. Then Γ : ker A → Z is injective and Γ−1 : Z → ker A, i.e. Γ is injective and everywhere solvable. Theorem 3 Let X, Y, Z be complex Banach spaces, A : X → Y a linear maximal,  is an injective and everywhere solvable restricgenerally not closed operator, and A tion of A defined by (44). Moreover, let X¯ A and the operator Γ as in (45) and Γ as in Lemma 2. Consider the functionals Fi ∈ Y ∗ , i = 1, . . . , n, Ψ j ∈ X ∗A , j = 1, . . . , m, and the vectors F = col(F1 , . . . , Fn ), Ψ = col(Ψ1 , . . . , Ψm ) and suppose that the components of vectors g = (g1 , . . . , gn ) ∈ Y n and v = (v1 , . . . , vm ) ∈ Z m are linearly independent. Then:

312

I. N. Parasidis and E. Providas

(i) The operator B : X → Y defined by Bu = Au − g F(Au) = f, f ∈ Y, D(B) = {u : u ∈ D(A), Γ u = vΨ (u)},

(46)

is injective and everywhere solvable if and only if (11) holds. (ii) If (i) is true then for all f ∈ Y the unique solution of (46) is given by (12). −1 and Γ−1 are bounded on the whole Y (iii) If in addition to (ii), the operators A  and Γ are correct and hence B is correct whereas and Z , respectively, then A for all f ∈ Y the unique solution of ( 46) is provided by (12). Theorem 4 Let X, Y, Z be complex Banach spaces, A : X → Y a linear maxi is an injective and everywhere solvmal, generally not closed operator, and A able restriction of A defined by (44). Moreover, let X¯ A and the operator Γ as in (45) and Γ as in Lemma 2. Consider the functionals G i ∈ X ∗A , i = 1, . . . , n, Ψ j ∈ X ∗A , j = 1, . . . , m, Fk ∈ Y ∗ , k = 1, . . . , l, and the vectors G = col(G 1 , . . . , G n ), Ψ = col(Ψ1 , . . . , Ψm ), F = col(F1 , . . . , Fl ). Suppose that the components of the vectors g = (g1 , . . . , gn ) ∈ Y n and v = (v1 , . . . , vm ) ∈ Z m are linearly independent and that w = (w1 , . . . , wl ) ∈ Z l . Then: (i) The operator B1 : X → Y defined by B1 u = Au − gG(u) = f, D(B1 ) ={u : u ∈ D(A), Γ u = vΨ (u) + wF(Au)},

(47)

is injective and everywhere solvable if and only if (26) is true. (ii) If (i) holds then for every f ∈ Y the unique solution of (47) is given by (27). −1 and Γ−1 are bounded on the whole Y (iii) If in addition to (ii), the operators A   and Z , respectively, then A and Γ are correct and hence B1 is correct, while for all f ∈ Y the unique solution of (47) is given by (27). The proofs of Lemma 2 and Theorems 3, 4 are analogous to proofs of their counterparts Lemma 1 and Theorems 1, 2, respectively.

4 Application to Perturbed Partial Differential Problems We apply the method developed in Section 3 to boundary value problems for perturbed partial differential equations with nonlocal or integral boundary conditions. Problem 1 We consider first the following boundary value problem for a partial integro-differential equation of Fredholm type with nonlocal perturbations of its boundary conditions, viz.

Exact Solutions to Problems with Perturbed Differential and Boundary Operators

u x y − (x + y)

11 0

0

yu x y (x, y)d xd y = 3x 2 − 21 (x + y),

313

0 ≤ x, y ≤ 1,

u x (x, 0) = xu(1, 1),   u(0, y) = 152 (5y + 4)u 21 , 21 .

(48)

¯ → C(Ω) ¯ be defined as Let Ω¯ = [0, 1] × [0, 1], the maximal operator A : C(Ω) Au = u x y ,

¯ ¯ : u x , u x y ∈ C(Ω)}, D(A) = {u : u ∈ C(Ω)

(49)

and 

1

F(Au) = 0



1

yu x y (x, y)d xd y,

0

g = x + y, 1 f = 3x 2 − (x + y). 2

(50)

In addition,

Γ u(x, y) =

u x (x, 0) u(0, y)







1 1 0 x u( , ), = u(1, 1) + 2 0 (5y + 4) 2 2 15

(51)

  where X A = D(A), || · || X A , ||u|| X A = ||u||C + ||u x y ||C , Z = R(Γ ) = C[0, 1] ⊗ C[0, 1] and for every h = col(h 1 (x), h 2 (y)) ∈ Z the norm ||h|| Z = ||h 1 (x)||C + ||h 2 (y)||C . From (51), we get

v = (v1 , v2 ) =

Ψ1 (u) Ψ (u) = Ψ2 (u)

x 0 

 0 , 2 (5y + 4) 15 

u(1, 1) . = u( 21 , 21 )

(52)

The boundary value problem (48) can be now formulated as in (46) where the operator ¯ → C(Ω). ¯ B : C(Ω) The unperturbed problem is defined by  = u x y (x, y) = f, Au

 = {u : u ∈ D(A), Γ u = 0}, D( A)

(53)

 ⊂ A is correct, and its unique solution is given by the formula where A −1 f = u=A



x 0



y

¯ f (t, s)dsdt, for all f ∈ C(Ω).

(54)

0

The boundary operator Γ ⊂ Γ by Lemma 2 is injective and everywhere solvable and for every h = col(h 1 (x), h 2 (y)) ∈ Z the problem

314

I. N. Parasidis and E. Providas

Γu = h =

 h 1 (x) , h 2 (y)

(55)

which is equivalent to the problem u x y = 0, u x (x, 0) = h 1 (x), u(0, y) = h 2 (y),

(56)

admits the unique solution u = Γ−1 h =



x

h 1 (s)ds + h 2 (y).

(57)

0

Therefore,

  Γ−1 v = Γ−1 v1 , Γ−1 v2 =

 x2 2 , (5y + 4) . 2 15

(58)

Moreover, the operator Γ−1 is bounded on Z , since for every h ∈ Z from (57) we have ||Γ−1 h||C ≤ ||h 1 (x)||C + ||h 2 (y)||C = ||h|| Z , and so Γ is correct. The functionals F, Ψ1 , Ψ2 are bounded too. We compute 

1



1

7 , 12 0 0  1

−1 v1 ) Ψ1 (Γ−1 v2 ) Ψ ( Γ 1 −1 = 21 Ψ (Γ v) = Ψ2 (Γ−1 v1 ) Ψ2 (Γ−1 v2 ) F(g) =

yg(x, y)d xd y =

18 15 13 8 15

 ,

(59)

and construct the matrices 5 , 12    V = I2 − Ψ (Γ−1 v) =

W = [I1 − F(g)] =

1 2

− 18 15

− 18

2 15

 .

(60)

5 17 Since det W = 12 = 0, det V = 60 = 0, it follows from Theorem 3, case (iii), that the problem (48) is correct and the solution is given by (12). By first computing



x



y

1 x y(4x 2 − x − y), 4 0 0  x y 1 −1  A g= g(t, s)dsdt = yx(x + y), 2 0 0  1 1 5 F( f ) = , y f (x, y)d xd y = 24 0 0

−1 f = A

f (t, s)dsdt =

Exact Solutions to Problems with Perturbed Differential and Boundary Operators

 1 −1 f ) Ψ1 ( A 2 −1 f ) = 0 , Ψ2 ( A  

−1 −1 g) = Ψ1 ( A−1 g) = 11 , Ψ(A  g) Ψ2 ( A 8

−1 f ) = Ψ(A

315

(61)

and then substituting into (12) we get the solution u(x, y) =

1 (4x 3 y − 5x 2 − 5y − 4). 4

(62)

Problem 2 As a second example we contemplate a partial differential equation perturbed with a load, namely − u(x) − g(x)u(x0 ) = f (x), x ∈ Ω,

(63)

subject to an integral perturbation of the boundary condition  u(x)|∂Ω = v(t)

 Ω

u(x)d x − w(t)

Ω

φ(x) u(x)d x, t ∈ ∂Ω,

(64)

where Ω = {x : x ∈ R3 , |x| < 1}, ∂Ω = {x : x ∈ R3 , |x| = 1}, x0 ∈ Ω and f (x), φ(x) ∈ L 2 (Ω) are given. Examine for which functions g(x), v(t), w(t) the problem is everywhere solvable and find its unique solution. We put Au = − u, D(A) = X¯ A = W22 (Ω),  = Au, D( A)  = {u : u ∈ D(A), u|∂Ω = 0}, Au Γ u = u|∂Ω = u(t), t ∈ ∂Ω, Γu = Γ u, D(Γ) = {u : u ∈ D(A), u = 0},   G(u) = u(x0 ), Ψ (u) = u(x)d x, F(Au) = − φ(x) u(x)d x. Ω

(65)

Ω

Note that Γ u(x) is the trace of u(x) on the sphere |x| = 1 and by the embedding 3/2 theorem for each u(x) ∈ W22 (Ω) follows that Γ u(x) = u(t) ∈ W2 (∂Ω). Then Z = 3/2 R(Γ ) = W2 (∂Ω). Thus, according to Theorem 4, g(x) ∈ L 2 (Ω) and v(t), w(t) ∈ Z . The problem (63), (64) can be now expressed as in (47) where B : L 2 (Ω) → L 2 (Ω). The unperturbed problem − u(x) = f (x), u(x) ∈ W22 (Ω), u(x)|∂Ω = 0,

f (x) ∈ L 2 (Ω), (66)

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I. N. Parasidis and E. Providas

is the well known Dirichlet problem for the Poisson equation and it is known that  corresponding to (66) is injective and everywhere solvable and the the operator A, unique solution of (66) is given by −1 f (x) = u(x) = A

 Ω

G (x, y) f (y)dy, ∀ f ∈ L 2 (Ω),

(67)

where G (x, y) is the Green’s function. The problem Γu = h(t), h(t) ∈ Z is equivalent to

u(x) = 0, u(x)|∂Ω = h(t), u(x) ∈ W22 (Ω), h(t) ∈ Z , (68) which is the Dirichlet problem for the Laplace equation and it is known that the operator Γ, corresponding to (68) is injective and everywhere solvable and the unique solution of (68) is given by u(x) = Γ−1 h(t) =

 P(x, t)h(t)dt, ∀ h(t) ∈ Z ,

∂Ω

(69)

¯ where P(x, t) is the Poisson’s kernel. Notice that F ∈ L ∗2 (Ω) and Ψ , G ∈ C ∗ (Ω). 2 ¯ Because of the embedding W2 (Ω) ⊂ C(Ω), we can suppose that for each u(x) ∈ ¯ and there exist the embedding [C(Ω)] ¯ ∗ ⊂ [W22 (Ω)]∗ . Then W22 (Ω) holds u(x) ∈ C(Ω) 2 ∗ ∗ Ψ , G ∈ [W2 (Ω)] = X A . Now Theorem 4 can be applied. For any g(x) ∈ L 2 (Ω) and v(t), w(t) ∈ Z , we find  −1  G (x, y)g(y)dy, A g= Ω Γ−1 v = P(x, t)v(t)dt, ∂Ω  Γ−1 w = P(x, t)w(t)dt, (70) ∂Ω

and then −1 g) = G( A G(Γ−1 v) = G(Γ−1 w) =

 Ω 

G (x0 , y)g(y)dy,

∂Ω

P(x0 , t)v(t)dt,

P(x0 , t)w(t)dt, ∂Ω −1 g) = Ψ(A G (x, y)g(y)d yd x, Ω Ω Ψ (Γ−1 v) = P(x, t)v(t)dtd x, Ω

∂Ω

Exact Solutions to Problems with Perturbed Differential and Boundary Operators

Ψ (Γ−1 w) = F(g) =

317

  Ω Ω

∂Ω

P(x, t)w(t)dtd x,

φ(x)g(x)d x.

(71)

Substitution into (26) yields det L 1 . If and only if det L 1 = 0, then the problem (63), (64) has a unique solution which can be delivered in closed form by (27).

References 1. Dhage, B.C.: Quadratic perturbations of periodic boundary value problems of second ordinary differential equations. Differ. Equ. Appl. 2(4), 465–486 (2010) 2. Kato, T.: On the perturbation theory of closed linear operators. J. Math. Soc. Jpn. 4(3–4), 323–337 (1952) 3. Kato, T.: Perturbation Theory for Linear Operators, Springer, Berlin (1995). https://doi.org/10. 1007/978-3-642-66282-9 4. Kokebaev, B.K., Otelbaev, M., Shynybekov, A.N.: About restrictions and extensions of operators. D.A.N. SSSR 271, 1307–1310 (1983) [Russian] 5. Oinarov, R.O., Parasidis, I.N.: Correct extensions of operators with finite defect in Banach spaces. Izv. Akad. Kaz. SSR 5, 42–46 (1988) [Russian] 6. Parasidis, I.N., Providas, E.: Extension operator method for the exact solution of integrodifferential equations. In: Pardalos, P.M., Rassias T.M. (eds.) Contributions in Mathematics and Engineering, pp. 473–496. Springer, Berlin (2016). https://doi.org/10.1007/978-3-319-313177 7. Parasidis, I.N., Providas, E.: On the exact solution of nonlinear integro-differential equations: In: Rassias, T.M. (ed.) Applications of Nonlinear Analysis, pp. 591–609. Springer, Cham (2018).https://doi.org/10.1007/978-3-319-89815-5 8. Sadybekov, M.A., Imanbaev, N.S.: A regular differential operator with perturbed boundary condition. Math. Notes 101(5), 878–887 (2017). https://doi.org/10.1134/S0001434617050133

On a Few Equivalent Statements of a Hilbert-Type Integral Inequality in the Whole Plane with the Hurwitz Zeta Function Themistocles M. Rassias and Bicheng Yang

On the occasion of the 100th birthday of Tosio Kato

Abstract In the present paper, we prove some equivalent statements of a Hilberttype integral inequality in the whole plane with intermediate variables. In our theorems, the constant factor is associated to the Hurwitz zeta function and we prove that it is the best possible. We also derive various special cases and applications. Mathematics Subject Classification 26D15 · 47A07

1 Introduction 

If 0
1, 1p + q1 = 1, f (x), g(y) ≥ 0, 

∞

0
1, 1p + q1 = 1, h(u) > 0,  φ(σ ) =

∞

h(u)u σ −1 du ∈ R+ ,

0

then 

∞



0

1 < φ( ) p

∞

h(x y) f (x)g(y)d xd y

0



∞

x 0

p−2

f (x)d x p

 1p 

∞

g (y)dy q

 q1

,

(3)

0

where the constant factor φ( 1p ) is the best possible (cf. [3], Theorem 350). In 1998, by introducing an independent parameter λ > 0, Yang proved an extension of (1) with the kernel 1 , (x + y)λ (cf. [4, 5]).

On a Few Equivalent Statements of a Hilbert-Type Integral …

321

In 2004, by introducing another pair of conjugate exponents (r, s) (r > 1,

1 1 + = 1), r s

Yang [6] obtained an extension of (2) with the kernel xλ

1 (λ > 0). + yλ

One year later, Yang et al. [7] extended also (2) and the result of [4] with the kernel 1 (λ > 0). (x + y)λ Krnic´ et al. [8–16] obtained some extensions of (1), (2) and (3). Kato, in various works [17–21], proved various other very interesting types of inequalities and operators. In 2009, Yang proved the following extension of (2) (cf. [22, 23]): If λ1 + λ2 = λ ∈ R = (−∞, ∞), kλ (x, y) is a nonnegative homogeneous function of degree −λ, satisfying kλ (ux, uy) = u −λ kλ (x, y)(u, x, y > 0), 

∞

k(λ1 ) =

kλ (u, 1)u λ1 −1 du ∈ R+ = (0, ∞),

0

then for p > 1, 

∞

1 p

+



∞

0

1 q

= 1, we have

kλ (x, y) f (x)g(y)d xd y

0



∞

< k(λ1 )

x p(1−λ1 )−1 f p (x)d x

 1p 

0

∞

y q(1−λ2 )−1 g q (y)dy

∞



0

∞

< φ(σ )

∞

x 0

(4)

1 p

+

1 q

= 1, the

h(x y) f (x)g(y)d xd y

0



,

0

where the constant factor k(λ1 ) is the best possible; for 0 < p < 1, reverse of (4) is satisfied. The following extension of (3) was given: For p > 1, 1p + q1 = 1, we have 

 q1

p(1−σ )−1

f (x)d x p

 1p 

∞

y 0

q(1−σ )−1 q

g (y)dy

 q1

,

(5)

322

T. M. Rassias and B. Yang

where the constant factor φ(σ ) is the best possible; for 0 < p < 1, 1p + q1 = 1, the reverse of (5) is satisfied (cf. [24]). In 2007, Yang [25] proved the following Hilbert-type integral inequality in the whole plane with the exponent function as the intermediate variable: 

∞



∞

f (x)g(y) d xd y (1 + e x+y )λ −∞ −∞  ∞  21  ∞ λ λ e−λx f 2 (x)d x e−λy g 2 (y)dy , < B( , ) 2 2 −∞ −∞

(6)

where the constant factor B( λ2 , λ2 ) (λ > 0, B(u, v) stands for the beta function) is proved to be the best possible. He et al. [15, 26–36] proved some new Hilbert-type integral inequalities in the whole plane with the best possible constant factors. Some inequalities equivalent to (4) and (5) have been considered in [23]. In 2013, Yang [24] also studied the equivalency between (4) and (5). In 2017, Hong [37] obtained an equivalent condition between (4) and some parameters. Several authors continue to investigate this topic for other types of integral inequalities and operators (cf. [38–43]). In the present paper, we prove some equivalent statements of a Hilbert-type integral inequality in the whole plane with intermediate variables (see Theorem 1). In our theorems, the constant factor is associated to the Hurwitz zeta function and we prove that it is the best possible (see Theorem 2). We also consider some particular cases in part of Theorem 2 and Remark 1, operator expressions in Theorems 3 and 4, as well as equivalent reverses with the best possible constant factor in Theorem 5.

2 An Example and a Few Lemmas Example 1 For γ > −1, ρ, μ, σ > 0, μ + σ = λ, setting h(u) :=

| ln ρu λ |γ (u > 0), 1 + ρu λ

we obtain (γ ) kλ (σ )



∞

:= 0

σ −1

h(u)u  ∞



∞

du = 0

| ln ρu λ |γ σ −1 u du 1 + ρu λ

| ln v|γ σ −1 1 v λ dv (v = ρu λ ) = σ/λ λρ 1 + v 0  1 μ (− ln v)γ σ −1 1 = (v λ + v λ −1 )dv λρ σ/λ 0 1+v

On a Few Equivalent Statements of a Hilbert-Type Integral …

= =

1 λρ σ/λ 1 λρ σ/λ



1

(− ln v)γ

0



∞ 

323 μ

σ

(−1)k vk (v λ −1 + v λ −1 )dv

k=0 1

(− ln v)γ

0

∞ 

σ

μ

(v2k − v2k+1 )(v λ −1 + v λ −1 )dv.

k=0

By the Lebesgue term by term theorem (cf. [44]), it follows that (γ ) kλ (σ )

∞  μ 1  1 σ = (− ln v)γ (v2k+ λ −1 + v2k+ λ )dv λρ σ/λ k=0 0  1 ∞ μ 1  σ k (−1) (− ln v)γ (vk+ λ −1 + vk+ λ −1 )dv = σ/λ λρ 0 k=0 ∞   ∞ ∞  (−1)k  (−1)k 1 −t γ e t dt + = λρ σ/λ 0 (k + σλ )γ +1 k=0 (k + μλ )γ +1 k=0 σ Γ (γ + 1)  μ ξ(γ + 1, = ) + ξ(γ + 1, ) ∈ R+ , λρ σ/λ λ λ



where Γ (s) :=

∞

(7)

e−v vs−1 dv (Re s > 0)

0

stands for the gamma function and ∞  (−1)k ξ(s, a) := (Re s > 0; a > 0) (k + a)s k=0

(cf. [45]). In particular: (1) for γ = 0, setting v = ρu λ , we have 

∞

u σ −1 du 1 = σ/λ λ 1 + ρu λρ 0 π = ; λρ σ/λ sin( πσ ) λ

kλ(0) (σ ) =

 0

∞

σ

v λ −1 dv 1+v

(2) for γ > 0, we have ∞ 

=

k=0

(−1)k (k + σλ )γ +1

∞ 

1

k=0

(k + σλ )γ +1

−2

∞  k=0

1 (2k + 1 + σλ )γ +1

(8)

324

T. M. Rassias and B. Yang

=

∞ 

1

k=0

(k + σλ )γ +1

= ζ (γ + 1, where ζ (s, a) :=

−

∞ 1 1  λ+σ γ 2 k=0 (k + 2λ )γ +1

1 σ λ+σ ) − γ ζ (γ + 1, ), λ 2 2λ

∞  (−1)k (Re s > 1; 0 < a ≤ 1) (k + a)s k=0

is the Hurwitz zeta function (cf. [45]). Then it follows that (γ )

kλ (σ ) =

 σ 1 Γ (γ + 1) λ+σ ζ (γ + 1, ) − γ ζ (γ + 1, ) λρ σ/λ λ 2 2λ  μ λ+μ 1 + ζ (γ + 1, ) − γ ζ (γ + 1, ) . λ 2 2λ

(9)

For δ ∈ {−1, 1}, α, β ∈ (−1, 1), we set xα := |x| + αx, yβ := |y| + βy (x, y ∈ R), E δ := {t ∈ R; |t|δ ≥ 1}, E −δ = {t ∈ R; |t|δ ≤ 1}. Lemma 1 For c > 0, η = α, β, we have 

1 1 1 , = + c (1 + η)cδ+1 (1 − η)cδ+1 E 

δ 1 1 1 cδ−1 ; tη dt = + c (1 + η)−cδ+1 (1 − η)−cδ+1 E −δ



tη−cδ−1 dt

for c ≤ 0, it follows that  Eδ

tη−cδ−1 dt =

 E −δ

tηcδ−1 dt = ∞.

Proof Setting E δ+ := {t ∈ R+ ;t δ ≥ 1}, E δ− := {−t ∈ R+ ; (−t)δ ≥ 1}, we have that E δ = E δ+ ∪ E δ− and  Eδ

tη−cδ−1 dt =

 E δ+

[(1 + η)t]−cδ−1 dt +

 E δ−

[(1 − η)(−t)]−cδ−1 dt

(10) (11)

On a Few Equivalent Statements of a Hilbert-Type Integral …

=

1 1 + (1 + η)cδ+1 (1 − η)cδ+1

325

 E δ+

t −cδ−1 dt.

Setting u = t δ (or t = u δ ), we derive that 1

 E δ+

t −cδ−1 dt =

1 |δ|



∞

u δ (−cδ−1) u δ −1 du = 1

1



1

∞

u −c−1 du.

1

Hence, for c > 0, (10) follows and for c ≤ 0, it holds  tη−cδ−1 dt = ∞. Eδ





Since

E −δ

tηcδ−1 dt

= E (−δ)

tη−c(−δ)−1 dt,

in view of (10), for c > 0, (11) follows, and for c ≤ 0, we get  E −δ

tηcδ−1 dt = ∞. 

This completes the proof of the lemma. In the sequel, we assume that p > 0 (= 1),

1 1 + = 1, δ ∈ {−1, 1}, α, β ∈ (−1, 1), γ > −1, p q ρ, μ, σ > 0, μ + σ = λ, σ1 ∈ R+ ,

(γ )

kλ (σ ) is as in (7), and (γ )

K α,β (σ ) :=

(γ )

2kλ (σ ) . 2 (1 − α )1/q (1 − β 2 )1/ p

(12)

For p > 1, n ∈ N = {1, 2, · · · }, E −1 = [−1, 1], x ∈ E δ , we define the following expressions: I

(−)



0



−1 1

(x) :=

| ln ρ(xαδ yβ )λ |γ σ + qn1 −1 y dy, 1 + ρ(xαδ yβ )λ β

| ln ρ(xαδ yβ )λ |γ σ + qn1 −1 y dy, δ λ β 0 1 + ρ(x α yβ )  | ln ρ(xαδ yβ )λ |γ σ + qn1 −1 I (x) := I (−) (x) + I (+) (x) = y dy. δ λ β E −1 1 + ρ(x α yβ )

I (+) (x) :=

326

T. M. Rassias and B. Yang

Since yβ = (sgn(y) + β)y, where ⎧ ⎨ −1, y < 0 0, y = 0 , sgn(y) := ⎩ 1, y > 0 it follows that xαδ = (1 + α · sgn(x))δ |x|δ ≥ min (1 ± |α|)δ (x ∈ E δ ), δ∈{−,,1}

1 − |α| ≤ (1 + |α|)−1 ≤ 1 + |α| ≤ (1 − |α|)−1 , and (1 ± β)xαδ ≥ m α,β := (1 − |β|)(1 − |α|) > 0 (x ∈ E δ ).

(13)

For fixed x ∈ E δ , setting u = xαδ yβ , we obtain that )



1 −δ(σ + qn ) xα



−δ(σ +

I

(−)

≥

I

(+)

1

qn xα (x) = 1−β

0

1−β

1 −δ(σ + qn )



1 −δ(σ + qn ) xα



xα (x) = 1+β

(1−β)xαδ

m α,β

0 (1+β)xαδ

0

| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ

| ln ρu λ |γ σ + qn1 −1 u du, 1 + ρu λ

| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ

| ln ρu λ |γ σ + qn1 −1 u du, 1+β 1 + ρu λ 0  (1−β)xαδ 1 −δ(σ + qn ) | ln ρu λ |γ σ + qn1 −1 1 I (x) = xα u du 1−β 0 1 + ρu λ   (1+β)xαδ | ln ρu λ |γ σ + qn1 −1 1 + u du 1+β 0 1 + ρu λ ≥

−δ(σ +

1

qn 2xα ≥ 1 − β2

)

m α,β



m α,β

0

| ln ρu λ |γ σ + qn1 −1 u du. 1 + ρu λ

For p > 1, n ∈ N, x ∈ Fδ , we define the following expressions: J (−) (x) := J (+) (x) :=



−1



−∞ ∞ 1

| ln ρ(xαδ yβ )λ |γ σ − qn1 −1 y dy, |1 − ρ(xαδ yβ )λ | β

| ln ρ(xαδ yβ )λ |γ σ − qn1 −1 y dy, |1 − ρ(xαδ yβ )λ | β

(14)

On a Few Equivalent Statements of a Hilbert-Type Integral …

327

J (x) := J (−) (x) + J (+) (x)  | ln ρ(xαδ yβ )λ |γ σ − qn1 −1 = y dy. δ λ β E 1 |1 − ρ(x α yβ ) | Since for x ∈ E −δ , xαδ = (1 + α · sgn(x))δ |x|δ

≤ max {(1 ± |α|)δ } = (1 − |α|)−1 , δ∈{−1,1}

we have

Mα,β := (1 + |β|)(1 − |α|)−1 ≥ (1 ± β)xαδ (x ∈ E −δ ).

(15)

For fixed x ∈ E −δ , setting u = xαδ yβ , we derive that −δ(σ −



)

1

qn xα J (−) (x) = 1−β



1 −δ(σ − qn )

≥

xα 1−β

−δ(σ −

1

qn xα J (+) (x) = 1+β

)

1 −δ(σ − qn )

 

∞ (1−β)xαδ ∞ Mα,β

∞ (1+β)xαδ

| ln ρu λ |γ σ − qn1 −1 u du 1 + ρu λ

| ln ρu λ |γ σ − qn1 −1 u du, 1 + ρu λ

| ln ρu λ |γ σ − qn1 −1 u du 1 + ρu λ

∞

| ln ρu λ |γ σ − qn1 −1 u du, λ Mα,β 1 + ρu   ∞ 1 −δ(σ − qn ) 1 | ln ρu λ |γ σ − qn1 −1 J (x) = xα u du 1 − β (1−β)xαδ 1 + ρu λ   ∞ 1 | ln ρu λ |γ σ − qn1 −1 + u du 1 + β (1+β)xαδ 1 + ρu λ ≥

xα 1+β

−δ(σ −

1

qn 2xα ≥ 1 − β2

)



∞ Mα,β

| ln ρu λ |γ σ − qn1 −1 u du. 1 + ρu λ

In view of (14) and (16), the following lemma follows: Lemma 2 For p > 1, the following inequalities hold true: 

1 δ(σ1 − pn )−1

I1 :=

I (x)xα Eδ

dx

(16)

328

T. M. Rassias and B. Yang



2 ≥ 1 − β2

−δ(σ −σ1 + n1 )−1 xα dx Eδ



J (x)xα Fδ

≥

2 1 − β2

| ln ρu λ |γ σ + qn1 −1 u du, 1 + ρu λ

(17)

| ln ρu λ |γ σ − qn1 −1 u du. 1 + ρu λ

(18)

m α,β 0

1 δ(σ1 + pn )−1

J1 :=





dx

δ(σ1 −σ + n1 )−1

xα



∞

dx Mα,β

E −δ

Lemma 3 If p > 1, there exists a constant M, such that for any nonnegative measurable functions f (x) and g(y) in R, satisfying 

∞ −∞

xαp(1−δσ1 )−1

 f (x)d x < ∞, and

∞

p

−∞

q(1−σ )−1 q

yβ

g (y)dy < ∞,

the following inequality 



∞

∞

| ln ρ(xαδ yβ )λ |γ f (x)g(y)d xd y δ λ −∞ −∞ 1 + ρ(x α yβ )

 ∞  1p  ∞  q1 q(1−σ )−1 q p(1−δσ1 )−1 p ≤ M xα f (x)d x yβ g (y)dy

I :=

−∞

−∞

holds true, then we have σ1 = σ. Proof If σ1 > σ, then for n ≥

1 σ1 −σ

(n ∈ N), we set the following functions:



1 δ(σ1 − pn )−1

xα

f n (x) :=  gn (y) :=

, x ∈ Eδ , 0, x ∈ R\E δ

σ+

1

−1

yβ qn , y ∈ E −1 . 0, y ∈ R\E −1

By (10) and (11), we obtain that J1 :=



∞ −∞



xαp(1−δσ1 )−1 f np (x)d x − nδ −1

=

xα

 1p 

= n

E −1

1 δ

(1 + α) n +1

+

∞

−∞ 1 n −1

dx

Eδ

 1p 

yβ

 q1

dy  1p

1 δ

(1 − α) n +1

q(1−σ )−1 q yβ gn (y)dy

 q1

(19)

On a Few Equivalent Statements of a Hilbert-Type Integral …

 ×

1

+

(1 + β)− n +1 1

329

 q1

1

< ∞.

(1 − β)− n +1 1

By (17) and (19) (for f = f n , g = gn ), we have 2 1 − β2  ≤ I1 =



−δ(σ −σ1 + n1 )−1 xα dx

Eδ ∞  ∞

−∞

−∞

≤ M J1 < ∞. Since for any n ≥ that

1 σ1 −σ



| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ

m α,β

0

| ln ρ(xαδ yβ )λ |γ f n (x)gn (y)d xd y 1 + ρ(xαδ yβ )λ

(n ∈ N), we have σ − σ1 + 

−δ(σ −σ1 + n1 )−1

xα

1 n

≤ 0, by Lemma 1, it follows

d x = ∞.

Eδ

In view of



m α,β 0

| ln ρu λ |γ σ + qn1 −1 u du > 0, 1 + ρu λ

we deduce that

∞ ≤ M J1 < ∞,

which is a contradiction. If σ > σ1 , then for n ≥

1 σ −σ1

(n ∈ N), we set the functions 

 f n (x) :=

1 δ(σ1 + pn )−1

xα 

, x ∈ E −δ , 0, x ∈ R\E −δ

σ−

1

−1

yβ qn , y ∈ E 1 , 0, y ∈ R\E 1

 gn (y) :=

and thus by (10) and (11), we obtain that J2 :=



∞ −∞



xαp(1−δσ1 )−1  f np (x)d x δ n −1

=

xα

 1p  dx

E −δ

= n

E1

1 δ

(1 + α)− n +1

+

 1p 

∞

−∞

− 1 −1 yβ n dy

 q1  1p

1 δ

(1 − α)− n +1

q(1−σ )−1 q yβ  gn (y)dy

 q1

330

T. M. Rassias and B. Yang

 ×

1 (1 + β) n +1 1

+

 q1

1

< ∞.

(1 − β) n +1 1

gn ), we have In view of (18) and (19) (for f =  fn , g =  2 1 − β2  ≤ J1 =



δ(σ1 −σ + n1 )−1

xα

≤ M J2 < ∞. 1 σ −σ1

∞ Mα,β

E −δ ∞  ∞

−∞

Since for n ≥

 dx

| ln ρ(xαδ yβ )λ |γ δ λ −∞ 1 + ρ(x α yβ )

 gn (y)d xd y f n (x)

(n ∈ N), we have σ1 − σ + 

δ(σ1 −σ + n1 )−1

xα

| ln ρu λ |γ σ − qn1 −1 u du 1 + ρu λ

1 n

≤ 0, by Lemma 1, it follows that

d x = ∞.

E −δ

In view of



∞ Mα,β

we have

| ln ρu λ |γ σ − qn1 −1 u du > 0, 1 + ρu λ ∞ ≤ M J2 < ∞,

which is a contradiction. Hence, we conclude that σ1 = σ. This completes the proof of the lemma. For σ1 = σ, the following lemma also holds true: Lemma 4 If p > 1, there exists a constant M, such that for any nonnegative measurable functions f (x) and g(y) in R, satisfying 

∞

−∞

xαp(1−δσ )−1

 f (x)d x < ∞, and p

∞ −∞

q(1−σ )−1 q

yβ

g (y)dy < ∞,

the following inequality 

∞



∞

| ln ρ(xαδ yβ )λ |γ f (x)g(y)d xd y δ λ −∞ −∞ 1 + ρ(x α yβ )

 ∞  1p  ∞  q1 q(1−σ )−1 q p(1−δσ )−1 p ≤M xα f (x)d x yβ g (y)dy −∞

−∞

(γ )

holds true, then we have K α,β (σ ) ≤ M.

(20)

On a Few Equivalent Statements of a Hilbert-Type Integral …

331

Proof By (14), we have 

1 δ(σ − pn )−1

I1 =

I (x)xα

dx =

Eδ

L (−) + L (+) K (−) + K (+) + , 1−β 1+β

where L L

(+)

(−)

 :=  :=

K (+) := K (−) :=

 

E δ+

− δ −1 xα n



(1−β)xαδ

| ln ρu λ |γ σ + qn1 −1 u dud x, 1 + ρu λ

(1+β)xαδ

| ln ρu λ |γ σ + qn1 −1 u dud x, 1 + ρu λ

(1+β)xαδ

| ln ρu λ |γ σ + qn1 −1 u dud x. 1 + ρu λ

0



xα

0

− nδ −1

E δ−

| ln ρu λ |γ σ + qn1 −1 u dud x, 1 + ρu λ

xα

− nδ −1

E δ+

(1−β)xαδ

0

− nδ −1

E δ−





xα

0

By Fubini’s theorem (cf. [44]), we obtain that L

(+)

= (1 + α)

− nδ −1

− nδ −1

 

E δ+

= (1 + α)

δ

= (1 + α)− n −1

x

− nδ −1

(1−β)(1+α)δ x δ

| ln ρu λ |γ σ + qn1 −1 u dud x 1 + ρu λ

(1−β)(1+α)δ y

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ

0

∞

y 1



− n1 −1



0 ∞

y − n −1 dy 1

∞

+

y − n −1 1



− nδ −1

= (1 + α)  + =

(1−β)(1+α)δ y

(1−β)(1+α)δ

1

  n 

∞

(1−β)(1+α)δ

n δ

(1 + α) n +1 1 n

| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ 

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ

(1−β)(1+α)δ 0 ∞

u (1−β)(1+α)δ



(1−β)(1+α)δ

0

1





| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ   | ln ρu λ |γ σ + qn1 −1 − n1 −1 y dy u du 1 + ρu λ

(1−β)(1+α)δ 0

+ (1 − β) (1 + α)

δ n



| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ

∞ (1−β)(1+α)δ

 | ln ρu λ |γ σ − pn1 −1 u du , 1 + ρu λ

(21)

332

T. M. Rassias and B. Yang

L

(−)

= (1 − α)



− nδ −1



δ

= (1 − α)− n −1

E δ−

(−x)

∞

y − n −1 1

= (1 − α)

(1−β)(1−α)δ y 0

∞

y

− n1 −1

δ

+(1 − α)− n −1

∞



y − n −1 1

δ

δ

= (1 + α)− n −1 δ

= (1 + α)− n −1

 

δ

x − n −1

E δ+

+

∞

y − n −1 1

(1+β)(1+α)δ x δ

| ln ρu λ |γ σ + qn1 −1 u dud x 1 + ρu λ

(1+β)(1+α)δ y

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ



0 ∞

y − n −1 1





y − n −1 dy 1

=



n δ

(1 + α) n +1

0

+ (1 + β) (1 + α)

δ

δ

= (1 − α)− n −1 = (1 − α)

− nδ −1

 



δ n

(1+β)(1+α)δ

δ

E δ−

∞

1

y − n −1 1



y

− n1 −1

+

∞

y − n −1 1

1

=



n (1 − α)



δ n +1

1 n

 0

(1+β)(1−α)δ (1+β)(1−α)δ 0 δ n



| ln ρu λ |γ σ + qn1 −1 u dud x 1 + ρu λ

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ

(1+β)(1−α)δ

dy

(1+β)(1−α)δ y

+ (1 + β) (1 − α)

(1+β)(1−α)δ (−x)δ

(1+β)(1−α)δ y

1



 | ln ρu λ |γ σ − pn1 −1 u du , 1 + ρu λ

0

 0

∞

| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ

∞

(−x )− n −1

| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ 

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ

(1+β)(1+α)δ

1 n

K (−) = (1 − α)− n −1

(1+β)(1+α)δ

0

(1+β)(1+α)δ y (1+β)(1+α)δ

1

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ

0

∞

1



1



| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ

(1−β)(1−α)δ

1

K (+) = (1 + α)− n −1

(1−β)(1−α)δ 0 (1−β)(1−α)δ y



| ln ρu λ |γ σ + qn1 −1 u dud x 1 + ρu λ

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ

dy

1



(1−β)(1−α)δ (−x)δ

0



1



− nδ −1



− nδ −1

| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ 

| ln ρu λ |γ σ + qn1 −1 u dudy 1 + ρu λ

| ln ρu λ |γ σ + qn1 −1 u du 1 + ρu λ

∞ (1+β)(1−α)δ

 | ln ρu λ |γ σ − pn1 −1 u du . 1 + ρu λ

On a Few Equivalent Statements of a Hilbert-Type Integral …

333

By (19) (for f = f n , g = gn ), we have 1 1 I1 = n n



L (−) + L (+) K (−) + K (+) + 1−β 1+β

 ≤

1  M J1 . n

For n → ∞, by Fatou’s lemma (cf. [44]) and the present results, we obtain that (γ )

2k (σ ) 2 · λ ≤M 1 − β 2 1 − α2



namely,

2 1 − α2

 1p 

2 1 − β2

 q1

,

(γ )

(γ )

K α,β (σ ) =

2kλ (σ ) ≤ M. (1 − α 2 )1/q (1 − β 2 )1/ p 

This completes the proof of the lemma. Lemma 5 We define the following weight functions: ωδ (σ1 , y) :=

yβσ



δ (σ, x) := xαδσ1

∞

| ln ρ(xαδ yβ )λ |γ δσ1 −1 x d x (y ∈ R), 1 + ρ(xαδ yβ )λ α

−∞  ∞

| ln ρ(xαδ yβ )λ |γ σ −1 y dy (x ∈ R). 1 + ρ(xαδ yβ )λ β

−∞

(22) (23)

Then we have (γ )

2kλ (σ1 ) σ −σ1 y , 1 − α2 β (γ ) 2k (σ ) δ (σ, x) = λ 2 xαδ(σ1 −σ ) (x, y ∈ R\{0}). 1−β

ωδ (σ1 , y) =

Proof For fixed y ∈ R\{0}, setting u = xαδ yβ , we find

ωδ (σ1 , y) = +yβσ =



0 σ −σ1 2yβ 1 − α2

∞



yβσ



0 −∞

| ln ρ(xαδ yβ )λ |γ δσ1 −1 x dx 1 + ρ(xαδ yβ )λ α

| ln ρ(xαδ yβ )λ |γ δσ1 −1 x dx 1 + ρ(xαδ yβ )λ α

0

∞

(γ )

| ln ρu λ |γ σ1 −1 2k (σ1 ) u du = λ 2 yβσ −σ1 ; 1 + ρu λ 1−α

for fixed x ∈ R\{0}, setting u = xαδ yβ , it follows that

(24)

334

T. M. Rassias and B. Yang

δ (σ, x) = 

+xαδσ1 =

∞

0

2xαδ(σ1 −σ ) 1 − β2



xαδσ1



0

−∞

| ln ρ(xαδ yβ )λ |γ σ −1 y dy 1 + ρ(xαδ yβ )λ β

| ln ρ(xαδ yβ )λ |γ σ −1 y dy 1 + ρ(xαδ yβ )λ β ∞ 0

(γ )

| ln ρu λ |γ σ −1 2k (σ ) u du = λ 2 xαδ(σ1 −σ ) . 1 + ρu λ 1−β

Hence, we have (24). This completes the proof of the lemma.



3 Main Results Theorem 1 If p > 1, M is a constant, then the following statements (i), (ii) and (iii) are equivalent: (i) For any f (x) ≥ 0, satisfying 

∞

−∞

xαp(1−δσ1 )−1 f p (x)d x < ∞,

the following inequality holds true:

 J :=

∞ −∞



≤ M

∞

−∞



∞

| ln ρ(xαδ yβ )λ |γ f (x)d x δ λ −∞ 1 + ρ(x α yβ )  1p xαp(1−δσ1 )−1 f p (x)d x .

pσ −1

yβ

p

 1p dy (25)

(ii) For any f (x), g(y) ≥ 0, satisfying 

∞ −∞

xαp(1−δσ1 )−1 f p (x)d x < ∞, and



∞ −∞

q(1−σ )−1 q

yβ

g (y)dy < ∞,

the following inequality holds true: 

∞



∞

| ln ρ(xαδ yβ )λ |γ f (x)g(y)d xd y δ λ −∞ −∞ 1 + ρ(x α yβ )

 ∞  1p  ∞  q1 q(1−σ )−1 q p(1−δσ1 )−1 p ≤M xα f (x)d x yβ g (y)dy .

I =

−∞

−∞

(γ )

(iii) σ1 = σ, and K α,β (σ ) ≤ M.

(26)

On a Few Equivalent Statements of a Hilbert-Type Integral …

335

Proof (i) ⇒ (ii) By Hölder’s inequality (cf. [46]), we have  I =



∞

−∞



≤ J

∞

−∞



∞

| ln ρ(xαδ yβ )λ |γ f (x)d x δ λ −∞ 1 + ρ(x α yβ )  q1 q(1−σ )−1 q yβ g (y)dy .

σ− 1 yβ p



−σ + 1 yβ p g(y)

 dy (27)

Then by (25), we obtain (26). (ii) ⇒ (iii) By Lemma 1, we have σ1 = σ. Then by Lemma 2, we derive that (γ )

K α,β (σ ) ≤ M. (iii) ⇒ (i) For σ1 = σ, by Hölder’s inequality with weight (see [46]) and (22), we have  ∞ p | ln ρ(xαδ yβ )λ |γ f (x)d x δ λ −∞ 1 + ρ(x α yβ )   (σ −1)/ p   (δσ −1)/q   p ∞ f (x) xα | ln ρ(xαδ yβ )λ |γ yβ = dx (δσ −1)/q (σ −1)/ p δ y )λ 1 + ρ(x −∞ xα yβ α β  ∞ σ −1 | ln ρ(xαδ yβ )λ |γ yβ ≤ f p (x)d x (δσ −1) p/q δ λ −∞ 1 + ρ(x α yβ ) x α   p/q ∞ | ln ρ(xαδ yβ )λ |γ x δσ −1 × dx (σ −1)q/ p δ λ −∞ 1 + ρ(x α yβ ) yβ  p−1  ∞ | ln ρ(x δ y )λ |γ y σ −1 f p (x)  β q(1−σ )−1 α β = ωδ (σ, y)yβ dx (δσ −1) p/q δ λ −∞ 1 + ρ(x α yβ ) xα  (γ )  p−1  ∞ σ −1 2kλ (σ ) | ln ρ(xαδ yβ )λ |γ yβ f p (x) − pσ +1 = y d x. (28) β (δσ −1) p/q δ λ 1 − α2 −∞ 1 + ρ(x α yβ ) xα By Fubini’s theorem, (28) and (23), we have  J ≤  = ×

(γ )

 q1 

(γ )

 q1

2kλ (σ ) 1 − α2

2kλ (σ ) 1 − α2   ∞

∞

−∞

−∞

∞ −∞



∞ −∞

σ −1 | ln ρ(xαδ yβ )λ |γ yβ f p (x) d xd y 1 + ρ(xαδ yβ )λ xα(δσ −1) p/q

σ −1 | ln ρ(xαδ yβ )λ |γ yβ dy 1 + ρ(xαδ yβ )λ xα(δσ −1) p/q



 1p f (x)d x p

 1p

336

T. M. Rassias and B. Yang



 q1   1p (γ ) ∞ 2kλ (σ ) p(1−δσ )−1 p = δ (σ, x)xδ f (x)d x 1 − α2 −∞

 ∞  1p (γ ) p(1−δσ )−1 p = K α,β (σ ) xδ f (x)d x . −∞

(γ )

For K α,β (σ ) ≤ M, we have (25) (for σ1 = σ ). Therefore, Statements (i), (ii) and (iii) are equivalent. This completes the proof of the theorem.



For σ1 = σ, we have the following theorem: Theorem 2 If p > 1, M is a constant, then the following statements (i), (ii) and (iii) are equivalent: (i) For any f (x) ≥ 0, satisfying  0
0, we have the following inequality:

342

T. M. Rassias and B. Yang

||T2 f || p,ψ 1− p < M|| f || p,φ .

(43)

(ii) For any f (x), g(y) ≥ 0, f ∈ L p,φ (R), g ∈ L q,ψ (R), || f || p,φ , ||g||q,ψ > 0, we have the following inequality: (T2 f, g) < M|| f || p,φ ||g||q,ψ .

(44)

(γ )

(iii) K α,β (σ ) ≤ M.

(γ )

Moreover, if the statement (iii) holds true, then the constant factor M = K α,β (σ ) in (43) and (44) is the best possible, namely, (γ )

||T2 || = ||T1 || = K α,β (σ ). Remark 1 (1) In particular, for α = β = 0 in (31) and (32), we have the following equivalent inequalities:



∞ −∞

< 

∞



|y|

(γ ) 2kλ (σ )

pσ −1





∞

−∞

∞

| ln ρ|x y|λ |γ f (x)d x λ −∞ 1 + ρ|x y|  1p p(1−σ )−1 p |x| f (x)d x ,

p

 1p dy (45)

∞

| ln ρ|x y|λ |γ f (x)g(y)d xd y λ −∞ −∞ 1 + ρ|x y|

 ∞  1p  (γ ) |x| p(1−σ )−1 f p (x)d x < 2kλ (σ ) −∞

∞

−∞

|y|q(1−σ )−1 g q (y)dy

 q1

, (46)

(γ )

where 2kλ (σ ) is the best possible constant factor. If f (−x) = f (x), g(−y) = g(y) (x, y ∈ R+ ), then we have the following equivalent inequalities:



∞



∞

| ln ρ(x y)λ |γ y f (x)d x 1 + ρ(x y)λ 0 0

 ∞  1p (γ ) p(1−σ )−1 p < kλ (σ ) x f (x)d x , pσ −1

p

 1p dy (47)

0



∞



∞

| ln ρ(x y)λ |γ f (x)g(y)d xd y 1 + ρ(x y)λ 0 0

 ∞  1p  (γ ) p(1−σ )−1 p < kλ (σ ) x f (x)d x 0

0

∞

y

q(1−σ )−1 q

g (y)dy

 q1

,

(48)

On a Few Equivalent Statements of a Hilbert-Type Integral …

343

(γ )

where kλ (σ ) is the best possible constant factor. (2) For α = β = 0 in (35) and (36), we have the following equivalent inequalities:



∞ −∞


≥



∞ −∞

q(1−σ )−1 q

(γ ) K α,β (σ )

∞

−∞

g (y)dy = J p = I

yβ



∞

−∞

xαp(1−δσ )−1

q(1−σ )−1 q yβ g (y)dy

 1p

f (x)d x p

=J≥

 1p 

(γ ) K α,β (σ )

∞

−∞



q(1−σ )−1 q yβ g (y)dy

∞

−∞

xαp(1−δσ )−1

 q1

f (x)d x p

> 0,  1p

,

namely, (53) follows. (iii) ⇒ (ii) Similarly, setting f (x) := x qδσ −1



∞ −∞

| ln ρ(xαδ yβ )λ |γ g(y)dy 1 + ρ(xαδ yβ )λ

q−1 (x ∈ R),

by (55), we obtain (54). Since (53) is true, then the statements (i), (ii) and (iii) are true and equivalent. 1 For n ≥ pσ (n ∈ N), we set the following functions:  f n (x) =

1 δ(σ − pn )−1

xα

, x ∈ Eδ , 0, x ∈ R\E δ

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σ+

−1

1

yβ qn , y ∈ E −1 . 0, y ∈ R\E −1

gn (y) =

(γ )

If there exists a constant M(≥ K α,β (σ )), such that (55) is valid when replacing (γ ) K α,β (σ )

by M, then for f = f n , g = gn , we have 



∞

∞

| ln ρ(xαδ yβ )λ |γ f (x)gn (y)d xd y δ λ n −∞ −∞ 1 + ρ(x α yβ )

 ∞  1p  ∞  q1 q(1−σ )−1 q p(1−δσ )−1 p ≥ M xα f n (x)d x yβ gn (y)dy .

In :=

−∞

−∞

We obtain

 Jn :=

∞

−∞

xαp(1−δσ )−1 f np (x)d x

= n ×

1

(1 + α)  1

δ n +1

+

(1 + β)− n +1 1

 1p 

∞

−∞

q(1−σ )−1 q yβ gn (y)dy

 q1

 1p

1 δ

(1 − α) n +1

 q1

1

+

,

(1 − β)− n +1 1

 1  

1 (σ − pn ) | ln ρ(xαδ yβ )λ |γ δ(σ − pn1 )−1 1 1 −1 yβ In = xα d x yβn dy δ λ n n E−1 E δ 1 + ρ(x α yβ )  1 1 1 −1 , y)yβn dy ≥ ωδ (σ − n E−1 pn   (γ ) 1 2kλ (σ − pn ) 1 1 + = 1 1 1 − α2 (1 + β)− n +1 (1 − β)− n +1 ≥

1 M Jn . n

(58)

Setting σ0 ∈ (0, σ ), n ≥ (γ )

kλ (σ −

1 )= pn ≤ ≤

1 , p(σ −σ0 )

 

1

it follows that

h(u)u σ − pn −1 du + 1

0 1

h(u)u σ0 −1 du +

0 (γ ) kλ (σ0 )



∞

1 ∞



1

h(u)u σ −1 du

1 (γ )

h(u)u σ − pn −1 du

+ kλ (σ ) < ∞.

On a Few Equivalent Statements of a Hilbert-Type Integral …

347

By the Lebesgue control convergence theorem (cf. [44]), we have 1 (γ ) ) → kλ (σ ) (n → ∞). pn

(γ )

kλ (σ −

Setting n → ∞ in (58), we have (γ )

2kλ (σ ) 2 ≥M 1 − α2 1 − β 2



2 1 − α2

namely

 1p 

2 1 − β2

 q1

,

(γ )

(γ )

K α,β (σ ) =

2kλ (σ ) ≥ M. 2 (1 − α )1/q (1 − β 2 )1/ p

(γ )

Hence, M = K α,β (σ ) is the best possible constant factor of (55). The constant factor in (53), (54) is still the best possible. Otherwise, by (56), (57) we would reach the contradiction that the constant factor in (55) is not the best possible. This completes the proof of the lemma.  Lemma 7 If 0 < p < 1, there exists a constant M, such that for any nonnegative measurable functions f (x) and g(y) in R, satisfying 

∞



xαp(1−δσ )−1

−∞

f (x)d x < ∞, and

∞

p

−∞

q(1−σ )−1 q

g (y)dy < ∞,

yβ

the following inequality 

∞



∞

| ln ρ(xαδ yβ )λ |γ f (x)g(y)d xd y δ λ −∞ −∞ 1 + ρ(x α yβ )

 ∞  1p  ∞  q1 q(1−σ )−1 q ≥M xαp(1−δσ1 )−1 f p (x)d x yβ g (y)dy

I =

−∞

(59)

−∞

(γ )

holds true, then we have σ1 = σ, and M ≤ K α,β (σ ). Proof By the reverse Hölder inequality and (24), we have  I =

∞



−∞

 ≥

∞ −∞

 =2

∞

−∞

| ln ρ(xαδ yβ )λ |γ 1 + ρ(xαδ yβ )λ



 1p 

(γ ) kλ (σ1 ) 1 − α2

f (x)



(δσ1 −1)/q

xα

δ (σ, x)xαp(1−δσ1 )−1

(γ ) kλ (σ ) 1 − β2

(σ −1)/ p

yβ

f (x)d x

 q1

p

 1p 

(δσ1 −1)/q

xα

g(y)

(σ −1)/ p

yβ ∞ −∞

 d xd y

q(1−σ )−1 q ω(σ1 , y)yβ g (y)dy

 q1

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T. M. Rassias and B. Yang

 ×

∞ −∞

xαδ(σ1 −σ ) xαp(1−δσ1 )−1 f p (x)d x

 1p 

∞ −∞

q(1−σ )−1 q

yβσ −σ1 yβ

g (y)dy

 q1

.

In view of (59), we obtain that xαδ(σ1 −σ ) = Constant and yβσ −σ1 = Constant , namely, σ1 = σ. Then we obtain that 

(γ )

k (σ ) 2 λ 1 − β2

 1p 

(γ )

kλ (σ1 ) 1 − α2

 q1

(γ )

= K α,β (σ ).

(γ )

By Lemma 6, since the constant factor K α,β (σ ) in (55) is the best possible, it follows (γ )

that M ≤ K α,β (σ ). This completes the proof of the lemma.



In view of Lemmas 6 and 7, we obtain the following theorem: Theorem 5 If 0 < p < 1, M is a constant, then the following statements (i), (ii), (iii) and (iv) are equivalent: (i) For any f (x) ≥ 0, satisfying  0
M

∞ −∞



∞

| ln ρ(xαδ yβ )λ |γ f (x)d x δ λ −∞ 1 + ρ(x α yβ )  1p xαp(1−δσ1 )−1 f p (x)d x .

pσ −1 yβ

p

 1p dy (60)

(ii) For any g(y) ≥ 0, satisfying  0
M yβ g (y)dy . qδσ1 −1

−∞

q

 q1 dx (61)

On a Few Equivalent Statements of a Hilbert-Type Integral …

349

(iii) For any f (x), g(y) ≥ 0, satisfying  0
5 limit point, limit circle methods show that for C sufficiently large −Δ − Cr −2 is not essentially self-adjoint on C0∞ and r −2 lies in all L p + L ∞ with p < ν/2. The longer paper includes further discussion of these higher dimensional analogs and also discusses Kato’s application in his 1951 paper to the Coulomb-Dirac-Hamiltonian.

4 The Adiabatic Theorem In 1950, Kato published a paper [25] in a physics journal (denoted as based on a presentation in 1948) on the quantum adiabatic theorem. It is his only paper on the subject but has strongly impacted virtually all the huge literature on the subject and related subjects ever since (there are more Google Scholar citations of this paper than of the one on self-adjointness of atomic Hamiltonians). We will begin by describing his theorem and its proof which introduced what he called adiabatic dynamics and

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I’ll call the Kato dynamics. We’ll see that the Kato dynamics defines a notion of parallel transport on the natural vector bundle over the manifold of all k-dimensional subspaces of a Hilbert space, H , and so a connection. This connection is called the Berry connection and its holonomy is the Berry phase (when k = 1). All this Berry stuff was certainly not even hinted at in Kato’s work but it is implicit in the framework. Then I’ll say something about the history before Kato. The adiabatic theorem considers a family of time-dependent Hamiltonians, H (s), 0 ≤ s ≤ 1 and imagines changing them slowly, i.e., looking at H (s/T ), 0 ≤ s ≤ T for T very large. Thus, we look for U˜ T (s) solving d ˜ UT (s) = −i H (s/T )U˜ T (s), 0 ≤ s ≤ T ; ds

U˜ T (0) = 1

(1)

Letting UT (s) = U˜ T (sT ), 0 ≤ s ≤ 1, we see that UT (s), 0 ≤ s ≤ 1 solves d UT (s) = −i T H (s)UT (s), 0 ≤ s ≤ 1; ds

UT (0) = 1

(2)

Here is Kato’s adiabatic theorem Theorem 4 (Kato [25]) Let H (s) be a C 2 family of bounded self-adjoint operators on a (complex, separable) Hilbert space, H . Suppose there is a C 2 function, λ(s), so that for all s, λ(s) is an isolated point in the spectrum of H (s) and so that α ≡ inf dist(λ(s), σ (H (s)) \ {λ(s)}) > 0 0≤s≤1

(3)

Let P(s) be the projection onto the eigenspace for λ(s) as an eigenvalue of H (s). Then (4) lim (1 − P(s))UT (s)P(0) = 0 T →∞

uniformly in s in [0, 1]. Remarks 1. Thus if ϕ0 ∈ ran P(0), this says that when T is large, UT (s)ϕ0 is close to lying in ran P(s). That is as T → ∞, the solution gets very close to the “curve” {ran P(s)}0≤s≤1 . 2. If there is an eigenvalue of constant multiplicity near λ(0) for s small, it follows from the contour representation of P(s) that P(s) and λ(s) are C 2 . 3. Kato made no explicit assumptions on regularity in s saying “Our proof given below is rather formal and not faultless from the mathematical point of view. Of course, it is possible to retain mathematical rigor by a detailed argument based on clearly defined assumptions, but it would take us too far into an unnecessary complication and obscure the essentials of the problem.” It is hard to imagine the Kato of 1960 using such language! In any event, the proof requires that P(s) be C 2.

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4. As we’ll see, the size estimate for (4) is O(1/T ). Kato’s wonderful realization is that there is an explicit dynamics, W (s) for which (4) is exact, i.e., (1 − P(s))W (s)P(0) = 0 (5) He not only constructs it but proves the theorem by showing that (this formula only holds in case λ(s) ≡ 0; see (10) below) lim [UT (s) − W (s)]P(0) = 0

T →∞

(6)

The W (s) that Kato constructs, he called the adiabatic dynamics. It is sometimes called Kato’s adiabatic dynamics. We call it the Kato dynamics. Here is the basic result: Theorem 5 (Kato dynamics [25]) Let W (s) solve d W (s) = i A(s)W (s), 0 ≤ s ≤ 1; ds

W (0) = 1

i A(s) ≡ [P (s), P(s)]

(7) (8)

Then W (s) is unitary and obeys W (s)P(0)W (s)−1 = P(s)

(9)

The proof is not hard (see the longer paper for details). Using P(s)2 = P(s) and its derivative, one shows that W (s)−1 P(s)W (s) has zero derivative. The proof of Theorem 4 depends on proving that UT (s)P(0) − e−i T

s 0

λ(s) ds

W (s)P(0) = O(1/T )

(10)

Equation (10) says a lot more than (4). Equation (4) says that as T → ∞, UT (s) maps ran P(0) to ran P(s). Equation (4) actually tells you what the precise limiting map is! One fancy pants way of describing this is as follows. Fix k ≥ 1 in Z. Let M be the manifold of all k-dimensional subspaces of some Hilbert space, H . We want dim(H ) ≥ k, but it could be finite. Or M might be a smooth submanifold of the set of all such subspaces. For each ω ∈ M , we have the projection P(ω). There is a natural vector bundle of k-dimensional spaces over M , namely, we associate to ω ∈ M , the space ran P(ω). If k = 1, we get a complex line bundle. The Kato dynamics, W (s), tells you how to “parallel transport” a vector v ∈ ran P(γ (0)) along a curve γ (s); 0 ≤ s ≤ 1 in M . In the language of differential geometry, it defines a connection and such a connection has a holonomy and a curvature. In less fancy terms, consider the case k = 1. Suppose γ is a closed curve. Then W (1) is a unitary map of ran P(0) to itself, so multiplication by ei B (γ ) . Returning to UT , it says that the phase change over a closed curve isn’t what one might

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T 1 naively expect, namely exp(−i 0 λ(s/T ) ds) = exp(−i T 0 λ(s) ds). There is an additional term, exp(i B ). This is the Berry phase discovered by Berry [4] in 1983 (it was discovered in 1956 by Pancharatnam [64], but then forgotten). I realized [80] that this was just the holonomy of a natural bundle connection and that, moreover, this bundle and connection is precisely the one whose Chern integers are the TKN2 integers of Thouless et al. [85] (as discussed by Avron–Seiler–Simon [3]). Thouless got a recent physics Nobel Prize in part for the discovery of the TKN2 integers. The holonomy, i.e. Berry’s phase, is an integral of the Kato connection [P, d P]. As usual, this line integral over a closed curve is the integral of its differential [d P, d P] over a bounding surface. This quantity is the curvature of the bundle and has come to be called the Berry curvature (even though Berry did not use the differential geometric language). Naively [d P, d P] would seem to be zero but it is shorthand for the two-form  ∂ P ∂ P dsi ∧ ds j , (11) ∂si ∂s j i= j This formula of Avron–Seiler–Simon for the Berry curvature is a direct descendant of formulae in Kato’s paper, although, of course, he did not consider the questions that lead to Berry’s phase. Finally, a short excursion into the history of adiabatic theorems. “Adiabatic” first entered into physics as a term in thermodynamics meaning a process with no heat exchange. In 1916, Ehrenfest [15] discussed the “adiabatic principle” in classical mechanics. The basic example is the realization (earlier than Ehrenfest) that while the energy of a harmonic oscillator is not conserved under time dependent change of the underlying parameters, the action (energy divided by frequency) is fixed in the limit that the parameters are slowly changed (the reader should figure out what Kato’s adiabatic theorem says about a harmonic oscillator with slowly varying frequency). Interestingly enough, many adiabatic processes in the thermodynamic sense are quite rapid, so the Ehrenfest use has, at best, a very weak connection to the initial meaning of the term! Ehrenfest used these ideas by asserting that in old quantum theory, the natural quantum numbers were precisely these adiabatic invariants. Once new quantum mechanics was discovered, Born and Fock [11] in 1928 discussed what they called the quantum adiabatic theorem, essentially Theorem 4 for simple eigenvalues with a complete set of (normalizable) eigenfunctions. It was 20 years before Kato found his wonderful extension (and then more than 30 years before Berry made the next breakthrough). The longer article has a discussion on the considerable further mathematical literature of the quantum adiabatic theorem.

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5 Kato’s Inequality This section will discuss a self-adjointness method that appeared in a 1972 paper [39] of Kato based on a remarkable distributional inequality. Its consequences are a subject to which Kato returned often with at least seven additional papers [12, 41, 43, 44, 46–48]. It is also his work that most intersected my own—I [73] motivated his initial paper and it, in turn, motivated several of my later papers [74–79]. To explain the background, recall that in Section 3, we defined p to be ν-canonical (ν is dimension) if p = 2 for ν ≤ 3, p > 2 for ν = 4 and p = ν/2 for ν ≥ 5. For now, we focus on ν ≥ 5 so that p = ν/2. As we saw, if V ∈ L p (Rν ) + L ∞ (Rν ), then −Δ + V is esa-ν. The example V (x) = −C|x|−2 for C sufficiently large shows that p = ν/2 is sharp. That is, for any 2 ≤ q ≤ ν/2, there is a V ∈ L q (Rν ) + L ∞ (Rν ), so that −Δ + V is defined on but not esa on C0∞ (Rν ). In these counterexamples, though, V is negative. It was known since the late 1950s that while the negative part of V requires some global hypothesis for esaν, the positive part does not (e.g., −Δ − x 4 is not esa-ν while −Δ + x 4 is esa-ν). But when I started looking at these issues around 1970, there was presumption that for local singularities, there was no difference between the positive and negative parts. In retrospect, this shouldn’t have been the belief! After all, limit point–limit 2 so that circle methods show that if V (x) = |x|−α with α < ν/2 (to make V ∈ L loc ∞ ν −Δ + V is defined on C0 (R )) then −Δ + V is esa-ν although, if α > 2, −Δ − V is not. (Limit point–limit circle methods apply for −Δ + V for any α if we look at C0∞ (Rν \ {0}) but then only when α < ν/2, we can extend the conclusion to C0∞ (Rν ).) This example shows that the conventional wisdom was faulty but people didn’t think about separate local conditions on V+ (x) ≡ max(V (x), 0);

V− (x) = max(−V (x), 0)

(1)

Kato’s result shattered the then conventional wisdom: 2 (Rν ), then −Δ + V is esa-ν. Theorem 6 (Kato [39]) If V ≥ 0 and V ∈ L loc

Kato’s result was actually a conjecture that I made on the basis of a slightly weaker result that I had proven: Theorem 7 (Simon [73]) If V ≥ 0 and V ∈ L 2 (Rν , e−cx d ν x) for some c > 0, then −Δ + V is esa-ν. 2

Of course, this covers pretty wild growth at infinity but Theorem 6 is the definitive 2 result since one needs that V ∈ L loc (Rν ) for −Δ + V to be defined on all functions ∞ ν in C0 (R ). I found Theorem 7 because I was also working at the time in constructive quantum field theory which was then studying the simplest interacting field models ϕ24 and P(ϕ)2 (the subscript 2 means two space–time dimensions). I was able to use results in the theory of hypercontractive semigroups that seemed very different from what Kato used although connections were later found as well as a semigroup proof of Theorem 6. In my preprint proving Theorem 7, I conjectured Theorem 6 and

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B. Simon

Kato’s response was sent within a few weeks of my mailing him my preprint. The longer paper describes both my proof and these later developements but I want to focus here on Kato’s arguments. Kato proved: 1 Theorem 8 (Kato’s inequality [39]) Let u ∈ L loc (Rν ) be such that its distributional 1 ν Laplacian, Δu is also in L loc (R ). Define

sgn(u)(x) =

u(x)/|u(x|), if u(x) = 0 0, if u(x) = 0

(2)

(so u sgn(u) = |u|). Then as distributions   Δ|u| ≥ Re sgn(u)Δu

(3)

Remarks 1. What we call sgn(u), Kato calls sgn(u). ¯ 2. We should pause to emphasize what a surprise this was. Kato was a longestablished master of operator theory. He was 55 years old. Seemingly from left field, he pulled a distributional inequality out of his hat. It is true, like other analysts, that he’d been introduced to distributional ideas in the study of PDEs, but no one had ever used them in this way. Truly a remarkable discovery. The proof is not hard. By replacing u by u ∗ h n with h n a smooth approximate identity and taking limits (using sgn(u ∗ h n )(x) → sgn(u)(x) for a.e. x and using a suitable dominated convergence theorem), we can suppose that u is a C ∞ function. ¯ + ε2 )1/2 . From u 2ε = uu ¯ + ε2 , we get that In that case, for ε > 0, let u ε = (uu − → − → 2u ε ∇ u ε = 2 Re(u¯ ∇ u)

(4)

which implies (since |u| ¯ ≤ u ε ) that

Applying

→ 1− ∇· 2

− → − → | ∇ u ε | ≤ | ∇ u|

(5)

− → − → ¯ + | ∇ u|2 u ε Δu ε + | ∇ u ε |2 = Re(uΔ(u))

(6)

to (4), we get that

¯ ε , we get that Using (5) and letting sgnε (u) = u/u Δu ε ≥ Re(sgnε (u)Δu)

(7)

Taking ε ↓ 0 yields (3). Once we have (3), here is Kato’s proof [39] of Theorem 6. Consider T , the operator closure of −Δ + V on C0∞ (Rν ). T ≥ 0, so, by a simple argument, it suffices to show

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that ran(T + 1) = H or equivalently, that T ∗ u = −u ⇒ u = 0. So suppose that u ∈ L 2 (Rν ) and that (8) T ∗ u = −u Since T ∗ is defined via distributions, (3) implies that Δu = (V + 1)u

(9)

2 1 , we conclude that Δu ∈ L loc so by Kato’s inequalSince u and V + 1 are both in L loc ity Δ|u| ≥ (sgn(u))(V + 1)u = |u|(V + 1) ≥ |u| (10)

Convolution with nonnegative functions preserves positivity of distributions, so for any non-negative h ∈ C0∞ (Rν ), we have that Δ(h ∗ |u|) = h ∗ Δ|u| ≥ h ∗ |u|

(11)

Since u ∈ L 2 , h ∗ |u| is a C ∞ function with classical Laplacian in L 2 , so h ∗ |u| ∈ D(−Δ). (−Δ + 1)−1 has a positive integral kernel, so (11)⇒ (−Δ + 1)(h ∗ |u|) ≤ 0 ⇒ h ∗ |u| ≤ 0 ⇒ h ∗ |u| = 0. Taking h n to be an approximate identity, we have that h n ∗ u → u in L 2 , so u = 0 completing the proof. At first sight, Kato’s proof seems to have nothing to do with the semigroup ideas used in the proof of Theorem 7 and the proof of Theorem 6 that I found using semigroup methods. But in trying to understand Kato’s work, I found the following abstract result: Theorem 9 (Simon [77]) Let A be a positive self-adjoint operator on L 2 (M, dμ) for a σ -finite, separable measure space (M, , dμ). Then the following are equivalent: (a) (e−t A is positivity preserving) ∀u ∈ L 2 , u ≥ 0, t ≥ 0 ⇒ e−t A u ≥ 0 (b) (Beurling–Deny criterion) u ∈ Q(A) ⇒ |u| ∈ Q(A) and q A (|u|) ≤ q A (u)

(12)

(c) (Abstract Kato Inequality) u ∈ D(A) ⇒ |u| ∈ Q(A) and for all ϕ ∈ Q(A) with ϕ ≥ 0, one has that A1/2 ϕ, A1/2 |u| ≥ Reϕ, sgn(u)Au

(13)

In his original paper [39], Kato proved more than (3). He showed that  − → → Δ|u| ≥ Re sgn(u)( ∇ − i − a )2 u

(14)

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B. Simon

→ In his initial paper, he required that − a to be C 1 (Rν ) and he then followed his arguments to get Theorem 6 with −Δ + V replaced by −(∇ − ia)2 + V when a ∈ 2 (Rν ), V ≥ 0. But there was a more important consequence of (14) C 1 (Rν ), V ∈ L loc than a self-adjointness result. I noted [76] that (14) implies, by approximating |u| by positive ϕ ∈ C0∞ (Rν ), that |u|, Δ|u| ≥ u, D 2 u which implies that u, (−D 2 + V )u ≥ |u|, (−Δ + V )|u|

(15)

This, in turn, implies that turning on any, even non-constant, magnetic field always increases the ground-state energy (for spinless bosons), something I called universal diamagnetism. If one thinks of this as a zero temperature result, it is natural to expect a finite temperature result (that is, for, say, finite matrices, one has that lim β→∞ −β −1 Tr(e−β A ) = inf σ (A) which in statistical mechanical terms is saying that as the temperature goes to zero, the free energy approaches a ground-state energy). Tr(e−t H (a,V ) ) ≤ Tr(e−t H (a=0,V ) )

(16)

H (a, V ) = −(∇ − ia)2 + V

(17)

where

This suggested to me the inequality |e−t H (a,V ) ϕ| ≤ e−t H (a=0,V ) |ϕ|

(18)

I mentioned this conjecture at a brown bag lunch seminar when I was in Princeton. Ed Nelson remarked that formally, it followed from the Feynman–Kac–Ito formula for semigroups in magnetic fields,  →which says that adding a magnetic field → with gauge, − a , adds a factor exp(i − a (ω(s)) · dω) to the Feynman–Kac formula (the integral is an Ito stochastic integral). Equation (18) is immediate from → | exp(i − a (ω(s)) · dω)| = 1 and the positivity of the rest of the Feynman–Kac integrand. Some have called (18) the Nelson–Simon inequality but the name I gave it, namely diamagnetic inequality, has stuck. The issue with Nelson’s proof is that at the time, the Feynman–Kac–Ito was only known for smooth a’s. One can obtain the Feynman–Kac–Ito for more general a’s by independently proving a suitable core result. After successive improvements by me and then Kato, I proved that → 1 (Rν ) and − a ∈ Theorem 10 (Simon [78]) Equation (18) holds for V ≥ 0, V ∈ L loc 2 L loc .

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2 4 2 The optimal self-adjointness result (V ≥ 0, V ∈ L loc , a ∈ L loc , div a ∈ L loc ) was proven by Leinfelder–Simader [59]. As with Theorem 9, there is an abstract two-operator Kato inequality result (originally conjectured in Simon), which was proven by Hess–Schader-Uhlenbrock [19] and Simon [79]. For more details as well as a discussion of the Kato class which Kato introduced in his original Kato inequality note [39], see the longer paper.

6 Kato–Rosenblum and Kato–Birman Starting with Rutherford’s 1911 discovery of the atomic nucleus, scattering has been a central tool in fundamental physics, so it isn’t surprising that one of the first papers in the new quantum theory was by Born [10] in 1926 on scattering. While scattering is at a The Kato group, late 1950s. deep level a S.T. Kuroda (standing), T. Kato, T. Ikebe, H. Fujita, Y. time-dependent Nakata phenomenon, Born used eigenfunctions and time-independent ideas. In the early 1940s, the theoretical physics community first considered time-dependent approaches to scattering. Wheeler [90] and Heisenberg [18] defined the S-matrix and Møller [61] introduced wave operators as limits (with no precision as to what kind of limit). It was Friedrichs in a prescient 1948 paper [16], who first considered the invariance of the absolutely continuous spectrum under sufficiently regular perturbations. Friedrichs was Rellich’s slightly older contemporary. Both were students of Courant at Göttingen in the late 1920s (in 1925 and 1929 respectively). By 1948, Friedrichs was a professor at Courant’s Institute at NYU. He looked at several simple models and for one with purely a.c. spectrum, he could prove the perturbed models were unitarily equivalent to unperturbed models. While Friedrichs neither quoted Møller nor ever wrote down the explicit formulae Ω ± (H, H0 ) = s − limt→∓∞ eit H e−it H0

(1)

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(the strange ± versus ∓ convention that we use is universal in the theoretical physics community and uncommon among mathematicians and is not the convention that Kato used), he did prove something equivalent to showing that the limit Ω + existed and was the unitary he constructed with time-independent methods. Motivated in part by Friedrichs, in 1957, Kato published two papers [31, 32] that set out the basics of the theory we will discuss in this section. In the first, he had the important idea of defining Ω ± (A, B) = s − limt→∓∞ eit A e−it B Pac (B)

(2)

where Pac (B) is the projection onto Hac (B), the set of all ϕ ∈ H for which the spectral measure of B and ϕ is absolutely continuous with respect to Lebesgue measure. If these strong limits exist, we say that the wave operators Ω ± (A, B) exist. By replacing t by t + s, one sees that if Ω ± (A, B) exist then eis A Ω ± = Ω ± eis B . Since Ω ± are unitary maps, U ± , of Hac (B) to their ranges, we see that U ± B  Hac (B)(U ± )−1 = A  ran Ω ± . In particular, ran Ω ± are invariant subspaces for A and lie in Hac (A). It is thus natural to define: Ω ± (A, B) are said to be complete if ran Ω + (A, B) = ran Ω − (A, B) = Hac (A)

(3)

In the first of the 1957 papers, Kato proved the following. Theorem 11 (Kato [31]) Let Ω ± (A, B) exist. Then they are complete if and only if Ω ± (B, A) exist. The proof is almost trivial. It depends on noting that ψ = lim ei At e−it B ϕ ⇐⇒ ϕ = lim eit B e−it A ψ t→∞

t→∞

(4)

since ψ − ei At e−it B ϕ = eit B e−it A ψ − ϕ. That said, it is a critical realization because it reduces a completeness result to an existence theorem. In particular, it implies that symmetric conditions which imply existence also imply completeness. We’ll say more about this below. To show the importance of this idea, motivated by it in 1977, Deift and Simon [13] proved that completeness of multichannel scattering for N -body scattering was equivalent to the existence of geometrically define “inverse” wave operators. All proofs of asymptotic completeness for N -body systems prove it by showing the existence of these Deift–Simon wave operators in support of Kato’s Theorem 11. One consequence of Theorem 11 is that a symmetric condition for existence implies completeness also. Using this idea, in his first 1957 paper, Kato proved Theorem 12 (Kato [31]) Let H0 be a self-adjoint operator and V a (bounded) self-adjoint finite rank operator. Then,H = H0 + V is a self-adjoint operator and the wave operators Ω ± (H, H0 ) exist and are complete. Later in 1957, Kato proved

Tosio Kato’s Work on Non-relativistic Quantum Mechanics: A Brief Report

371

Theorem 13 (Kato–Rosenblum Theorem [32, 71]) The conclusions of Theorem 12 remain true if V is a (bounded) trace class operator. In a sense this theorem is optimal. It is a result of Weyl [89]–von Neumann [88] that if A is a self-adjoint operator, one can find a Hilbert–Schmidt operator, C, so that B = A + C has only pure point spectrum. Kato’s student, S. T. Kuroda [52], shortly after Kato proved Theorem 13, extended this result of Weyl–von Neumann to any trace ideal strictly bigger than trace class. So within trace ideal perturbations, one cannot do better than Theorem 13. The name given to this theorem comes from the fact that before Kato proved Theorem 13, Rosenblum [71] proved a special case that motivated Kato: namely, if A and B have purely a.c. spectrum and A − B is trace class, then Ω ± (A, B) exist and are unitary (so complete). I’d always assumed that Rosenblum’s paper [71] was a rapid reaction to Kato’s finite rank paper [31] which, in turn, motivated Kato’s trace class paper S. Kuroda, T. Ikebe, H. Fujita [32]. But I recently learned recently that this assumption is not correct. Rosenblum was a graduate student of Wolf at Berkeley, who submitted his thesis in March 1955. It contained his trace class result under some additional technical hypotheses; a Dec. 1955 Berkeley technical report had the result as eventually published without the extra technical assumption. Rosenblum submitted a paper to the American Journal of Mathematics which took a long time refereeing it before rejecting it. In April 1956, Rosenblum submitted a revised paper to the Pacific Journal in which it eventually appeared (this version dropped the technical condition; I’ve no idea what the original journal submission had). Kato’s finite rank paper was submitted to J. Math. Soc. Japan on March 15, 1957 and was published in the issue dated April 1957(!). The full trace class result was submitted to Proc. Japan Acad. on May 15, 1957. Kato’s first paper quotes an abstract of a talk Rosenblum gave to an A.M.S. meeting, but I don’t think that abstract contained many details. This finite rank paper has a note added in proof thanking Rosenblum for sending the technical report to Kato, quoting its main result

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B. Simon

and saying that Kato had found the full trace class results (“Details will be published elsewhere.”). That second paper used some technical ideas from Rosenblum’s paper. I’ve heard that Rosenblum always felt that he’d not received sufficient credit for his trace class paper. There is some justice to this. The realization that trace class is the natural class is important. As I’ve discussed, trace class is maximal in a certain sense. Kato was at Berkeley in 1954 when Rosenblum was a student (albeit some time before his thesis was completed) and Kato was in contact with Wolf. However, there is no indication that Kato knew anything about Rosenblum’s work until shortly before he wrote up his finite rank paper when he became aware of Rosenblum’s abstract. My surmise is that both, motivated by Friedrichs, independently became interested in scattering. The longer paper describes further developments and some applications of the trace class theory, some of them due to Kato himself. To me, the heroes of this later work are Kuroda [53], Pearson [66] and especially Birman [5–9].

7 Kato Smoothness In this final section, we discuss the theory of Kato smoothness which is based primarily on two papers of Kato published in 1966 [35] (when Kato was 49) and 1968 [36]. The first is the basic one with four important results: the equivalence of many conditions giving the definition, the connection to spectral analysis, the implications for existence and completeness of wave operators and, finally, a perturbation result. The second paper concerns the Putnam–Kato theorem on positive commutators. To me, the 1951 self-adjointness paper is Kato’s most significant work (with the adiabatic theorem paper a close second), Kato’s inequality his deepest and the subject of this section his most beautiful. One of the things that is so beautiful is that there isn’t just a relation between the time-independent and time-dependent objects— there is an equivalence! Part of the equivalence depends on the simple formula that holds when H is a self-adjoint operator on a (complex) Hilbert space H (where R(μ) = (H − μ)−1 )

∞

e−εt eitλ e−it H ϕ dt = −i R(λ + iε)ϕ

(1)

0

∞ for any ϕ ∈ H because 0 e−εt ei(λ−x)t dt = −i(x − λ − iε)−1 . Here, is the set of equivalent definitions: Theorem 14 (Kato (1966) [35]) Let H be a self-adjoint operator and A a closed operator. The following are all equal: 1 sup 2 4π ϕ=1 ε>0

∞ −∞

  A R(λ + iε)ϕ2 + A R(λ − iε)ϕ2 dλ

(2)

Tosio Kato’s Work on Non-relativistic Quantum Mechanics: A Brief Report

1 ϕ=1 2π

∞

sup

−∞

Ae−it H ϕ2 dt

373

(3)

P(a,b) (H )A∗ ϕ2 b−a ϕ=1, ϕ∈D(A∗ )

(4)

1 ∗ ¯ ϕ| |A∗ ϕ, [R(μ) − R(μ)]A μ∈R, / ϕ∈D(A∗ ) 2π

(5)

1 R(μ)A∗ ϕ2 |Imμ| μ∈R, / ϕ∈D(A∗ ) π

(6)

sup

−∞ 0. By definition, ψ is densely defined when {a ∈ A+ : ψ(a) < ∞} is dense in A+ and lower semicontinuous when {a ∈ A+ : ψ(a) ≤ α} is closed for all α ≥ 0. We will use [20–22] as our source of information on weights, and as in [22] we say that a weight is proper when it is nonzero, densely defined, and lower semicontinuous. Let ψ be a proper weight on A. Set Nψ = {a ∈ A : ψ(a ∗ a) < ∞} and note that   Nψ∗ Nψ = Span a ∗ b : a, b ∈ Nψ is a dense ∗-subalgebra of A, and that there is a unique well-defined linear map Nψ∗ Nψ → C which extends ψ : Nψ∗ Nψ ∩ A+ → [0, ∞). We denote also this densely defined linear map by ψ. Let α : R → Aut A be a point-wise norm-continuous one-parameter group of automorphisms on A. Let β ∈ R. Following [23], we say that a proper weight ψ on A is a β-KMS weight for α when (i) ψ ◦ αt = ψ for all t ∈ R, and (ii) for every pair a, b ∈ Nψ ∩ Nψ∗ there is a continuous and bounded function F defined on the closed strip Dβ in C consisting of the numbers z ∈ C whose imaginary part lies between 0 and β, and is holomorphic in the interior of the strip and satisfies that F(t) = ψ(aαt (b)), F(t + iβ) = ψ(αt (b)a) for all t ∈ R.2 that we apply the definition from [23] for the action α−t in order to use the same sign convention as in [1], for example.

2 Note

The Factor Type of Conservative KMS-Weights on Graph C ∗ -Algebras

381

A β-KMS-weight ψ with the property that sup {ψ(a) : 0 ≤ a ≤ 1} = 1 will be called a β-KMS-state. The following is Theorem 2.4 in [19]. Theorem 1 Let α : R → Aut A be a point-wise norm-continuous one-parameter group of automorphisms on a C ∗ -algebra A. Let p be a projection in the fixed point algebra of α such that p is full in A. For all β ∈ R, the map ψ → ψ( p)−1 ψ| p Ap is a bijection between the set of rays of β-KMS-weights for α and the β-KMS-states for the restriction of α to p Ap. Given a proper weight ψ on a C ∗ -algebra A, there is a GNS-type construction consisting of a Hilbert space Hψ , a linear map Λψ : Nψ → Hψ with dense range and a nondegenerate representation πψ of A on Hψ such that • ψ(b∗ a) = Λψ (a), Λψ (b) , a, b ∈ Nψ , and • πψ (a)Λψ (b) = Λψ (ab), a ∈ A, b ∈ Nψ , cf. [20–22]. A β-KMS-weight ψ on A is extremal when the only β-KMS-weights ϕ on A with the property that ϕ(a) ≤ ψ(a) for all a ∈ A+ are scalar multiples of ψ, viz. ϕ = sψ for some s > 0. The following is Lemma 4.9 in [14]. Lemma 1 Let A be a separable C ∗ -algebra and α a point-wise norm-continuous one-parameter group of automorphisms on A. Let ψ be an extremal β-KMS-weight for α. Then πψ (A)

is a factor. It is shown in Section 2.2 of [21] that a β-KMS-weight ψ extends to a normal semi-finite faithful weight ψ˜ on πψ (A)

such that ψ = ψ˜ ◦ πψ , and that the modular group on πψ (A)

corresponding to ψ˜ is the one-parameter group θ on πψ (A)

given by θt = α˜ −βt , where α˜ is the σ -weakly continuous extension of α defined such that α˜ t ◦ πψ = πψ ◦ αt . By construction α˜ t = Ad Ut , where Ut ∈ B(Hψ ) is defined by Ut Λψ (a) = Λψ (αt (a)) . In the setting of Theorem 1, let (πϕ , Hϕ , ξϕ ) be the GNS-representation of the state ϕ on p Ap defined such that ϕ(x) = ψ( p)−1 ψ(x) .

382

K. Thomsen

The modular automorphism group θ on πϕ ( p Ap)

corresponding to the vector state defined by ξϕ is given by       θt πϕ ( pap) = πϕ α−βt ( pap) = πϕ pα−βt (a) p .

Lemma 2 In the setting of Theorem 1 there is an ∗-isomorphism πϕ ( p Ap)

 πψ ( p)πψ (A)

πψ ( p)

(1)

of von Neumann algebras which is equivariant with respect to θ and θ . Proof Let q ∈ B(Hψ ) be the orthogonal projection on Λψ ( p Ap) and define a unitary W : Hϕ → q Hψ such that W πϕ (x)ξϕ = ψ ( p)− 2 Λψ (x) 1



for x ∈ p Ap. Conjugation  by W gives an isomorphism πϕ ( p Ap)  qπψ (A) q. Λψ ( p) Since q Hψ , πψ , √ψ( p) is the GNS-triple of ϕ it follows from Corollary 5.3.9 in [1] that Λψ ( p) is separating for πψ ( p)πψ (A)

πψ ( p). Since q commutes with πψ ( p)πψ (A)

πψ ( p) and q Hψ contains Λψ ( p), the map m → mq is an isomorphism

πψ ( p)πψ (A)

πψ ( p) → qπψ (A)

q . We obtain then the isomorphism (1) as the composition of two isomorphisms, both of which are equivariant. When M is a σ -finite von Neumann algebra factor every normal faithful semifinite weight on M comes together with a modular automorphism group θ = (θt )t∈R and the Connes-invariant Γ (M) is the intersection Γ (M) =



Sp(q Mq) ,

q

where we take the intersection over all projections q ∈ M fixed by θ , and Sp(q Mq) denotes the Arveson spectrum of the restriction of θ to q Mq. In more detail, Sp(q Mq) is defined as follows. For f ∈ L 1 (R) define a linear map θ f : q Mq → q Mq such that f (t)θt (a) dt . θ f (a) = R

Then Sp(q Mq) =



 Z ( f ) : f ∈ L 1 (R), θ f (q Mq) = {0} ,

The Factor Type of Conservative KMS-Weights on Graph C ∗ -Algebras

where

383



eitr f (t) dt = 0 . Z( f ) = r ∈ R : R

See [24]. Inparticular, when ψ is an extremal β-KMS-weight on A we can calculate Γ πψ (A)

by using the automorphism group θt = α˜ −βt and we get the following immediate corollary to Lemma 2. Corollary 1 In the setting of Lemma 2,     Γ πϕ ( p Ap)

= Γ πψ (A)

.

2.2 Generalized Gauge Actions on Graph C ∗ -Algebras Let G be a countable directed graph with vertex set GV and arrow set G Ar . For an arrow a ∈ G Ar we denote by s(a) ∈ GV its source and by r (a) ∈ GV its range. A vertex v which does not emit any arrow is a sink, while a vertex v which emits infinitely many arrows is called an infinite emitter. The set of sinks and infinite emitters in G is denoted by V∞ . An infinite path in G is an element p ∈ (G Ar )N such n ∈ (G Ar )n is that r ( pi ) = s( pi+1 ) for all i. A finite path μ = a1 a2 · · · an = (ai )i=1 defined similarly. The number of edges in μ is its length and we denote it by |μ|. A vertex v ∈ GV will be considered as a finite path of length 0. We let P(G ) denote the (possibly empty) set of infinite paths in G and P f (G ) the set of finite paths in G . The set P(G ) is a complete metric space when the metric is given by d( p, q) =

∞

2−i δ( pi , qi ) ,

(2)

i=1

where δ(a, a) = 0 and δ(a, b) = 1 when a = b. We extend the source map to P(G ) ∞ ∈ P(G ), and the range and source maps to such that s( p) = s( p1 ) when p = ( pi )i=1 P f (G ) such that s(μ) = s(a1 ) and r (μ) = r (a|μ| ) when |μ| ≥ 1, and s(v) = r (v) = v when v ∈ GV . Associated to the finite path μ ∈ P f (G ) is the cylinder set   ∞ ∈ P(G ) : p j = a j , j = 1, 2, · · · , |μ| Z (μ) = ( pi )i=1 which is an open and closed set in P(G ). In particular, when μ has length 0 and hence is just a vertex v, Z (v) = { p ∈ P(G ) : s( p) = v} . The C ∗ -algebra C ∗ (G ) of the graph G was introduced in this generality in [25] as the universal C ∗ -algebra generated by a collection Sa , a ∈ G Ar , of partial

384

K. Thomsen

isometries and a collection Pv , v ∈ GV , of mutually orthogonal projections subject to the conditions that (1) (2) (3) (4)

Sa∗ Sa = Pr (a) , ∀a ∈ G Ar , Sa Sa∗ ≤ Ps(a), ∀a ∈ G Ar ,

Sa Sa∗ ≤ Pv for every finite subset F ⊆ s −1 (v) and all v ∈ GV , and a∈F Pv = a∈s −1 (v) Sa Sa∗ , ∀v ∈ GV \V∞ . |μ|

For a finite path μ = (ai )i=1 ∈ P f (G ), we set Sμ = Sa1 Sa2 Sa3 · · · Sa|μ| . The elements Sμ Sν∗ , μ, ν ∈ P f (G ), span a dense ∗-subalgebra A in C ∗ (G ). The projections Pμ = Sμ Sμ∗ will play an important role in the following. Lemma 3 C ∗ (G ) is a nuclear C ∗ -algebra and π(C ∗ (G ))

is a hyperfinite von Neumann algebra for all nondegenerate representations π of C ∗ (G ). Proof The nuclearity of C ∗ (G ) follows from [26] when G is row-finite in the sense that #s −1 (v) < ∞ for all v ∈ GV , and the general case follows then from [27]. The second statement is a well-known consequence of the first and goes back to [28] and [29]. We describe next the necessary and sufficient conditions which G must satisfy for C ∗ (G ) to be simple. These conditions were identified by Szymanski in [30]. A loop in G is a finite path μ ∈ P f (G ) of positive length such that r (μ) = s(μ). We will say that a loop μ has an exit then #s −1 (v) ≥ 2 for at least one vertex v in μ. A subset H ⊆ GV is hereditary when a ∈ G Ar , s(a) ∈ H ⇒ r (a) ∈ H , and saturated when v ∈ GV \V∞ , r (s −1 (v)) ⊆ H ⇒ v ∈ H . In the following we say that G is cofinal when the only nonempty subset of GV , which is both hereditary and saturated is GV itself. Theorem 2 (Theorem 12 in [30].) C ∗ (G ) is simple if and only if G is cofinal and every loop in G has an exit. A function F : G Ar → R will be called a potential on G. Using  it we can define a point-wise norm-continuous one-parameter group α F = αtF t∈R on C ∗ (G ) such that αtF (Sa ) = ei F(a)t Sa for all a ∈ G Ar and

αtF (Pv ) = Pv

The Factor Type of Conservative KMS-Weights on Graph C ∗ -Algebras

385

for all v ∈ GV . An action of this sort is called a generalized gauge action; the gauge action itself being the one-parameter group corresponding to the constant function F = 1. To describe the KMS-weights for a generalized gauge action, extend F to a map F : P f (G ) → R such that F(v) = 0 when v ∈ GV , and F(μ) =

n

F(ai )

i=1

  n ∈ (G Ar )n . For β ∈ R, define the matrix A(β) = A(β)v,w v,w∈G V when μ = (ai )i=1 over GV such that A(β)v,w = e−β F(a) a

where we sum over arrows a ∈ G Ar with s(a) = v and r (a) = w. As in [19] we say that a nonzero nonnegative vector ψ = (ψv )v∈G V is almost A(β)-harmonic when

• w∈G V A(β)v,w ψw ≤ ψv , ∀v ∈ GV , and • w∈G V A(β)v,w ψw = ψv , ∀v ∈ GV \V∞ . The almost A(β)-harmonic vectors ψ for which

• w∈G V A(β)v,w ψw = ψv , ∀v ∈ GV , will be called A(β)-harmonic. In particular, when G is row-finite without sinks an almost A(β)-harmonic vector is automatically A(β)-harmonic. An almost A(β)harmonic vector which is not A(β)-harmonic will be said to be a proper almost A(β)-harmonic vector. It was shown in [19] that an almost A(β)-harmonic vector ϕ gives rise to a β-KMS- weight Wϕ characterized by the properties that Sμ∗ ∈ NWϕ and    0, μ = ν ∗ Wϕ Sμ Sν = −β F(μ) e ϕr (μ) , μ = ν when μ, ν ∈ P f (G ), and that all gauge invariant KMS-weights arise like this. Generally there can be KMS-weights that are not gauge invariant, but as shown in Proposition 5.6 of [31] all KMS-weights are gauge invariant when C ∗ (G ) is simple. Therefore, in the case that concerns us here, all KMS-weights arise from almost A(β)-harmonic vectors. Borrowing terminology from harmonic analysis we say that an almost A(β)-harmonic vector ϕ is minimal when it only dominates multiples of itself, i.e., when every almost A(β)-harmonic vector ϕ with the property that ϕv ≤ ϕv for all v ∈ GV is of the form ϕ = λϕ for some λ > 0. Thus, the minimal almost A(β)-harmonic vectors are those for which the corresponding β-KMS-weight Wϕ is extremal. The minimal almost A(β)-harmonic vectors can be subdivided into various ways. Here we shall consider three fundamental classes. The first class consists of the minimal proper almost A(β)-harmonic vectors. The other two classes consist of the A(β)-harmonic vectors and are distinguished by the properties of the measures they

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K. Thomsen

define on P(G ). To explain this observe that by Lemma 3.7 in [19] an A(β)-harmonic vector ϕ defines a Borel measure m ϕ on P(G ) such that m ϕ (Z (μ)) = e−β F(μ) ϕr (μ)

(3)

for all μ ∈ P f (G ). The Borel measures m on P(G ) that arise from A(β)-harmonic vectors in this way are characterized by the two properties that • m(Z (v)) < ∞ for all v ∈ GV , and • m (σ (B ∩ Z (a))) = eβ F(a) m(B ∩ Z (a)) for every edge a ∈ G Ar and every Borel subset B of P(G ). Here σ denotes the shift on P(G ), defined such that σ ( p)i = pi+1 . Nonzero Borel measures on P(G ) with these two properties are called eβ F -conformal, [32]. They are the measures that were called harmonic β-KMS measures in [19]. Let G be a cofinal graph. As in [14, 19] we say that a vertex v ∈ GV is non-wandering when there is a finite path μ ∈ P f (G ) of positive length such that v = s(μ) = r (μ). When the set N WG of non-wandering vertexes is not empty it is a hereditary subset of GV , and together with the arrows N W Ar = {a ∈ G Ar : s(a) ∈ N WG } they constitute a strongly connected subgraph of G which we also denote by N WG , cf. Proposition 4.3 in [19]. Set P(G )r ec =



{ p ∈ P(G ) : pi = a for infinitely many i}

a∈N W Ar

and P(G )wan =



{ p ∈ P(G ) : # {i ∈ N : s( pi ) = v} < ∞ } .

v∈G V

We say that an A(β)-harmonic vector ϕ is conservative when m ϕ is concentrated on P(G )r ec and that ϕ is dissipative when m ϕ is concentrated on P(G )wan . When N WG is empty, P(G )r ec = ∅ and P(G )wan = P(G ), and hence all A(β)-harmonic vectors are dissipative. To see that we have introduced a genuine dichotomy we need the following. For strongly connected graphs it is Theorem 4.10 in [19]. Theorem 3 Let G be a cofinal digraph such that N WG = ∅. Every eβ F -conformal measure m is concentrated either on P(G )r ec or on P(G )wan , and • m is concentrated on P(G )r ec if and only if ∞

A(β)nv,v = ∞

n=0

for one, and hence all v ∈ N WG , and

The Factor Type of Conservative KMS-Weights on Graph C ∗ -Algebras

387

• m is concentrated on P(G )wan if and only if ∞

A(β)nv,v < ∞

n=0

for one, and hence all v ∈ N WG . Proof Consider an eβ F -conformal measure m on P(G ). For every μ ∈ P f (G ), set Z (μ)P(N WG ) =

  ∞ ∞ ( pi )i=1 ∈ Z (μ) : ( pi )i=|μ|+1 ∈ P(N WG ) .

Note that since m is eβ F -conformal, m (Z (μ)P(N WG )) = e−β F(μ) m (P(N WG ) ∩ Z (r (μ))) . Combined with the observation that P(G ) =



Z (μ)P(N WG )

(4)

μ∈P f (G )

by Proposition 4.3 in [19], it follows that m(P(N WG )) = 0. In short, no eβ F conformal measure annihilates P(N WG ). Therefore, all conclusions follow from (4) above and Theorem 4.10 in [19]. It follows that when G is cofinal every minimal A(β)-harmonic vector is either • a proper almost A(β)-harmonic vector, • a conservative A(β)-harmonic vector, or • a dissipative A(β)-harmonic vector. In this paper, we focus on the conservative minimal A(β)-harmonic vectors. We call the corresponding β-KMS-weights conservative. The adjective “minimal” is superfluous in connection with conservative β-KMS-weights because of the following. Theorem 4 Assume that C ∗ (G ) is simple. There is a conservative A(β)-harmonic vector if and only if • N WG = ∅, n • ∞ n=0 A(β)v,v = ∞, and  1 • lim supn A(β)nv,v n = 1 for one and hence all v ∈ N WG . When it exists it is unique up to multiplication by scalars and it is minimal. Proof As observed above, there can not be any conservative eβ F -conformal measure on P(G ) unless N WG = ∅. Therefore, all statements follow by combining Theorem 3 above with Proposition 4.9 and Theorem 4.14 in [19].

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K. Thomsen

In particular, when C ∗ (G ) is simple the existence of a conservative KMS-weight for α F depends only on the restriction of F to the strongly connected subgraph N WG . In Appendix 10 in [32], it is shown that there is at most one value of β for which the conditions of Theorem 4 hold. When G is strongly connected and F = 1, in which case α F is the gauge action, this value is the Gurevich entropy of G and the conditions will hold if and only if G is recurrent in the sense of Ruette, [33].

3 The Factor Type of a Conservative β-KMS-Weight In the setting of Section 2.2, assume that N WG = ∅ and pick a vertex v in N WG . Then 

β F(μ) − β F(μ ) : μ, μ ∈ P f (G ), r (μ) = r (μ ) = s(μ) = s(μ ) = v



is a subgroup of R which does not depend on the vertex v ∈ N WG . Let RG ,F be the closure in R of this subgroup. Lemma 4 Assume that G is cofinal and that N WG = ∅. Let ψ be an extremal βKMS weight for α F . Then πψ (C ∗ (G ))

is a hyperfinite factor and   Γ πψ (C ∗ (G ))

⊆ RG ,F .

(5)

Proof M = πψ (C ∗ (G ))

is hyperfinite by Lemma 3. In the following proof, we suppress πψ in the notation and consider C ∗ (G ) as a subalgebra of M. We will show that R\RG ,F ⊆ R\Γ (M). Let, therefore, r ∈ R\RG ,F and choose a function f ∈ L 1 (R) whose Fourier transform fˆ satisfies that fˆ(t) = 0 for all t ∈ RG ,F and fˆ(r ) = 0. Fix a vertex v ∈ N WG and let μ, ν ∈ P f (G ) be finite paths with s(μ) = s(ν) = v. We assume that r (μ) = r (ν) since Sμ Sν∗ is zero otherwise. Since v is wandering and N WG is strongly connected there is a finite path δ in G such that s(δ) = r (μ) = r (ν) and r (δ) = v. Then β F(μ) − β F(ν) = β F(μδ) − β F(νδ) ∈ RG ,F . It follows that θ f (Sμ Sν∗ ) =

R

f (t)θt (Sμ Sν∗ ) dt = fˆ(β(F(μ) − F(ν)))Sμ Sν∗ = 0

because β(F(μ) − F(ν)) ∈ RG ,F . Since the elements of the form Sμ Sν∗ for some μ, ν ∈ P f (G ) with s(μ) = s(ν) = v span a strongly dense subspace of Pv M Pv we conclude that θ f (Pv M Pv ) = {0}. Since fˆ(r ) = 0 we conclude that r ∈ / Sp(Pv M Pv ), and hence r ∈ / Γ (M).

The Factor Type of Conservative KMS-Weights on Graph C ∗ -Algebras

389

Lemma 5 Assume C ∗ (G ) is simple and that there is a β-KMS-weight for α F . Let μ, ν, δ be finite paths in G such that |δ| > max{|μ|, |ν|} and F(μ) = F(ν). Then Pδ Sμ Sν∗ Pδ

 0, when μ = ν = Pδ Pμ , when μ = ν .

Proof If |μ| = |ν| and Sδ∗ Sμ Sν∗ Sδ = 0, the relations defining C ∗ (G ) imply that a piece of μ or a piece of ν will be a loop κ of positive length in G such that F(κ) = 0. By Lemma 10.4 in [32], the existence of a β-KMS-weight rules out the existence of such a loop. It follows that Sδ∗ Sμ Sν∗ Sδ can only be nonzero when |μ| = |ν|. But Sδ∗ Sμ = 0 implies that μ must be the initial piece of δ and similarly Sν∗ Sδ = 0 implies that ν must also be the initial piece of δ. Therefore Sδ∗ Sμ Sν∗ Sδ = 0 implies that μ = ν, in which case Pδ Sμ Sν∗ Pδ = Sδ Sδ∗ Sμ Sμ∗ Sδ Sδ∗ = Pδ Pμ . Theorem 5 Assume that C ∗ (G ) is simple and N WG not empty. Let ψ be a conservative β-KMS- weight for the generalized gauge action α F . Then Γ (πψ (C ∗ (G ))

) = RG ,F . Proof The proof is a further development of the proofs of Proposition 4.11 in [14] and Theorem 4.1 in [16]. As in the proof of Lemma 4 we suppress πψ in the notation and consider C ∗ (G ) as a subalgebra of M. The modular automorphism group θ on M defined by ψ˜ is given by θt = α˜F −βt , where α˜F is the σ -weakly continuous extension of α F . Note that β = 0 by Proposition 2.4 in [32]. Let N ⊆ M be the fixed point algebra for θ and consider a vertex v ∈ N WG . By Lemma 5, it suffices to show that RG ,F ⊆ Γ (M), and from Lemme 2.3.3 and Proposition 2.2.2 in [24] we see that it suffices for this to consider a nonzero central projection q ∈ Pv N Pv for some vertex v ∈ N WG , and show that β F(l) ⊆ Sp(q Mq) when l is a loop in G such that s(l) = r (l) = v. Let ω be the ˜ Then ω is a faithful normal state state on Pv M Pv given by ω(a) = ψ(Pv )−1 ψ(a). which is a trace on Pv N Pv and we consider the corresponding 2-norm av =

 ω(a ∗ a) .

By Kaplansky’s density theorem, there is an element f ∈ Pv A Pv such that 0 ≤ f ≤ 1 and q − f v is as small as we want. Then f is a linear combination of elements of the form Sμ Sν∗ where μ, ν ∈ P f (G ) and s(μ) = s(ν) = v. Note that 1 lim R→∞ R

0

R

 θt (Sμ Sν∗ ) dt =

with convergence in norm, and that

Sμ Sν∗ 0

when F(μ) = F(ν) when F(μ) = F(ν) ,

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   R  R   1  q − 1   θt ( f ) dt  =  θt (q − f ) dt    ≤ q − f v R 0 R 0 v v by Kadisons Schwarz inequality. By exchanging lim R→∞ may assume that N λi Sμi Sν∗i , f =

1 R

R 0

θt ( f ) dt for f , we

i=1

where 0 ≤ λi ≤ 1 and F(μi ) = F(νi ) for all i. Let m ψ be the eβ F -conformal measure on P(G ) defined by ψ and let m be the restriction of the measure m ψ (Z (v))−1 m ψ to Z (v). Then m is a Borel probability measure on Z (v) ⊆ P(G ) such that   ω Pμ = m(Z (μ)) for all μ ∈ P f (G ) with s(μ) = v. For each k ∈ N, we let Mk be the set of paths δ in G such that s(δ) = v and |δ| = k. Then δ∈Mk

ω(Pδ ) =



m(Z (δ)) = 1 ,

δ∈Mk

which implies that 1 = δ∈Mk Pδ , where the sum converges with respect to the 2norm and hence also in the strong operator topology. It follows that we can define Q k : Pv M Pv → Pv M Pv such that

Q k (m) =

Pδ m Pδ .

δ∈Mk

Then Q k is a positive linear map of norm one and Q k (q) = q. When k > max{|μ|, |ν|} it follows from Lemma 5 that    Pμ , when μ = ν ∗ Q k Sμ Sν = 0, when μ = ν . Thus, for some k large enough, we have that Q k ( f ) is a linear combination

Qk ( f ) =

N

λi Pμi ,

i=1

where 0 ≤ λi ≤ 1 for all i. Using Kadisons Schwarz inequality again we find that q − Q k ( f )v = Q k (q − f )v ≤

    ω Q k (q − f )2 = q − f v

The Factor Type of Conservative KMS-Weights on Graph C ∗ -Algebras

391

since ω ◦ Q k = ω for all k. By exchanging f with Q k ( f ) for some k large enough we may assume that N λi Pμi . (6) f = i=1

Now, observe that since ψ is conservative by assumption it follows that m is concentrated on { p ∈ Z (v) : s( pi ) = v for infinitely many i} . Fix one of the paths μi appearing in (6). Let H j denote the set of finite paths δ of length j such that s(δ) = r (μi ), r (δ) = v. Then, up to an m-null set, ∞  

 Z (μi δ) : δ ∈ H j = Z (μi ) .

j=1

We can, therefore, find a finite set K i ⊆

∞ j=1

H j such that

  m Z (μi )\ δ∈K i Z (μi δ) is as small as we want. Note that            Pμ − m Z (μi )\ δ∈K i Z (μi δ) P = μi δ   i   δ∈K i v

is then also small. Therefore, by exchanging δ∈K i Pμi δ for Pμi we may assume that s(μi ) = r (μi ) = v for all v. Finally, since q is a projection, a standard argument, as in the proof of Lemma 12.2.3 in [34], allows us to select a subset of the μi ’s and arrange, after a renumbering, that p=

M

Pμi

(7)

i=1

is a projection in Pv A Pv such that q − pv ≤ ε , where ε > 0 can be chosen as small as we need. We choose ε > 0 so small that ω(q) − e F(l)β ε − 3ε > 0 . For each μi from (7) we let wi ∈ Pv A Pv be the elements

(8)

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wi = Sμi Sμ∗ i l . Each wi is a partial isometry such that (a) wi wi ∗ = Pμi , (b) wi ∗ wi = Pμi l ≤ Pμi , and (c) αtF (wi ) = e−i F(l)t wi for all t ∈ R.

M Set w = i=1 wi and note that w is a partial isometry. It follows from (b) that wp = w and therefore from (8), (c), (b), and (a) that ω(qwqw∗ q) ≥ ω(wqw∗ ) − 2ε = eβ F(l) ω(qw ∗ w) − 2ε ≥ eβ F(l) ω( pw ∗ w) − eβ F(l) ε − 2ε = eβ F(l) ω(w ∗ w) − eβ F(l) ε − 2ε = ω(ww∗ ) − eβ F(l) ε − 2ε = ω( p) − eβ F(l) ε − 2ε ≥ ω(q) − eβ F(l) ε − 3ε . The choice of ε ensures that ω(qwqw∗ q) > 0 and hence that qwq = 0. Since θt (qwq) = e−itβ F(l) qwq for all t ∈ R, it follows from Lemme 2.3.6 in [24] that β F(l) ∈ Sp(q Mq), as desired. In combination with Corollary 1, we get the following Corollary 2 Assume that C ∗ (G ) is simple and that N WG = ∅. Let

ϕ be a β-KMSn state for the restriction of α F to Pv C ∗ (G )Pv for some v ∈ GV . When ∞ n=0 A(β)w,w = ∗

∞for some (and hence  all) w ∈ N WG , the Connes invariant of πϕ (Pv C (G )Pv ) is ∗

Γ πϕ (Pv C (G )Pv ) = RG ,F . In an Appendix in [32], it is shown that in the setting of Theorem 5 the subgroup RG ,F is never {0}. Therefore, thanks to Connes’ classification in [29] and Haagerup’s result in [35], we get the following. Corollary 3 In the setting of Theorem 5 and Corollary 2 the factors πψ (C ∗ (G ))

and πϕ (Pv C ∗ (G )Pv )

are isomorphic; they are both isomorphic to the hyperfinite factor of type I I Iλ for some 0 < λ ≤ 1. Example 1 (Generalized gauge actions on O∞ .) The Cuntz algebra O∞ , [36], is the graph C ∗ -algebra C ∗ (G ) when G is the countable digraph with one vertex and infinitely many arrows, an , n = 1, 2, 3, · · · . Since O∞ is unital all proper weights are bounded and can be normalized to states. Let {tn }∞ n=1 be a sequence of real numbers and define F : G Ar → R such that F(an ) = tn for all n. The gauge action, where tn = 1 for all n, was considered by Olesen and Pedersen who showed in [37] that there are no KMS-states for the gauge action. In general, it follows from [19] that a −βtn ≤ 1 and that it is unique. There is a conservative β-KMS-state exists iff ∞ n=1 e β0 -KMS-state for α F iff ∞ e−β0 tn = 1 . n=1

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393

By Theorem 5 the Connes invariant of the corresponding factor is the closed subgroup ∞ . It follows that the factor is always of type of R generated by the numbers {β0 ti }i=1 I I I and never of type I I I0 while all hyperfinite factors of type I I Iλ for 0 < λ ≤ 1 can occur by varying the sequence {tn }. The KMS-states for α F that are not conservative −βtn < 1 and they correspond to minimal occur for values of β for which ∞ n=1 e proper almost A(β)-harmonic vectors, albeit of a somewhat degenerate kind since the vectors only have one coordinate. Acknowledgements I am grateful to Johannes Christensen for discussions and help to eliminate mistakes. The work was supported by the DFF-Research Project 2 ‘Automorphisms and Invariants of Operator Algebras’, no. 7014-00145B.

References 1. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I + II. Texts and Monographs in Physics. Springer, New York (1979, 1981) 2. Haag, R., Winnink, M., Hugenholtz, N.M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967) 3. Enomoto, M., Fujii, M., Watatani, Y.: KMS states for gauge action on O A . Math. Jpn. 29, 607–619 (1984) 4. Bost, J.-B., Connes, A.: Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Math. (N.S.) 1, 411–457 (1995) 5. Okayasu, R.: Type III factors arising from Cuntz-Krieger algebras. Proc. Am. Math. Soc. 131, 2145–2153 (2003) 6. Barreto, S.D., Fidaleo, F.: On the structure of KMS states of disordered systems. Commun. Math. Phys. 250, 1–21 (2004) 7. Izumi, M., Kajiwara, T., Watatani, Y.: KMS states and branched points. Ergod. Theory Dyn. Syst. 27, 1887–1918 (2007) 8. Neshveyev, S.: von Neumann algebras arising from Bost-Connes type systems. Int. Math. Res. Not. IMRN 2011, 217–236 (2011) 9. Laca, M., Neshveyev, S.: Type I I I1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbere. J. Funct. Anal. 261, 169–187 (2011) 10. Yang, D.: Type III von Neumann algebras associated with 2-graphs. Bull. Lond. Math. Soc. 44, 675–686 (2012) 11. Laca, M., Larsen, N., Neshveyev, S., Sims, A., Webster, S.B.G.: Von Neumann algebras of strongly connected higher-rank graphs. Math. Ann. 363, 657–678 (2015) 12. Carey, A., Phillips, J., Putnam, I., Rennie, A.: Families of type III KMS states on a class of C ∗ -algebras containing On and Q N . J. Funct. Anal. 260, 1637–1681 (2011) 13. Kajiwara, T., Watatani, Y.: KMS states on finite-graph C ∗ -algebras. Kyushu J. Math. 67, 83–104 (2013) 14. Thomsen, K.: KMS weights on groupoid and graph C ∗ -algebras. J. Funct. Anal. 266, 2959– 2988 (2014) 15. Thomsen, K.: Exact circle maps and KMS states. Isr. J. Math. 205, 397–420 (2015) 16. Thomsen, K.: Phase transition in O2 . Commun. Math. Phys. 349, 481–492 (2017) 17. Yang, D.: Factoriality and type classification of k-graph von Neumann algebras. Proc. Edinb. Math. Soc. (2) 60, 499–518 (2017) 18. Izumi, M.: The flow of weights and the Cuntz-Pimsner algebras. Commun. Math. Phys., To appear 19. Thomsen, K.: KMS weights on graph C ∗ -algebras. Adv. Math. 309, 334–391 (2017)

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20. Kustermans, J.: KMS-weights on C ∗ -algebras. arXiv:9704008v1 21. Kustermans, J., Vaes, S.: Weight theory for C ∗ -algebraic quantum groups. arXiv:990163 22. Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Scient. Éc. Norm. Sup. 33, 837–934 (2000) 23. Combes, F.: Poids associé à une algèbre hilbertienne à gauche. Compos. Math. 23, 49–77 (1971) 24. Connes, A.: Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup. 6, 133–252 (1973) 25. Bates, T., Hong, J.H., Raeburn, I., Szymanski, W.: The ideal structure of the C ∗ -algebras of infinite graphs. Illinois J. Math. 46, 1159–1176 (2002) 26. Kumjian, A., Pask, D.: C ∗ -algebras of directed graphs and group actions. Ergod. Theory Dyn. Syst. 19, 1503–1519 (1999) 27. Drinen, D., Tomforde, M.: The C ∗ -algebras of arbitrary graphs. Rocky Mt. J. Math. 35, 105–135 (2005) 28. Choi, M.D., Effros, E.G.: Separable nuclear C ∗ -algebras and injectivity. Duke Math. J. 43, 309–322 (1976) 29. Connes, A.: Classification of injective factors. Cases I I1 , I I∞ , I I Iλ , λ = 1. Ann. Math. (2) 104, 73–115 (1976) 30. Szymanski, W.: Simplicity of Cuntz-Krieger algebras of infinite matrices. Pac. J. Math. 122, 249–256 (2001) 31. Christensen, J., Thomsen, K.: Diagonality of actions and KMS weights. J. Oper. Theory 76, 449–471 (2016) 32. Thomsen, K.: KMS weights, conformal measures and ends in digraphs. arXiv:1612.04716v2 33. Ruette, S.: On the Vere-Jones classification and existence of maximal measure for countable topological Markov chains. Pac. J. Math. 209, 365–380 (2003) 34. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II. Academic, London (1986) 35. Haagerup, U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type I I I1 . Acta Math. 158, 95–148 (1987) 36. Cuntz, J.: Simple C ∗ -algebras generated by isometries. Commun. Math. Phys. 57, 173–185 (1977) 37. Olesen, D., Pedersen, G.K.: Some C ∗ -dynamical systems with a single KMS-state. Math. Scand. 42, 111–118 (1978)

Trotter–Kato Product Formulae in Dixmier Ideal Valentin A. Zagrebnov

On the occasion of the 100th birthday of Tosio Kato

Abstract It is shown that for a certain class of the Kato functions, the Trotter–Kato product formulae converge in Dixmier ideal C1,∞ in topology, which is defined by the  · 1,∞ -norm. Moreover, the rate of convergence in this topology inherits the error-bound estimate for the corresponding operator-norm convergence.

1 Preliminaries. Symmetrically-Normed Ideals Let H be a separable Hilbert space. For the first time, the Trotter–Kato product formulae in Dixmier ideal C1,∞ (H ) were shortly discussed in conclusion of the paper [19]. This remark was a programme addressed to extension of results, which were known for the von Neumann–Schatten ideals C p (H ), p ≥ 1 since [14, 24]. Note that a subtle point of this programme is the question about the rate of convergence in the corresponding topology. Since the limit of the Trotter–Kato product formula is a strongly continuous semigroup, for the von Neumann–Schatten ideals, this topology is the trace norm  · 1 on the trace-class ideal C1 (H ). In this case, the limit is a Gibbs semigroup [25]. For self-adjoint Gibbs semigroups, the rate of convergence was estimated for the first time in [7, 9]. The authors considered the case of the Gibbs–Schrödinger semigroups. They scrutinised, in these papers, a dependence of the rate of converV. A. Zagrebnov (B) Institut de Mathématiques de Marseille (UMR 7373) - AMU, Centre de Mathématiques et Informatique - Technopôle Château-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. M. Rassias and V. A. Zagrebnov (eds.), Analysis and Operator Theory, Springer Optimization and Its Applications 146, https://doi.org/10.1007/978-3-030-12661-2_18

395

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gence for the (exponential) Trotter formula on the smoothness of the potential in the Schrödinger generator. The first abstract result in this direction was due to [19]. In this paper, a general scheme of lifting the operator-norm rate convergence for the Trotter–Kato product formulae was proposed and advocated for estimating the rate of the trace-norm convergence. This scheme was then improved and extended in [2] to the case of nonself-adjoint Gibbs semigroups. The aim of the present note is to elucidate the question about the existence of other than the von Neumann–Schatten proper two-sided ideals I(H ) of L (H ) and then to prove the (non-exponential) Trotter–Kato product formula in topology of these ideals together with an estimate of the corresponding rate of convergence. Here, a particular case of the Dixmier ideal C1,∞ (H ) [4, 6] is considered. To specify this ideal we recall in Section 2, the notion of singular trace and then of the Dixmier trace [3, 5] in Section 3. Main results about the Trotter–Kato product formulae in the Dixmier ideal C1,∞ (H ) are collected in Section 4. There the arguments based on the lifting scheme [19] (Theorem 5.1) are refined for proving the Trotter–Kato product formulae convergence in the  · 1,∞ -topology with the rate, which is inherited from the operator-norm convergence. To this end, in the rest of the present section, we recall an important auxiliary tool: the concept of symmetrically-normed ideals, see e.g. [8, 22]. ∞ Let c0 ⊂ l ∞ (N) be the subspace of bounded sequences ξ = {ξ j }∞ j=1 ∈ l (N) of real numbers, which tend to zero. We denote by c f the subspace of c0 consisting of all sequences with finite number of non-zero terms (finite sequences). Definition 1 A real-valued function φ : ξ → φ(ξ ) defined on c f is called a norming function if it has the following properties: φ(ξ ) > 0, ∀ξ ∈ c f , ξ = 0, φ(αξ ) = |α|φ(ξ ), ∀ξ ∈ c f , ∀α ∈ R, φ(ξ + η) ≤ φ(ξ ) + φ(η), ∀ξ, η ∈ c f ,

(1.1) (1.2) (1.3)

φ(1, 0, . . .) = 1.

(1.4)

A norming function φ is called to be symmetric if it has the additional property φ(ξ1 , ξ2 , ..., ξn , 0, 0, . . .) = φ(|ξ j1 |, |ξ j2 |, ..., |ξ jn |, 0, 0, . . .)

(1.5)

for any ξ ∈ c f and any permutation j1 , j2 , . . . , jn of integers 1, 2, . . . , n. It turns out that for any symmetric norming function and for any elements ξ, η from the positive cone c+ of non-negative, non-increasing sequences such that ξ, η ∈ c f obey ξ1 ≥ ξ2 ≥ . . . ≥ 0, η1 ≥ η2 ≥ . . . ≥ 0 and n  j=1

ξj ≤

n  j=1

ηj,

n = 1, 2, . . . ,

(1.6)

Trotter–Kato Product Formulae in Dixmier Ideal

397

one gets the Ky Fan inequality [8] (Sect. 3, §3) : φ(ξ ) ≤ φ(η).

(1.7)

Moreover, (1.7) together with the properties, (1.1), (1.2) and (1.4) imply ξ1 ≤ φ(ξ ) ≤

∞ 

ξ ∈ c+f := c f ∩ c+ .

ξj,

(1.8)

j=1

Note that the left- and right-hand sides of (1.8) are the simplest examples of symmetric norming functions on domain c+f : φ∞ (ξ ) := ξ1 and φ1 (ξ ) :=

∞ 

ξj .

(1.9)

j=1

By Definition 1, the observations (1.8) and (1.9) yield φ∞ (ξ ) := max |ξ j | , φ1 (ξ ) := j≥1

∞ 

|ξ j | ,

(1.10)

j=1

φ∞ (ξ ) ≤ φ(ξ ) ≤ φ1 (ξ ) , for all ξ ∈ c f . ξ1∗ ξ1∗

We denote by ξ ∗ := {ξ1∗ , ξ2∗ , . . . } a decreasing rearrangement: ξ1∗ = sup j≥1 |ξ j | , + ξ2∗ = supi = j {|ξi | + |ξ j |}, . . . , of the sequence of absolute values {|ξn |}n≥1 , i.e. ≥ ξ2∗ ≥ . . . . Then, ξ ∈ c f implies ξ ∗ ∈ c f and by (1.5) one obtains also that φ(ξ ) = φ(ξ ∗ ),

ξ ∈ cf .

(1.11)

Therefore, any symmetric norming function φ is uniquely defined by its values on the positive cone c+ . Now, let ξ = {ξ1 , ξ2 , . . .} ∈ c0 . We define ξ (n) := {ξ1 , ξ2 , . . . , ξn , 0, 0, . . . } ∈ c f . Then, if φ is a symmetric norming function, we set cφ := {ξ ∈ c0 : sup φ(ξ (n) ) < +∞}. n

Therefore, one gets c f ⊆ cφ ⊆ c0 . Note that by (1.5)–(1.7) and (1.12), one gets

(1.12)

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φ(ξ (n) ) ≤ φ(ξ (n+1) ) ≤ sup φ(ξ (n) ) , for any ξ ∈ cφ . n

Then, the limit

φ(ξ ) := lim φ(ξ (n) ) , n→∞

ξ ∈ cφ ,

exists and φ(ξ ) = supn φ(ξ (n) ), i.e. the symmetric norming function φ is a normal functional on the set cφ (1.12), which is a linear space over R. By (1.3) and (1.10) one also gets that any symmetric norming function is continuous on c f : |φ(ξ ) − φ(η)| ≤ φ(ξ − η) ≤ φ1 (ξ − η) , ∀ξ, η ∈ c f . Suppose that X is a compact operator, i.e. X ∈ C∞ (H ). Then by s(X ) := {s1 (X ), s2 (X ), . . . }, we denote the sequence of singular values of X counting multiplicities. We always assume that s1 (X ) ≥ s2 (X ) ≥ . . . ≥ sn (X ) ≥ . . . . To define symmetrically-normed ideals of the compact operators C∞ (H ), we introduce the notion of a symmetric norm. Definition 2 Let I be a two-sided ideal of C∞ (H ). A functional  · sym : I → R+ 0 is called a symmetric norm if besides the usual properties of the operator norm  · : X sym > 0,

∀X ∈ I,

X = 0,

α X sym = |α|X sym , ∀X ∈ I, ∀α ∈ C, X + Y sym ≤ X sym + Y sym , ∀X, Y ∈ I, it verifies the following additional properties: AX Bsym ≤ AX sym B, X ∈ I, A, B ∈ L (H ), α X sym = |α|X  = |α| s1 (X ), for any one − rank operator X ∈ I.

(1.13) (1.14)

If the condition (1.13) is replaced by U X sym = XU sym = X sym , X ∈ I ,

(1.15)

for any unitary operator U on H , then, instead of the symmetric norm, one gets definition of an invariant norm  · inv . First, we note that the ordinary operator norm  ·  on any ideal I ⊆ C∞ (H ) is evidently a symmetric norm as well as an invariant norm.

Trotter–Kato Product Formulae in Dixmier Ideal

399

Second, we observe that, in fact, every symmetric norm is invariant. Indeed, for any unitary operators U and V one gets by (1.13) that U X V sym ≤ X sym , X ∈ I .

(1.16)

Since X = U −1 U X V V −1 , we also get X sym ≤ U X V sym , which together with (1.16) yield (1.15). Third, we claim that X sym = X ∗ sym . To this aim, let X = U |X | be the polar representation of the operator X ∈ I. Since U ∗ X = |X |, then by (1.13), we obtain X sym = |X |sym . The same line of reasoning applied to the adjoint operator X ∗ = |X |U ∗ yields X ∗ sym = |X |sym , which proves the claim. Now, we can apply the concept of the symmetric norming functions to describe the symmetrically-normed ideals of the unital algebra of bounded operators L (H ), or in general, the symmetrically-normed ideals generated by symmetric norming functions. Recall that any proper two-sided ideal I(H ) of L (H ) is contained in compact operators C∞ (H ) and contains the set K (H ) of finite-rank operators, see e.g. [20, 22]: (1.17) K (H ) ⊆ I(H ) ⊆ C∞ (H ). To clarify the relation between symmetric norming functions and the symmetricallynormed ideals, we mention that there is an obvious one-to-one correspondence between functions φ (Definition 1) on the cone c+ and the symmetric norms  · sym on K (H ). To proceed with a general setting, one needs definition of the following relation. Definition 3 Let cφ be the set of vectors (1.12) generated by a symmetric norming function φ. We associate with cφ a subset of compact operators Cφ (H ) := {X ∈ C∞ (H ) : s(X ) ∈ cφ } .

(1.18)

This definition implies that the set Cφ (H ) is a proper two-sided ideal of the algebra L (H ) of all bounded operators on H . Setting X φ := φ(s(X )) ,

X ∈ Cφ (H ) ,

(1.19)

one obtains a symmetric norm:  · sym =  · φ , on the ideal I = Cφ (H ) (Definition 2) such that this symmetrically-normed ideal becomes a Banach space. Then, in accordance with (1.17) and (1.18), we obtain by (1.10) that K (H ) ⊆ C1 (H ) ⊆ Cφ (H ) ⊆ C∞ (H ) .

(1.20)

Here, the trace-class operators C1 (H ) := Cφ1 (H ), where the symmetric norming function φ1 is defined in (1.9), and X φ ≤ X 1 ,

X ∈ C1 (H ) .

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Remark 1 By virtue of inequality (1.7) and by definition of symmetric norm (1.19), the so-called dominance property holds: if X ∈ Cφ (H ), Y ∈ C∞ (H ) and n 

s j (Y ) ≤

n 

j=1

s j (X ) ,

n = 1, 2, . . . ,

j=1

then Y ∈ Cφ (H ) and Y φ ≤ X φ . Remark 2 To distinguish in (1.20) nontrivial ideals Cφ one needs some criteria based on the properties of φ or  · φ . For example, any symmetric norming function (1.11) defined by r  (r ) (ξ ) := ξ ∗j , ξ ∈ cf , φ j=1

generates for arbitrary fixed r ∈ N, the symmetrically-normed ideals, which are trivial in the sense that all Cφ (r ) (H ) = C∞ (H ). Criterion for an operator A to belong to a nontrivial ideal Cφ is M = sup Pm A Pm φ < ∞ ,

(1.21)

m≥1

where {Pm }m≥1 is a monotonously increasing sequence of the finite-dimensional orthogonal projectors on H strongly convergent to the identity operator [8]. Note that for A ∈ C∞ , the condition (1.21) is trivial. We consider now a couple of examples to elucidate the concept of the symmetricallynormed ideals Cφ (H ) generated by the symmetric norming functions φ and the rôle of the functional trace on these ideals. Example 1 The von Neumann–Schatten ideals C p (H ) [21]. These ideals of C∞ (H ) are generated by symmetric norming functions φ(ξ ) := ξ  p , where ⎛ ξ  p = ⎝

∞ 

⎞1/ p |ξ j | p ⎠

,

ξ ∈ cf,

j=1

for 1 ≤ p < +∞, and by ξ ∞ = sup |ξ j |,

ξ ∈ cf,

j

for p = +∞. Indeed, if we set {ξ ∗j := s j (X )} j≥1 , for X ∈ C∞ (H ), then the symmetric norm X φ = s(X ) p coincides with X  p and the corresponding symmetricallynormed ideal Cφ (H ) is identical to the von Neumann–Schatten class  C p (H ). By definition, for any X ∈ C p (H ) the trace: |X | → Tr|X | = j≥1 s j (X ) ≥ 0. The trace norm X 1 = Tr|X | is finite on the trace-class operators C1 (H ) and it is

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401

infinite for X ∈ C p>1 (H ). We say that for p > 1, the von Neumann–Schatten ideals admit no trace, whereas for p = 1 the map: X → Tr X exists and it is continuous in the  · 1 -topology. Note that by virtute of the Tr-linearity the trace norm: C1,+ (H )  X → X 1 is linear on the positive cone C1,+ (H ) of the trace-class operators. Example 2 Now, we consider symmetrically-normed ideals CΠ (H ). To this aim let + Π = {π j }∞ j=1 ∈ c be a non-increasing sequence of positive numbers with π1 = 1. We associate with Π the function ⎧ ⎫ n ⎨ ⎬  1 φΠ (ξ ) = sup n ξ ∗j , (1.22) ξ ∈ cf. ⎭ n ⎩ j=1 π j j=1

It turns out that φΠ is a symmetric norming function. Then, the corresponding (1.12) set cφΠ is defined by cφΠ :=

⎧ ⎨ ⎩

ξ ∈ c f : sup n n

n 

1

j=1

πj

j=1

⎫ ⎬ ξ ∗j < +∞ . ⎭

Hence, the two-sided symmetrically-normed ideal CΠ (H ) := CφΠ (H ) generated by symmetric norming function (1.22) consists of all those compact operators X that X φΠ := sup n n

n 

1

j=1

πj

s j (X ) < +∞ .

(1.23)

j=1

This equation defines a symmetric norm X sym = X φΠ on the ideal CΠ (H ), see Definition 2. Now, let Π = {π j }∞ j=1 , with π1 = 1, satisfy ∞ 

π j = +∞

and

j=1

lim π j = 0 .

j→∞

(1.24)

Then, the ideal CΠ (H ) is nontrivial: CΠ (H ) = C∞ (H ) and CΠ (H ) = C1 (H ), see Remark 2, and one has C1 (H ) ⊂ CΠ (H ) ⊂ C∞ (H ).

(1.25)

If in addition to (1.24), the sequence Π = {π j }∞ j=1 is regular, i.e. it obeys n  j=1

π j = O(nπn ) ,

n→∞,

(1.26)

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then X ∈ CΠ (H ), if and only if sn (X ) = O(πn ) ,

n→∞,

(1.27)

cf. condition (1.21). On the other hand, the asymptotics sn (X ) = o(πn ) ,

n→∞,

implies that X belongs to some ideal in between of C1 (H ) and C1+ε (H ). −α Remark 3 A natural choice of the sequence {π j }∞ , j=1 that satisfies (1.24) is π j = j ∞ 0 < α ≤ 1. Note that if 0 < α < 1, then the sequence Π = {π j } j=1 satisfies (1.26), i.e. it is regular for ε = 1 − α. Therefore, the two-sided symmetrically-normed ideal CΠ (H ) generated by symmetric norming function (1.22) consists of all those compact operators X , which singular values obey (1.27):

sn (X ) = O(n −α ), 0 < α < 1, n → ∞ .

(1.28)

Let α = 1/ p , p > 1. Then, the corresponding to (1.28) symmetrically-normed ideal is defined by C p,∞ (H ) := {X ∈ C∞ (H ) : sn (X ) = O(n −1/ p ), p > 1} , which is known as the weak-C p ideal [20, 22]. Whilst by virtue of (1.28) the weak-C p ideal admit no trace, definition (1.23) implies that for the regular case p > 1 a symmetric norm on C p,∞ (H ) is equivalent to n 1  s j (X ) , (1.29) X  p,∞ = sup 1−1/ p n n j=1 and C1 (H ) ⊂ C p,∞ (H ) ⊂ C∞ (H ).

2 Singular Traces Note that (1.29) implies: C1 (H )  A → A p,∞ < ∞, but any related linear, positive and unitarily invariant functional (trace), which is nontrivial on the ideal C p,∞ (H ) is zero on the set of finite-rank operators K (H ). We remind that these non-normal traces: Tr ω (X ) := ω({n −1+1/ p

n  j=1

s j (X )}∞ n=1 ) ,

(2.1)

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403

are called singular, [5, 13]. Here, ω is an appropriate linear positive normalised functional (state) on the Banach space l ∞ (N) of bounded sequences. Recall that the set of these states S (l ∞ (N)) ⊂ (l ∞ (N))∗ , where (l ∞ (N))∗ is dual of the Banach space l ∞ (N). Note that the singular trace (2.1) is continuous in topology defined by the norm (1.29). Remark 4 (a) The weak-C p ideal, which is defined for p = 1 by C1,∞ (H ) := {X ∈ C∞ (H ) :

n 

s j (X ) = O(ln(n)), n → ∞} ,

(2.2)

j=1

has a special interest. Note that since Π = { j −1 }∞ j=1 does not satisfy (1.26), the characterisation sn (X ) = O(n −1 ) is not true, see (1.27), (1.28). In this case, the equivalent norm can be defined on the ideal (2.2) as  1 s j (X ). 1 + ln(n) j=1 n

X 1,∞ := sup n∈N

(2.3)

By virtute of (1.25) and Remark 3 one gets that C1 (H ) ⊂ C1,∞ (H ) ⊂ C1+ε (H ) for any ε > 0 and that C1 (H )  A → A1,∞ = A1 ≥ A1+ε . (b) In contrast to the trace norm  · 1 on the positive cone C1,+ (H ), see Example 1, the map X → X 1,∞ on the positive cone C1,∞,+ (H ) is not linear although homogeneous. On the space l ∞ (N), there exists a state ω ∈ S (l ∞ (N)) such that the map n  s j (X )}∞ (2.4) X → Tr ω (X ) := ω({(1 + ln(n))−1 n=1 ) , j=1

is linear and verifies the properties of the (singular) trace for any X ∈ C1,∞ (H ). We construct ω in Section 3. This particular choice of the state ω defines the Dixmier trace on the space C1,∞ (H ), which is called, in turn, the Dixmier ideal, see e.g. [3, 4]. The Dixmier trace (2.4) is obviously continuous in topology defined by the norm (2.3). This last property is basic for discussion in Section 4 of the Trotter–Kato product formula in the  ·  p,∞ -topology, for p ≥ 1. Example 3 With non-increasing sequence of positive numbers π = {π j }∞ j=1 , π1 = 1, one can associate the symmetric norming function φπ given by φπ (ξ ) :=

∞ 

π j ξ ∗j ,

ξ ∈ cf .

j=1

The corresponding symmetrically-normed ideal is denoted by Cπ (H ) := Cφπ (H ). If the sequence π satisfies (1.24), then ideal Cπ (H ) does not coincide neither with C∞ (H ) nor with C1 (H ). If, in particular, π j = j −α , j = 1, 2, . . . , for 0 < α ≤ 1,

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then the corresponding ideal is denoted by C∞, p (H ), p = 1/α. The norm on this ideal is given by X ∞, p :=

∞ 

j −1/ p s j (X ) ,

p ∈ [1, ∞).

j=1

The symmetrically-normed ideal C∞,1 (H ) is called the Macaev ideal [8]. It turns out that the Dixmier ideal C1,∞ (H ) is dual of the Macaev ideal: C1,∞ (H ) = C∞,1 (H )∗ . Proposition 2.1 The space C1,∞ (H ) endowed with the norm  · 1,∞ is a Banach space. The proof is quite standard although tedious and long. We address the readers to the corresponding references, e.g. [8]. Proposition 2.2 The space C1,∞ (H ) endowed with the norm  · 1,∞ is a Banach ideal in the algebra of bounded operators L (H ). Proof To this end, it is sufficient to prove that if A and C are bounded operators, then B ∈ C1,∞ (H ) implies ABC ∈ C1,∞ (H ). Recall that singular values of the operator ABC verify the estimate s j (ABC) ≤ ACs j (B). By (2.3), it yields  1 s j (ABC) ≤ 1 + ln(n) j=1 n

ABC1,∞ = sup n∈N

(2.5)

 1 AC sup s j (B) = ACB1,∞ , n∈N 1 + ln(n) j=1 n

and consequently the proof of the assertion.



Recall that for any A ∈ L (H ) and all B ∈ C1 (H ) one can define a linear functional on C1 (H ) given by Tr H (AB). The set of these functionals {Tr H (A·)} A∈L (H ) is just the dual space C1 (H )∗ of C1 (H ) with the operator-norm topology. In other words, L (H ) = C1 (H )∗ , in the sense that the map A → Tr H (A·) is the isometric isomorphism of L (H ) onto C1 (H )∗ . With the help of the duality relation A|B := Tr H (AB) ,

(2.6)

one can also describe the space C1 (H )∗ , which is a predual of C1 (H ), i.e. its dual (C1 (H )∗ )∗ = C1 (H ). To this aim for each fixed B ∈ C1 (H ), we consider the functionals A → Tr H (AB) on L (H ). It is known that they are not all continuous linear functional on bounded operators L (H ), i.e. C1 (H ) ⊂ L (H )∗ , but they yield the entire dual only of compact operators, i.e. C1 (H ) = C∞ (H )∗ . Hence, C1 (H )∗ = C∞ (H ).

Trotter–Kato Product Formulae in Dixmier Ideal

405

Now, we note that under duality relation (2.6), the Dixmier ideal C1,∞ (H ) is the dual of the Macaev ideal: C1,∞ (H ) = C∞,1 (H )∗ , where C∞,1 (H ) = {A ∈ C∞ (H ) :

1 sn (A) < ∞} , n n≥1

(2.7)

see Example 3. By the same duality relation and by similar calculations one also (0) obtains that the predual of C∞,1 (H ) is the ideal C∞,1 (H )∗ = C1,∞ (H ), defined by n  (0) (H ) := {A ∈ C∞ (H ) : s j (A) = o(ln(n)), n → ∞} . (2.8) C1,∞ j≥1

By virtue of (2.2) (see Remark 4), the ideal (2.8) is not self-dual since (0) (0) C1,∞ (H )∗∗ = C1,∞ (H ) ⊃ C1,∞ (H ).

The problem which has motivated construction of the Dixmier trace in [5] was related to the question of a general definition of the trace, i.e. a linear, positive and unitarily invariant functional on a proper Banach ideal I(H ) of the unital algebra of bounded operators L (H ). Since any proper two-sided ideal I(H ) of L (H ) is contained in compact operators C∞ (H ) and contains the set K (H ) of finite-rank operators ((1.17), Section 1), domain of definition of the trace has to coincide with the ideal I(H ). Remark 5 The canonical trace Tr H (·) is nontrivial only on domain, which is the trace-class ideal C1 (H ), see Example 1. We recall that it is characterised by the property of normality: Tr H (supα Bα ) = supα Tr H (Bα ), for every directed increasing bounded family {Bα }α∈ of positive operators from C1,+ (H ). Note that every nontrivial normal trace on L (H ) is proportional to the canonical trace Tr H (·), see e.g. [6, 20]. Therefore, the Dixmier trace (2.4) : C1,∞  X → Tr ω (X ), is not normal. Definition 4 A trace on the proper Banach ideal I(H ) ⊂ L (H ) is called singular if it vanishes on the set K (H ). Since a singular trace is defined up to trace-class operators C1 (H ), then by Remark 5 it is obviously not normal.

3 Dixmier Trace Recall that only the ideal of trace-class operators has the property that on its positive cone C1,+ (H ) := {A ∈ C1 (H ) : A ≥ 0} the trace norm is linear since A + B1 = Tr (A + B) = Tr (A) + Tr (B) = A1 + B1 for A, B ∈ C1,+ (H ), see

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Example 1. Then, the uniqueness of the trace norm allows to extend the trace to the whole linear space C1 (H ). Imitation of this idea fails for other symmetricallynormed ideals. This problem motivates the Dixmier trace construction as a certain limiting procedure involving the  · 1,∞ -norm. Let C1,∞,+ (H ) be a positive cone of the Dixmier ideal. One can try to construct on C1,∞,+ (H ) a linear, positive and unitarily invariant functional (called trace T ) via extension of the limit (called Lim) of the sequence of properly normalised finite sums of the operator X singular values:  1 s j (X ) , X ∈ C1,∞,+ (H ). 1 + ln(n) j=1 n

T (X ) := Limn→∞

(3.1)

First, we note that since for any unitary U : H → U , the singular values of X ∈ C∞ (H ) are invariant: s j (X ) = s j (U X U ∗ ), it is also true for the sequence σn (X ) :=

n 

s j (X ) , n ∈ N .

(3.2)

j=1

Then, the Lim in (3.1) (if it exists) inherits the property of unitarity. Now, we comment that positivity: X ≥ 0 implies the positivity of eigenvalues {λ j (X )} j≥1 and consequently: λ j (X ) = s j (X ). Therefore, σn (X ) ≥ 0 and the Lim in (3.1) is a positive mapping. The next problem with the formula for T (X ) is its linearity. To proceed, we recall that if P : H → P(H ) is an orthogonal projection on a finite-dimensional subspace with dim P(H ) = n, then for any bounded operator X ≥ 0 the (3.2) gives σn (X ) = sup {Tr H (X P) : dim P(H ) = n} .

(3.3)

P

As a corollary of (3.3), one obtains the Horn–Ky Fan inequality σn (X + Y ) ≤ σn (X ) + σn (Y ) , n ∈ N,

(3.4)

valid, in particular, for any pair of bounded positive compact operators X and Y . For dim P(H ) ≤ 2n, one similarly gets from (3.3) that σ2n (X + Y ) ≥ σn (X ) + σn (Y ) , n ∈ N .

(3.5)

Motivated by (3.1), we now introduce Tn (X ) :=

1 σn (X ) , X ∈ C1,∞,+ (H ) , 1 + ln(n)

(3.6)

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407

and denote by Lim{Tn (X )}n∈N := Limn→∞ Tn (X ) the right-hand side of the functional in (3.1). Note that by (3.6), the inequalities (3.4) and (3.5) yield for n ∈ N Tn (X + Y ) ≤ Tn (X ) + Tn (Y ) ,

1 + ln(2n) T2n (X + Y ) ≥ Tn (X ) + Tn (Y ). 1 + ln(n) (3.7)

Since the functional Lim includes the limit n → ∞, the inequalities (3.7) would give a desired linearity of the trace T : T (X + Y ) = T (X ) + T (Y ) ,

(3.8)

if one proves that the Limn→∞ in (3.1) exists and finite for X, Y as well as for X + Y. To this end, we note that if the right-hand side of (3.1) is finite, then one has (3.8), and hence the Lim{Tn (X )}n∈N is a positive linear map Lim : l ∞ (N) → R, which defines a state ω ∈ S (l ∞ (N)) on the Banach space of sequences {Tn (·)}n∈N ∈ l ∞ (N). To make this definition more precise, we impose on the state ω the following conditions: (a) ω(η) ≥ 0 , for ∀η = {ηn ≥ 0}n∈N , (b) ω(η) = Lim{ηn }n∈N = lim ηn , if {ηn ≥ 0}n∈N is convergent . n→∞

By virtue of (a) and (b), the definitions (3.1) and (3.6) imply that for X, Y ∈ C1,∞,+ (H ), one gets T (X ) = ω({Tn (X )}n∈N ) = lim Tn (X ) ,

(3.9)

T (Y ) = ω({Tn (Y )}n∈N ) = lim Tn (Y ) ,

(3.10)

T (X + Y ) = ω({Tn (X + Y )}n∈N ) = lim Tn (X + Y ) ,

(3.11)

n→∞

n→∞

n→∞

if the limits in the right-hand sides of (3.9)–(3.11) exist. Now, to ensure (3.8) one has to select ω in such a way that it allows to restore the equality in (3.7), when n → ∞. To this aim, we impose on the state ω the condition of dilation D2 -invariance. Let D2 : l ∞ (N) → l ∞ (N) be dilation mapping η → D2 (η): D2 : (η1 , η2 , . . . ηk , . . .) → (η1 , η1 , η2 , η2 , . . . ηk , ηk , . . .) , ∀η ∈ l ∞ (N). (3.12) We say that ω is dilation D2 -invariant if for any η ∈ l ∞ (N) it verifies the property (c)

ω(η) = ω(D2 (η)).

(3.13)

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We shall discuss the question of existence, the dilation D2 -invariant states (the invariant means), on the Banach space l ∞ (N) in Remark 6. Let X, Y ∈ C1,∞,+ (H ). Then, applying the property (c) to the sequence η = {ξ2n := T2n (X + Y )}∞ n=1 , we obtain ω(η) = ω(D2 (η)) = ω(ξ2 , ξ2 , ξ4 , ξ4 , ξ6 , ξ6 , . . .) .

(3.14)

Note that if ξ = {ξn = Tn (X + Y )}∞ n=1 , then the difference of the sequences: D2 (η) − ξ = (ξ2 , ξ2 , ξ4 , ξ4 , ξ6 , ξ6 , . . .) − (ξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 , . . .) , converges to zero if ξ2n − ξ2n−1 → 0 as n → ∞. Then, by virtue of (3.11) and (3.14), this would imply ω({T2n (X + Y )}n∈N ) = ω(D2 ({T2n (X + Y )}n∈N )) = ω({Tn (X + Y )}n∈N ) , or by (3.11): limn→∞ T2n (X + Y ) = limn→∞ Tn (X + Y ), which by estimates (3.7) would also yield lim Tn (X + Y ) = lim Tn (X ) + lim Tn (Y ).

n→∞

n→∞

n→∞

(3.15)

Now, summarising (3.9), (3.10), (3.11) and (3.15), we obtain the linearity (3.8) of the limiting functional T on the positive cone C1,∞,+ (H ) if it is defined by the corresponding D2 -invariant state ω or a dilation-invariant mean. Therefore, to finish the proof of linearity it rests only to check that limn→∞ (ξ2n − ξ2n−1 ) = 0. To this end, we note that by definitions (3.2) and (3.6), one gets  1 1 σ2n−1 (X + Y ) − ln(2n) ln(2n − 1) 1 + s2n (X + Y ). ln(2n) 

ξ2n − ξ2n−1 =

(3.16)

Since X, Y ∈ C1,∞,+ (H ), we obtain that limn→∞ s2n (X + Y ) = 0 and that σ2n−1 (X + Y ) = O(ln(2n − 1)). Then taking into account that (1/ln(2n) − 1/ ln(2n − 1)) = o(1/ln(2n − 1)) one gets that for n → ∞ the right-hand side of (3.16) converges to zero. To conclude our construction of the trace T (·) we note that by linearity (3.8) one can uniquely extend this functional from the positive cone C1,∞,+ (H ) to the real subspace of the Banach space C1,∞ (H ), and finally to the entire ideal C1,∞ (H ). Definition 5 The Dixmier trace Tr ω (X ) of the operator X ∈ C1,∞,+ (H ) is the value of the linear functional (3.1): Tr ω (X ) := Limn→∞

σn (X ) , 1 + ln(n)

(3.17)

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where Limn→∞ is defined by a dilation-invariant state ω ∈ S (l ∞ (N)) on l ∞ (N) that satisfies the properties (a), (b) and (c). Since any self-adjoint operator X ∈ C1,∞ (H ) has the representation: X = X + − X − , where X ± ∈ C1,∞,+ (H ), one gets Tr ω (X ) = Tr ω (X + ) − Tr ω (X − ). Then, for arbitrary Z ∈ C1,∞ (H ), the Dixmier trace is Tr ω (Z ) = Tr ω (ReZ ) + iTr ω (Im Z ). Note that if X ∈ C1,∞,+ (H ), then definition (3.17) of Tr ω (·) together with definition of the norm  · 1,∞ in (2.3) readily imply the estimate Tr ω (X ) ≤ X 1,∞ , which, in turn, yields the inequality for arbitrary Z from the Dixmier ideal C1,∞ (H ): |Tr ω (Z )| ≤ Z 1,∞ .

(3.18)

Remark 6 A decisive for construction of the Dixmier trace Tr ω (·) is the existence of the invariant mean ω ∈ S (l ∞ (N)) ⊂ (l ∞ (N))∗ . Here, the space (l ∞ (N))∗ is dual to the Banach space of bounded sequences. Then, by the Banach–Alaoglu theorem, the convex set of states S (l ∞ (N)) is compact in (l ∞ (N))∗ in the weak* topology. Now, for any φ ∈ S (l ∞ (N)), the relation φ(D2 (·)) =: (D∗2 φ)(·) defines the dual D∗2 -dilation on the set of states. By definition (3.12), this map is such that D∗2 : S (l ∞ (N)) → S (l ∞ (N)), as well as continuous and affine (in fact, linear). Then, by the Markov–Kakutani theorem, the dilation D∗2 has a fix point ω ∈ S (l ∞ (N)) : D∗2 ω = ω. This observation justifies the existence of the invariant mean (c) for D2 dilation. Note that Remark 6 has a straightforward extension to any Dk -dilation for k > 2, which is defined similar to (3.12). Since dilations for different k ≥ 2 commute, the extension of the Markov–Kakutani theorem yields that the commutative family F = {D∗k }k≥2 has in S (l ∞ (N)) the common fix point ω = D∗2 ω. Therefore, Definition 5 of the Dixmier trace does not depend on the degree k ≥ 2 of dilation Dk . For more details about different constructions of invariant means and the corresponding Dixmier trace on C1,∞ (H ), see e.g., [3, 13]. Proposition 3.1 The Dixmier trace has the following properties: (a) For any bounded operator B ∈ L (H ) and Z ∈ C1,∞ (H ) one has Tr ω (Z B) = Tr ω (B Z ). (b) Tr ω (C) = 0 for any operator C ∈ C1 (H ) from the trace-class ideal, which is the closure of finite-rank operators K (H ) for the  · 1 -norm. (c) The Dixmier trace Tr ω : C1,∞ (H ) → C is continuous in the  · 1,∞ -norm. Proof (a) Since every operator B ∈ L (H ) is a linear combination of four unitary operators, it is sufficient to prove the equality Tr ω (ZU ) = Tr ω (U Z ) for a unitary operator U and moreover only for Z ∈ C1,∞,+ (H ). Then, the corresponding equality follows from the unitary invariance: s j (Z ) = s j (ZU ) = s j (U Z ) = s j (U ZU ∗ ) of singular values of the positive operator Z for all j ≥ 1.

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(b) Since C ∈ C1 (H ) yields C1 < ∞, definition (3.2) implies σn (C) ≤ C1 for any n ≥ 1. Then, by Definition 5 one gets Tr ω (C) = 0. Proof of the last part of the statement is standard. (c) Since the ideal C1,∞ (H ) is a Banach space and Tr ω : C1,∞ (H ) → C a linear functional, it is sufficient to consider continuity at X = 0. Then, let the sequence {X k }k≥1 ⊂ C1,∞ (H ) converges to X = 0 in  · 1,∞ -topology, i.e. by (2.3) lim X k 1,∞ = lim sup

k→∞

k→∞ n∈N

1 σn (X k ) = 0 . 1 + ln(n)

(3.19)

Since (3.18) implies |Tr ω (X k )| ≤ X k 1,∞ , the assertion follows from (3.19).



Therefore, the Dixmier construction gives an example of a singular trace in the sense of Definition 4.

4 Trotter–Kato Product Formulae in the Dixmier Ideal Let A ≥ 0 and B ≥ 0 be two non-negative self-adjoint operators in a separable Hilbert that space H and let the subspace H0 := dom(A1/2 ) ∩ dom(B 1/2 ). It may happen . dom(A) ∩ dom(B) = {0}, but the form-sum of these operators: H = A + B is well-defined in the subspace H0 ⊆ H . Kato proved in [12] that under these conditions the Trotter product formula  n s − lim e−t A/n e−t B/n = e−t H P0 , n→∞

t ≥ 0,

(4.1)

converges in the strong operator topology uniformly in t ∈ [0, T ], 0 < T < +∞ , where P0 denotes the orthogonal projection from H onto H0 . Moreover, in [12] it was also shown that the product formula is true not only for the exponential function e−x , x ≥ 0, but for a whole class of Borel measurable functions f (·) and g(·), which are defined on R+ 0 := [0, ∞) and satisfy the conditions: 0 ≤ f (x) ≤ 1, 0 ≤ g(x) ≤ 1,

f (0) = 1, g(0) = 1,

f  (+0) = −1, g  (+0) = −1.

(4.2) (4.3)

Namely, the main result of [12] says that besides (4.1) one also gets convergence τ − lim ( f (t A/n)g(t B/n))n = e−t H P0 , n→∞

t ≥ 0,

(4.4)

uniformly in t ∈ [0, T ], 0 < T < +∞, if topology τ = s. Product formulae of the type (4.4) are called the Trotter–Kato product formulae for functions (4.2), (4.3), which are called the Kato functions K . Note that K is closed with respect to the products of Kato functions.

Trotter–Kato Product Formulae in Dixmier Ideal

411

For some particular classes of the Kato functions we refer to [15, 25]. In the following, it is useful to consider instead of f (x)g(x) two Kato functions: g(x/2) f (x)g(x/2) and f (x/2)g(x) f (x/2) that produce the self-adjoint operator families F(t) := g(t B/2) f (t A)g(t B/2) and T (t) := f (t A/2)g(t B) f (t A/2), t ≥ 0. (4.5) Since [14] it is known that the lifting of the topology of convergence in (4.4) to the operator norm τ =  ·  needs more conditions on operators A and B as well as on the key Kato functions f, g ∈ K . One finds a discussion and more references on this subject in [25]. Here, we quote a result that will be used below for the Trotter–Kato product formulae in the Dixmier ideal C1,∞ (H ). Consider the class Kβ of Kato functions, which is defined in [10, 11] as given below:  (i) Measurable functions 0 ≤ h ≤ 1 on R+ 0 , such that h(0) = 1 and h (+0) = −1. (ii) For ε > 0 there exists δ = δ(ε) < 1, such that h(s) ≤ 1 − δ(ε) for s ≥ ε and

[h]β := sup s>0

|h(s) − 1 + s| < ∞ , for 1 < β ≤ 2 . sβ

The standard examples are h(s) = e−s and h(s) = (1 + a −1 s)−a , a > 0. Below we consider the class Kβ and a particular case of generators A and B, such that for the Trotter–Kato product formulae the estimate of the convergence rate is optimal. Proposition 4.1 ([11]) Let f, g ∈ Kβ with β = 2, and let A, B be non-negative selfadjoint operators in H such that the operator sum C := A + B is self-adjoint on domain dom(C) := dom(A) ∩ dom(B). Then, the Trotter–Kato product formulae converge for n → ∞ in the operator norm: [ f (t A/n)g(t B/n)]n − e−tC  = O(n −1 ) , [g(t B/n) f (t A/n)]n − e−tC  = O(n −1 ) ,

F(t/n)n − e−tC  = O(n −1 ) ,

T (t/n)n − e−tC  = O(n −1 ).

Note that corresponding to each formula, error bounds O(n −1 ) are equal up to coefficients {Γ j > 0}4j=1 and that each rate of convergence Γ j ε(n) = O(n −1 ), j = 1, . . . 4, is optimal. The first lifting lemma yields sufficient conditions that allow to strengthen the strong operator convergence to the  · φ -norm convergence in the symmetricallynormed ideal Cφ (H ). Lemma 4.2 Let self-adjoint operators: X ∈Cφ (H ), Y ∈ C∞ (H ) and Z ∈ L (H ). If {Z (t)}t≥0 is a family of self-adjoint bounded operators such that s − lim Z (t) = Z , t→+0

(4.6)

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then lim sup (Z (t/r ) − Z )Y X φ = lim sup X Y (Z (t/r ) − Z )φ = 0 , (4.7)

r →∞ t∈[0,τ ]

r →∞ t∈[0,τ ]

for any τ ∈ (0, ∞). Proof Note that (4.6) yields the strong operator convergence s − limr →∞ Z (t/r ) = Z , uniformly in t ∈ [0, τ ]. Since Y ∈ C∞ (H ), this implies lim sup (Z (t/r ) − Z )Y  = 0 .

(4.8)

r →∞ t∈[0,τ ]

Since Cφ (H ) is a Banach space with symmetric norm (1.13) that verifies Z X φ ≤ Z X φ , one gets the estimate (Z (t/r ) − Z )Y X φ ≤ (Z (t/r ) − Z )Y X φ ,

(4.9) 

which together with (4.8) give the proof of (4.7).

The second lifting lemma allows to estimate the rate of convergence of the Trotter– Kato product formula in the norm (1.19) of symmetrically-normed ideal Cφ (H ) via the error bound ε(n) in the operator norm due to Proposition 4.1. Lemma 4.3 Let A and B be non-negative self-adjoint operators on the separable Hilbert space H that satisfy the conditions of Proposition 4.1. Let f, g ∈ K2 be such that F(t0 ) ∈ Cφ (H ) for some t0 > 0. If Γt0 ε(n), n ∈ N is the operator-norm error bound away from t0 > 0 of the φ Trotter–Kato product formula for { f (t A)g(t B)}t≥0 , then for some Γ2t0 > 0 the function εφ (n) := {ε([n/2]) + ε([(n + 1)/2])}, n ∈ N defines the error bound away from 2t0 of the Trotter–Kato product formula in the ideal Cφ (H ): φ

[ f (t A/n)g(t B/n)]n − e−tC φ = Γ2t0 εφ (n) , n → ∞.

t ≥ 2t0 .

(4.10)

Here [x] := max{l ∈ N0 : l ≤ x}, for x ∈ R+ 0. Proof To prove the assertion for the family { f (t A)g(t B)}t≥0 , we use decompositions n = k + m, k ∈ N and m = 2, 3, . . . , n ≥ 3, for representation ( f (t A/n)g(t B/n))n − e−tC =   ( f (t A/n)g(t B/n))k − e−ktC/n ( f (t A/n)g(t B/n))m   + e−ktC/n ( f (t A/n)g(t B/n))m − e−mtC/n .

(4.11)

Since by conditions of lemma F(t0 ) ∈ Cφ (H ), definition (4.5) and representation f (t A/n)g(t B/n))m = f (t A/n)g(t B/n)1/2 F(t/n)m−1 g(t B)1/2 yield

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( f (t A/n)g(t B/n))m φ ≤ F(t0 )φ ,

(4.12)

for t such that t0 ≤ (m − 1)t/n ≤ (m − 1)t0 and m − 1 ≥ 1. Note that for self-adjoint operators e−tC and F(t) by Araki’s log-order inequality for compact operators [1], one gets for kt/n ≥ t0 the bound of e−ktC/n in the  · φ norm: (4.13) e−ktC/n φ ≤ F(t0 )φ . Since by Definitions 2 and 3 the ideal Cφ (H ) is a Banach space, and from (4.11)– (4.13), we obtain the estimate ( f (t A/n)g(t B/n))n − e−tC φ ≤ F(t0 )φ ( f (t A/n)g(t B/n))k − e−ktC/n 

(4.14)

+F(t0 )φ ( f (t A/n)g(t B/n))m − e−mtC/n  , for t such that (1 + (k + 1)/(m − 1))t0 ≤ t ≤ nt0 , m ≥ 2 and t ≥ (1 + m/k)t0 . Now, by conditions of lemma Γt0 ε(·) is the operator-norm error bound away from t0 , for any interval [a, b] ⊆ (t0 , +∞). Then, there exists n 0 ∈ N such that ( f (t A/n)g(t B/n))k − e−ktC/n  ≤ Γt0 ε(k) ,

(4.15)

for kt/n ∈ [a, b] ⇔ t ∈ [(1 + m/k)a, (1 + m/k)b] and ( f (t A/n)g(t B/n))m − e−mtC/n  ≤ Γt0 ε(m) ,

(4.16)

for mt/n ∈ [a, b] ⇔ t ∈ [(1 + k/m)a, (1 + k/m)b] for all n > n 0 . Setting m := [(n + 1)/2] and k = [n/2], n ≥ 3, we satisfy n = k + m and m ≥ 2, as well as, limn→∞ (k + 1)/(m − 1) = 1, limn→∞ m/k = 1 and limn→∞ k/m = 1. Hence, for any interval [τ0 , τ ] ⊆ (2t0 , +∞), we find that [τ0 , τ ] ⊆ [(1 + (k + 1)/ (m − 1))t0 , nt0 ] for sufficiently large n. Moreover, choosing [τ0 /2, τ/2] ⊆ (a, b) ⊆ (t0 , +∞), we satisfy [τ0 , τ ] ⊆ [(1 + m/k)a, (1 + m/k)b] and [τ0 , τ ] ⊆ [(1 + k/ m)a, (1 + k/m)b] again for sufficiently large n. Thus, for any interval [τ0 , τ ] ⊆ (2t0 , +∞) there is n 0 ∈ N such that (4.14)–(4.16) hold for t ∈ [τ0 , τ ] and n ≥ n 0 . Therefore, (4.14) yields the estimate ( f (t A/n)g(t B/n))n − e−tC φ ≤ Γt0 F(t0 )φ {ε([n/2]) + ε([(n + 1)/2])} , φ

(4.17)

φ

for t ∈ [τ0 , τ ] ⊆ (2t0 , +∞) and n ≥ n 0 . Hence, Γ2t0 := Γt0 F(t0 )φ and Γ2t0 εφ (·) is an error bound in the Trotter–Kato product formula (4.10) away from 2t0 in Cφ (H ) for the family { f (t A)g(t B)}t≥0 . The lifting Lemma 4.2 allows to extend the proofs for other approximants:  {g(t B) f (t A)}t≥0 , {F(t)}t≥0 and {T (t)}t≥0 .

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Now, we apply Lemma 4.3 in Dixmier ideal Cφ (H ) = C1,∞ (H ). This concerns the norm convergence (4.10), but also the estimate of the convergence rate for Dixmier traces: (4.18) |Tr ω (e−tC ) − Tr ω (F(t/n)n )| ≤ Γ ω εω (n). In fact, it is the same (up to Γ ω ) for all Trotter–Kato approximants: {T (t)}t≥0 , { f (t)g(t)}t≥0 and {g(t) f (t)}t≥0 . Indeed, since by inequality (3.18) and Lemma 4.3 for t ∈ [τ0 , τ ] and n ≥ n 0 , one has φ

|Tr ω (e−tC ) − Tr ω (F(t/n)n )| ≤ e−tC − F(t/n)n 1,∞ ≤ Γ2t0 ε1,∞ (n) , (4.19) we obtain for the rate in (4.18): εω (·) = ε1,∞ (·). Therefore, the estimate of the convergence rate for Dixmier traces (4.18) and for  · 1,∞ -convergence in (4.19) are entirely defined by the operator-norm error bound ε(·) from Lemma 4.3 and have the following form: ε1,∞ (n) := {ε([n/2]) + ε([(n + 1)/2])} , n ∈ N .

(4.20)

Note that for the particular case of Proposition 4.1, these arguments yield for (4.17) the explicit convergence rate asymptotics O(n −1 ) for the Trotter–Kato formulae and consequently, the same asymptotics for convergence rates of the Trotter–Kato formulae for the Dixmier trace (4.18), (4.19). Therefore, we proved in the Dixmier ideal C1,∞ (H ) the following assertion. Theorem 1 Let f, g ∈ Kβ with β = 2, and let A, B be non-negative self-adjoint operators in H such that the operator sum C := A + B is self-adjoint on domain dom(C) := dom(A) ∩ dom(B). If F(t0 ) ∈ C1,∞ (H ) for some t0 > 0, then the Trotter–Kato product formulae converge for n → ∞ in the  · 1,∞ -norm: [ f (t A/n)g(t B/n)]n − e−tC 1,∞ = O(n −1 ) , [g(t B/n) f (t A/n)]n − e−tC 1,∞ = O(n −1 ) , F(t/n)n − e−tC 1,∞ = O(n −1 ) ,

T (t/n)n − e−tC 1,∞ = O(n −1 ) ,

away from 2t0 . The rate O(n −1 ) of convergence is optimal in the sense of [11]. By virtue of (4.19), the same asymptotics O(n −1 ) of the convergence rate are valid for convergence of the Trotter–Kato formulae for the Dixmier trace: |Tr ω ([ f (t A/n)g(t B/n)]n ) − Tr ω (e−tC )| = O(n −1 ) ,

|Tr ω (F(t/n)n ) − Tr ω (e−tC )| = O(n −1 ) , |Tr ω (T (t/n)n ) − Tr ω (e−tC )| = O(n −1 ) ,

away from 2t0 . Optimality of the estimates in Theorem 1 is a heritage of the optimality in Proposition 4.1. Recall that, in particular, this means that in contrast to the Lie product formula for

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bounded generators A and B, the symmetrisation of approximants { f (t)g(t)}t≥0 , and {g(t) f (t)}t≥0 by {F(t)}t≥0 and {T (t)}t≥0 , does not yield (in general) the improvement of the convergence rate, see [11, 23] and discussion in [26]. We resume that the lifting Lemma 4.2 and 4.3 are a general method to study the convergence in symmetrically-normed ideals Cφ (H ) as soon as it is established in L (H ) in the operator-norm topology. The crucial is to check that for any of the key Kato functions (e.g. for {F(t)}t≥0 ) there exists t0 > 0 such that F(t)|t≥t0 ∈ Cφ (H ). Sufficient conditions for that one can be found in [16–18] or in [25]. Acknowledgements I am thankful to referee for useful remarks and suggestions.

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