Mathematics Of Open Quantum Systems, The: Dissipative And Non-unitary Representations And Quantum Measurements 9811241228, 9789811241222

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Control Number: 2021059101 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

THE MATHEMATICS OF OPEN QUANTUM SYSTEMS Dissipative and Non-Unitary Representations and Quantum Measurements Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-124-122-2 (hardcover) ISBN 978-981-124-123-9 (ebook for institutions) ISBN 978-981-124-124-6 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12395#t=suppl Typeset by Stallion Press Email: [email protected] Printed in Singapore

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In respectful memory of our beloved teachers M. Livˇsic and B. Pavlov. This book is dedicated to the memory of the remarkable Human Beings and Mathematicians Michail Samoilovich Livˇsic (M.S.) and Boris Sergeevich Pavlov (B.S.). Their pioneering research in the theory of nonself-adjoint operators and applications to scattering problems and system theory have attracted many researchers and made the present book possible. The Apostolic service to students, colleagues and the mathematical community provided by M.S. and B.S. was enormous and their good deeds will never be forgotten. The light of scientific accomplishments of M.S. and B.S. shine brightly and is succinctly described by the poetic word of Galina Volchek: Uhod ostavte Svet! to bolxe, qem ostats... to luqxe, qem prowats i vane˘ i, qem dat sovet... Uhod ostavte Svet - pered nim otstupit holod! Svet sobo˘ i zapolnit gorod... Dae esli vas tam net...

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ACKNOWLEDGMENTS

We are very grateful to M. Ashbaugh, W. Banks and S. Belyi for the invaluable help in the preparation of this monograph for publication. K.A.M. is indebted to A. B. Plachenov for stimulating discussions. The authors are also grateful to the three referees for their valuable comments and suggestions. K.A.M. is grateful to his wife Marina and son Konstantin for their patience and continuous support. E.T. is very thankful to his son Vladislav, daughter-in-law Elena, and granddaughters H and R for overwhelming continuous help, support and care. Research of K.A.M. was partially supported by the Simons collaboration grant 00061759.

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PREFACE

The main goal of this monograph is to develop a mathematical framework that accommodates an adequate description of the results of continuous observation of quantum systems. It is well known that a quantum observation/measurement always affects the system subject to it, and therefore the system can no longer be considered completely isolated. Instead, it should be treated as part of a more general system in which the presence of an observer/measuring device is taken into account. As such, the initial system should be regarded as an open quantum system that interacts with a part of the larger system. In the abstract setting, dealing with open systems assumes the presence of communication channels through which the interaction is carried out, both between parts of the system in question and with the outside world. A special case of open systems is the class of dissipative systems, the dynamics of which are governed by a strongly continuous semigroup of contractions. By applying the canonical dilation procedure, such open systems can always be viewed as a part of a larger closed system: the resulting Hamiltonian of the dilated system is chosen to be the self-adjoint dilatation of the generator of the semigroup, while the space of (pure) states of the larger system can be identified with the extended Hilbert space where the dilated operator has been realized as a self-adjoint operator. Despite its mathematical attractiveness, this dilation method is incompatible with the physical requirement that the total energy of the obtained large isolated system must be bounded from below. The transition in the opposite direction is usually associated with the reduction of the unitary evolution onto a subspace, in most cases lacks the semigroup property and needs a special consideration. The situation

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changes dramatically if the reduced description of the evolution is accompanied by continuous monitoring of the system. Under certain circumstances, the exponential decay of the states under continuous monitoring can be justified even if the energy distribution of the state is semi-bounded from below and the spectrum of the system is discrete. Therefore, within the continuous monitoring paradigm, one can bypass applying the Weisskopf-Wigner method that can only give an approximate description of the decay processes only. In the same time, the exponential decay under the continuous monitoring scenario fills in the gap between the quantum Zeno effect (Turing’s paradox) and the anti-Zeno phenomenon, and what is also important is, it gives a fresh look at the descent of dissipative operators and opens up new perspectives for their applications in quantum theory. In the present monograph we discuss two closely related topics. In the first part, based on the study of unitary invariants of symmetric operators, we provide the complete classification of the simplest dissipative solutions of the Heisenberg commutation relations (in the Weyl form). The second part of the monograph deals with mathematical problems of continuous monitoring of general quantum systems initially prepared in a pure state. Special attention is paid to the discussion of the behavior of massless particles on a ring under continuous monitoring.

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CONTENTS

Acknowledgments

vii

Preface

ix

Part 1: Representations of Operator Commutation Relations

2

1. Introduction

3

2. Preliminaries and Basic Definitions

13

3. The Commutation Relations and Character-Automorphic Functions

19

4. The Differentiation Operator on Metric Graphs

29

5. The Magnetic Hamiltonian

37

(z) 6. The Livˇsic Function s(D,D ˙ Θ)

49

7. The Weyl-Titchmarsh Function M(D,D (z) ˙ Θ)

57

8. The Model Dissipative Operators

65

9. The Characteristic Function S(D, ˙ D,D  Θ ) (z)

73

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10. The Transmission Coefficient and the Characteristic Function

81

11. Uniqueness Results

87

12. Dissipative Solutions to the CCR

93

13. Main Results

97

14. Unitary Dynamics on the Full Graph

115

Part 2: Continuous Monitoring, Quantum Measurements

119

15. Continuous Monitoring of the Quantum Systems 15.1 15.2 15.3 15.4

Quantum Zeno Effect . . . . Anti-Zeno Effect . . . . . . The Exponential Decay . . Frequent Measurements and Uncertainty Principle . . . . 15.5 Presto . . . . . . . . . . . .

. . . . . . the . . . .

. . . . . . . . . . . . . . . . . . . . . . . . Time-Energy . . . . . . . . . . . . . . . .

121 . . . . . . . 122 . . . . . . . 123 . . . . . . . 125 . . . . . . . 130 . . . . . . . 131

16. The Quantum Zeno versus Anti-zeno Effect Alternative

137

17. The Quantum Zeno Effect versus Exponential Decay Alternative

155

18. Preliminaries: Probabilities versus Amplitudes

177

19. Massless Particles on a Ring

179

20. Continuous Monitoring with Interference

181

20.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 21. Continuous Monitoring with No Interference

189

21.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 22. The Self-adjoint Dilation

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Contents

23. General Open Quantum Systems on a Ring

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xiii

201

23.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 23.2 Random Phase Method . . . . . . . . . . . . . . . . . . . 208 24. Operator Coupling Limit Theorems

211

Appendix A The Characteristic Function for Rank-One Perturbations

221

Appendix B

Prime Symmetric Operators

229

Appendix C

A Functional Model of a Triple

237

Appendix D The Spectral Analysis of the Model Dissipative Operator

241

Appendix E Transformation Laws

249

Appendix F The Invariance Principle

257

Appendix G The Operator Coupling and the Multiplication Theorem

263

Appendix H Stable Laws

269

References

273

Index

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One might still like to ask: “How does it work? What is the machinery behind the law?” No one has found any machinery behind the law. No one can “explain” any more than we have just “explained.” No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced. Richard Feynman, The Feynman Lectures on Physics, Volume III

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Chapter 1

INTRODUCTION

The classical Stone-von Neumann theorem [27, 83, 125, 127] states that the unitary representations of the canonical commutation relations (CCR) of Quantum Mechanics in the Weyl form [139] Ut Vs = eist Vs Ut

(1.1)

for strongly continuous, one-parameter groups of unitary operators Ut and Vs in a separable Hilbert space H are unitarily equivalent to a direct sum of copies of the unique irreducible system in the Hilbert space H = L2 (R) with (Vs f )(x) = f (x − s). For the history of the subject we refer to [115] where one can find a thorough discussion of the further generalizations initiated by G. W. Mackey in his ground-breaking paper [83], and the subsequent development in number theory due to A. Weil [137]. We refer also to the series of publications [13, 21, 38, 57, 72, 81, 113, 114] where the interested reader can find a truly extensive body of information on the subject. The CCR (1.1) can be reformulated in an equivalent infinitesimal form (see [27], [125]) as a relation for the self-adjoint generator A of the group Vs = eisA (Ut f )(x) = exp(ixt)f (x)

Ut AUt∗ = A + tI

and

on Dom(A),

t ∈ R,

(1.2)

or as the equality invoking the spectral measure E(dλ) of the self-adjoint operator A Ut E(δ)Ut∗ = E(δ − t), t ∈ R,

δ a Borel set.

It is worth mentioning that rewriting relations (1.1) in its infinitesimal (semi-Weyl) form (1.2) opens a way for the further developments and generalizations in various directions. 3

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In this book we choose a presentation line taking into account the following observation: the Stone-von Neumann uniqueness result [98] implies that if a self-adjoint operator A satisfies (1.2), then A always admits a symmetric restriction A˙ ⊂ A with deficiency indices (1, 1) such that the same commutation relations ˙ t∗ = A˙ + tI Ut AU

on

˙ Dom(A),

t ∈ R,

(1.3)

hold. Given the commutation relations for a symmetric operator (1.3), see Hypothesis 3.1, the following natural problems can be posed: (I) a) Characterize such symmetric operator solutions A˙ up to unitary equivalence; b) Provide an intrinsic characterization of those solutions.  to the infinitesimal Weyl (II) Find the maximal dissipative solutions A relations of the form  ∗=A  + tI Ut AU t

on

 Dom(A)

(1.4)

 ⊂ (A) ˙ ∗. such that A˙ ⊂ A Problem (I) b) was posed in [52] and we will refer to it as the JørgensenMuhly problem.  Notice that in this situation the semigroup Vs = eisA , s ≥ 0, generated  and the unitary group Ut = eitB , generated by the dissipative operator A by a self-adjoint operator B = B ∗ , satisfy the restricted Weyl commutation relations Ut Vs = eist Vs Ut ,

t ∈ R,

s ≥ 0.

(1.5)

More generally, one can ask to provide the complete classification (up to mutual unitary equivalence) of the pairs of corresponding generators  B) under the solely assumption that the generator A  is an extension of (A, a symmetric operator with arbitrary deficiency indices (m, n). Much progress has been achieved in this area of research (see [9, 50, 51, 52, 53, 54, 55, 121, 122], also see [88]). For instance, it is known that a semi-group satisfies the restricted Weyl relations if and only if the characteristic function of its generator has a particularly simple form [55, Theorem 20]. Moreover, in this case, the restricted Weyl system can be dilated to a canonical Weyl system in an extended Hilbert space [55, Theorem 15]. However, to the best of our knowledge, the complete classification of irreducible representations of the restricted commutation relations (1.5), even in the simplest case of deficiency indices (1, 1), has not been obtained yet.

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In this book we give the complete solution to problem (II) under the  of the semi-group Vs is a dissipative quasiassumption that the generator A selfadjoint extension [3] of a prime symmetric operator A˙ with deficiency indices (1, 1). As in the Stone-von Neumann theorem, we show that the  B) of generators is mutually unitarily equivalent to the “canonipair (A,  Q) on a metric graph Y, finite or infinite. Here P  stands for a cal” pair (P, dissipative differentiation (momentum) operator on Y with appropriate vertex boundary conditions and Q is the self-adjoint multiplication (position) operator on the graph Y. In contrast to the Stone-von Neumann uniqueness theorem, where the corresponding graph is just the real axis (with no reference vertices), the graph geometry of Y is more varied (see Definition 13.6 for the classification). Moreover, the knowledge of the complete set of unitary invariants of the solutions to the commutation relations determines not only the geometry of the metric graph Y but also the location of the central vertex of the graph. For instance, given a solution of the commutation relations on a metric graph, one obtains new series of unitarily inequivalent solutions by shifting the graph. Our approach is based on the detailed study of unitary invariants of operators such as the Livˇsic and/or Weyl-Titchmarsh functions associated with the pair of a symmetric operator and its self-adjoint (reference) extension as well as the characteristic function of a dissipative triple of operators. A comprehensive study of the concept of a characteristic function associated with various classes of non-selfadjoint operators, in particular, with applications to scattering theory, system theory and boundary value problems one can find in [10, 72, 73, 77, 79, 82, 95, 100, 105, 106] as well as in [1, 4, 7, 14, 15, 17, 18, 19, 24, 25, 68, 69, 78, 80, 81, 94, 96, 97, 99, 104, 107, 111, 118, 128, 129, 134, 140, 142]. The departure point for our study of commutation relations is structure Theorem 3.5. This result states that in the situation in question the characteristic function of a dissipative triple is either (i) a constant, or (ii) a singular inner function in the upper half-plane with “mass at infinity”, or (iii) the product of those two. The examples of differentiation operators (more precisely, a triple of those) with either a constant or entire characteristic function are known (see, e.g., [3, Ch. IX]). The construction of the model differentiation operator/triple in the general case (iii) can be achieved in the framework of operator coupling theory [87]. Notice that those examples of differentiation operators are the building blocks in our approach. In particular, addressing Problem (I) a), we obtain the complete

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classification of the symmetric operators with deficiency indices (1, 1) that solve the infinitesimal relations (1.3) up to unitary equivalence. We also provide an intrinsic characterization of the corresponding symmetric operator solutions to the commutation relations, thus giving a comprehensive answer to the Jørgensen-Muhly problem (I) b) [52] (see Remark 12.3). In the second part of the book we address the problem of how a closed quantum system becomes an open one under continuous monitoring. We refer to a selected list of monographs [11, 12, 17, 27, 29, 46, 74, 75] where different aspects of open/closed quantum systems theory are discussed. In addition, within the continuous monitoring paradigm, we study in detail theoretical foundations for complementarity of the Quantum Zeno and Exponential Decay scenarios in frequent quantum measurements experiments. Notice that considering the corresponding open dissipative quantum system as part of a larger closed system within the self-adjoint dilation scheme generates solutions to the canonical commutation relations (see Section 15.5 for more details). For the relevant background material, we refer to [6, 58, 93] and [36, 66, 138], respectively, and the references therein. In this context, we also want to mention the revolutionary paper by Gamow [39] who was the first to introduce quantum states with “complex” energies and, based on this concept, gave an explanation for the decay law for a quasi-stationary state. In the framework of our formalism we give a justification for the exponential decay scenario (under continuous monitoring) by recognizing the phenomenon as a variant of the Gnedenko-Kolmogorov 1-stable limit theorem. Having this link in mind, we obtain several principal results in quantum measurement theory. In particular, we show that a “typical smooth” state of a material (massive) particle under continuous monitoring is either a Zeno state or an anti-Zeno state (see Theorems 16.3 and 16.4). In contrast to that, for the systems of massless particles (fields) the situation is quite different: if the Hamiltonian of the system is given by the first order differentiation operator, then the quantum Zeno and exponential decay scenarios are complementary instead. In addition, it turns out that for that kind of systems, the decay rate is rather sensitive to the choice of a selfadjoint realization of the Hamiltonian on the metric graph, especially if the graph is not simply connected. From the point of view of physics, this phenomenon is a manifestation of the Aharonov-Bohm effect. That is, in the absence of the magnetic field, the magnetic potential by itself affects the magnitude of the decay rate in this case (see Theorem 17.4 and Chapter 18, eq. (20.14)). We also notice that the existence of states the decay

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rate of which is independent of the Aharonov-Bohm field is closely related to the search for dissipative operator solutions of the infinitesimal Weyl commutation relations (1.4). As an illustration, within continuous monitoring paradigm we discuss a Gedankenexperiment where the renowned exclusive and interference measurement alternatives in quantum theory can be rigorously analyzed. In addition, on the basis of an explicitly soluble model, we present a variant of the celebrated “double-slit experiment” in a way that is accessible for mathematicians (see Theorem 20.1, eq. (20.3), and Theorem 21.1, eq. (21.3)). We conclude our treatise by the discussion of limit theorems in the framework of operator coupling theory. More specifically, we introduce a new mode of convergence for dissipative operators (in distribution) and show that basic dissipative solutions to the commutation relations can be considered analogs of stable distributions in the orthodox probability theory. This observation, in our opinion, sheds some light on the foundations of dissipative quantum theory of open systems which our dear teachers M.S. and B.S. dedicated their scientific life to. The book is organized as follows. • In Chapter 2, following our work [86], we recall the concept of the Weyl-Titchmarsh and the Livˇsic functions associated with the pair of oper˙ A) where A˙ is a symmetric operator with deficiency indices (1, 1) ators (A, and A its self-adjoint extension. Given a dissipative quasi-selfadjoint exten of A, ˙ we also introduce a characteristic function associated with the sion A ˙ A,  A) and provide a characterization of the projection of the defitriple (A, ˙ ∗ − zI) along the ˙ ∗ − zI) onto the subspace Ker((A) ciency subspace Ker((A)  domain of the dissipative operator A in terms of the characteristic function of the triple (see Proposition 2.2). • In Chapter 3 we study symmetric operators with deficiency indices (1, 1) that satisfy the semi-Weyl commutation relations (1.3) (see Hypothesis 3.1). We show that if A˙ admits a self-adjoint extension A that solves the same commutation relations as the operator A˙ does, then the Livˇsic ˙ A) is identically zero in the upper halffunction associated with the pair (A, plane (see Theorem 3.3). In this case the corresponding Weyl-Titchmarsh √ function is z-independent and coincides with i = −1 on the whole upper half-plane. On the other hand, if A˙ has no self-adjoint extension that solves the same commutation relations but does have a dissipative extension solving (1.4), then Theorem 3.5 asserts that the characteristic function of the corresponding triple is periodic with a real period and has a particularly

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simple form. Notice that in the case where A˙ admits both a self-adjoint A  that solve (1.4), the characteristic function and a dissipative extension A ˙ A,  A) is a constant from the open unit disk. of the corresponding triple (A, As a corollary to Theorems 3.3 and 3.5 one gets an explicit representation for the Livˇsic function of an arbitrary symmetric operator satisfying Hypothesis 3.1 (see Corollary 3.7 for a precise statement). • In Chapter 4 we discuss first order symmetric differential operators on a metric graph Y assuming that the graph is in one of the following three cases: the graph (i) is the real axis with a reference point, (ii) is a finite interval, and finally, (iii) is obtained by attaching a finite interval to the real axis. Notice that the symmetric operators in question solve the semi-Weyl commutation relations (1.3) (see Remark 4.6). • In Chapter 5 we give a useful parameterization for the family of all selfadjoint as well as quasi-selfadjoint extensions of a symmetric differentiation discussed in Chapter 4. We also show that any self-adjoint realization DΘ of the symmetric differentiation operator D˙ on the graph Y in Case (iii) serves as a minimal self-adjoint dilation of an appropriate quasi-selfadjoint dissipative differentiation operator on the metric graph in Case (ii) (see Theorem 5.7). • In Chapter 6 we examine the Livˇsic functions associated with the pair ˙ DΘ ) where D˙ is the symmetric operator on the metric graph Y (in (D, Cases (i)–(iii)) and DΘ is its arbitrary reference self-adjoint extension. In particular, we provide a complete solution of Problem (I) a) (see Corollary 6.4). • Chapter 7 is devoted to the comprehensive study of the corresponding Weyl-Titchmarsh functions. • In Chapter 8, given the graph Y in one of the Cases (i)–(iii), we introduce maximal dissipative differentiation (model) operators the domains of which are invariant with respect to the group of gauge transformations. Those are the prototypes of general dissipative solutions to the commutation relations (1.4). Notice that in Cases (i) and (iii) the corresponding boundary conditions are determined not only by the geometry of the metric graph Y but also by its peculiar “conductivity” exponent k, 0 ≤ k < 1, which we call the quantum gate coefficient. We also give an explicit informal description of the associated contraction semi-groups generated by these operators. A visualization of the corresponding dissipative dynamics is shown on Figures 1–3. • In Chapter 9 we evaluate the characteristic function of the triples associated with the model dissipative operators discussed in Chapter 8 and prove the converse of important structure Theorem 3.5 (see Theorem 9.7).

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• In Chapter 10, following the line of research initiated by Livˇsic [78], who was the first to discover the connection between the Heisenberg scattering matrix and the characteristic function of a dissipative operator (also see [1, 14, 72]), we focus on the discussion of general quasi-selfadjoint dissipative differentiation operators d on the metric graph Y in Case (ii). In particular, we relate the characteristic function of the corresponding triple with the reciprocal of the transmission coefficient in a scattering problem  the magnetic Hamiltonian. Using the fundafor a self-adjoint dilation of d, mental relation (C.1) in Appendix C between the characteristic and Livˇsic functions, we also get a representation for the transmission coefficient via the Livˇsic function and the von Neumann parameter of the triple (see Corollary 10.4 combined with eq. (10.3)). Notice that both the differentiation operator d and the magnetic Hamiltonian solve commutation relations 1.4 with respect to a discrete subgroup of the unitary group Ut . • In Chapter 11 we discuss uniqueness results for symmetric operators that commute with a unitary. In particular, we show that the unitary group Ut from (1.4) is uniquely determined (up to a character) unless the dissipative solution to the commutation relations has point spectrum filling in the whole upper half-plane. In the latter (exceptional) case the representation t → Ut is reducible and splits into the orthogonal sum of two irreducible representations uniquely determined up to a unitary character. • In Chapter 12 we characterize maximal dissipative solutions to the semi-Weyl commutation relations (1.4) that extend a prime symmetric operator satisfying Hypothesis 3.1 (see Theorem 12.1). As a by-product of these considerations, we provide an intrinsic characterization of these symmetric operators thus solving the Jørgensen-Muhly problem in the particular case of the deficiency indices (1, 1). • In Chapter 13 we study solutions to the restricted Weyl commutation  relations (1.5) for a unitary group Ut = eiBt and a semi-group Vs = eiAs of  of Vs is a quasicontractions. Under the assumption that the generator A selfadjoint extension of a prime symmetric operator with deficiency indices  B) up to mutual unitary (1, 1) we characterize the pairs of generators (A,  and B can be equivalence. In particular, we show that the generators A  realized as the dissipative differentiation operator P on the metric graph Y in one of the Cases I∗ , I–III (see Definition 13.6 for the classification) with appropriate boundary conditions at the vertex of the graph and the multiplication operator Q on the graph, respectively (see Theorem 13.12). In contrast to the classic Stone-von Neumann uniqueness result, the pairs  Q) are not unitarily equivalent for different choices of the center of the (P, graph. However, the uniqueness theorem in the self-adjoint case can be

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adopted to fit the format of its non-selfadjoint counterpart (see Theorem 13.14). In particular, we get the full version of the Stone-von Neumann Theorem (see Corollary 13.16). • In Chapter 14 we consider a family of unitary solutions to the canonical Weyl commutation relations (1.1) on the full metric graph X obtained by composing two identical copies of the metric graph (−∞, μ) (μ, ∞), μ ∈ R is a parameter. The corresponding self-adjoint momentum differentiation operator P, the generator of the group Vs of shifts, is determined by the boundary conditions at the vertex μ with the bond S-matrix given by √   k − 1 − k2 S= √ for some 0 ≤ k < 1, 1 − k2 k and the generator Q of the second group Ut is just the position operator  of Vs is a quasi-selfadjoint on the graph X. As long as the generator A extension of a prime symmetric operator with deficiency indices (1, 1), the structure Theorem 14.1 shows that up to mutual unitary equivalence any solution to the restricted Weyl commutation relations (1.5) can be obtained by an appropriate compression of the unitary groups Ut and Vs onto some subspace K of the Hilbert space L2 (X). In fact, the subspace K coincides with the Hilbert space L2 (Y) where Y ⊂ X is a subgraph of X in one of the three canonical cases discussed above. Notice that the subspace K reduces the multiplication group Ut = eitQ and is coinvariant for the group of shifts Vs = eisP . For a pictorial description of the corresponding unitary dynamics on the full metric graph X we refer to Figure 4. In the second part of the book we discuss applications to decay phenomena in quantum systems theory. • In Chapter 15 we recall the concept of continuous monitoring of a quantum system and describe possible ex-post monitoring scenarios: the Quantum Zeno and Anti-Zeno effects as well as the Exponential Decay phenomenon in frequent measurements theory. We also provide an example of unstable (pure) states of the quantum oscillator that decay exponentially under continuous monitoring of the system which eventually confirms the conclusions of the phenomenological Weisskopf-Wigner theory of decay [138]. • In Chapter 16 we discuss the quantum Zeno versus Anti-Zeno effect alternative for massive particles. Applying 1/2- and 3/2-stable limit theorems we also show that continuous monitoring in nonlinear time scales leads to exponential decay of some appropriate states in a quantum systems the Hamiltonian of which is given by the free Sch¨odinger operator on

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the semi-axis with (various) Robin type boundary conditions at the origin (see Theorems 16.3 and 16.4). • In Chapter 17 we examine the exponential decay phenomenon for “zero-mass” systems where the exponential decay typically alternates with the quantum Zeno scenario (see Theorem 17.4). • In Chapter 18 we recall main concepts for the “exclusive” versus “interference” alternatives theory going back to the celebrated two slit experiment in quantum mechanics. • In Chapter 19 we consider a model of a quantum system on a ring that describes a motion of a (relativistic) massless particle and set the stage for monitoring such systems within the continuous observation paradigm. • In Chapter 20 we show that in some cases continuous monitoring of the model quantum system triggers emission of particle, see Theorem 20.1. For instance, this phenomenon occurs if the initial state has a unique jumppoint discontinuity on the ring. The magnitude of the state decay in this case can be theoretically predicted as if the quantum particle were a wave. That is, the particle interferes with itself at the point of observations (where the wave functions has a jump) and the results of this interference can informally be explained in the framework of the “interference” alternatives theory. We also illustrate some features in the state decay under continuous monitoring of a massless particle moving along the Aharonov-Bohm ring (see Corollary 20.2 and eq. (20.14), cf. [59, 67]). • In Chapter 21 we discuss results of continuous monitoring of open quantum systems on a ring under the hypothesis that the time evolution of the system is governed by a semi-group of contractions. If the initial state of the system satisfies the radiation condition (21.4), that is, it belongs to the domain of the evolution generator, then the decay rate can easily be computed using purely classical considerations, as if the quantum particle were a classical particle (see Theorem 21.1, Corollary 21.2 and the related discussion). • In Chapter 22 we explicitly describe the self-adjoint dilation of the dissipative generator of the open system on the ring. • In Chapter 23 we consider more general states of the open quantum system on the ring. The main result (see Theorem 23.1) states that the emission rate splits into the sum of two terms. One of the terms is due to the interference of the particle with itself at the point of observations. The second source of emission is caused by inelastic collision of the quantum particle with the point “defect” (membrane) at the observation point where the corpuscular nature of the quantum particle is fully manifested.

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• In Chapter 24 we introduce the concept of convergence of dissipative operators in distribution and prove several limit theorems with respect to multiple coupling of an operator with itself. We also show that the generator of the nilpotent semi-group, one of the building blocks of the restricted commutation relations theory, can be obtained as the limit of appropriately normalized n-fold couplings of almost arbitrary dissipative operator (see Theorem 24.3). The book has eight appendices where, for the reader’s convenience, one can find relevant information scattered in the literature. Some of the results presented there are new. In Appendix A we recall the notion of the Weyl-Titchmarsh functions as well as a characteristic function for rank-one dissipative perturbations of a self-adjoint operator. We also provide the corresponding uniqueness result (see Theorem A.5). In Appendix B we collect necessary background material from the theory of symmetric operators. In Appendix C we present a functional model for triples of operators following our work [86] (also see [4, Ch. 10, Sec. 10.4, p. 357] and [68, 89, 116, 117, 133]). Appendix D contains a discussion aimed at the spectral analysis of model dissipative triple of operators. In Appendix E we study the dependence of the Weyl-Titchmarsh, Livˇsic, and characteristic function under affine transformation of the operators. In Appendix F we discuss the invariance principle for affine transformations of a dissipative operator. In Appendix G we recall the concept of an operator coupling of two dissipative operators and discuss the corresponding multiplication theorem. In Appendix H one can find a brief discussion of stable laws in probability theory and the formulation of the general Gnedenko-Kolmogorov limit theorem. Some words about notation: The domain of a linear operator K is denoted by Dom(K), its range by Ran(K), and its kernel by Ker(K). The restriction of K to a given subset C of Dom(K) is written as K|C . We write ρ(K) for the resolvent set of a closed operator K on a Hilbert space, and K ∗ stands for the adjoint operator of K if K is densely defined.

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Chapter 2

PRELIMINARIES AND BASIC DEFINITIONS Let A˙ be a densely defined symmetric operator with deficiency indices (1, 1) and A its self-adjoint (reference) extension. Following [23, 41, 43, 73, 86] recall the concept of the Weyl-Titchmarsh ˙ A). and Livˇsic functions associated with the pair (A, ˙ ∗ ∓ iI), Suppose that (normalized) deficiency elements g± ∈ Ker((A) g±  = 1, are chosen in such a way that g+ − g− ∈ Dom(A).

(2.1)

Consider the Weyl-Titchmarsh function M (z) = ((Az + I)(A − zI)−1 g+ , g+ ),

z ∈ C+ ,

(2.2)

and the Livˇsic function s(z) =

z − i (gz , g− ) · , z + i (gz , g+ )

z ∈ C+ ,

(2.3)

˙ A). Here gz ∈ Ker((A) ˙ ∗ − zI), z ∈ C+ . Paying associated with the pair (A, tribute to historical justice it is worth mentioning that the function M (z) has been introduced by Donoghue in [23]. However, as one can see from [41, eq. (5.42)], it is elementary to express M (z) in terms of the classical Weyl-Titchmarsh function which explains the terminology we use. Clearly, the Weyl-Titchmarsh function M (z) does not depend on the concrete choice of the normalized deficiency element g+ . We also remark that if  ˙ ∗ ∓ iI), g± ∈ Ker((A)

13

 g±  = 1,

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is any deficiency elements such that   − g− ∈ Dom(A), g+  = Θg± for some unimodular factor Θ. Therefore, from then necessarily g± (2.3) it follows that the Livˇsic function does not depend on the choice of the deficiency elements g± (whenever (2.1) holds). However it may and in most of the cases does depend on the reference operator A. As far as the Weyl-Titchmarsh functions are concerned we also refer to the related concept of a Q-function introduced in [64] and discussed in [65]. Recall the important relationship between the Weyl-Titchmarsh and Livˇsic functions [86]

s(z) =

M (z) − i , M (z) + i

z ∈ C+ .

(2.4)

 = (A)  ∗ is a maximal dissipative extension of A, ˙ Next, suppose that A  f ) ≥ 0, f ∈ Dom(A).  Im(Af,  is automatically quasiSince A˙ is symmetric, its dissipative extension A selfadjoint [112, 129] (also see [4, 86]), that is,  ⊂ (A) ˙ ∗, A˙ ⊂ A and hence,  g+ − κg− ∈ Dom(A)

for some |κ| < 1.

(2.5)

By definition, we call κ the von Neumann parameter of the triple ˙  A). (A, A, Remark 2.1. Likewise, one can think of the von Neumann parameter κ  and the pair {g+ , g− } of being determined by the dissipative operator A ∗ ˙  and normalized deficiency elements g± ∈ Ker((A) ∓ iI). Indeed, given A {g+ , g− } there are a unique κ satisfying (2.5) and a unique self-adjoint refer˙ A,  A) ence extension A of A˙ such that (2.1) holds. Therefore, the triple (A,  is uniquely determined by the knowledge of A and {g+ , g− }. Cleary, κ coinwhich proves cides with the von Neumann parameter of the triple κ(A, ˙ A,A)  the claim. Given (2.1) and (2.5), consider the characteristic function S(z) = ˙ A,  A) (see [86], cf. [77]) (z) associated with the triple (A, S(A, ˙ A,A)  S(z) =

s(z) − κ , κ s(z) − 1

z ∈ C+ ,

(2.6)

˙ A). (z) is the Livˇsic function associated with the pair (A, where s(z) = s(A,A) ˙

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By (2.4) and (2.6), one also gets the representation for the characteristic function via the Weyl-Titchmarch function as S(z) = −

1+κ 1 − κ M (z) − i 1−κ · . 1+κ 1 − κ M (z) + i 1−κ

(2.7)

˙ A,  A), one can always find a basis g± We remark that given a triple (A, ∗ ∗ ˙ Ker(A˙ + iI) such that in the subspace Ker(A˙ − iI)+ g±  = 1,

˙ ∗ ∓ iI), g± ∈ Ker((A)

g+ − g− ∈ Dom(A) and  g+ − κg− ∈ Dom(A)

for some |κ| < 1.

In this case, the von Neumann parameter κ can explicitly be evaluated ˙ A,  A) as in terms of the characteristic function of the triple (A, κ = S(A, (i). ˙ A,A) 

(2.8)

Hence, as it follows from (2.6), the Livˇsic function associated with the pair ˙ A) admits the representation (A, s(A,A) (z) = ˙

S(z) − κ , κ S(z) − 1

z ∈ C+ .

(2.9)

In particular, S(A, (z) = −s(A,A) (z), ˙ ˙ A,A) 

z ∈ C+ ,

˙ A,  A) vanishes. whenever the von Neumann parameter κ of the triple (A, The following proposition provides a curious characterization for the ˙ ∗ − zI) onto the subspace projection of the deficiency subspace Ker((A) ∗  ˙ Ker((A) − zI) along Dom(A) in terms of the characteristic function of the triple. ˙ A,  A) be a triple. Suppose that the deficiency Proposition 2.2. Let (A, ∗ ˙ − zI) and gz ∈ Ker((A) ˙ ∗ − zI), gz  = gz  = 0 elements gz ∈ Ker((A) are chosen in such a way that  gz − γ(z)gz ∈ Dom(A),

z ∈ C+ .

Then |γ(z)| = |S(z)|,

z ∈ C+ ,

˙ A,  A). where S(z) is the characteristic function of the triple (A,

(2.10)

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˙ A,  A) is mutuProof. By Theorem C.1 in Appendix C, the triple (A, ˙ B,  B) in the Hilbert space ally unitarily equivalent to the model triple (B, L2 (R; dμ) given by (C.6), (C.7) and (C.8) in Appendix C. Here μ(dλ) is the measure from the representations    1 λ − (2.11) M (z) = dμ(λ), z ∈ C+ , λ − z λ2 + 1 R ˙ A). for the Weyl-Titchmarsh function associated with the pair (A, In view of Remark C.2 in Appendix C it suffices to show that if 1 1  −α ∈ Dom(B), λ−z λ−z

(2.12)

then α = |S(z)|,

z ∈ C+ .

(2.13)

We claim that λ M (z) 1 = 2 + + f (λ), λ−z λ + 1 λ2 + 1 ˙ where f ∈ Dom(B). Indeed,    f (λ)dμ(λ) = R

R

1 λ − 2 λ−z λ +1

= M (z) − M (z) = 0,



 dμ(λ) − M (z)

R

dμ(λ) λ2 + 1

z ∈ C+ .

Here we have used (2.11) and the normalization condition (C.5) in Appendix C. From the characterization of the domain of the symmetric operator B˙ ˙ proving the given by (C.7) in Appendix C, it follows that f ∈ Dom(B), claim. Next, we have 1 (1 − α)λ M (z) − αM (z) 1 −α = 2 + + h(λ), λ−z λ−z λ +1 λ2 + 1 ˙ where h ∈ Dom(B). Recall that by (C.8) in Appendix C,

(2.14)

1 1  −κ ∈ Dom(B), λ−i λ+i ˙ B,  B). Since the where κ is the von Neumann parameter of the triple (B, ˙  ˙  triples (A, A, A) and (B, B, B) are mutually unitarily equivalent, κ coincides ˙ A,  A) as well. with von Neumann parameter of the triple (A,

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Since (1 − α)λ M (z) − αM (z) 1 1 + = [(1 − α) − i(M (z) − αM (z)] λ2 + 1 λ2 + 1 2 λ−i 1 1 , + [(1 − α) − i(M (z) − αM (z)] 2 λ+i in view of (2.14), the requirement (2.12) connecting the characteristic functions S(z) of the triple and the Weyl-Tichmarsch function yields the relation −κ =

(1 − α) + i(M (z) − αM (z) , (1 − α) − i(M (z) − αM (z)

z ∈ C+ .

Hence, α=

F (z) , F (z)

where F (z) = 1 + κ + iM (z)(1 − κ). On the other hand, from the relation (2.7) it follows that S(z) =

F (z) F (z)

and therefore |α| = |S(z)|,

z ∈ C+ ,

which proves (2.13) and completes the proof of the proposition.



Remark 2.3. Notice that given the deficiency elements gz  = gz  = 0, z ∈ C+ , one can always find γ(z) such that (2.10) holds. Indeed, from the  it follows that one can find definition of the quasi-selfadjoint extension A α and β such that  0 = αgz + βgz ∈ Dom(A).  and Therefore, it suffices to show that α = 0. Otherwise, gz ∈ Dom(A) hence ˙ ∗ gz , gz ) = Im(zgz , gz ) = Im(z)gz 2 < 0,  z , gz ) = Im((A) Im(Ag  is a dissipative operator. which contradicts the requirement that A Definition 2.4. We call the harmonic function ΓA(z) = log |γ(z)| = log |S(z)|,

z ∈ C+ \ {zk | S(zk ) = 0},

(2.15)

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 the von Neumann (logarithmic) potential of the dissipative operator A. Here γ(z) is the function referred to in Proposition 2.2. Remark 2.5. We remark that under the assumption that the symmetric operator A˙ is prime, both the Weyl-Titchmarsh function M (z) and ˙ A). the Livˇsic function s(z) are complete unitary invariants of the pair (A, Moreover, in this case, the characteristic function S(z) is a complete unitary ˙ A,  A) (see [86], also see Theorem C.1 in Appendix invariant of the triple (A, ˙ A,  A) in the C). Notice that if the symmetric operator A˙ from a triple (A,  ˙ Hilbert space H is not prime and A is its prime part in a subspace H ⊂ H,  H , A|H ) have the same character˙ A,  A) and (A| ˙ H , A| then the triples (A, istic function (see Theorem B.2 in Appendix B for details). As it follows from Lemma E.1 and Proposition E.2 in Appendix E, the absolute value of the Livˇsic function |s(z)| is a complete unitary invarinat of the symmetric operator A˙ while |S(z)| is a (complete) unitary invariant of  In particular, in view of (2.8), the von the maximal dissipative operator A. ˙ A,  A), is a unitary invariant of the triple (A, Neumann parameter κ(A, ˙ A,A)  while its absolute value  = |κ ˙  | κ (A) (A,A,A)

(2.16)

 is a well defined unitary invariant of the dissipative operator A. We also refer to [73] where it was shown that the knowledge or s(z) and S(z) (up to a unimodular constant factor), equivalently, |s(z)| and |S(z)| characterizes the symmetric operator A˙ and is maximal dissipative  respectively, up to unitary equivalence (whenever A˙ is a prime extension A, operator). We also notice that the knowledge of the von Neumann logarithmic  up to unitary equivpotential ΓA(z) determines the dissipative operator A alence.

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Chapter 3

THE COMMUTATION RELATIONS AND CHARACTER-AUTOMORPHIC FUNCTIONS

Throughout this chapter we assume the following hypothesis. Hypothesis 3.1. Assume that A˙ is a symmetric operator with deficiency indices (1, 1) and R  t → Ut a strongly continuous unitary group. Suppose, ˙ is Ut -invariant and the commutation relations in addition, that Dom(A) ˙ t = A˙ + tI Ut∗ AU

on

˙ Dom(A),

t ∈ R,

(3.1)

hold. It is well known that under Hypothesis 3.1 the symmetric operator A˙ has either a) a self-adjoint A or/and b) a non-selfadjoint maximal dissipative  satisfying the same commutation relations (cf. [52, Theorem extension A ] 15 , also see [88, Theorem 5.4]). It is worth mentioning that the existence of a quasi-selfadjoint extension of A˙ that solves (3.1) required in Hypothesis 3.1 is a consequence of the Lefschetz fixed point theorem for flows on manifolds: Proposition 3.2 (see, e.g., [136, Theorem 6.28]). If M is a closed oriented manifold such that the Euler characteristic χ(M) of M is not zero, then any flow on M has a fixed point.  is a maximal dissipative Indeed, if A˙ satisfies Hypothesis 3.1 and A ∗  ˙ Since the set of maximal ˙  extension of A, then At = Ut AUt also extends A. 19

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dissipative extensions of A˙ is in one-to-one correspondence with the closed unit disk D, and  → A t A

(3.2)

determines a flow ϕ(t, ·) on D (the continuity of the flow can easily be established (see, e.g., [88])). By the Lefschetz theorem, the flow ϕ(t, ·) has a fixed point either on the boundary of the unit disk or in its interior. First, consider the case where the flow ϕ(t, ·) has a fixed point on the boundary of the unit disk and therefore the commutation relations (3.1) have a self-adjoint solution. Theorem 3.3. Assume Hypothesis 3.1. Suppose that A is a self-adjoint extension of the symmetric operator A˙ such that Ut∗ AUt = A + tI

on

Dom(A).

(3.3)

˙ A) has the Then the Weyl-Titchmarsh function M (z) of the pair (A, form M (z) = i,

z ∈ C+ .

˙ A) Equivalently, the Livˇsic function s(z) associated with the pair (A, vanishes identically in the upper half-plane, s(z) = 0,

z ∈ C+ .

Proof. Introducing the family of bounded operators Bt = Ut (A − iI)(A − iI + tI)−1 ,

t ∈ R,

it is easy to see that the family R  t → Bt forms a strongly continuous (commutative) group. Indeed, using the commutation relation (3.3) for the self-adjoint operator A one obtains Bt Bs = Ut (A − iI)(A + (−i + t)I)−1 Us (A − iI)(A + (−i + s)I)−1 = Ut Us Us∗ (A − iI)(A − iI + tI)−1 Us (A − iI)(A + (−i + s)I)−1 = Ut+s (A − iI + sI)(A − iI + (t + s)I)−1 (A − iI)(A − iI + sI)−1 = Ut+s (A − iI + (t + s)I)−1 (A − iI) = Bt+s ,

s, t ∈ R.

˙ ∗ − iI), g+  = 1, be a normalized deficiency element Let g+ ∈ Ker((A) ˙ Since of A. ˙ ∗ − (i − t)I) (A − iI)(A − (i − t)I)−1 g+ ∈ Ker((A)

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21

and ˙ ∗ − (i − t)I) = Ker((A) ˙ ∗ − iI), Ut Ker((A) ˙ ∗ − iI) is invariant for one concludes that the deficiency subspace Ker((A) Bt , t ∈ R. Therefore, the restriction Bt on the deficiency subspace is a continuous one-dimensional representation (a one-dimensional representation of a strongly continuous group is continuous). Hence, Bt g + = b t g +

for some

b ∈ C.

From the definition of the Weyl-Titchmarsh function (2.2) it follows that d M (z) = ((A2 + I)(A − zI)−2 g+ , g+ ), dz One computes M  (z) :=

z ∈ C+ .

|b|2t M  (i − t) = |b|2t ((A2 + I)(A − iI + tI)−2 g+ , g+ ) = ((A2 + I)(A − iI + tI)−2 Bt g+ , Bt g+ ) = (Bt∗ (A2 + I)(A − iI + tI)−2 Bt g+ , g+ ) = ((A + iI)(A + iI + tI)−1 Ut∗ (A2 + I) × (A − iI − tI)−2 Bt g+ , g+ ) = ((A + iI)(A + iI + tI)−1 Ut∗ (A2 + I)(A − iI − tI)−2 Ut × (A − iI)(A − iI + tI)−1 g+ , g+ ) = ((A + iI)(A + iI + tI)−1 ((A + tI)2 + I)(A − iI)−2 (A − iI) × (A − iI + tI)−1 g+ , g+ ) = ((A + iI)(A − iI)−1 g+ , g+ ) = (g+ , (A − iI)(A + iI)−1 g+ ). Hence, M  (i − t) = |b|−2t (g+ , g− ),

t ∈ R,

where ˙ ∗ + iI). g− = (A − iI)(A + iI)−1 g+ ∈ Ker((A) Denote by μ(dλ) the spectral measure of the element g+ , that is, μ(dλ) = (EA (dλ)g+ , g+ ),

(3.4)

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where EA (dλ) is the projection-valued spectral measure of the self-adjoint operator A from the spectral decomposition  λdEA (λ). A= R

Since

 M (z) =

and therefore M  (i − t) =

R

 R

λz + 1 dμ(λ), λ−z λ2 + 1 dμ(λ), (λ − t − i)2

one gets the estimate  λ2 + 1  dμ(λ) |M (i − t)| ≤ 2 R (λ + t) + 1   λ2 + 1 λ2 + 1 dμ(λ) + dμ(λ) = 2 2 {|λ|≤2|t|} (λ + t) + 1 {|λ|>2|t|} (λ + t) + 1   1 2 ≤ (4t + 1)μ{|λ| ≤ |t|} + + 1 μ{|λ| > |t|}. (1 − 12 )2 Therefore, M  (i − t) = O(t2 ) as

t → ∞.

(3.5)

Combining (3.5) with (3.4) shows that either (g+ , g− ) = 0 or |b| = 1. In the first case, i.e. (g+ , g− ) = 0, M (z) is a constant function and hence M (z) = M (i) = i,

z ∈ C.

If |b| = 1, we have M  (i − t) = (g+ , g− ),

t ∈ R.



In particular, M (z) = (g+ , g− ) for all z ∈ C+ and hence M (z) = (g+ , g− )z + C,

z ∈ C+ ,

for some constant C. We have, see [56], M (iy) lim = 0, y→∞ y which implies (g+ , g− ) = 0 and again shows that M (z) is a constant function in the upper half-plane and M (z) = C = M (i) = i, which completes the proof.

z ∈ C+ , 

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Remark 3.4. In connection with our main hypothesis of this chapter, we remark that if a self-adjoint operator A solves commutation relations (3.3), then one can always find a symmetric restriction A˙ that satisfies Hypothesis 3.1. Indeed, the commutation relations (3.3) imply that the one-parameter group Vs = eisA generated by A satisfies commutation relations in the Weyl form (see, e.g., [27, Ch. 3, Sect. 1, Theorem 5] or [125]) and then the existence of such a symmetric operator A˙ is an immediate corollary of the Stone-von Neumann uniqueness theorem. Next, we threat the case where the flow ϕ(t, ·) associated with the transformation (3.2) has a fixed point in the interior of the unit disk. That is, A˙ admits a maximal dissipative “invariant” extension that is not selfadjoint (cf. [55, Theorem 20]). We present the corresponding result in a slightly stronger form. In particular, in this case the requirement (3.1) can be relaxed.  is a quasi-selfadjoint dissipative extension Theorem 3.5. Suppose that A of a closed symmetric operator A˙ with deficiency indices (1, 1) and A is a ˙ (reference) self-adjoint extension of A. Suppose that the commutation relation  t=A  + tI Ut∗ AU

on

 Dom(A)

(3.6)

hold. ˙ A,  A) Then the characteristic function S(z) associated with the triple (A, admits the representation S(z) = keiz ,

z ∈ C+ ,

(3.7)

for some |k| ≤ 1 and  ≥ 0. Furthermore, if  = 0, then necessarily |k| < 1 and if |k| = 1, then  > 0. ˙ A,  A) is In particular, the von Neumann parameter κ of the triple (A, given by κ = ke− .  is a quasi-selfadjoint extension of A, ˙ we have Proof. Since A  A˙ = A|   ∗). Dom(A)∩Dom(( A)

(3.8)

We claim that  ∗ Ut = (A)  ∗ + tI Ut∗ (A)

on

 ∗ ). Dom((A)

(3.9)

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 and f ∈ Dom((A)  ∗ ). Then To see that assume that g ∈ Dom(A)  f ) = (U ∗ AU  t U ∗ g, f ) = ((A  Ut f ) = (U ∗ Ag,  + tI)U ∗ g, f ) (Ag, t t t t  ∗ + tI)f ) = (g, Ut ((A)  ∗ + tI)f ). = (Ut∗ g, ((A)

(3.10)

 is Ut -invariant for all t ∈ R Here we have used that the domain Dom(A) ∗   one and therefore Ut g ∈ Dom(A). Since (3.10) holds for all g ∈ Dom(A), ∗  ensures that Dom((A) ) is Ut -invariant and  ∗ Ut f = Ut ((A)  ∗ + tI)f, (A)

 ∗ ), f ∈ Dom((A)

which proves (3.9).  ∗ )) = Dom((A)  ∗ ), from (3.8) one concludes that the Since Ut (Dom((A) commutation relations ˙ ∗ Ut = (A) ˙ ∗ + tI Ut∗ (A)

on

˙ ∗) Dom((A)

(3.11)

hold. Taking into account that  + tI)Ut∗  = Ut (A A and A˙ = Ut (A˙ + tI)Ut∗ , and also observing that the operator At given by At = Ut (A + tI)Ut∗ ˙ we see that the is a self-adjoint extension of the symmetric operator A, ˙  ˙  triples (A, A, At ) and (A + tI, A + tI, A + tI) are mutually unitarily equivalent. In particular, (z) = S(A, S(A+tI, ˙  ˙ A,A  t ) (z), A+tI,A+tI)

z ∈ C+ .

˙ by Lemma E.1 (see (E.4)) in Since At is a self-adjoint extension of A, Appendix E, we have (1)

(z), S(A, ˙ A,A  t ) (z) = Θt S(A, ˙ A,A)  (1)

(1)

z ∈ C+ ,

for some unimodular constant Θt , |Θt | = 1, which is a continuous function of the parameter t (see [88] for the proof of continuity).

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By Theorem F.1 in Appendix F, (2)

S(A+tI, (z) = Θt S(A, (z − t), ˙  ˙ A,A)  A+tI,A+tI)

z ∈ C+ ,

(2)

where Θt is another continuous unimodular function in t. Therefore, the functional equation (z − t) = Θt S(A, (z), S(A, ˙ A,A)  ˙ A,A) 

t ∈ R,

holds, where (1)

(2)

Θt = Θt Θt . From the functional equation it also follows that Θt is a continuous unimodular solution of the equation Θt+s = Θt Θs ,

s, t ∈ R,

and therefore (see, e.g., [32, XVII, 6]) Θt = eit

for some

 ∈ R.

In particular, this proves that the characteristic function S(A, (z) is ˙ A,A)  a character-automorphic function with respect to the shifts, that is (z + t) = eit S(A, (z), S(A, ˙ A,A)  ˙ A,A) 

t ∈ R.

(3.12)

Since S(A, (z) is a contractive analytic function on C+ , it admits the ˙ A,A)  representation (z) = θB(z)eiM(z) , S(A, ˙ A,A)  where |θ| ≤ 1, B is the Blaschke product associated with the (possible) zeros (z) in the upper half-plane, and M (z) is a Herglotz-Nevanlinna of S(A, ˙ A,A)  function. (z) is not identically Suppose that the characteristic function S(A, ˙ A,A)  zero and thus θ = 0. Then, from the functional equation (3.12) it follows (z) has no zeros in C+ , and hence that S(A, ˙ A,A)  (z) = θeiM(z) , S(A, ˙ A,A) 

z ∈ C+ .

(z) is character-automorphic, one concludes that the funcSince S(A, ˙ A,A)  tional equation M (z + t) = t + M (z) holds.

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Next, we have d M (z + t)|t=0 = M  (z) = . dt Taking into account that M (z) maps the upper half-plane into itself, we obtain that M (z) = z + b, where  ≥ 0 and Im(b) ≥ 0. Therefore, S(A, (z) = θeib eiz , ˙ A,A)  which proves (3.7) with k = θeib .



Remark 3.6. Notice that in the situation of Theorem 3.5 we have that  A˙ = A|  ∗ ) . Dom(A)∩Dom( A Therefore the symmetric operator A˙ is uniquely determined by the gener In this case we will call A˙ the symmetric part of A.  ator A. We also remark that if under the hypothesis of Theorem 3.5 there is no self-adjoint extension satisfying the commutation relations (3.3), the  is unique (see, e.g., [88, corresponding maximal dissipative extension A Theorem 6.3]). The following corollary is the first step towards a complete classification up to unitary equivalence of “invariant” symmetric operators from Hypothesis 3.1 (see Problem (I) a) in Introduction). Corollary 3.7. Assume Hypothesis 3.1. Then there exists a self-adjoint extension A of A˙ such that the Livˇsic function associated with the pair ˙ A) admits the representation (A, (z) = k s(A,A) ˙

eiz − e− , −1

k 2 e− eiz

z ∈ C+ ,

for some 0 ≤ k ≤ 1 and  > 0. Proof. If A˙ admits a self-adjoint extension A that satisfies the same commutation relations as A˙ does, then by Theorem 3.3 s(A,A) (z) = 0, ˙

z ∈ C+ ,

which proves the claimed representation with k = 0.

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 that If A˙ admits a maximal (non-selfadjoint) dissipative extension A satisfies the same commutation relations, then by Theorem 3.5 there exists a (reference) self-adjoint extension A of A˙ such that the characteristic ˙ A,  A ) is of the form function of the triple (A, S(z) = keiz ,

z ∈ C+ .

Therefore, by (2.9) s(A,A ˙  ) (z) =

S(z) − S(i) S(i) S(z) − 1

=k

eiz − e− . |k|2 e− eiz − 1

By Lemma E.1 in Appendix E, one can always find a possibly different self-adjoint extension A of A˙ such that s(A,A) (z) = |k| ˙ and the claim follows.

eiz − e− , |k|2 e− eiz − 1

z ∈ C+ , 

B1948

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Chapter 4

THE DIFFERENTIATION OPERATOR ON METRIC GRAPHS Let Y be a directed metric graph (see, e.g., [8, 59]). We will distinguish the following three cases. Case (i): Y = (−∞, 0)  (0, ∞), with (−∞, 0) the incoming and (0, ∞) outgoing bonds; Case (ii): Y = (0, ),

the outgoing bond;

Case (iii): Y = (−∞, 0)  (0, ∞)  (0, ), with (−∞, 0) the incoming and both (0, ∞) and (0, ) the outgoing bonds. d the differentiation operator on the metric graph Y Denote by D˙ = i dx ˙ of functions f ∈ W21 (Y) in Cases (i)–(iii) defined on the domain Dom(D) with the following boundary conditions on the vertices of the graph, in Case (i):

f∞ (0+) = f∞ (0−) = 0;

29

(4.1)

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in Case (ii): f (0) = f () = 0; in Case (iii): ⎧ f∞ (0+) = kf∞ (0−) ⎪ ⎪ ⎨ √ f (0) = 1 − k 2 f∞ (0−) ⎪ ⎪ ⎩ =0 f ()

for some

(4.2)

0 < k < 1.

(4.3)

Here we have used the following notation. If the graph Y is in Cases (i) and (ii), the functions from the Hilbert space L2 (Y) are denoted by f∞ and f , respectively. If the metric graph Y is in Case (iii), in view of the natural identification of L2 (Y) with the orthogonal sum L2 (R) ⊕ L2 ((0, )), it is convenient to represent an arbitrary element f ∈ L2 (Y) as the two-component vectorfunction   f∞ . f= f (Here L2 (Y) denotes the Hilbert space of square-integrable functions with respect to Lebesgue measure on the edges of the metric graph Y.) Notice that if the graph Y is in Case (iii) and k = 0 in (4.3), then the boundary conditions (4.3) can be rewritten as ⎧ f (0+) = 0 ⎪ ⎪ ⎨ ∞ f (0) = f∞ (0−) ⎪ ⎪ ⎩ f () = 0. In this case, the operator D˙ splits into the orthogonal sum of the symmetric differentiation operators on the semi-axes (−∞, ) and (0, ∞) with the Dirichlet boundary conditions at the end-points, respectively. Therefore, if k = 0, then the operator D˙ is unitarily equivalent to the symmetric differentiation in Case (i). Remark 4.1. In Cases (i) and (iii) the metric graph Y is not finite. However, one can assign two additional vertices to the external edges at ±∞. Under this hypothesis, in all Cases (i)–(iii) the Euler characteristic χ(Y) of the graph Y, the number of vertices minus the number of edges, equals one. Therefore, the corresponding first Betti number β(Y) = −χ(Y) + 1 of the graph Y, the number of edges that have to be removed to turn the graph into a connected tree, vanishes.

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Lemma 4.2. The operator D˙ on a metric graph Y is Cases (i)–(iii) is symmetric. Moreover, ⎧ ⎨W 1 ((−∞, 0)) ⊕ W 1 ((0, ∞)), in Case (i) 2 2 ˙ ∗) = . Dom((D) ⎩W 1 ((0, )), in Case (ii) 2 ˙ ∗ ) consists of the vector-functions In Case (iii), the domain Dom((D) h = (h∞ , h )T ∈ (W21 (R− ) ⊕ W21 (R+ )) ⊕ W21 ((0, )) satisfying the “boundary condition” h∞ (0−) − k h∞ (0+) −

 1 − k 2 h (0) = 0.

(4.4)

Proof. The corresponding result in Cases (i) and (ii) is well known. ˙ the In Case (iii), from (4.3) it follows that for f = (f∞ , f )T ∈ Dom(D) “quantum Kirchhoff rule” |f∞ (0−)|2 = |f∞ (0+)|2 + |f (0)|2 holds. Since also f () = 0, integration by parts ∞  0   ˙ if∞ (x)f∞ (x)dx + if∞ (x)f∞ (x)dx + if (x)f (x)dx (Df, f ) = −∞



=−



0

0



−∞

−

0

0

 (x)dx − if∞ (x)f∞

∞

0

 (x)dx if∞ (x)f∞

if (x)f (x)dx

 + i |f∞ (0−)|2 − |f∞ (0+)|2 − |f (0)|2 ,

˙ f ∈ Dom(D),

˙ f ) is real and therefore the operator D˙ shows that the quadratic form (Df, is indeed symmetric. Similar computations show that ∞  0 ˙ )= ih∞ (x)f− (x)dx + ih∞ (x)f+ (x)dx + ih (x)f (x)dx (h, Df −∞



=−

−∞

−

0

0

0



 (x)dx − ih∞ (x)f∞

0

0

∞

 (x)dx ih∞ (x)f∞

ih (x)f (x)dx + i h∞ (0−)f∞ (0−)

− h∞ (0+)f∞ (0+) − h (0)f (0) ,

˙ f ∈ Dom(D).

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˙ ∗ ) if and only if Therefore, h ∈ Dom((D) h∞ (0−)f∞ (0−) − h∞ (0+)f∞ (0+) − h (0)f (0) = 0

˙ for all f ∈ Dom(D).

Taking into account the boundary conditions (4.3), we have

 h∞ (0−) − h∞ (0+)k − h (0) 1 − k 2 f∞ (0−) = 0

(4.5)

˙ Since f∞ (0−) may be chosen arbitrarily, (4.4) follows for all f ∈ Dom(D). from (4.5).  The following lemma introduces a natural (standard) basis in the subspace ˙ ∗ + iI). ˙ ∗ − iI)+ ˙ Ker((D) N = Ker((D) ˙ ∗ ∓ iI) of the symmetric Lemma 4.3. The deficiency subspaces Ker((D) operator D˙ on the metric graph Y is Cases (i)–(iii) are spanned by the following normalized deficiency elements g± . Here, in Case (i), √ √ g+ (x) = 2ex χ(−∞,0) (x) and g− (x) = 2e−x χ(0,∞) (x), x ∈ R, (4.6) in Case (ii), √ 2 ex and g+ (x) = √ e2 − 1 (4.7) √ 2 −x e , x ∈ [0, ]. g− (x) = √ e2 − 1 Finally, in Case (iii), ⎧√ √ ⎨ 1 − k 2 χ(−∞,0) (x), 2 ex g+ (x) = √ e2 − k 2 ⎩1,

x ∈ (−∞, 0)  (0, ∞) x ∈ [0, ]

,

(4.8) √ 2 g− (x) = √ e 2 e − k2

⎧ √ ⎨− 1 − k 2 χ(0,∞) (x), −x ⎩k,

x ∈ (−∞, 0)  (0, ∞) x ∈ [0, ]

.

(4.9) In particular, D˙ is a symmetric operator with deficiency indices (1, 1).

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33

Proof. The deficiency subspaces of the symmetric operator D˙ in Cases (i) and (ii) can be easily calculated. Indeed, in Case (i), we have ˙ ∗ − zI) = span{gz }, Ker((D) where gz (x) =

⎧ ⎨e−izx , x < 0 ⎩0,

x≥0

and gz (x) =

⎧ ⎨0,

x 0, (4.15) −∞

0

and

 2 − 1−k

0

∞

f∞ (x)e

−sx

dx + k



0

f (x)e−sx dx = 0

Therefore, from (4.15) it follows that  h(x)esx dx = 0 −∞

where h(x) =

for all s > 0. (4.16)

for all s > 0,

⎧√ ⎨ 1 − k 2 f∞ (x),

x 0.

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By the uniqueness theorem f∞ (x) = 0 for x ≥ 0 as well. That is, f = (f∞ , f )T = 0. Thus, (4.14) implies f = 0 and therefore, by Theorem B.2 in Appendix B, the differentiation operator D˙ in Case (iii) is a prime symmetric operator as well.  Remark 4.5. We remark that the symmetric operator D˙ in Case (iii) determined by the boundary conditions (4.3) with k = 0 is also a prime operator: in this case D˙ is unitarily equivalent to the symmetric differential operator in Case (i), which is a prime operator by Lemma 4.4. Remark 4.6. It is easy to see that the prime symmetric differentiation operator D˙ on the metric graph Y in Cases (i)–(iii) satisfies the semi-Weyl commutation relations in the form (cf. Hypothesis 3.1) ˙ t = D˙ + tI Ut∗ DU

on

˙ Dom(D),

t ∈ R,

(4.17)

where Ut = e−itQ is the unitary group generated by the operator Q of multiplication by independent variable on the graph Y. To show that the commutation relations (4.17) hold we proceed as follows. Let A(x) denote a real-valued piecewise continuous function on Y. We remark that the operators D˙ and D˙ + A(x) are unitarily equivalent. Indeed, let φ(x) be any solution to the differential equation φ (x) = A(x)

(4.18)

on the edges of the graph. Since the graph Y is a connected tree, the function φ(x) is determined up to a constant, and we may without loss require that φ vanishes at the origin of the graph Y, that is, φ(0) = 0.

(4.19)

Denote by V the unitary local gauge transformation (V f )(x) = eiφ(x) f (x),

f ∈ L2 (Y).

(4.20)

Taking into account the boundary conditions (4.1), (4.2) and (4.3) one concludes that the domain of D˙ is V -invariant, that is, ˙ = Dom(D). ˙ V (Dom(D)) Next, a simple computation shows that D˙ = V ∗ (D˙ + A(x))V.

(4.21)

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In the particular case of a constant (magnetic) potential A(x) ≡ t, t ∈ R, solving (4.18) with the boundary condition (4.19) on the graph Y, one immediately concludes that the unitary operator V from (4.20) is given by V = eitQ , and therefore (4.21) implies the commutation relations (4.17).

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Chapter 5

THE MAGNETIC HAMILTONIAN

In this chapter we explicitly describe the set of all self-adjoint (reference) and, more generally, quasi-selfadjoint extensions of the differentiation symmetric operators D˙ on the metric graph Y in Cases (i)–(iii) introduced in Chapter 4. Theorem 5.1. Suppose that the metric graph Y is in one of the Cases (i)-(iii) and let D˙ be the symmetric differentiation operator given by (4.1), (4.2) and (4.3), respectively. Then the one-parameter family of differentiation operators DΘ , |Θ| = 1 on the graph Y in Cases (i)–(iii) with boundary conditions f∞ (0+) = −Θf∞ (0−), 

(5.1)

f (0) = −Θf(),     √ f∞ (0+) k f∞ (0−) 1 − k2 Θ = √ , 1 − k 2 −kΘ f (0) f ()

(5.2) (5.3)

respectively, coincides with the set of all self-adjoint extensions of the sym˙ metric operator D. Moreover, let g± be the deficiency elements of D˙ referred to in Lemma 4.3. Then g = g+ − κg− ∈ Dom(DΘ ), if and only if Θ = F (κ),

37

|κ| = 1,

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where ⎧ ⎪ κ, ⎪ ⎪ ⎪ ⎪ ⎨ κ − e− F (κ) = − e− κ − 1 , ⎪ ⎪ ⎪ ⎪ κ − ke− ⎪ ⎩− , ke− κ − 1

in Case (i) in Case (ii)

(5.4)

in Case (iii).

Proof. If Y is in Cases (i)–(ii), the first assertion of the theorem is well known (see, e.g., [3]). If the graph Y is in Case (iii) and Y = (−∞, 0)  (0, ∞)  (0, ), one can identify the right endpoint of the edge [0, ] of the graph Y with its origin thus making number of incoming outgoing bonds equal.  f and  f the ∞ (0−) ∞ (0+) and outgoing data are related by Since the incoming f (0)  () the unitary matrix σ with   √ k 1 − k2 Θ σ= √ , 1 − k2 −kΘ from [8, Theorem 2.2.1] it follows that the operator D is self-adjoint. Recall d on an that this theorem states that the differentiation operator D = i dx oriented graph is self-adjoint if and only if for each (finite) vertex v the numbers of incoming and outgoing bonds are equal and the vectors F in (v) and F out (v) composed from the values of f ∈ Dom(D) attained by f from the incoming and outgoing bonds satisfy the condition F out (v) = σ(v)F in (v), ˙ the boundary conditions where σ is a unitary matrix. Next, if f ∈ Dom(D), (4.3) imply that the boundary conditions (5.3) also hold, and therefore the ˙ self-adjoint operator D extends D. ˙ by [8, Theorem 2.2.1] Conversely, if D is a self-adjoint extension of D, the boundary conditions     f (0−) f∞ (0+) ∞ = σ , (5.5) f (0) f ()

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hold, where σ  is a unitary matrix. The requirement that the self-adjoint operator D extends D˙ shows that σ  has to be of the form   k α  σ = √ 1 − k2 β for some α and β. Since σ  is unitary, we have that   √ k 1 − k2 Θ  σ = √ 1 − k2 −kΘ for some Θ, |Θ| = 1, which completes the proof of the first assertion of the theorem. To prove (5.4) we argue as follows. We use the following notation g = g∞ in Case (i), g = g in Case (ii), and finally, g = (g∞ , g )T in Case (iii) as introduced in Chapter 4. From the representation (4.6) we get that √ √ g∞ (x) = 2ex χ(−∞,0) (x) − κ 2e−x χ(0,∞) (x), x ∈ R, so that

√ g∞ (0+) = −κ 2

while

g∞ (0−) =

√ 2.

That is, the element g satisfied boundary condition (5.1) with Θ = κ which proves (5.4) in Case (i). In Case (ii), we use (4.7) to see that √ 2 (1 − κe ) g (0) = √ 2 e −1 and √ 2 g () = √ (e − κ), 2 e −1 which shows that the requirement g+ − κg− ∈ Dom(DΘ ) means that κ − e− 1 − κe = − − .  e −κ κe − 1 In Case (iii), from the representations for the deficiency elements g± (4.8) and (4.9) it follows

g∞ (0+) = a 1 − k 2 e κ,

g∞ (0−) = a 1 − k 2 , Θ=−

g () = a(1 − ke κ),

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where a= √

√ 2 . 2 e − k2

Since g ∈ Dom(DΘ ), by (5.3) we have g∞ (0+) = kg∞ (0−) +

1 − k 2 Θg (),

which implies



1 − k 2 e κ = k 1 − k 2 + Θ 1 − k 2 (1 − ke κ). Therefore Θ=

κ − ke− e κ − k = − . 1 − ke κ ke− κ − 1



Remark 5.2. Notice that in Case (i) the self-adjoint operator DΘ satisfies the semi-Weyl commutation relations Ut∗ DΘ Ut = DΘ + tI

on

Dom(DΘ ),

t ∈ R,

|Θ| = 1.

(5.6)

Here Ut = e−itQ is the unitary group generated by the self-adjoint operator Q of multiplication by independent variable on the graph Y. However, if the graph Y is in Cases (ii) and (iii), then we only have the commutation relations with respect to a discrete subgroup Z  n → Un 2π  of the group Ut . That is, Un∗ 2π DΘ Un 2π = DΘ + n  

2π I 

on

Dom(DΘ ),

n ∈ Z,

|Θ| = 1. (5.7)

This phenomenon has the following topological explanation: the set of self-adjoint extensions DΘ , |Θ| = 1, is in one-to-one correspondence with the unit circle T. The map DΘ → DΘt = Ut∗ DΘ Ut , determines the flow Θ → Θt on T. Using boundary conditions (5.1)-(5.3) it is straightforward to see that ⎧ ⎨1, in Case (i) Θt = Θ ⎩eit , in Cases (ii) and (iii),

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so that in Cases (ii) and (iii) Dom(DΘ ) is not invariant with respect to the , n ∈ Z. In particular, whole group Ut , t ∈ R, but only to its subgroup Un 2π  the flow Θ → Θt has no fixed point, whenever the graph Y is in Cases (ii) and (iii) (notice that the Euler characteristic of T is zero and hence Proposition 3.2 is not applicable). In this regard, it is worth mentioning the fall to the center “catastrophe” in Quantum Mechanics [71, 108, 109]. For a related discussion of the Efimov Effect in three-body systems see [26, 31, 92] where the collapse in a three-body system with point interactions has been discovered, also see [85] and references therein. More generally, suppose that A(x) is a real-valued piecewise continuous function on Y. Prescribing the magnetic potential A(x) to all edges of the graph, consider the magnetic differentiation operator DΘ + A(x). If the graph Y is in Case (i), the local gauge transformation f (x) −→ eiφ(x) f (x),

f ∈ L2 (Y),

where φ(x) is a solution to the differential equation φ (x) = A(x),

x ∈ Y,

φ(0) = 0, eliminates the magnetic potential and one shows that the self-adjoint operators DΘ and DΘ + A(x) are unitarily equivalent, with the unitary equivalence performed by the unitary operator (V f )(x) = eiφ(x) f (x),

f ∈ L2 (Y).

(5.8)

Clearly Dom(DΘ ) is V -invariant, that is, V (Dom(DΘ )) = Dom(DΘ ), and therefore DΘ = V ∗ (DΘ + A(x))V.

(5.9)

If the graph Y is in Cases (ii) and (iii), the gauge transformation still eliminates the magnetic potential but changes the boundary conditions. That is, V (Dom(DΘ )) = Dom(DΘ·e−iΦ ),

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where

 Φ= 0



A(s)ds,

the flux of the magnetic field,

and hence DΘ·e−iΦ = V ∗ (DΘ + A(x))V.

(5.10)

Notice, that in the particular case of a constant potential A(x) ≡ t, one gets the commutation relations (5.6) and (5.7) as a corollary of (5.9) and (5.10), respectively. Having in mind the unitary equivalences (5.9) and (5.10) we adopt the following definition. Definition 5.3. The self-adjoint differentiation operator DΘ for |Θ| = 1 referred to in Theorem 5.1 will be called the magnetic Hamiltonian. Notice that in Cases (ii) and (iii), the boundary conditions (5.2) and (5.3) are not local vertex conditions. Bearing in mind applications in quantum mechanics, in Cases (ii) and (iii) one can identify the end points of the interval [0, ] to get a one-cycle graph Y. As it has been explained in Remark 4.1, in Case (iii) one can also assign two additional vertices to the external edges at ±∞ of the one-cycle graph Y, so that the one-cycle graphs in Case (ii) and (iii) have the Euler characteristic χ(Y) zero with the corresponding first Betti numbers equal to one. In this case, the graph Y can be considered to be the Aharonov-Bohm ring, the configuration space for the quantum system with the magnetic Hamiltonian DΘ . This system describes a (massless) quantum particle moving on the edges of the graph and the argument of the parameter Θ that determines the magnetic Hamiltonian DΘ can be interpreted to be the flux of the (zero) magnetic field through the cycle (see (5.10)). For a related information about graphs with Euler characteristic zero in the context of the inverse scattering theory we refer to [67]. Our next goal is to obtain an explicit description of all quasi-selfadjoint extensions of the symmetric differentiation operators D˙ introduced in Chapter 4. Theorem 5.4. The differentiation operators DΘ , Θ ∈ C∪{∞} with |Θ| = 1 referred to in Theorem 5.1 with boundary conditions (5.1)–(5.3) is in one to one correspondence with the set of all quasi-selfadjoint extensions of the ˙ symmetric operator D.

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43

Remark 5.5. If Θ = ∞, the boundary conditions (5.1)–(5.3) in Cases (i)–(iii) should be understood as follows f∞ (0−) = 0,

(5.11)

f () = 0,     √

f∞ (0−) f∞ (0+) k 1 − k2 = , 0 0 f (0) f ()

(5.12) (5.13)

respectively. Notice that the boundary condition (5.13) can be justified as follows. Rewrite (5.3) as −1      √ f∞ (0+) f∞ (0−) 1 − k2 Θ k √ = 1 − k2 −kΘ f (0) f () and observe that   √ 1 − k2 k 0

0

 = lim

Θ→∞

k

√ 1 − k2

1 = lim Θ→∞ −Θ

−1 √ 1 − k2 Θ −kΘ

√

−kΘ − 1 − k2 Θ √ − 1 − k2 k



to get (5.13) as a limiting case. Notice that the boundary conditions (5.13) can also be rewritten as

kf∞ (0+) + 1 − k 2 f (0) = f∞ (0−), f () = 0. Proof. If Y is in Case (i) or (ii), the corresponding result is well known (see, e.g., [3]). Suppose that the metric graph Y is in Case (iii). We will describe the required one to one correspondence explicitly. Denote by Dκ (κ ∈ C, |κ| = 1) a quasi-selfadjoint extension of D˙ such that ˙ + span{g+ − κg− }, Dom(Dκ ) = Dom(D) where the deficiency elements g± are given by (4.8) and (4.9).

(5.14)

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If

 f=

f∞



f

∈ Dom(Dκ ),

(5.15)

then



f∞ (x) = α 1 − k 2 ex χ− (x) + ακ 1 − k 2 e−x χ+ (x) + h∞ (x), f (x) = αex − ακke−x + h (x),

for some α ∈ C and some h= In particular,

  h∞ h

x ∈ R,

x ∈ [0, ),

˙ ∈ Dom(D).

f∞ (0−) = α 1 − k 2 + f∞ (0−),

f∞ (0+) = ακ 1 − k 2 e + h∞ (0+),

and f (0) = α(1 − kκe ) + h (0), f () = α(e − kκ) + h (). ˙ the boundary conditions (4.3) hold and therefore Since h ∈ Dom(D),

f∞ (0−) = α 1 − k 2 + h∞ (0−),

f∞ (0+) = ακ 1 − k 2 e + kh∞ (0−),

f (0) = α(1 − kκe ) + 1 − k 2 h∞ (0−), f () = α(e − kκ). Equivalently,   f∞ (0+) f (0) and

=

 √ κ 1 − k 2 e

  f∞ (0−) f ()

1 − kκe

=

k



√ 1 − k2

√  1 − k2 1 e − kκ 0

α



h∞ (0−)

α



h∞ (0−)

.

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If e − kκ = 0, one obtains that   f∞ (0+) f (0) where

 =S

 √ κ 1 − k 2 e

k

 f∞ (0−) f ()

45

,

(5.16)

 √ −1 1 − k2 1

√ 1 − k2 1 − kκe   √ k 1 − k2 Θ = √ . 1 − k2 −kΘ

S=

page 45

e − kκ 0

Combining (5.15), (5.16) and (5.3) shows that D κ = DΘ , where Θ=−

κe − k . kκ − e

(5.17)

If e − kκ = 0, then necessarily f () = 0 and 



kf∞ (0+) + 1 − k 2 f (0) = k ακ 1 − k 2 e + kh∞ (0−) 



+ 1 − k 2 α(1 − kκe ) + 1 − k 2 h∞ (0−)

= α 1 − k 2 + h∞ (0−) = f∞ (0−), which shows that boundary conditions (5.13) holds (Θ = ∞, formally). It remains to consider the case of the quasi-selfadjoint extension D∞ defined on ˙ + span{g− }, Dom(D∞ ) = Dom(D)

(5.18)

which corresponds to the infinite value of the von Neumann parameter κ (κ = ∞). If (5.18) holds (κ = ∞), a similar computation shows that the corresponding quasi-selfadjoint extension corresponds to the boundary condition (5.3) with e , k which is well defined (k =  0 by the hypothesis). Θ=−

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Notice that (5.17) gives the link between the boundary condition parameter Θ and the von Neumann extension parameter κ from (5.14) and thus establishes the required correspondence.  Remark 5.6. Observe that in Case (i) the operator DΘ satisfies the semiWeyl commutation relations Ut∗ DΘ Ut = DΘ + tI

on

Dom(DΘ ),

t ∈ R,

(5.19)

for all Θ ∈ C ∪ {∞}, where Ut = e−itQ is the unitary group generated by the self-adjoint operator Q of multiplication by independent variable on the graph Y. If the graph Y is in Cases (ii) and (iii), the commutation relations (5.19) hold only if Θ = 0 or Θ = ∞. Otherwise, we only have the commu, tation relations with respect to the discrete subgroup Z  n → Un 2π  cf. Remark 5.2, = DΘ + n Un∗ 2π DΘ Un 2π  

2π I 

on Dom(DΘ ),

n ∈ Z,

|Θ| = 1. (5.20)

It is interesting to notice that the metric graph Y = (−∞, 0)  (0, ∞)  (0, ) in Case (iii) serves as the configuration space for a minimal self-adjoint dilation of almost all (with the only one exception) maximal dissipative differentiation operators on the finite interval (0, ). Actually, the corresponding self-adjoint dilations coincide with the set of magnetic Hamiltonians DΘ , |Θ| = 1, in Case (iii). Theorem 5.7 (cf. [101]). The self-adjoint operator DΘ , |Θ| = 1, on the metric graph Y = (−∞, 0)  (0, ∞)  (0, ) in Case (iii) with the boundary conditions (5.3) (0 < k < 1) is a (minimal) self-adjoint dilation of the maximal dissipative differentiation operator dΘ on its subgraph K = (0, ) determined by the boundary condition Dom(dΘ ) = {f ∈ W21 ((0, )) | f (0) = −kΘf ()}.

(5.21)

That is, P (DΘ − zI)−1 |K = (dΘ − zI)−1 ,

z ∈ C− ,

(5.22)

where P is the orthogonal projection from L2 (Y) onto the subspace K = L2 (K) = L2 ((0, )). In particular, 

eitdΘ = P eitDΘ |K ,

t ≥ 0.

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47

Proof. Let g = (g∞ , g )T ∈ L2 (Y) and f = (DΘ − zI)−1 g,

z ∈ C− .

T

Since f = (f∞ , f ) ∈ Dom(DΘ ), the boundary conditions      √ f∞ (0+) 1 − k2 Θ k f∞ (0−) = √ f (0) f () 1 − k2 −kΘ

(5.23)

hold. We have d i f∞ (x) − zf∞ (x) = g∞ (x), x ∈ (−∞, 0) ∪ (0, ∞) ⊂ Y, dx and d i f (x) − zf (x) = g (x), x ∈ (0, ) ⊂ Y. (5.24) dx If g ∈ K, then g∞ = 0 and hence d f∞ (x) − zf∞ (x) = 0, x ∈ (−∞, 0) ∪ (0, ∞) ⊂ Y. dx Since z ∈ C− , the function f∞ has to vanish on the negative real-axis. In particular, f∞ (0−) = 0. From (5.23) it follows that i

f (0) = −kΘf (). Therefore the boundary condition (5.21) holds and hence f ∈ Dom(dΘ ). Combined with (5.24) this means that f = (dΘ − zI)−1 g ,

z ∈ C− ,

which proves (5.22) and eventually shows that DΘ is a self-adjoint dilation  of dΘ . Remark 5.8. Theorem 5.7 does not say anything about the dilation of the (exceptional) maximal dissipative differentiation operator d defined on  = {f ∈ W 1 ((0, )) | f (0) = 0} Dom(d) 2 (it is explicitly assumed that k = 0 in the boundary condition (5.21)). In fact, the corresponding self-adjoint dilation coincides with the selfd on the metric graph adjoint realization of i dx Y = (−∞, 0)  (0, )  (, ∞), which can be identified with the real axis. Therefore, in the exceptional case the configuration space of the dilation can be identified with the graph Y in Case (i).

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Indeed, to treat the exceptional case, assume that g ∈ L2 (R) is supported by the finite interval [0, ]. Then the element f = (D − zI)−1 g,

z ∈ C,

is supported by the positive semi-axis and its continuous representative satisfies the boundary condition f (0) = 0. In particular, i

d f (x) − zf (x) = g(x), dx

x ∈ [0, ],

f (0) = 0,

and therefore the compressed resolvent of D in the lower half-plane coincides  with the resolvent of d proving that D dilates the dissipative operator d.

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Chapter 6

ˇ THE LIVSIC FUNCTION s(D,D (z) ˙ Θ) The main goal of this and the forthcoming chapter is to describe those unitary invariants of the prime symmetric operator D˙ that characterize the operator up to unitary equivalence. Here D˙ is the symmetric differentiation operator on the metric graph Y in one of the Cases (i)–(iii) with boundary conditions (4.1), (4.2) and (4.3), respectively. To do so, we need to fix a (reference) self-adjoint extension of the ˙ We choose as such an extension the self-adjoint realization operator D. D = DΘ |Θ=1 of the differentiation operator referred to in Theorem 5.1. Recall that the domain of the self-adjoint operator D = D1 is characterized by the following boundary conditions f∞ (0+) = −f∞ (0−), 

f (0) = −f (),     √ 1 − k2 f∞ (0+) k f∞ (0−) = √ , 1 − k2 −k f (0) f ()

(6.1) (6.2) (6.3)

whenever the graph Y is in Cases (i)–(iii), respectively. We start with the following important observation. Lemma 6.1. Let g± be the deficiency elements of the symmetric operator D˙ referred to in Lemma 4.3. Then f = g+ − g− ∈ Dom(D). Proof. In Case (i) we have the representation √ f (x) = 2(ex χ(−∞,0) (x) − e−x χ(0,∞) (x)) √ so that f (0−) = 2 = −f (0+) and therefore f ∈ Dom(D). 49

(6.4)

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In Case (ii),

√ 2

f (x) = √ (ex − e−x ), e2 − 1 which implies

√

2

x ∈ [0, ],

√ 2



f (0) = √ (1 − e ) = − √ (e − 1) = −f () e2 − 1 e2 − 1 thus showing that f ∈ Dom(D) as well. Finally, in Case (iii), from (4.8) and (4.9) it follows that the element f admits the representation √ 2 (f∞ , f )T , f= √ e2 − k 2 where   f∞ (x) = 1 − k 2 ex χ(−∞,0) (x) + 1 − k 2 e−x χ(0,∞) (x), x ∈ R, and f (x) = ex − ke−x , We have f∞ (0−) =

 1 − k2 ,

x ∈ [0, ].

f∞ (0+) =

 1 − k 2 e ,

and f (0) = 1 − ke ,

f () = e − k.

Here k, 0 < k < 1, is the parameter from the boundary conditions (4.3) and ˙ and Dom(D) in Case (iii), respec(6.3) describing the domains Dom(D) tively.  f (0−)  ∞ and outgoing F in = As a consequence, the incoming F in = f ()  f (0+)  ∞ boundary data are related as f (0)     √  √  √ f∞ (0+) k 1 − k 2 e 1 − k2 1 − k2 = √ = 1 − ke 1 − k2 −k e − k f (0)   f∞ (0−) = S(1) , f () where the bond scattering matrix S(1) is given by (5.3) for Θ = 1, which shows that f ∈ Dom(D(1)) = Dom(D).



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Based on Lemma 6.1, now we are ready to evaluate the Livˇsic function ˙ D), which is one of the unitary invariants that associated with the pair (D, ˙ characterizes the pair (D, D) up to unitary equivalence. ˙ D) admits Lemma 6.2. The Livˇsic function associated with the pair (D, the representation ⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎨ eiz − e− (z) = e− eiz − 1 , s(D,D) ˙ ⎪ ⎪ ⎪ ⎪ eiz − e− ⎪ ⎩k , k 2 e− eiz − 1

in Case (i) in Case (ii)

(6.5)

in Case (iii).

Here k, 0 < k < 1, is the parameter from the boundary conditions (4.3) and ˙ and Dom(D) in Case (iii). (6.3) describing the domains Dom(D) Proof. Denote by g± the deficiency elements of the symmetric operator D˙ referred to in Lemma 4.3. By Lemma 6.1, g+ − g− ∈ Dom(D)

(6.6)

in all Cases (i)–(ii). As long as (6.6) is established, in accordance with ˙ D) can be definition (2.3), the Livˇsic function associated with the pair (D, evaluated as (z) = s(D,D) ˙

z − i (gz , g− ) · , z + i (gz , g+ )

z ∈ C+ .

˙ ∗ − zI), z ∈ C \ R are given by Here the deficiency elements gz ∈ Ker((D) (4.10), (4.11), (4.12) and (4.13) in Cases (i)–(iii), respectively. In Case (i), one observes that gz ⊥ g− , z ∈ C+ , and hence (z) = 0, s(D,D) ˙

z ∈ C+ .

In Case (ii), one computes   (−iz−1)x e dx z − i (gz , g− ) e(−iz−1) − 1 z − i s(D,D) = e (−iz+1) (z) = · =e · 0 ˙ z + i (gz , g+ ) z+i e −1 e(−iz+1)x dx 0 =

eiz − e− . e− eiz − 1

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Finally, in Case (iii), we have s(D,D) (z) = ˙

z − i (gz , g− ) · z + i (gz , g+ )

 k 0 e(−iz−1)x dx z−i · e =   z + i (1 − k 2 ) 0 e(−iz+1)x dx +  e(−iz+1)x dx −∞ 0 =

k(e(−iz−1) − 1) e (1 − k 2 ) + e(−iz+1) − 1

=

k(e−iz − e ) eiz − e− = k e e−iz − k 2 k 2 e− eiz − 1

=k

eiz − e− . −1

k 2 e− eiz

Combing these results proves (6.5).



Remark 6.3. The representation (6.5) in Cases (i) and (ii) is known (see, e.g., [3]). The following corollary provides a complete characterization of prime symmetric operators with deficiency indices (1, 1) satisfying the commutation relations (1.3) (see Problem (I) a) in the Introduction). Corollary 6.4. Let A˙ be a symmetric operator referred to in Hypothesis 3.1. Suppose that A˙ is a prime operator. Then A˙ is unitarily equivalent to d one of the differentiation operators D˙ = i dx on a metric graph Y in one of the Cases (i)–(iii) introduced in Chapter 4 (see eqs. (4.1), (4.2) and (4.3)). Proof. By Corollary 3.7, A˙ admits a self-adjoint extensions such that ˙ A) coincides with the one the Livˇsic function associated with the pair (A, ˙ referred to in Lemma 6.2. Since A is a prime operator, by the Uniqueness Theorem C.1 in Appendix C, the operator A˙ is unitarily equivalent to the symmetric differentiation operator on the metric graph Y in one of the cases Cases (i)–(iii).  ˙ DΘ ), More generally, the Livˇsic function associated with the pair (D, |Θ| = 1, where DΘ is the self-adjoint realization of the differentiation oper(z) evaluator referred to in Theorem 5.1, differs from the function s(D,D) ˙ ated above in Lemma 6.2 by a constant unimodular factor. For the sake of completeness, we present the following result.

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53

Theorem 6.5. Suppose that |Θ| = 1. The Livˇsic function s(D,D (z) asso˙ Θ) ˙ DΘ ) admits the representation ciated with the pair (D, ⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎨ eiz − e− −2iα , s(D,D (z) = e ˙ e− eiz − 1 Θ) ⎪ ⎪ ⎪ ⎪ ⎪ eiz − e− ⎩ k 2 − iz , k e e −1

in Case (i) in Case (ii)

(6.7)

in Case (iii).

Here k, 0 < k < 1, is the parameter from the boundary conditions (4.3) ˙ and Dom(DΘ ) in Case (iii), and (5.3) describing the domains Dom(D) respectively. Here α and the boundary condition parameter Θ are related as follows

e2iα

⎧ Θ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Θ + e− = e− Θ + 1 , ⎪ ⎪ ⎪ ⎪ Θ + e−(+ ) ⎪ ⎩ , e−(+ ) Θ + 1

in Case (i) in Case (ii) ,

1 with  = ln . k

(6.8)

in Case (iii)

In particular, s(D,D (z) = e−2iα s(D,D) (z), ˙ ˙ Θ)

(6.9)

where D = DΘ |Θ=1 . Proof. From Theorem 5.1 it follows that F = g+ − e2iα g− ∈ Dom(DΘ ),

(6.10)

where g± are the deficiency elements referred to in Lemma 4.3 and e2iα is given by (6.8). Now (6.7) follows from (6.5) by Lemma E.1 in Appendix E.  We conclude this chapter with several remarks of analytic character.

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Remark 6.6. (i) One observes that the Livˇsic function sII (z; ) =

eiz − e− , e− eiz − 1

z ∈ C+ ,

given by (6.7) in Case (ii) admits the representations sin(z − i) 2 eiz − e− = e− eiz − 1 sin(z + i) 2 

z−i 2  1 − 2πn  z − i n∈Z  . · = z + i 1 − z+i 2 2πn

sII (z; ) =

n∈Z

Therefore, sII (z; ) is a pure Blaschke product with zeroes zn located on the lattice zn = i +

2π n, 

n ∈ Z.

(ii) A direct computation shows that the Livˇsic function sIII (z; k, ) = k

eiz − e− , k 2 e− eiz − 1

z ∈ C+ ,

0 < k < 1,

(cf. (6.7) in Case (iii)) can be obtained by an analytic continuation of sII (z; ) with an appropriate identification of the parameters. That is, 

sIII (z; e− , ) = sII



 z + i  , ;  +   + 

z ∈ C+ .

(6.11)

(iii) In the inner-outer factorization of the Livˇsic function in Case (iii) sIII (z; k, ) = k

eiz − e− III = sIII in (z) · sout (z) k 2 e− eiz − 1

the inner factor sIII sic function in Case (ii), i.e., in (z) coincides with the Livˇ II sIII in = s (z; ) =

eiz − e− . e− eiz − 1

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Indeed,

s

III

(z; e

−

 sin(z − i) 2   , ) =   sin (z + i) + i 2  sin(z + i) 2 .  = s (z; ) ·  sin (z + i) + i 2 II



(In particular, the functions sIII (z; e− , ) and sII (z; ) have the same set of zeros). To complete the proof of the claim it remains to show that the function t(z) =

sin(z + i) 2    sin (z + i) + i 2

(6.12)

is an outer function. First, one observes that t(z) is a contractive function in the upper halfplane. Next, let t(z) = tin (z)tout (z) be its inner-outer factorization. Since t(z) does not vanish in the upper half-plane, the inner factor of t(z) is necessarily a singular inner function. Since t(z) admits an analytic continuation into a strip in the lower halfplane, the singular measure in the exponential representation of tin (z) does not charge bounded sets and therefore tin (z) = eiLz for some L ≥ 0, “mass” at infinity. In particular, lim tin (iy) = 0

y→∞

unless L = 0. However, from (6.12) it follows that 

lim t(iy) = e− ,

y→∞

which implies that L = 0. Therefore tin (z) = 1 and hence t(z) is an outer function.

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(iv) In fact, for the outer factor sIII out (z) = t(z) one gets the representation  sinh  sIII out (z) = sinh( + 2 )      i 1 λ + × exp ρ(λ)dλ , (6.13) 2π R λ + z 1 + λ2 where the density is given by ρ(λ) = log

Pe−−2 (λ) . Pe− (λ)

Here Pr (θ) =

1 − r2 1 + r2 − 2r cos θ

is the Poisson kernel. Indeed, since t(z) is an outer function in the upper half-plane, we have the representation [60]      1 λ i + log |t(λ)|dλ . (6.14) t(z) = exp π R λ + z 1 + λ2 Using (6.12) one computes that    − eiz − e−   log |t(λ)| = log e e−2 e− eiz − 1  = − +

(cos λ − e− )2 + sin2 λ 1 log −−2 2 (e cos λ − 1)2 + e−2−4 sin2 λ

1 − 2 cos λe− + e−2 1 log 2 1 − 2e−−2 cos λ + e−2−4    1 1 − e−2 Pe−−2 (λ) = log e−2 · · 2 1 − e−2−4 Pe− (λ) = − +

= and since i π

sinh  1 log + ρ(λ), 2 sinh( + 2 )   R

1 λ + λ+z 1 + λ2

 dλ = 1,

representation (6.14) simplifies to (6.13).

z ∈ C+ ,

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Chapter 7

THE WEYL-TITCHMARSH FUNCTION M(D,D (z) ˙ Θ) Along with the Livˇsic function, the Weyl-Titchmarsh function associated with a pair consisting of a prime symmetric operator and its self-adjoint extension characterizes the pair up to mutual unitary equivalence. So that our next goal is to evaluate the Weyl-Titchmarsh function associated with the symmetric differentiation D˙ on the metric graph Y and its self-adjoint reference extension. Suppose that Y is the metric graph Y in one of the Cases (i)–(iii). As it associated follows from Lemma 6.2, the Weyl-Titchmarsh function M(D,D) ˙ ˙ with the pair (D, D) has the form ⎧ ⎪ i, in Case (i) ⎪ ⎪ ⎪ ⎪ ⎪   ⎨ coth tan z, in Case (ii) M(D,D) (z) = , (7.1) 2 2 ˙ ⎪ ⎪   ⎪ ⎪    +  ⎪ ⎪ ⎩coth tan z+i , in Case (iii) 2 2 2 where  = ln

1 k

(0 < k < 1)

and k, 0 < k < 1, is the parameter from the boundary conditions (4.3) and ˙ and Dom(D) in Case (iii). (6.3) describing the domains Dom(D)

57

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Indeed, in Case (iii) one observes that (z) + 1 ˙ 1 1 s(D,D) = (z) = M(D,D) ˙ i s(D,D) (z) − 1 i ˙

keiz −ke− k2 e− eiz −1 keiz −ke− k2 e− eiz −1

+1 −1

1 1 + ke− keiz − 1  +  = · · = coth · tan i 1 − ke− keiz + 1 2



  z+i 2 2

 .

Case (ii) then follows by setting  = 0, equivalently k = 1, and the corresponding representation in Case (i) is obvious (formally take the limit as  → ∞ in Case (ii)). More generally, we have the following result. Theorem 7.1. Let DΘ , |Θ| = 1, be the one-parameter family of self-adjoint reference operators referred to in Theorem 5.1. Then the Weyl-Titchmarsh ˙ DΘ ) admits the representaassociated with the pair (D, function M(D,D ˙ Θ) tion ⎧ ⎪ i, in Case (i) ⎪ ⎪ ⎪   ⎪ ⎪  Φ sin Φ ⎨ A(Φ) tan z− , in Case (ii) + , (z) = M(D,D ˙ 2 2 sinh  ) Θ ⎪ ⎪   ⎪  ⎪   Φ sin Φ ⎪ ⎪ z+ i− , in Case (iii) + ⎩A(Φ) tan 2 2 2 sinh( +  ) (7.2) where Φ = arg Θ,

 = log

1 k

(0 < k < 1)

and k, 0 < k < 1, is the parameter from the boundary conditions (4.3) and ˙ and Dom(DΘ ) in Case (iii). (5.3) describing the domains Dom(D) Here in Case (iii) the amplitude A(Φ) is given by the convex combination A(Φ) = cos2

 +   +  Φ Φ · coth + sin2 · tanh , 2 2 2 2

(7.3)

and in Case (ii) A(Φ) is given by the same expression with  = 0. Proof. In Case (i) there is nothing to prove, since s(D,D (z) = 0 and ˙ Θ) (z) = i for all z ∈ C . hence M(D,D ˙ + Θ) In Case (ii), by Theorem 6.5 (see eq. (6.9)), we have s(D,D (z) = e−2iα s(D,D) (z), ˙ ˙ Θ)

(7.4)

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where D = DΘ |Θ=1 and α and Θ are related as in (6.8). From (7.4) and Lemma E.1 in Appendix E it follows M(D,D (z) = ˙ Θ)

(z) − tan α M(D,D) ˙ 1 + tan α · M(D,D) (z) ˙

.

(7.5)

By (7.1), M(D,D) (z) = coth ˙

  tan z = m tan ζ =: M(D,D) (ζ), ˙ 2 2

(7.6)

where m = coth

 2

and ζ =

 z, 2

(7.7)

and therefore M(D,D (z) = ˙ Θ)

m tan ζ − tan α . 1 + m tan α tan ζ

Using the identity tan ζ − m tan α 1 + tan2 α m tan ζ − tan α m = 1 + m tan α tan ζ 1 + m2 tan2 α 1 + m tan α tan ζ +

m2 tan α − tan α , 1 + m2 tan2 α

we obtain the following representation M(D,D (z) = ˙ Θ)

1 + tan2 α M ˙ (ζ − t) 1 + m2 tan2 α (D,D) +

m2 tan α − tan α , 1 + m2 tan2 α

where tan t = m tan α. Therefore, (7.8) can be rewritten as M(D,D (z) = ˙ Θ)

1 1 + m12 tan2 t m− m M tan t. (ζ − t) + ˙ (D,D) 1 + tan2 t 1 + tan2 t

(7.8)

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In view of (7.6) and (7.7), we have     2 2 (z) = coth cos t + tanh sin t M(D,D ˙ Θ) 2 2    1 × tan z−t + sin 2t. 2 sinh 

(7.9)

From (6.8) it follows that 1 tan α = i

Θ−e− e− Θ−1 Θ−e− e− Θ−1

−1

 = tanh · tan 2 +1



so that  tan t = m tan α = coth · tan α = tan 2

 1 arg Θ , 2



 1 arg Θ . 2

In particular, 1 1 arg Θ = Φ, 2 2 which along with (7.9) shows that    Φ sin Φ M(D,D z − , (z) = A(Φ) tan + ˙ Θ) 2 2 sinh  t=

(7.10)

proving (7.2) with A(Φ) given by (7.3) in Case (ii). To prove (7.2) in Case (iii), in a similar way one gets (cf. (7.5))      +  tan z+i M(D,D) (z) = coth ˙ 2 2 2 and establishes that (7.8) holds where now ζ=

  z+i , 2 2

m = coth

 +  2

and tan t = m tan α. Observing that 

1 tan α = i

Θ+e−− e−− Θ+1 Θ+e−− e−− Θ+1

−1 +1

= tanh

 +  tan 2



 1 arg Θ , 2

one justifies (7.10) in Case (iii) as well. Literally repeating the reasoning above one obtains the representation (7.2) in Case (iii). 

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Our last result in this chapter shows that the spectral measure of the reference operator DΘ , |Θ| = 1, is rather sensitive to the magnitude of the “flux” Φ = arg Θ. (z) Corollary 7.2. The Weyl-Titchmarsh function MΘ (z) = M(D,D ˙ Θ) ˙ DΘ ) admits the representation associated with the pair (D,  

MΘ (z) =

R

λ 1 − λ−z 1 + λ2



dμΘ (λ),

|Θ| = 1,

where μΘ (dλ), the spectral measure, is (i) the absolutely continuous measure with a constant density μΘ (dλ) =

1 dλ π

in Case (i);

(7.11)

(ii) the discrete pure point measure μΘ (dλ) =

2 A(arg Θ) δ (2k+1)π+Φ (dλ)  

in Case (ii),

(7.12)

k∈Z

with δx (dλ) the Dirac mass at x and A(Φ) = cos2

  Φ Φ · coth + sin2 · tanh ; 2 2 2 2

(iii) the absolutely continuous measure with a periodic density μΘ (dλ) =

1 A(arg Θ)Pe− (λ − π − arg Θ) dλ. π

(7.13)

Here Pr (ϕ) =

1 − r2 1 + r2 − 2r cos ϕ

is the Poisson kernel, 1  = log , k with k, 0 < k < 1, the parameter from the boundary conditions (4.3), (5.3), and A(Φ) = cos2

 +   +  Φ Φ · coth + sin2 · tanh . 2 2 2 2

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Proof. Indeed, since i=

1 π

  R

λ 1 − λ−z 1 + λ2

 dλ,

z ∈ C+ ,

(7.11) follows from the equality MΘ = i in Case (i). To check (7.12), we use the representation  ∞  1 1 tan(z) = − + z − (k + 12 )π z + (k + 12 )π k=0    1 λ − = dν(λ), λ−z 1 + λ2 R where ν(dλ) is a discrete point measure δ(k+ 12 )π (dλ), ν(dλ) =

(7.14)

(7.15)

k∈Z

which proves (7.12) in the particular case of  = 2, and Θ = 1, and then the general case follows by making a simple change of variables. Using the explicit representation (7.2) for the Weyl-Titchmarsh function MΘ (z) in Case (iii) one observes that MΘ (z) is bounded in the upper half-plane. Therefore, the representing measure μΘ (dλ) in Case (iii) is absolutely continuous with the density given by μΘ (dλ) =

=

1 1 Im MΘ (λ + i0)dλ = A(arg Θ) π π   λ +  i − arg Θ · Im tan dλ 2 1 A(arg Θ)Pe− (λ − π − arg Θ) dλ, π

(7.16)

which proves (7.13). Here we have used the representation for the imaginary part of the tangent function of a complex argument via the the Poisson kernel, 1 ei(λ+iτ ) − e−i(λ+iτ ) Im tan (λ + iτ ) = Im · i(λ+iτ ) i e + e−i(λ+iτ ) = −Re =

e2τ

eiλ e−τ − e−iλ eτ e−iλ e−τ + eiλ eτ · eiλ e−τ + e−iλ eτ e−iλ e−τ + eiλ eτ

e2τ − e−2τ = Pe−2τ (2λ − π), + 2 cos 2λ + e−2τ λ ∈ R,

τ > 0.



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Remark 7.3. Observing that 1 Pr (λ)dλ → δn (dλ) 2π n

as r ↑ 1,

we get the following spectral hierarchy of the representing measures given by (7.13), (7.12) and (7.11): 1 A(arg Θ) · Pe− (λ − π − arg Θ) dλ π ↓ ( → 0) 2 A(arg Θ)· δ (2πk+1)π+Φ (dλ)   k∈Z

↓

( → ∞)

1 dλ, π with the limits as  → 0 and  → ∞ taken in the sense of the weak convergence of the measures. Here we used the inequality (see (7.3)) tanh

 +   +  ≤ A(Φ) ≤ coth 2 2

on the last step that ensures that the amplitude A(Φ) approaches 1 as  → ∞.

B1948

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Chapter 8

THE MODEL DISSIPATIVE OPERATORS Given a metric graph Y in one of the Cases (i)–(iii) and a real parameter k, 0 ≤ k < 1, we construct a family of model maximal dissipative differentiation operators on Y. In what follows we will refer to the parameter k as the quantum gate coefficient on the graph Y. If the metric graph Y is in Case (i), Y = (−∞, 0)  (0, ∞), denote by  =D  I (k) = i d , 0 ≤ k < 1, D dx the maximal dissipative differentiation operator with the boundary condition at the origin f∞ (0+) = kf∞ (0−),

0 ≤ k < 1.

(8.1)

Notice that the case k = 0 is exceptional in the sense that the point  I (0) fills in the entire (open) upper spectrum of the dissipative operator D half-plane.  I (k) The corresponding strongly continuous semi-group generated by D describes the motion from left to right of a (quantum) particle which is emitted outside (see Fig. 8.1) with probability 1 − k 2 through the quantum gate at the origin and keeps moving along the axes with probability k 2 . If the metric graph Y is in Case (ii) and Y = (0, )

65

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k

Fig. 8.1. Dynamics on the Metric Graph Y in Case (i) with the Quantum Gate Coefficient k

Fig. 8.2. Dynamics on the Metric Graph Y in Case (ii) with the Quantum Gate Coefficient k = 0

for some  > 0, denote by  =D  II (k, ) = i d , D dx

(8.2)

the maximal dissipative differentiation operator determined by the boundary condition f (0) = kf (),

0 ≤ k < 1.

(8.3)

Notice that the case k = 0 is also exceptional. That is, the dissipative  II (0, ) corresponding to the boundary condition differentiation operator D f (0) = 0

(8.4)

has no spectrum.  II (0, ) describes the motion The (nilpotent) semi-group generated by D of a particle which is emitted with probability one at the right end-point of the finite interval [0, ] (see Fig. 8.2). Finally, if the graph Y is in Case (iii), Y = (−∞, 0)  (0, ∞)  (0, ),

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k

√ 1 − k2 Fig. 8.3. Dynamics on the Metric Graph Y in Case (iii) with the Quantum Gate Coefficient k

 = D  III (k, ) = i d the maximal dissipative differentiation denote by D dx operator on Y with the boundary conditions      k 0 f∞ (0−) f∞ (0+) = √ , 0 < k < 1. (8.5) 1 − k2 0 f (0) f () The dynamics associated with the strongly continuous semi-group gen III (k, ) describes the motion a wave-packet moving from left erated by D to right (see Fig. 8.3). If the packet is initially supported by the negative semi-axis, after the interaction with the scatterer located at the center of the graph, the particle continues its rightward motion along the real axis with its initial shape amplified by the factor k while a copy of the wave-packet amplified by √ 1 − k 2 turns right onto an appendix of length  attached to the obstacle. When the wave-packet approaches the right end of the interval [0, ] the wave is terminated. From the boundary conditions (8.5) it follows that the quantum Kirchhoff rule (at the junction) |f∞ (0−)|2 = |f∞ (0+)|2 + |f (0)|2

(8.6)

holds. Taking into account wave-particle duality, one can also say that the corresponding particle with probability k 2 keeps moving along the real axis and with probability 1 − k 2 enters the appendix. Then, assuming that the initial profile of the wave-packet was supported by the interval [−L, 0), the particle is emitted with probability one after time t =  + L has elapsed.

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Notice that a wave-packet initially supported to the right of the obstacle moves to the right without changing its shape regardless whether the wave is supported by the semi-axis (−∞, 0] or by the finite interval [0, ]. To complete the description of the dynamics in the general case, one applies the superposition principle. Remark 8.1. Notice that the boundary conditions (8.1), (8.4) and (8.5) (but not (8.3) with k = 0) are the local vertex conditions, which means that different vertices do not interact. In particular, the domains of the corre are invariant with respect to the group sponding dissipative operators D of local gauge transformations. As a corollary, the dissipative operators  =D  I,II,III satisfy the commutation relations D  t=D  + tI Ut∗ DU

on

 Dom(D),

t ∈ R,

(8.7)

where Ut = e−itQ is the unitary group generated by the operator Q of multiplication by independent variable on the graph Y. This can be justified immediately but it also follows from a more general considerations below (cf. Remark 4.6). Let A(x) denote a real-valued piecewise continuous function on Y. We  and D+A(x)  remark that the operators D are unitarily equivalent. Indeed, let φ(x) be any solution to the differential equation φ (x) = A(x), on the edges of the graph and continuous at the origin {0} ∈ Y. Denote by V the unitary local gauge transformation (V f )(x) = eiφ(x) f (x),

f ∈ L2 (Y).

(8.8)

Then, taking into account the boundary conditions (8.1), (8.4) and (8.5)  is V -invariant one concludes that the domain of D  = Dom(D),  V (Dom(D)) and moreover,  = V ∗ (D  + A(x))V. D In particular, (8.7) holds. Remark 8.2. Notice that the model dissipative differentiation operators  extend the symmetric differentiation operator D˙ on the graph Y with D

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the boundary conditions (4.1), (4.2) and (4.3), respectively. Moreover, the  and symmetric operator D˙ is uniquely determined by D D D˙ = D|

 ∩ Dom((D)  ∗ ). where D = Dom(D)

 is in Case (i), then the corresponding symmetric operator D˙ admits If D a quasi-selfadjoint extension the point spectrum of which fills in the entire upper half-plane. Notice that this property characterizes the operator up to unitary equivalence. That is, any prime closed symmetric operator with deficiency indices (1, 1) that admits a quasi-selfadjoint extension with point spectrum filling in the whole upper half-plane is unitarily equivalent to the operator D˙ in Case (i) [3, Ch. IX, Sec. 114]. Apparently, any point from  I (0). C+ is an eigenvalue for the extension D Moreover, the symmetric operator D˙ has a relatively poor family of unitarily inequivalent (dissipative) quasi-selfadjoint extensions. The reason is that any two (dissipative) extensions with the same absolute value of the von Neumann parameter k, (0 ≤ k ≤ 1), are unitarily equivalent to the  I (k). Recall that the absolute value of the von Neumann paramoperator D eter of the dissipative operators in question is well defined (see Remark 2.5). This property also characterizes D˙ up to unitary equivalence: any prime closed symmetric operator with deficiency indices (1, 1) that admits two distinct unitarily equivalent quasi-selfadjoint extensions is unitarily equivalent to the operator D˙ [5, Theorem 2].  =D  II (0, ) in Case (ii) given by (8.2) and The dissipative operator D (8.3) is the only dissipative extension of D˙ II () whose resolvent set coincides with the whole complex plane C. Moreover, any dissipative quasiselfadjoint extension of a prime closed symmetric operator with deficiency indices (1, 1) without spectrum is unitarily equivalent to the symmetric differentiation operator D˙ II () on a finite interval of length  [73, Theorem 14]. We remark that in contrast to Case (i), in Cases (ii) and (iii) the max with the boundary conditions (8.1) and (8.4), imal dissipative operator D respectively, is the only one maximal dissipative extension of the symmetric differentiation operator D˙ with a gauge invariant domain. Indeed, the boundary conditions (5.2) and (5.3) are gauge invariant if either Θ = 0 or Θ = ∞. If Θ = ∞, the corresponding quasi-selfadjoint extension is not dissipative, which proves the claim. The following structure theorem shows that the differentiation  operator D,  =D  III (k, ), D

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in Case (iii) can be obtained as the result of an operator coupling (spectral synthesis) of the more “elementary” dissipative differentiation operators  =D  I (k) in Case (i) and D  =D  II (0, ) in Case (ii). For the concept of D an operator coupling we refer to Appendix G.  III (k, ) on the metric graph Theorem 8.3. The differentiation operator D YIII = (−∞, 0]  [0, ∞)  [0, ] with the quantum gate coefficient k is an operator coupling of the differen I (k) on the edge tiation operator D YI = (−∞, 0]  [0, ∞)  II (0, ) on the with the same quantum gate coefficient k and the operator D remaining edge YII = [0, ] with the quantum gate coefficient 0, respectively. That is, YIII = YI  YII and  III (k, ) = D  I (k) D  II (0, ). D

(8.9)

Proof. To see that, set H1 = L2 ((−∞, ∞)) and H2 = L2 ((0, )). One observes that  III (k, )|  D  III (k,))∩H1 = DI (k) Dom(D and hence the requirement (i) in the definition G.4 (see Appendix G) of a coupling of two dissipative operators is met.  II ())∗ ) consists of the three-component  I (0, k))⊕ Dom((D Next, Dom(D T functions f = (f− , f+ , f ) , with f− ⊕ f+ ⊕ f ∈ W21 ((−∞, 0)) ⊕ W21 ((0, ∞)) ⊕ W21 ((0, )) such that f+ (0+) = kf− (0−)

and

f (−) = 0

and hence ˙ ⊂ Dom(D  I (k)) ⊕ Dom((D  II (0, ))∗ ). Dom(D)

(8.10)

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In particular,  I (k) ⊕ (D  II (0, ))∗ D˙ ⊂ D and hence the requirement (ii) in the definition of an operator coupling is met as well. Therefore, (8.9) holds.  For the further references it is convenient to adopt the following Hypothesis. Hypothesis 8.4. Assume that the metric graph Y is in one of the Cases  is the model dissipative operator with the quantum gate (i)–(iii) and D coefficient k on Y given by the boundary conditions (8.1), (8.3) and (8.5),  on D = Dom(D)  ∩ Dom((D)  ∗ ). respectively. Let D˙ be the restriction of D Assume that DΘ is the self-adjoint reference extension of D˙ referred to in Theorem 5.1 (see (5.1), (5.2), and (5.3)). If the graph Y is in Case (ii) assume that k = 0 and if Y is in Case (iii) we require that k = 0. Definition 8.5. Under Hypothesis 8.4 suppose that k = 0 whenever the graph Y is in Case (ii) and that k = 0 whenever Y is in Case (iii). Under ˙ D,  DΘ ) this assumption we call the triple of differentiation operators (D, the model triple on Y with the quantum gate coefficient k. More explicitly, each of the differentiation operators from the triple ˙  DΘ ) in the Hilbert L2 (Y) is given by the differential expression (D, D, τ =i

d dx

(on the edges of the graph Y) initially defined on the Sobolev space W21 (Y), ⎧ 1 W ((−∞, 0)) ⊕ W21 ((0, ∞)) ⎪ ⎪ ⎨ 2 W21 (Y) = W21 ((0, )) ⎪ ⎪ ⎩ 1 (W2 ((−∞, 0)) ⊕ W21 ((0, ∞)) ⊕ W21 ((0, )). In Case (i), the metric graph has the form Y = (−∞, 0)  (0, ∞), and ˙ = {f∞ ∈ W 1 (Y) |f∞ (0+) = f∞ (0−) = 0}, Dom(D) 2  = {f∞ ∈ W 1 (Y) | f∞ (0+) = kf∞ (0−)}, Dom(D) 2 Dom(DΘ ) = {f∞ ∈ W21 (Y) | f∞ (0+) = −Θf∞ (0−)}, 0 ≤ k < 1.

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In Case (ii), Y = (0, ), ˙ = {f ∈ W 1 (Y) |f (0) = f () = 0}, Dom(D) 2  = {f ∈ W21 (Y) |f (0) = 0}, Dom(D) Dom(DΘ ) = {f ∈ W21 (Y) |f (0) = −Θf ()}. In Case (iii), Y = (−∞, 0)  (0, ∞)  (0, ), and ⎧ ⎧ ⎫ kf∞ (0−) ⎪ ⎪ ⎪ ⎨ ⎨ f∞ (0+) = √ ⎬ 2 f (0−) ˙ = f∞ ⊕ f ∈ W 1 (Y) | , Dom(D) (0+) = 1 − k f  ∞ 2 ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ f () = 0

f∞ (0+) = kf∞ (0−) 1  √ Dom(D) = f∞ ⊕ f ∈ W2 (Y) | , f (0+) = 1 − k 2 f∞ (0−)

√ f∞ (0+) = kf∞ (0−) + 1 − k 2 Θf () 1 √ , Dom(DΘ ) = f∞ ⊕ f ∈ W2 (Y) | f (0+) = 1 − k 2 f∞ (0−) − kΘf () 0 < k < 1.

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Chapter 9

THE CHARACTERISTIC FUNCTION S(D,  Θ ) (z) ˙ D,D

Now we are ready to evaluate the characteristic function S(D, ˙ D,D  Θ ) (z) of ˙ D,  DΘ ) of differentiation operators on a metric graph Y in the triple (D, Cases (i)–(iii). ˙ D,  D) for First, we evaluate the characteristic function of the triple (D, a particular choice of the reference self-adjoint operator D referred to in Theorem 5.1, in Cases (i), (ii) and (iii) with Θ = 1, respectively. Recall that the operator D is determined by the boundary conditions f∞ (0+) = −f∞ (0−), f (0) = −f (), ⎧ √ ⎨f∞ (0+) = kf∞ (0−) + 1 − k 2 f () √ ⎩f (0+) = 1 − k 2 f∞ (0−) − kf () in Cases (i)–(iii), respectively. ˙ D,  D) be the model triple of the differentiation operLemma 9.1. Let (D, ators on the metric graph Y in one of the Cases (i)–(iii) as above. Then ˙ D,  D) admits the representation the characteristic function of the triple (D, ⎧ in Case (i) ⎪ ⎪k, ⎨ iz (z) = e , (9.1) z ∈ C+ , S(D, in Case (ii) ˙ D,D)  ⎪ ⎪ ⎩ iz ke , in Case (iii), where 0 ≤ k < 1 (in Case (i)) and 0 < k < 1 (in Case (iii)). 73

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Proof. To check (9.1) we proceed as follows. Let g± be the deficiency elements given by (4.6) in Case (i), (4.7) in Case (ii) and (4.8), (4.9) in Case (iii). We claim that  f = g+ − S(i)g− ∈ Dom(D),

(9.2)

where we have used the shorthand notation S(z) = S(D, (z). ˙ D,D)  It suffices to check that f satisfies the boundary conditions (8.1), (8.4) and (8.5) in Cases (i)–(iii), respectively, and therefore  f = g+ − S(i)g− ∈ Dom(D). Indeed, in Case (i), S(i) = k,

(9.3)

and hence f∞ (x) =

√ x 2(e χ(−∞,0) (x) + ke−x χ(0,∞) (x)).

Clearly, f∞ (0−) = kf∞ (0+),  and therefore f∞ ∈ Dom(D). In Case (ii), S(i) = e− ,

(9.4)

and therefore the element √ 2

f (x) = √ (ex − e− e−x ), e2 − 1

x ∈ [0, ],

satisfies the Dirichlet boundary condition f (0) = 0 which proves that f ∈  Dom(D). Finally, in Case (iii) we have that S(i) = ke− .

(9.5)

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Therefore, the element f = g+ − S(i)g− = g+ − ke− g− admits the representation (see (4.8) and (4.9)) √ 2 √ f= (f∞ , f )T , 2 e − k2 where f∞ (x) =

  1 − k 2 ex χ(−∞,0) (x) + ke− 1 − k 2 e−x χ(0,∞) (x),

x ∈ R,

and f (x) = ex − (ke− )ke−x , We have

x ∈ [0, ].



1 − k2 ,  f∞ (0+) = k 1 − k 2 ,

f∞ (0−) =

f (0) = 1 − k 2 , which shows that the boundary conditions (8.5) hold. Therefore f ∈  Dom(D). Since g+ − g− ∈ Dom(D) and  g+ − S(i)g− ∈ Dom(D), one computes S(D, (z) = ˙ D,D) 

(z) − S(i) s(D,D) ˙ S(i)s(D,D) (z) − 1 ˙

=

(z) − ke− s(D,D) ˙ ke− s(D,D) (z) − 1 ˙

.

It remains to remark that since the Livˇsic function s(D,D) (z) is given by ˙ (6.5), one gets (9.1) by a direct computation.  Remark 9.2. Notice that Lemma 9.1, in particular, states that the char˙ D,  D) in Case (iii) is the product of the acteristic function of the triple (D, characteristic functions in Cases (i) and (ii), respectively. That is, (z) = k · eiz . S(D, ˙ D,D) 

(9.6)

In view of Theorem 8.3, the rule (9.6) can also be obtained as a corollary of the Multiplication Theorem G.5 in Appendix G.

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Also, comparing (9.3), (9.4) and (9.5), one observes that the von Neu III (k, ) associated with the bases mann parameter ke− of the coupling D (4.8) and (4.9) is the product of the von Neumann parameters k and e− of  II () with respect to the bases (4.6)  I (k) and D the dissipative operators D and (4.7), respectively (see Remark 2.1 for the terminology). This observation illustrates the multiplicativity property for the absolute values of the von Neumann extension parameters under coupling (see [87, Theorem 5.4] or Theorem G.5 in Appendix G). Recall that the concept of absolute value of the von Neumann parameter is well defined by Remark 2.5. The more general result below can be understood as the solution of the following inverse problem: find a triple with a prescribed characteristic function referred to in Theorem 3.5 (cf. [55, Theorem 20]). Theorem 9.3. The characteristic function S(D, ˙ D,D  Θ ) (z) of the model triple ˙ D,  DΘ ), |Θ| = 1, admits the representation (D, ⎧ k, in Case (i) ⎪ ⎪ ⎨ −2iα S(D, (9.7) eiz , in Case (ii) , z ∈ C+ , ˙ D,D  Θ ) (z) = e ⎪ ⎪ ⎩ iz ke , in Case (iii) where 0 ≤ k < 1 (in Case (i)) and 0 < k < 1 (in Case (iii)). Here α and the boundary condition parameter Θ are related as follows ⎧ ⎪ Θ, in Case (i) ⎪ ⎪ ⎪ ⎪ ⎪ − ⎨ Θ+e in Case (ii) e2iα = e− Θ + 1 , (9.8) ⎪ ⎪  ⎪ ⎪ Θ + e−(+ ) ⎪ ⎪ ⎩ , in Case (iii), with  = ln k1 . e−(+ ) Θ + 1 In particular, −2iα S(D, S(D, (z). ˙ D,D  Θ ) (z) = e ˙ D,D) 

(9.9)

Proof. In view of Lemma E.1 in Appendix E, the assertion of the theorem is a direct consequence of Theorem 6.5.  Remark 9.4. Observe that in Case (ii) the characteristic function of the ˙ D,  DΘ ) is a singular inner function with “mass at infinity.” triple (D, ˙ D,  DΘ ), |Θ| = 1, and (D, ˙ D,  DΘ ), Corollary 9.5. The model triples (D,   |Θ | = 1, Θ = Θ, are not mutually unitarily equivalent unless the graph

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Y is in Case (i) and the point spectrum of the dissipative differentiation  fills in the whole upper half-plane C+ . In the latter case the operator D triples in question are mutually unitarily equivalent to one another for any Θ and Θ (|Θ| = |Θ | = 1). Proof. Combining (9.7), (9.8) and (9.9) shows that the characteristic func(z) are different for Θ = Θ unless tions S(D, ˙ D,D  Θ ) (z) and S(D, ˙ D,D  ) Θ

S(D, ˙ D,D  Θ ) (z) = 0

for all z ∈ C+

 has no regular for some (and therefore for all) |Θ| = 1. In the latter case D ˙  points in the upper half-plane and the triples (D, D, DΘ ), |Θ| = 1, and ˙ D,  DΘ ), |Θ | = 1, are mutually unitarily equivalent by the uniqueness (D, Theorem C.1 in Appendix C.  Remark 9.6. Let A(x) be a real-valued piecewise continuous function on the metric graph Y in Cases (i)–(iii). Combining Remarks 4.6, 5.2 and 8.1 + imply that if the graph Y is in Case (i), then the triple (D˙ + A(x), D ˙ D,  DΘ ). A(x), DΘ + A(x)) is mutually unitarily equivalent to the triple (D,  + A(x), DΘ + A(x)) If the graph is in Cases (ii) or (iii), then (D˙ + A(x), D ˙ D,  DΘe−iΦ ), where is mutually unitarily equivalent to the triple (D,   Φ= A(x)dx. 0

Moreover, the corresponding unitary equivalence is given by a gauge transformation. ˙ D,  DΘ ), The knowledge of the characteristic function of the triple (D, which is its complete unitary invariant, enables us to obtain the converse to structure Theorem 3.5. ˙ A,  A) be a triple of operators in a Hilbert space H Theorem 9.7. Let (A, ˙  is where A is a prime symmetric operator with deficiency indices (1, 1), A its dissipative quasi-selfadjoint extension and A is a reference self-adjoint ˙ extension of A. (z) of the triple Suppose that the characteristic function S(z) = S(A, ˙ A,A)  ˙  (A, A, A) admits the representation S(z) = keiz ,

z ∈ C+ ,

(9.10)

for some |k| ≤ 1 and  ≥ 0. We also assume that if  = 0, then necessarily |k| < 1 and if |k| = 1, then  > 0.

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˙ Then there exists a unitary group Ut such that the domains Dom(A)  are Ut -invariant and and Dom(A) ˙ t = A˙ + tI Ut∗ AU

˙ on Dom(A),

(9.11)

 t=A  + tI Ut∗ AU

 on Dom(A).

(9.12)

˙ A,  A) Proof. By the uniqueness Theorem C.1 in Appendix C, the triple (A, ˙ D,  DΘ ) referred to in Theis mutually unitarily equivalent to the triple (D, orem 9.3 for some choice of the extension parameter Θ. By (4.17) and (8.7), ˙ −itQ = D˙ + tI eitQ De

˙ on Dom(D),

t ∈ R,

 −itQ = D  + tI eitQ De

 on Dom(D),

t ∈ R,

and

where Q the self-adjoint operator of multiplication by independent variable on the graph Y. Pulling back the group e−itQ in L2 (Y) to the original Hilbert space H proves the assertion.  It is worth mentioning that the choice of the orientation of the graph Y was ad hoc from the very beginning. To complete the exposition, along with the graph Y, consider the metric graph Y∗ obtained from Y by reversing  ∗ on Y∗ the orientation. The corresponding differentiation operator −(D) extends the (symmetric) differentiation operator d −D˙ = −i , dx and its domain is determined by the following boundary conditions in Case (i): f∞ (0−) = kf∞ (0+);

(9.13)

f () = 0;

(9.14)

in Case (ii):

in Case (iii):

⎧ √ ⎨f∞ (0−) = kf∞ (0+) + 1 − k 2 f (0) ⎩f ()

=0

.

(9.15)

Notice that the graph Y∗ in Case (iii) (as opposed to the graph Y) has only two incoming and only one outgoing bonds which is reflected in a

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slightly different way of posing boundary conditions (cf. (8.5) and (9.15)).  ∗ solve the commutation relations (8.7). ˙ ∗ and −(D) Meanwhile, both −(D) On the algebraic level, it can be seen by observing that the relations  t=A  + tI Ut∗ AU

 on Dom(A),

and  ∗ Vt = (−A)  ∗ + tI Vt∗ (−A)

 on Dom(A),

with Vt = U−t , imply one another. In fact, reversing the orientation of the graph Y does not lead to the new solutions as far as the classification up to unitary equivalence is concerned. ˙ −(D)  ∗ , −D ) are mutually ˙ D,  DΘ ) and (−D, Lemma 9.8. The triples (D, Θ unitarily equivalent. Proof. From Theorem 9.3 it follows that −2iα keiz , S(D, ˙ D,D  Θ ) (z) = e

where α is given by (6.8). Applying Lemma F.2 in Appendix F, we have that −2iα keiz = S(D, S(−D,−( ˙  ∗ ,−D ) (z) = S(D, ˙ D,D  ˙ D,D  Θ ) (z), D) ) (−z) = e Θ

Θ

which ensures a mutual unitary equivalence of the triples in question by the uniqueness Theorem C.1 in Appendix C (the symmetric operator D˙ is ˙ prime by Lemma 4.4, so is the operator −D). 

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Chapter 10

THE TRANSMISSION COEFFICIENT AND THE CHARACTERISTIC FUNCTION Recall that if the metric graph Y is in Cases (ii) or (iii), the differentiation operators DΘ , Θ ∈ C ∪ {∞}, referred to in Theorem 5.4 satisfy the commutation relations 2π Z, Ut∗ DΘ Ut = DΘ + tI, t ∈  with respect to a discrete subgroup of one-parameter strongly continuous group of unitary operators Ut . On fact, one can choose Ut = e−itQ ,

t ∈ R,

where Q is the multiplication operator by independent variable on the graph Y. However, in the exceptional cases Θ = ∞, the semi-Weyl relations Ut∗ DΘ Ut = DΘ + tI,

t ∈ R,

(Θ = 0

or Θ = ∞),

hold (D∞ = −D0∗ ). Suppose that the metric graph Y is in Case (ii), that is, Y = (0, ). Our fist goal is to evaluate the characteristic function of a dissipative triple on the graph Y.

81

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˙ dΘ , d) be the triple, where d˙ is the symmetric To be more specific, let (d, differentiation on ˙ = {f ∈ W 1 ((0, )) | f (0) = f () = 0}, Dom(d) 2  II (−kΘ, ) (0 < k < 1) dΘ = D is the maximal dissipative differentiation operator on Dom(dΘ ) = {f ∈ W21 ((0, )) | f (0) = −kΘf ()}, and d is the self-adjoint differentiation operator defined on Dom(d) = {f ∈ W21 ((0, )) | f (0) = −f ()}. Lemma 10.1. The characteristic function S(d,˙ dΘ ,d) (z) of the triple ˙ dΘ , d) has the form (d, S(d,˙ dΘ ,d) (z) =

Θ + e− k · B(z), ke− Θ + 1

z ∈ C+ ,

(10.1)

where 

cos z−arg2 Θ−i eiz + kΘ = B(z) =  , Θ + eiz k cos z−arg2 Θ+i

z ∈ C+ ,

is a pure Blaschke product with simple zeros zn given by zn = i

π arg Θ  + (2πn + 1) + ,   

with

1  = log , k

(10.2)

0 < k < 1. In particular, the characteristic function S(d,˙ dΘ ,d) (z) of the triple ˙  (d, dΘ , d) is a periodic function   2π S(d,˙ dΘ ,d) z + = S(d,˙ dΘ ,d) (z), z ∈ C+ ,  with the minimal period T =

2π  .

Proof. Let g± be the deficiency elements of the symmetric operator d˙ given by (4.7).

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˙ its domain can be repreSince dΘ is a quasi-selfadjoint extension of d, sented as ˙ +span{g ˙ Dom(dΘ ) = Dom(d) + − κg− }, for some |κ| < 1. In particular the function g+ −κg− satisfies the boundary condition (g+ − κg− )(0) = −kΘ(g+ − κg− )(), which allows to relate the von Neumann extension parameter κ and the coefficient k as 1 − κe = −kΘ(e − κ). Therefore, κ=

kΘ + e− . kΘe− + 1

Since g+ − g− ∈ Dom(d), one computes (see (2.9)) that S(z) = S(d,˙ dΘ ,d) (z) =

s(z) − κ , κs(z) − 1

(10.3)

˙ d) where s(z) = s(d,d) sic function associated with the pair (d, ˙ (z) is the Livˇ given by Lemma 6.2 as s(z) = s(d,d) ˙ (z) =

eiz − e− . e− eiz − 1

We claim that the inner functions B(z) and S(z) have the same set of (simple) roots. Indeed, if S(z0 ) = 0, then s(z0 ) =

eiz0 − e− = κ. e− eiz0 − 1

(10.4)

Therefore, eiz0 =

e− − κ 1 − e κ = −kΘ, =  − 1 − κe e −κ

which implies that B(z0 ) = 0, and vice versa. Since both B(z) and S(z) are Blaschke products, we get B(z) =

B(i) S(z). S(i)

(10.5)

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To complete the proof it remains to observe that B(i) B(i) e− + kΘ kΘe− + 1 kΘe− + 1 = = · . = − − S(i) κ Θ + e k kΘ + e Θ + e− k

(10.6) 

Next, recall that by Theorem 5.7 the self-adjoint magnetic Hamiltonian DΘ , |Θ| = 1, on the metric graph Y = (−∞, 0)  (0, ∞)  (0, ) in Case (iii) referred to in Theorem 5.7 dilates the maximal dissipative differentiation operator dΘ . Define the transmission coefficient t(λ) in the scattering problem on the graph Y (obtained from Y by identifying the right vertex of the edge [0, ] with the vertex at the origin) as the amplitude of the generalized eigenfunction of the Hamiltonian DΘ , the solution f to the equation i

d f = λf dx

on

Y = (−∞, 0)  (0, ∞)  (0, )

(10.7)

that coincides with e−iλx on the incoming edge (−∞, 0) of the graph Y, equals t(λ)e−iλx on the outgoing edge (0, ∞), and f = (f∞ , f ) satisfies the boundary conditions (5.3),      √ f∞ (0−) 1 − k2 Θ k f∞ (0+) = √ . (10.8) f (0) f () 1 − k2 −kΘ The analytic counterpart of the dilation Theorem (5.7) is as follows. Theorem 10.2. The transmission coefficient t(λ) in the scattering problem (10.7), (10.8) has the form t(λ) =

Θ + eiλ k , eiλ + kΘ

λ ∈ R.

(10.9)

Proof. Let f be the solution to the scattering problem (10.7). We have f∞ (λ, x) = e−iλx χ(−∞,0) (x) + t(λ)e−iλx χ(0,∞) (x), and f (λ, x) = a(λ)e−iλx ,

x ∈ (0, ).

x ∈ R,

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85

From (5.3) it follows that      √ 1 t(λ) k 1 − k2 Θ . = √ a(λ)e−iλ a(λ) 1 − k2 −kΘ Solving for a(λ) we get that

√

a(λ) =

1 − k2 1 + kΘe−iλ

and hence   t(λ) = k + 1 − k 2 Θe−iλ a(k) = k + 1 − k 2 Θe−iλ =

Θ + eiλ k , eiλ + kΘ

√

1 − k2 1 + kΘe−iλ

λ ∈ R.



Remark 10.3. We observe that if one sets Θ = 1 in (10.9), then t(λ) =

1   , s    λ

where  = log k1 and 

s (z) =



eiz − e− e− eiz − 1

˙ D) on the metric graph is the Livˇsic function associated with the pair (D, Y = (0,  ) in Case (ii) referred to in Lemma 6.2. Corollary 10.4. Let t(λ) be the transmission coefficient in the scattering problem (10.7), (10.8). Then, t(λ) =

Θ + e− k · S −1 (λ + i0), ke− Θ + 1

λ ∈ R,

(10.10)

˙ dΘ , d) in Case (ii). where S(z) is the characteristic function of the triple (d, In particular, the poles of the analytic (meromorphic) continuation of the transmission coefficient t(λ) to the upper half-plane coincide with the eigenvalues of the dissipative operator dΘ .

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Proof. Representation (10.10) follows from (10.1) and (10.9). Since S(z) is analytic in C+ , the transmission coefficient t(λ) can be meromorphically continued on the whole complex plane. The (simple) poles of this continuation are located at the zeroes of the characteristic function S(z) which  are the eigenvalues of the dissipative operator dΘ . ˙ d,  d), where d is the Remark 10.5. The exceptional case of the triple (d, differentiation operator on  = {f ∈ W 1 ((0, )) | f (0) = 0} Dom(d) 2

(10.11)

deserves a special discussion. Notice that d is the only one dissipative quasiselfadjoint extension of the symmetric differentiation d˙ which is not in the family (5.21) with k = 0 (cf. Frostman’s observation of general character: S−κ is a pure Blaschke product for almost if S is an inner function, then 1−κS all κ ∈ D). By Lemma 9.1, the characteristic function of the triple = eiz , S(d,˙ d,d) 

z ∈ C+ ,

is a singular inner function (see Remark 9.4). On the other hand, the transmission coefficient of the self-adjoint dilation D of the dissipative operator d on the one-cycle graph can be evaluated by solving the equation i

d f = λf dx

on the metric graph Y with boundary conditions (5.3) with k = 0, and Θ = 1,      0 1 f∞ (0−) f∞ (0+) = 1 0 f (0) f () to get an analog of (10.10). In this case, t(λ) = e−iz = S −1 ˙  (λ), (d,d,d)

λ ∈ R.

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Chapter 11

UNIQUENESS RESULTS

So far we were interested in characterizing solutions to the commutation relations (1.3) under the assumption that the unitary group Ut is given. Our next goal is to show that if the symmetric operator A˙ (from Hypothesis 3.1) is a prime symmetric operator, then the commutation relations (1.3) uniquely determine the group Ut up to a character t → eitμ (with the only one exception). We start with a preliminary observation that a prime symmetric operator with deficiency indices (1, 1) has a rather poor set of symmetries. Lemma 11.1. Suppose that A˙ is a symmetric operator with deficiency indices (1, 1) and U is a unitary operator. Assume that the operator U commutes with A˙ in the sense that ˙ = Dom(A) ˙ U(Dom(A))

(11.1)

and ˙ ˙ AUf = U Af

˙ for all f ∈ Dom(A).

Then the subspaces ˙ ∗ − zI) H± = spanz∈C± Ker((A) are invariant for U. Moreover, the corresponding restrictions of U onto those subspaces are multiples of the identity. That is, U|H± = Θ± IH±

for some |Θ± | = 1.

87

(11.2)

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˙ ∗ ). Then Proof. Suppose that f ∈ Dom((A) ∗ ˙ f ) = (U ∗ AUU ˙ ˙ ˙ Uf ) = (U ∗ Ag, g, f ) for all g ∈ Dom(A). (Ag,

˙ g ∈ Dom(A). ˙ Therefore, From (11.1) it follows that U ∗ g ∈ Dom(A), ∗ ˙ ˙ ∗ g, f ) = (g, U(A) ˙ ∗ f ). g, f ) = (AU (U ∗ AUU

That is, ˙ ˙ Uf ) = (g, U(A) ˙ ∗ f ) for all g ∈ Dom(A), (Ag, which means that ˙ ∗) ˙ ∗ )) ⊂ Dom((A) U(Dom((A) and ˙ ∗ ˙ ∗ U = U(A) (A)

˙ ∗ ). on Dom((A)

(11.3)

Since (11.3) holds, the deficiency subspace ˙ ∗ − zI), Nz = Ker((A)

z ∈ C+ ,

is an eigensubspace of U. Therefore, the subspace ˙ ∗ − zI) H+ = spanz∈C+ Ker((A) is invariant for U. Next we claim that the deficiency subspaces Nz and Nζ are not orthogonal to each other for z, ζ ∈ C+ . ˙ Suppose that g+ ∈ Ni Indeed, let A be a self-adjoint extension of A. with g+ = 0. Then the element gz = (A − iI)(A − zI)−1 g+ generates the subspace Nz , Im(z) = 0. Therefore,   (gz , gζ ) = (A − iI)(A − zI)−1 g+ , (A − iI)(A − ζI)−1 g+  λ2 + 1 = dμ(λ) R (λ − z)(λ − ζ)    1 1 1 − (λ2 + 1)dμ(λ), = z−ζ R λ−z λ−ζ

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where μ(dλ) is the spectral measure of the element g+ associated with the self-adjoint operator A. That is, dμ(λ) = (dE(λ)g+ , g+ ), where E(λ) is the resolution of identity for the self-adjoint operator A,  A= λdE(λ). R

Clearly,

 Im

1 1 − λ−z λ−ζ

 > 0,

whenever z, ζ ∈ C+ ,

and therefore (gz , gζ ) = 0,

z, ζ ∈ C+ ,

which proves the claim. ˙ ∗ − zI) and Ker((A) ˙ ∗ − ζI) for z, ζ ∈ C+ are not Finally, since Ker((A) orthogonal to each other, the restrictions of U onto these subspaces have the same eigenvalues, proving that the restriction of U onto H+ is a multiple of the identity. The same reasoning shows that the restriction of U onto its invariant subspace ˙ ∗ − zI) H− = spanz∈C− Ker((A) is also a (possibly different) multiple of the identity as well. The proof is complete.



Lemma 11.2. Suppose that A˙ is a symmetric operator with deficiency indices (1, 1). Assume that Ut and Vt are strongly continuous unitary groups such that the commutation relations ˙ t = V ∗ AV ˙ t = A˙ + tI Ut∗ AU t

˙ on Dom(A)

hold. Then the subspaces ˙ ∗ − zI) H± = spanz∈C± Ker((A) reduce the groups Ut and Vt and Vt |H± = eitμ± Ut |H±

for some μ± ∈ R.

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Proof. In a similar way as in the proof of Lemma 11.1, one observes that the commutation relations ˙ ∗ Ut = V ∗ (A) ˙ ∗ Vt = (A) ˙ ∗ + tI Ut∗ (A) t

˙ ∗) on Dom((A)

˙ ∗ hold. for the adjoint operator (A) Since obviously ˙ ∗ − zI)) = Vt (Ker((A) ˙ ∗ − zI)) = (Ker((A) ˙ ∗ − (z − t)I)), Ut (Ker((A)

t ∈ R,

the subspaces H± are invariant for both Ut and Vt for all t, and therefore H± reduce the groups Ut and Vt . Since the unitary operator Ut = Ut∗ Vt ˙ by Lemma 11.1 one gets that commutes with A˙ on Dom(A), Ut∗ Vt |H± = Ut |H± = eiα± (t) IH± ,

t ∈ R,

(11.4)

for some continuous real-valued functions α± (t) (the continuity of the function α(t) follows from the hypothesis that the groups Ut and Vt are strongly continuous). That is, Vt = eiα± (t) Ut

on H± .

Since Ut and Vt are one-parameter groups, it follows that the functional equation α± (t + s) = α± (t) + α± (s) holds and hence, due to the continuity of α± , we conclude that α± (t) = μ± t, for some μ± ∈ R, which combined with (11.4) completes the proof. Theorem 11.3. Suppose prime symmetric operator Assume that Ut and Vt the commutation relations



 is a maximal dissipative extension of a that A ˙ A with deficiency indices (1, 1). are strongly continuous unitary groups such that

 t = V ∗ AV  t=A  + tI Ut∗ AU t

 on Dom(A)

(11.5)

hold.  is self-adjoint assume, in addition, that If A ˙ t = Vt∗ AV ˙ t = A˙ + tI Ut∗ AU

˙ on Dom(A).

(11.6)

 has a regular point in the upper half-plane, in particular, if A  is If A self-adjoint, then Vt = eitμ Ut

for some μ ∈ R.

(11.7)

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 fills in the Moreover, in the exceptional case when the spectrum of A whole upper half-plane, the subspaces ˙ ∗ − zI) H± = spanz∈C± Ker((A) are orthogonal, reduce the groups Ut and Vt and Vt |H± = eitμ± Ut |H±

for some μ± ∈ R.

 in the upper half-plane. As it has Proof. Let z be a regular point of A  is not self-adjoint, then been explained in the proof of Theorem 3.5, if A (11.6) holds automatically. Therefore, Lemma 11.2 is applicable and hence Ut g = eiμ− t g

for some μ− ∈ R,

˙ ∗ + iI) ⊂ H− . where Ut = Ut∗ Vt and 0 = g ∈ Ker((A) Set  + iI)(A  − zI)−1 g. f = (A ˙ ∗ − zI) ⊂ H+ , by Lemma 11.2, Since f ∈ Ker((A) Ut f = eiμ+ t f

for some μ+ ∈ R.

 we On the other hand, since the unitary operator Ut commutes with A, have  + iI)(A  − zI)−1 Ut g = (A  + iI)(A  − zI)−1 eiμ− t g = eiμ− t f. Ut f = (A Therefore, μ+ = μ− . Finally, taking into account that A˙ is a prime operator, it follows that the subspaces H± span the whole Hilbert space H, and the claim follows. In view of Lemma 11.2, to prove the last assertion it remains to show  fills in the that H± are mutually orthogonal whenever the spectrum of A upper half-plane.  has no regular points Since (11.5) holds and the dissipative operator A in the upper half-plane, one can apply Theorem 3.5 to conclude that for any ˙ A,  A) self-adjoint extension of A˙ the characteristic function of the triple (A, is identically zero. By Lemma 9.1, the characteristic function of the triple ˙ D  I (0), D) on the metric graph Y in Case (i) with quantum gate coef(D, ficient k = 0 also vanishes identically in the upper-half-plane. The operators A˙ and D˙ are prime symmetric operators, therefore A˙ is unitarily  is in the equivalent to D˙ on the metric graph Y in Case (i), where D exceptional case, that is, its point spectrum fills in C+ . In this case the

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˙ = spanz∈C Ker((D) ˙ ∗ − zI) = L2 (R± ) for the operator subspaces H± (D) ± ˙ D are orthogonal, so are the subspaces ˙ = spanz∈C Ker((A) ˙ ∗ − zI) H± (A) ± ˙ for the symmetric operator A.



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Chapter 12

DISSIPATIVE SOLUTIONS TO THE CCR

Now we are prepared to get a complete classification (up to unitary equiv to alence) of the simplest non-self-adjoint maximal dissipative solutions A the commutation relations (1.4). More generally, we have the following result. Theorem 12.1. Assume Hypothesis 3.1. Suppose, in addition, that A˙ is a  is a maxiprime operator and A its self-adjoint extension. Suppose that A mal dissipative extension of A˙ such that  t=A  + tI on Dom(A).  U ∗ AU t

 is self-adjoint, then there exists a unique Θ, |Θ| = 1, such that the (i) If A ˙ A,  A) is mutually unitarily equivalent to the triple (D, ˙ D, DΘ ) triple (A, on the metric graph Y in Case (i) and therefore Θ is a unitary invariant ˙ A,  A). of (A,  ˙ A,  A) is mutually unitarily (ii) If A is not self-adjoint, then the triple (A, ˙  equivalent to the model triple (D, D, DΘ ) in one of the Cases (i)–(iii)  has at least one regular point in the for some |Θ| = 1. In addition, if A upper half-plane, then the parameter Θ is uniquely determined by the ˙ A,  A) and therefore Θ is a unitary invariant of (A, ˙ A,  A) in triple (A,    ˙  ˙ this case. That is, if some triples (A, A, A) and (A , A , A ) are mutually unitarily equivalent, then the corresponding parameters Θ and Θ coincide.  is self-adjoint, we argue as follows. By Theorem 3.3, the Proof. (i) If A ˙ A)  coincides with i in the upper half-plane. Weyl-Titchmarsh function of (A, ˙ A)  is mutually unitarily Therefore, since A˙ is a prime operator, the pair (A, 93

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˙ D) on the metric graph Y in Case (i). Since A equivalent to the pair (D, ˙ the triple (A, ˙ A,  A) is mutually unitarily is a self-adjoint extension of A, ˙ D, DΘ ) for some |Θ| = 1, which proves the equivalent to the triple (D, existence of such a Θ. ˙ D, DΘ1 ) and To establish the uniqueness, suppose that the triples (D, ˙ D, DΘ2 ) in Case (i) (recall that D is self-adjoint here) are mutually (D, unitarily equivalent. In particular, there exists a unitary operator U such that ˙ = Dom(D)), ˙ U (Dom(D)

U (Dom(D)) = Dom(D),

˙ = DU ˙ f U Df

˙ for all f ∈ Dom(D),

U Df = DU f

for all f ∈ Dom(D)

and U ∗ D Θ1 U = D Θ2 . By Lemma 11.2, the subspaces L2 (R± ) are eigensubspaces for the unitary operator U , and since U (Dom(D)) = Dom(D), the corresponding eigenvalues of U coincide. Therefore, U is necessarily a (unimodular) multiple of the identity and hence D Θ2 = U ∗ D Θ1 U = D Θ1 so that Θ1 = Θ2 .  is not self-adjoint. By Theorem 3.5, the charac(ii) Suppose that A teristic function of the triple admits the representation (3.7). Combining Theorem 3.5 and Lemma 9.1 one concludes that there exists a (possibly ˙ A,  A ) and different) self-adjoint extension A of A˙ such that the triples (A, ˙  (D, D, D) are mutually unitarily equivalent. ˙ = D˙ and U −1 A U = D for some unitary operator. In particular, U −1 AU −1 Hence U AU is a self-adjoint extension of D˙ and hence U −1 AU = DΘ for some Θ. It is the unitary transformation U that establishes the required mutual unitary equivalence of the triples.  has at least one To prove the last assertion, one observes that since A regular point in the upper half-plane, the characteristic function of the triple is not identically zero. In particular, if A is another reference operator, ˙ A,  A) and (A, ˙ A,  A ) are mutually unitarily equivalent then the triples (A,  if and only if A and A coincide by Lemma E.1 in Appendix E.

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˙ A,  A) is mutually uniTherefore, taking into account that the triple (A, ˙ D,  DΘ ) for some |Θ| = 1, one concludes that in tarily equivalent to (D, this case the unimodular parameter Θ is uniquely determined by the triple ˙ A,  A). (A,   has no regular points in the upper half-plane, then Remark 12.2. If A  ˙ D,  DΘ ) for some (and ˙ A,  A ) is mutually unitarily equivalent to (D, (A,  therefore for all) Θ, |Θ| = 1, where D is in the exceptional case (Case (i) with k = 0) (see Corollary 9.5). Therefore, in this exceptional case, the ˙ A,  A ). parameter Θ is not determined uniquely by the triple (A, Remark 12.3. On account of the remarks that we made in Chapter 8, structure Theorem 8.3 combined with Theorem 12.1 provides the following intrinsic characterization of all symmetric operators satisfying Hypothesis 3.1 thus giving a complete solution of the Jørgensen-Muhly problem for symmetric operators in the case of deficiency indices (1, 1) (see Problem (I) b) in the Introduction): either i) A˙ admits a (dissipative) quasi-selfadjoint extension with the point spectrum filling in C+ , equivalently, A˙ admits a pair of distinct quasiselfadjoint extensions that are unitarily equivalent, or, ii) A˙ admits a quasi-selfadjoint extension with no spectrum, or, finally, iii) A˙ is the symmetric part of an operator coupling of a dissipative extension of the symmetric operator A˙ without point spectrum in case i) and the dissipative extension of A˙ with no spectrum in case ii) (see Remark 3.6 for the definition of the symmetric part of a dissipative operator in connection with Hypothesis 3.1).

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Chapter 13

MAIN RESULTS

In this chapter we provide the complete classification of the simplest solutions to the restricted Weyl commutation relations Vs Ut = eist Ut Vs ,

t ∈ R,

s ≥ 0,

for a strongly continuous group of unitary operators Ut and a strongly continuous semi-group of contractions Vs in a separable Hilbert space H. Hypothesis 13.1. Let (−∞, ∞)  t → Ut = eiBt be a strongly continuous  group of unitary operators and [0, ∞)  s → Vs = eiAs a semi-group of contractions in a separable Hilbert space H. Suppose that the restricted Weyl commutation relations Vs Ut = eist Ut Vs ,

t ∈ R,

s ≥ 0,

(13.1)

hold. We remark that in the light of Corollary 6.4 one could have started from the much weaker Hypothesis 3.1. The following two results characterize the simplest solutions to the restricted Weyl commutation relations. We start with the case where Vs is a semi-group of isometries. Theorem 13.2. Assume Hypothesis 13.1. Suppose, in addition, that the  of the semi-group Vs is a prime symmetric operator with defigenerator A ciency indices (0, 1). Then there exists a unique metric graph Y = (μ, ∞), μ ∈ R, such that  B) is mutually unitarily equivalent to the pair (P,  Q), where the pair (A,

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 = i d is the differentiation operator in L2 (Y) = L2 ((μ, ∞)) on P dx  = {f ∈ W 1 ((μ, ∞)) | f (μ) = 0} Dom(P) 2 and Q is the operator of multiplication by independent variable in L2 (Y).  is a generator of a semi-group, A  is a closed operator. Proof. Since A By the Stone-von Neumann uniqueness result (see, e.g., [3, Theorem 2, Ch. VIII, Sec. 104]) there exists a isometric map U from H onto L2 ((0, ∞))  −1 coincides with the differentiation operator in L2 ((0, ∞)) such that U AU with the Dirichlet boundary condition at the origin. Lemma 11.2 and Lemma B.5 in Appendix B show that there exists a  B) is mutually unitarily equivalent to the pair μ such that the pair (A, (P0 , Q0 + μI), where P0 = i

d dx

is the differentiation operator in L2 ((0, ∞)) on Dom(P0 ) = {f ∈ W21 ((0, ∞)) | f (0) = 0}, and Q0 is the operator of multiplication by independent variable in  Q) in the Hilbert spaces L2 ((0, ∞)). Clearly the pairs (P0 , Q0 +μI) and (P, 2 2 L ((0, ∞)) and L ((μ, ∞)), respectively, are mutually unitarily equivalent. By construction, the spectrum of the generator B coincides with the semiaxis [μ, ∞). Therefore, μ is a unitary invariant which is uniquely deter B). In particular, the graph Y = (μ, ∞) is also mined by the pair (A,  B). uniquely determined by the pair (A,  In a completely analogous way one proves the following result. Theorem 13.3. Assume Hypothesis 13.1. Suppose, in addition, that the  of the semi-group Vs is a maximal dissipative extension of a generator A prime symmetric operator with deficiency indices (1, 0). Then there exists a unique metric graph Y = (−∞, ν), ν ∈ R, such that  B) is mutually unitarily equivalent to the pair (P,  Q) where the pair (A, d 2 2  P = i dx is the differentiation operator in L (Y) = L ((−∞, ν)) on  = W21 ((−∞, ν)) Dom(P) and Q is the operator of multiplication by independent variable in L2 (Y).

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 is a prime symmetric operator with Remark 13.4. We remark that if A deficiency indices (0, 1) or (1, 0) in a Hilbert space H, then H is necessarily separable.  B) Remark 13.5. Theorems 13.2 and 13.3 are dual to each other: if (A, ∗  satisfies the hypotheses of Theorem 13.2, then (−A , −B) satisfies the  B) and (−A ∗ , −B) are hypotheses of Theorem 13.3, however the pairs (A, not mutually unitarily equivalent. In this case the self-adjoint operator B is semi-bounded from below but −B is semi-bounded from above. Moreover,  has no point spectrum while the point spectrum of (−A) ∗ the generator A fills in the entire open upper half-plane. Next we treat the case where the dissipative generator of the semigroup Vs is a quasi-selfadjoint extension of a prime symmetric operator with deficiency indices (1, 1). To formulate the corresponding uniqueness result, we need some preparations. Definition 13.6. Let Y be a metric graph in the following cases ⎧ ⎪ (−∞, ν)  (μ, ∞) Case I∗ (ν = μ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(−∞, μ)  (μ, ∞) Case I Y= ⎪ ⎪ (μ, ν) Case II (μ < ν) ⎪ ⎪ ⎪ ⎪ ⎩(−∞, μ)  (μ, ∞)  (μ, ν) Case III (μ < ν).

(13.2)

Given a real number k, 0 ≤ k < 1, define the position operator Q as the operator of multiplication by independent variable on the edges of the graph Y and  as the differentiation operator i d on the the momentum operator P dx edges of the graph Y,  )(x) = i (Pf

d f (x) a. e. x ∈ e on every edge e of Y dx

on   Dom(P)

 e⊂Y

W21 (e)  L2 (Y),

(13.3)

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the space of locally absolutely continuous functions on the edges with the following vertex boundary conditions

 = Dom(P)

⎧ ⎪ f− ⊕ f+ ∈ W21 ((−∞, ν)) ⊕ W21 ((μ, ∞)) | f+ (μ+) = 0 , ⎪ ⎪ ⎪ ⎪ ⎪ in Case I∗ ⎪ ⎪ ⎪ ⎪ ⎨f ∈ W 1 ((−∞, μ)) ⊕ W 1 ((μ, ∞)) | f (μ+) = kf (μ−) , ∞

2

2

∞

∞

⎪ in Case I ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ f ∈ W21 ((μ, ν)) | f (μ+) = 0 , ⎪ ⎪ ⎪ ⎩ in Case II.

Here, in Case I we require that 0 < k < 1 and in Case II we assume that ν = μ +  > μ.  consists of the two-component vector-functions In Case III, Dom(P) T f = (f∞ , f ) ,

 f∞ ⊕ f ∈ W21 ((−∞, μ)) ⊕ W21 ((μ, ∞)) ⊕ W21 ((μ, ν)) that satisfy the “boundary conditions” f∞ (μ+) = kf∞ (μ−) and f (μ+) =

1 − k 2 f∞ (μ−)

(Case III). (13.4)

 Q) is said to be the dissipative canonical pair By definition the pair (P, with the quantum gate coefficient k, 0 ≤ k < 1 on the metric graph Y. In Case III we always assume that k > 0 and formally set k = 0 whenever the ˙ P,  Q), where graph Y is in Case I∗ or in Case II. We also call the triple (P,  P˙ = P|  ∗ ) , Dom(P)∩Dom( P the canonical dissipative triple on Y with the quantum gate coefficient k. Remark 13.7. In Case I∗ the metric graph is “disconnected” whenever ν < μ, while if μ < ν, one may think that the edges of the graph eventually “overlap” over the finite interval [μ, ν]. Also, in Case III the boundary conditions (13.4) at the junction point μ of the graph Y, the center of the graph, yield the quantum Kirchhoff rule |f∞ (μ+)|2 + |f (μ+)|2 = |f∞ (μ−)|2 .

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Remark 13.8. It is easy to see that the spectrum of the position operator Q is given by ⎧ ⎪ (−∞, ν] ∪ [μ, ∞), in Case I∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(−∞, ∞), in Case I spec(Q) = . (13.5) ⎪ ⎪ [μ, ν], in Case II ⎪ ⎪ ⎪ ⎪ ⎩(−∞, ∞), in Case III From (13.5) it follows that if Y is in Case I∗ with ν > μ or in Case III, then the spectrum of the position operator Q has multiplicity 2 on the finite interval [μ, ν]. In Case I and II the position operator Q has simple Lebesgue spectrum filling in the whole real axis (−∞, ∞) and the finite interval [μ, ν] respectively. We also notice that the spectrum of the dissipative momentum operator  P is ⎧ ⎪ C+ ∪ (−∞, ∞), in Case I∗ and Case I with k = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(−∞, ∞), in Case I with k > 0  . (13.6) spec(P) = ⎪ ⎪ ∅, in Case II ⎪ ⎪ ⎪ ⎪ ⎩(−∞, ∞), in Case III Notice that Case I∗ and Case I with k = 0 are exceptional in the sense that  any point in the (open) upper half-plane is an eigenvalue of P. If a metric graph Y is in Case I, we also introduce the concept of the Weyl canonical triple on Y. Definition 13.9. Let Y be a metric graph in Case I, that is, Y = (−∞, μ)  (μ, ∞) for some μ ∈ R. d be the self-adjoint differentiation operator on Let P = i dx

Dom(P) = {f ∈ W21 (Y) | f (μ − 0) = f (μ + 0},

(13.7)

P˙ its symmetric restriction on ˙ = {f ∈ W 1 (Y) | f (μ − 0) = f (μ + 0) = 0}, Dom(P) 2 ˙ P, Q) the Weyl canonical and Q the position operator on Y. We call (P, triple on Y (centered at μ).

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 Q) = (P(k),  Remark 13.10. If (P, Q) is the dissipative canonical pair on the metric graph Y = (−∞, μ] ∪ [μ, ∞) with the quantum gate coefficient k, then  = P, s-lim P(k) k→1

where P is the self-adjoint differentiation operator defined on (13.7) and the limit is taken in the strong resolvent sense. Therefore, the Weyl canonical triple on the metric graph Y can be considered the limiting case of ˙ P,  Q) = (P(k), ˙  the dissipative triple (P, P(k), Q) with the quantum gate coefficient k as k → 1. Our first auxiliary result is that the pair (Y, k) is a unitary invariant of a dissipative canonical pair.  Lemma 13.11. Suppose that the canonical pairs (P(k), Q) and (P (k  ), Q ) with the quantum gate coefficients k and k  on metric graphs Y and Y , respectively, are mutually unitarily equivalent. Then Y = Y

and

k = k .

Proof. As it has been explained in Remark 13.8, there are two options: either the position operator Q has simple spectrum or Q has spectrum of multiplicity 2 filling in a finite interval. First, assume that the position operator Q has spectrum of multiplicity 2 supported by a finite interval [μ, ν], ν > μ. So does Q . Therefore the graphs Y and Y have the same vertices but may possibly be in different cases, in Case I∗ or in Case III only. Suppose that Y is in Case I∗ and therefore k = 0. Then the point spec = P(0)  trum of the dissipative momentum operator P fills in the whole   since upper half-plane C+ , so does the dissipative momentum operator P  and P   are unitarily equivalent. Therefore, Y is in Case I∗ with k  = 0 as P well. Analogously, if Y is in Case III, then C+ belongs to the resolvent set     (k  ) are unitarily equivalent, C+ belongs of P(k). Again, since P(k) and P    (k ) and then necessarily Y is in Case III. Thus Y to the resolvent set of P  and Y have the same vertices and are in the same cases. Therefore, Y = Y . It remains to treat the case where the multiplication operator Q has simple spectrum. There are two options: either both Y and Y are in Case II, or both of them are in Case I. If they are in Case II, the knowledge of the spectrum of Q (Q ) uniquely determines the location of the vertices of the graph Y (Y ) and the graph(s) itself.

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If both Y and Y are in Case I, we proceed as follows. Since the pairs  (P(k), Q) and (P (k  ), Q ) are mutually unitarily equivalent and Q = Q , there exists a unitary operator U commuting with the multiplication operator Q such that    (k  ) = U ∗ P(k)U. P Since Q has simple spectrum and the unitary operator U commutes with Q, the operator U is the multiplication operator by a unimodular function u. We have    (k  ))). Dom(P(k)) = U (Dom(P

(13.8)

Suppose that the vertices μ and μ of the graphs Y and Y are different, that is, μ = μ . From (13.8) it follows that the function u(x)f (x) is a contin  (k  )), so uous function in a neighborhood of the point μ for all f ∈ Dom(P is the function |f (x)| = |u(x)f (x)|, which is incompatible with the bound  (k  )), since k  < 1. ary condition f (μ +) = k  f (μ −) for all f ∈ Dom(P  Therefore, the vertices of the graphs Y and Y coincide, μ = μ , and hence Y = Y . To prove that k = k  , notice that if the metric graph Y and therefore Y is in Cases I∗ or II, then k = k  = 0 by definition. Suppose that Y = Y is in Cases I, Y = (−∞, μ)  (μ, ∞) = Y .  In this case, the absolute values of the von Neumann parameters of P(k) and    P (k ) (more precisely, of the corresponding triples) coincide with k and k  ,    (k  ) are unitarily equivalent. respectively. By the hypothesis, P(k) and P Therefore, k = k , since the absolute value of the von Neumann parameter is a unitary invariant of a dissipative operator by Remark 2.5. Next, assume that Y = Y is in Case III, Y = (−∞, μ)  (μ, ∞)  (μ, ν) = Y

(μ < ν).

By Theorem 9.3, the absolute values of the von Neumann parameters of    (k  ) are ke− and k  e− , respectively (see the relation (2.8)), P(k) and P where  = ν − μ.

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 Therefore, k = k  , since P(k) and P (k  ) are unitarily equivalent by the hypothesis.  Now we are ready to present the central result of the first part of the book. Theorem 13.12. Assume Hypothesis 13.1. Suppose, in addition, that the  of the semi-group Vs is not self-adjoint and that the restriction generator A  A˙ = A|   ∗) Dom(A)∩Dom(( A) is a prime symmetric operator with deficiency indices (1, 1). Then there exists a unique metric graph Y in one of the Cases I ∗ , I–III  B) (triple (A, ˙ A,  B)) is mutuand a unique k ∈ [0, 1) such that the pair (A,  ally unitarily equivalent to the canonical dissipative pair (P(k), Q) (triple ˙  (P, P(k), Q)) on Y, respectively. Proof. By the hypothesis the restricted Weyl relations (13.1) hold. Therefore (see [27, 125])  t=A  + tI Ut∗ AU

 on Dom(A),

t ∈ R.

(13.9)

As in the proof of Theorem 3.5 one shows that symmetric operator A˙ solves the commutation relations ˙ t = A˙ + tI Ut∗ AU

˙ on Dom(A).

(13.10)

Therefore, the operator A˙ satisfies Hypothesis 3.1. In this situation one can apply Theorem 12.1 (ii) to see that there is a metric graph Y0 in one of the Cases (i)–(iii) with the quantum gate coefficient k such that the  is unitarily equivalent to the one of the following dissipative operator A model dissipative differentiation operators:  =D  I (k) = i d on Y0 = (−∞, 0)  (0, ∞) with the boundary condi(i) D dx tion f∞ (0+) = kf∞ (0−),

0 ≤ k < 1;

(13.11)

or  =D  II (0, ) = i d on Y0 = (0, ) with the boundary condition (ii) D dx f (0) = 0;

(13.12)

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or  = D  III (k, ) = i d on Y0 = (−∞, 0)  (0, ∞)  (0, ) with the (iii) D dx boundary conditions

   f∞ (0+) k 0 f∞ (0−) = √ , 0 < k < 1. (13.13) 1 − k2 0 f (0) f () That is, there exists a unitary map W from H onto L2 (Y0 ) such that  −1 = D.  W AW

(13.14)

In particular, from (13.9) it follows that  t=D  + tI Wt∗ DW

 on Dom(D),

t ∈ R,

(13.15)

where Wt is the unitary group on L2 (Y) given by Wt = WUt W −1 ,

t ∈ R.

On the other hand,  −itQ0 = D  + tI eitQ0 De

 on Dom(D),

t ∈ R,

where Q0 is the operator of multiplication by independent variable on the graph Y0 .  one obtains Applying Theorem 11.3 to the dissipative operator D, Wt = WeiBt W −1 = e−iμt eiQ0 t

for some μ ∈ R,

(13.16)

 and therefore A,  has a regular point in the upper half-plane. whenever D, In this case, combining (13.14) and (13.16) one concludes that the pair  B) is mutually unitarily equivalent to the pair (D,  Q − μI) on the graph (A,  Q − μI) is in turn Y0 (with the quantum gate coefficient k). The pair (D,  mutually unitarily equivalent to the canonical dissipative pair (P(k), Q) with the quantum gate coefficient k on the metric graph Y centered at μ. Notice that Y can be obtained from the graph Y0 by a shift.  has no regular points in the upper halfIf the dissipative operator A  is unitarily equivalent the model difplane, Theorem 12.1 asserts that A   ferentiation operator D = DI (0) on the graph Y0 = (−∞, 0)  (0, ∞) in Case (i) with quantum gate coefficient k = 0. The same reasoning as above shows that in this exceptional case, the  B) is mutually unitarily equivalent to the pair (D  I (0), Q− μ− R− − pair (A, μ+ R+ ) on the graph Y0 = (−∞, 0)  (0, ∞) for some μ± ∈ R. Here R−

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and R+ are the orthogonal projections in L2 (Y0 ) = L2 ((−∞, 0)) ⊕ L2 ((0, ∞)) onto the subspace L2 ((−∞, 0)) and L2 ((0, ∞)), respectively.  I (0), Q−μ− R− −μ+ R+ ) = (D  I (0), Q−μI) If μ+ = μ− = μ, the pair (D  is mutually unitarily equivalent the canonical dissipative pair (P(0), Q) on the graph Y = (−∞, μ)  (μ, ∞) in Case I with quantum gate coefficient k = 0.  I (0), Q − μ− R− − μ+ R+ ) on the metric If μ+ = μ− , then the pair (D graph Y0 = (−∞, 0)(0, ∞) (in Case(i)) is mutually unitarily equivalent to  Q) on the graph Y = (−∞, μ− )  (μ+ , ∞) the canonical dissipative pair (P, ∗ in Case I . The uniqueness part of the statement is an immediate consequence of Lemma 13.11.  Remark 13.13. If in addition to the hypotheses of Theorem 13.12 one ˙ then we immeassumes that A is a self-adjoint (reference) extension of A, diately get that there exists a self-adjoint extension P of P˙ such that the ˙ A,  A, B) is mutually unitarily equivalent to the quadruple quadruple (A, ˙  (P, P(k), P, Q) on the metric graph Y in Cases I∗ , I–III with the quantum gate coefficient k = 0 for some k ∈ [0, 1). The extension P is determined ˙ A,  A, B) uniquely unless the graph Y is in Case I ∗ or by the quadruple (A, in Case I with the quantum gate coefficient k = 0. With a minor modification, the result of Theorem 13.12 extends to the  is self-adjoint. case where the generator A Theorem 13.14. Assume Hypothesis 13.1. Suppose, in addition, that  = A is self-adjoint and that A˙ is a prime symmetric restriction of A  A with deficiency indices (1, 1) such that ˙ t = A˙ + tI Ut∗ AU

˙ on Dom(A).

(13.17)

Then there exists a unique metric graph Y = (−∞, μ)  (μ, ∞) (in ˙ A, B) is mutually unitarily Case I) centered at μ ∈ R such that the triple (A, ˙ P, Q) on Y (see Definition 13.9). equivalent to the Weyl canonical triple (P, Proof. From the hypothesis it follows that in fact (unrestricted) Weyl commutation relations Vs Ut = eist Ut Vs ,

t ∈ R,

s ∈ R,

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hold and therefore Ut∗ AUt = A + tI

on Dom(A),

t ∈ R.

(13.18)

By Theorem 3.3, the Weyl-Titchmarsh function associated with the pair ˙ A) has the form (A, M (z) = i,

z ∈ C+ .

˙ P) on So does the Weyl-Titchmarsh function associated with the pair (P, the graph Y0 = (−∞, 0)  (0, ∞) in Case (i). ˙ A) is mutually unitarily Since A˙ and P˙ are prime operators, the pair (A, ˙ equivalent to the pair (P, P) on the graph Y0 . That is, there exists a unitary operator W : H → L2 (R) such that WAW −1 = P

and

˙ −1 = P. ˙ W AW

From (13.17) and (13.18) it follows that Wt∗ PWt = P + tI

on Dom(A)

˙ t = P˙ + tI Wt∗ PW

˙ on Dom(A),

and

where Wt = WUt∗ W −1 . ˙ P, Q), the commutaBy the definition of the Weyl canonical triple (P, tion relations eitQ Pe−itQ = P + tI

˙ −itQ = P˙ + tI and eitQ Pe

hold. Now one can apply Theorem 11.3 to see that there exists a μ ∈ R such that Wt = WUt W −1 = e−iμt eitQ = eit(Q−μI) . ˙ A, B) is mutually unitarily equivalent to the Therefore, the triple (A, ˙ P, Q − μI) on the metric graph Y0 . triple (P, ˙ P, Q − μI) is mutually unitarily equivalent to the In turn, the triple (P, ˙ Weyl canonical triple (Pμ , P, Q) on the metric graph Yμ = (−∞, μ)(μ, ∞) in Case I. (Recall that Dom(P˙ μ ) = {f ∈ W21 (R) | f (μ) = 0}.) To complete the proof of the theorem it remains to show that the Weyl triples (P˙ μ , P, Q) and (P˙ μ , P, Q) on the graphs Yμ and Yμ , respectively, are not mutually unitarily equivalent unless μ = μ . Indeed, assume that

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they are. Denote by U the unitary operator that establishes the mentioned mutual unitary equivalence. Since U commutes with P and Q, the operator U is a (unimodular) multiple of the identity. Therefore, U ∗ P˙ μ U = P˙ μ implies P˙ μ = P˙ μ and hence μ = μ , a contradiction.  Remark 13.15. Comparing the assumptions of Theorems 13.12 and 13.14 it is clearly seen that the main difference is that in Theorem 13.14 one has to ˙ while in the require the commutation relation for the symmetric operator A, case of Theorem 13.12 the corresponding relations hold automatically. As we have already mentioned in the Introduction, the existence of a symmetric operator A˙ with the required properties in the hypothesis of Theorem 13.14 follows from the Stone-von Neumann uniqueness result. d and Q the operIndeed, let (P, Q) be the canonical pair with P = i dx 2 ator of multiplication by independent variable in L (R). Suppose that W : H → L2 (R) is a unitary operator such that WAW −1 = P

and WBW −1 = Q.

(13.19)

Then the symmetric restriction P˙ μ of P on Dom(P˙ μ ) = {f ∈ W21 (R) | f (μ) = 0} has deficiency indices (1, 1) and satisfies the commutation relations eitQ P˙ μ e−itQ = P˙ μ + tI. It remains to choose A˙ = W −1 P˙μ W

(13.20)

and the existence of a restriction with the required properties as follows. From the uniqueness part of Theorem 13.14 it also follows that if a closed symmetric restriction P˙ of P (with deficiency indices (1, 1)) satisfies commutation relations ˙ −itQ = P˙ + tI, eitQ Pe then P˙ = P˙ μ for some μ ∈ R (cf. [50]). In this sense as far as the unitary equivalence (13.19) is established (based on the Stone-von Neumann uniqueness result), the choice of A˙ via (13.20) in the hypothesis of Theorem 13.14 is canonical. Notice that as long as the existence of such a restriction is established/required the reasoning above can be considered an independent proof of the Stone-von Neumann uniqueness result.

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The following corollary can be considered an important extension of the Stone-von Neumann uniqueness theorem. Corollary 13.16. Suppose that strongly continuous groups of unitary operators Vt = eiAt and Ut = eiBt in the Hilbert space H solve the Weyl commutation relations Vs Ut = eist Ut Vs ,

s, t ∈ R.

Assume that the self-adjoint operator A has simple spectrum. Without loss of generality suppose that A˙ is a closed symmetric restriction of A with deficiency indices (1, 1) such that ˙ t = A˙ + tI Ut∗ AU

˙ on Dom(A),

t ∈ R.

(13.21)

˙ then the Weyl commutaIf A is any other self-adjoint extension of A, tion relations Vs Ut = eist Ut Vs ,



with Vs = eisA ,

s, t ∈ R,

(13.22)

hold. Proof. By Theorem 13.14, there exists a unique metric graph Y = ˙ A, B) is mutually unitarily equiv(−∞, μ)  (μ, ∞) such that the triple (A, ˙ alent to the Weyl canonical triple (P, P, Q) on Y, so that ˙ −1 , A˙ = U PU

A = UPU −1

and B = UQU −1

for some unitary map U from L2 (Y) onto H. Let A be a self-adjoint exten˙ Therefore, P  = U −1 A U is a self-adjoint extension of P˙ on sion of A. Dom(P  ) = {f ∈ W21 ((−∞, μ)) ⊕ W21 ((μ, ∞)) | f (μ− ) = Θf (μ+)} for some |Θ| = 1. We have eitQ P  e−itQ = P  + tI

on Dom(P  ),

t ∈ R,

(13.23)

and therefore Ut∗ A Ut = A˙  + tI which in turn implies (13.22).

on Dom(A ),

t ∈ R,

(13.24) 

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Remark 13.17. In the situation in question one can state more, cf. Remark 13.13. For instance, there exists a unique μ ∈ R and a unique ˙ P, P  , Q) are ˙ A, A , B) and (P, Φ ∈ [0, 2π) such that the quadruples (A, mutually unitarily equivalent. ˙ P and P  are differentiation operators in L2 (R) defined on Here P, ˙ = {f ∈ W 1 (R) | f (μ) = 0}, Dom(P) 2 Dom(P) = W21 (R), Dom(P  ) = {f ∈ W21 ((−∞, 0)) ⊕ W21 ((0, ∞)) | f (μ+) = eiΦ f (μ−)}, respectively, and Q is the operator of multiplication by independent variable in L2 (R).  is a maximal dissipative extension of A, ˙ then the More generally, if A restricted Weyl commutation relations Vs Ut = eist Ut Vs ,

 with Vs = eisA ,

t ∈ R,

s ≥ 0,

hold. In this case, there exists a unique point (μ, Φ, k) ∈ R × [0, 2π) × [0, 1) ˙ P,  P  , Q) are mutually uni˙ A,  A , B) and (P, such that the quadruples (A,  ˙  is tarily equivalent. Here the operators P, P and Q are as above and P 2 differentiation operators in L (R) on  = {f ∈ W21 ((−∞, μ)) ⊕ W21 ((μ, ∞)) | f (μ+) = kf (μ−)}. Dom(P) To be complete, we provide the description of the dynamics associated  with the strongly continuous semi-group Vs = eiPs generated by the dis = P(k)  sipative momentum operator P with the quantum gate coefficient k ∈ [0, 1) on a metric graph Y in Cases I∗ , I–III (see Chapter 8 for a more informal description of the dynamics).  = P(k)  Theorem 13.18. Let P be the canonical dissipative momentum operator with the quantum gate coefficient 0 ≤ k < 1 on a metric graph Y in one of the Cases I*, I–III. Then the strongly continuous semi-group  in the Hilbert space L2 (Y) admits the Vs of contractions generated by P(k) following explicit description.

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In Case I*, the semigroup Vs acts as the right shift on the semi-axis [μ, ∞) and as the truncated right shift on (−∞, ν] (Vs F )+ (x) = χ[s+μ,∞) (x)f+ (x − s),

x ∈ [μ, ∞),

and (Vs F )− (x) = f− (x − s),

x ∈ (−∞, ν],

where F = (f− , f+ )T ∈ L2 (Y) = L2 ((−∞, ν)) ⊕ L2 ((μ, ∞)). In Case I, we have that (Vs F )(x) = (χ(−∞,μ) (x) + kχ[μ,∞) (x))f∞ (x − s), 2

2

x ∈ R,

2

F = f∞ ∈ L (Y) = L ((−∞, μ)) ⊕ L ((μ, ∞)). In Case II, the semi-group Vs is a nilpotent shift with index  = ν −μ > 0 (Vs = 0 for s ≥ ) (Vs F ) (x) = χ[μ+s,ν] (x)f (x),

x ∈ [μ, ν],

F = f ∈ L2 (Y) = L2 ((μ, ν)). In Case III, the action of the semi-group Vs is given by (Vs F )∞ (x) = (χ(−∞,μ) (x) + kχ[μ,∞) (x))f∞ (x − s), √ (Vs F ) (x) = χ[μ+s,ν] (x)f (x − s) + 1 − k 2 f∞ (x − s),

x ∈ R, x ∈ [μ, ν],

where F = (f∞ , f )T ∈ L2 (Y) = L2 ((−∞, μ)) ⊕ L2 ((μ, ∞)) ⊕ L2 ((μ, ν)) and  = ν − μ. Proof. If the metric graph Y is in Cases I* or II, there is nothing to prove. We provide a complete proof when Y is in Case I. Without loss we may assume that μ = 0. From the definition of the semi-group Vs it follows that  lim s−1 (Vs − I)f = −f  = iPf, s↓0

f ∈ C0∞ (R \ {0}).

(13.25)

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Introduce the functions ⎧ ⎨ke−x , x ≥ 0 g(x) = ⎩e x , x 0, the Exponential Decay. 121

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15.1. Quantum Zeno Effect Hypothesis 15.1. Suppose that H is a self-adjoint operator in the Hilbert space H expressed by  H= λdEH (λ), R

where EH (λ) is a resolution of the identity. Let φ be a unit vector (state) in H and νφ (dλ) denote the spectral measure of the state φ, νφ (dλ) = (EH (dλ)φ, φ). Assume that N (λ) is the corresponding right-continuous distribution function N (λ) = νφ ((−∞, λ]),

λ ∈ R.

(15.2)

Definition 15.2. We say that φ is a Zeno state under continuous monitoring of the quantum unitary evolution φ → eitH φ if lim |(eit/nH φ, φ)|2n = 1

n→∞

for all t ≥ 0.

Recall the following necessary and sufficient conditions for the Quantum Zeno effect to occur. Proposition 15.3 ([6]). Assume Hypothesis 15.1. Then the state φ is a Zeno state if and only if the light tails requirement lim λ(1 − N (λ) + N (−λ)) = 0

λ→∞

(15.3)

holds. In particular, if φ ∈ Dom(|H|1/2 ), then φ is a Zeno state. Remark 15.4. The discovery of the phenomenon that the evolution of quantum system can be eventually frozen under continuous monitoring is due to A. Turing (Turing’s paradox)1 and L. Khalfin [58] while the term the quantum Zeno effect was coined by B. Misra and E. C. G. Sudarshan [93]. 1 In his 1954 letter to M. H. A. Newman, Turing wrote: “it is easy to show using standard theory that if a system starts in an eigenstate of some observable, and measurements are made of that observable N times a second, then, even if the state is not a stationary one, the probability that the system will be in the same state after, say, 1 second tends to one as N tends to infinity; i.e. that continual observation will prevent motion...”, see [132].

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The necessary and sufficient condition for the occurrence of the quantum Zeno effect is due to H. Atmanspacher, W. Ehm and T. Gneiting [6] where the authors explored the well known fact (see [33, Theorem 1, p. 232]) that the light tails requirement (15.3) is necessary and sufficient for the weak law of large numbers to hold. We also refer to [119] for an excellent introduction to the subject. The quantum Zeno dynamics of a relativistic system is discussed in [91]. As for an experimental confirmation of the effect see [120]. Remark 15.5. Notice that the membership φ ∈ Dom(|H|1/2 ) means that the spectral (probability) measure (EH (dλ)φ, φ) has the first moment and hence φ is a Zeno state. This can also be seen directly (cf. [58]) as follows. Consider a sequence ξ1 , ξ2 , . . . of independent copies of a random variable ξ with the common distribution function N (λ) given by (15.2). By the strong law of large numbers, lim

n→∞

ξ1 + ξ2 + · · · + ξn =a n

almost surely,

where a = Eξ = (sgn(H)|H|1/2 φ, |H|1/2 φ) ∈ R is the mathematical expectation of the random variable ξ. Recall that φ ∈ Dom(|H|1/2 ) and therefore the right hand side (sgn(H)|H|1/2 φ, |H|1/2 φ) is well defined. Since the random variables ξ1 , ξ2 , . . . are independent and equidistributed, we have it/nξn 2 (eit/nH φ, φ)n = (Eeit/nξ )n = Eeit/nξ1 · Eeit/nξ  · · · · Ee  n times

ξ1 + ξ2 + · · · + ξn n → eita = Ee it

as n → ∞,

which shows that lim |(eit/nH φ, φ)|2n = 1.

n→∞

(15.4)

That is, φ is a Zeno state. 15.2. Anti-Zeno Effect Frequent observations can also accelerate the decay process and the corresponding phenomenon is known as the quantum anti-Zeno effect.

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Definition 15.6. We say that φ, φ = 1, is an anti-Zeno state under continuous monitoring of the quantum unitary evolution φ → eitH φ if lim |(eit/nH φ, φ)|2n = 0

n→∞

for all t > 0.

(15.5)

The following lemma provides a simple sufficient condition for a state to be an anti-Zeno state. We state the corresponding result using the language of the theory of limit distributions of sums of independent random variables (we refer to Appendix H for the terminology and a brief exposition of the theory). Lemma 15.7 (cf. [30]). Assume Hypothesis 15.1. Suppose that the distribution N (λ) of the state φ belongs to the domain of attraction of an α-stable law with 0 < α < 1. Then φ is an anti-Zeno state. Proof. The characteristic function of the distribution N (λ) coincides with (eitH φ, φ) and therefore admits the representation (by Remark H.2 in Appendix H)   t itH α˜ (e φ, φ) = exp −σ|t| h(t)(1 − iβ ω(t, α)) , |t| ˜ where h(t) is slowly varying as t → 0. In particular, α ˜ (1 + o(1))) |(eit/nH φ, φ)|2n = exp(−2cn1−α h(t/n)|t|

as n → ∞.

Since (see, e.g., [48, Appendix 1]) ˜ lim n1−α h(t/n) = +∞,

n→∞

one concludes that lim |(eit/nH φ, φ)|2n = 0,

n→∞

t = 0.



Remark 15.8. Necessary and sufficient conditions for the quantum antiZeno effect to occur can be found in [6, Theorem 2]. Remark 15.9. It is worth mentioning that the situation is quite different if the distribution function belongs to the domain of attraction of an α-stable law with 1 < α ≤ 2. In this case the probability measure νφ (dλ) has the first moment and therefore the state φ is a Zeno state.

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15.3. The Exponential Decay Next we turn to the borderline case of α-stable distributions with α = 1 which play an exceptional role in explanation of the exponential decay phenomenon under continuous monitoring. Definition 15.10. We say that φ is a resonant state under continuous monitoring of the quantum unitary evolution φ → eitH φ if lim |(eit/nH φ, φ)|2n = e−τ |t| ,

n→∞

for some τ > 0 and all t ≥ 0.

(15.6)

Remark 15.11. Notice that one can also consider exponentially decaying (resonant) states by requiring that the survival probability p(t) = |(eitH φ, φ)|2 tends to zero exponentially fast as |t| approaches infinity. In this case, however, the spectrum of the Hamiltonian H has to fill in the whole real axis which excludes from the consideration the quantum systems with semi-bounded Hamiltonians. It can be easily seen as follows. The requirement that the survival probability p(t) falls off exponentially implies that the survival probability amplitude (eitH φ, φ) is the Fourier transform of an absolute continuous measure with the density that is analytic in a strip containing the real axis. In d (EH (λ)φ, φ) of the spectral particular, the Radon-Nykodim derivative dλ measure of the element φ is positive almost everywhere which shows that spec(H) = R. If the Hamiltonian H has a gap in its spectrum, then there are no exponentially decaying states whatsoever unless the quantum system is under continuous monitoring. The geometric reason behind is that the unitary group eitH does not have orthogonal incoming and outgoing subspaces as it follows from the Hegerfeldt Theorem [45]. A sufficient condition for the exponential decay (15.6) is provided by the following corollary of the Gnedenko-Kolmorogov limit theorem (see Theorem H.1 in Appendix H). Theorem 15.12. Assume Hypothesis 15.1. Suppose, in addition, that lim λ(1 − N (λ)) =

λ→∞

1+β σ π

and

lim λN (−λ) =

λ→∞

1−β σ π

(15.7)

for some σ > 0 and β ∈ [−1, 1]. Then φ is a resonant state and lim |(eit/nH φ, φ)|2n = e−2σ|t| ,

n→∞

t ∈ R.

(15.8)

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Proof. By Theorem H.1 and Remark H.2 in Appendix H, the distribution N (λ) belongs to the domain of normal attraction of the 1-stable law. In particular, there are constants An such that    2 t it/nH n iAn log |t| , φ, φ) e = exp −σ|t| 1 + iβ lim (e n→∞ π |t| 

from which (15.8) follows.

Remark 15.13. If σ > 0 and therefore φ is a resonant state, the probability measure νφ (dλ) does not have the first moment and hence φ∈ / Dom(|H|1/2 ). In this case the “total energy” of the quantum system in the state φ is infinite. Introducing the “ultra-violet” cut-off Hamiltonian  λdEH (λ), HE = |λ|≤E

one observes that the “truncated” energy (|HE |φ, φ) of the state φ is logdivergent as the truncation parameter E approaches infinity. That is, 2 σ log E + o(log E) as E → ∞. (15.9) π In particular, the parameter σ determines the rate of convergence of the mean-value cut-off energy in the logarithmic scale as (|HE |φ, φ) =

σ=

π (|HE |φ, φ) lim . 2 E→∞ log E

Indeed, one gets that   (HE φ, φ) = λdN (λ) = [−E,E]



[−E,0)

λdN (λ) +

(0,E]

λdN (λ).

(15.10)

Integrating by parts (see, e.g., [37, Theorem 3.36]) one obtains   λdN (λ) = λd(N (λ) − 1) (0,E]

(0,E]

 = E(N (E) − 1) + 1+β σ+ =− π



0

E

(1 − N (λ))dλ

 1+β σ log E + o(log E) π

where we have used (15.7) on the last step.

as E → ∞,

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Therefore,  1+β σ log E + o(log E) as E → ∞. λdN (λ) = π (0,E] In a similar way one shows that  1−β σ log E + o(log E) as E → ∞, λdN (λ) = − π [−E,0)

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127

(15.11)

(15.12)

which together with (15.11) implies (15.9). Combining (15.10), (15.11) and (15.12), one also justifies the logarithmic divergence of the averaged truncated energy of the state (β = 0), that is, (HE φ, φ) =

2β σ log E + o(log E) as E → ∞. π

(15.13)

Example 15.14. The following example shows that neither the non-semiboundedness of the operator H nor the requirement of absolutely continuity of its spectrum is necessary for the existence of resonant states under continuous observation. Let H be a self-adjoint operator in the Hilbert space with simple discrete spectrum such that spec(H) = N and φn the corresponding eigenfunctions, Hφn = nφn , Let ψ be a state ψ=

n ∈ N,

φn  = 1.

an φn ,

n∈N

where {an }∞ n=1 is a sequence of complex numbers such that |an |2 = 1. n∈N

Then N (λ) =



|an |2 .

n 1.

Combining Lemma 15.7, Theorem 15.12 and Proposition 15.3 (cf. Remark 15.5) one concludes that the state ⎧ an anti-Zeno state, 12 < s < 1 ⎪ ⎪ ⎨ 1 1 . (15.15) ψs =

φn is a resonant state, s=1 ⎪ ζ(2s) n∈N ns ⎪ ⎩ a Zeno state, 1 0).

(15.24)

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In particular, the distribution function N (λ) of the state φ given by (15.2) belongs to the domain of the normal attraction of the 1-stable law, the density of which is given by the Breit-Wigner-Cauchy-Lorentz shape (dλ) =

1 σ dλ, 2π λ2 + 14 σ 2

with σ = Im(α).

By Theorem 15.12, lim λ(1 − N (λ)) = lim λN (−λ) =

λ→+∞

λ→+∞

σ > 0, π

which shows that the Hamiltonian H of the system is neither semibounded from below nor from above. As a corollary, one immediately concludes that for the quantum oscillator in Example 15.14 in the state φ given by (15.15) for s = 1, the limit T (t) = lim (U (t/n)φ, φ)n n→∞

does not exists although the limit p(t) = lim |(U (t/n)φ, φ)|2n = lim |(eit/nH φ, φ)|2n n→∞

n→∞

is well defined. Notice that in Example 15.14 the Hamiltonian H of the quantum oscillator is a positive operator with discrete spectrum and φ is a resonant state (see Definition 15.10) the energy distribution of which belongs to the domain of attraction of the Landau distribution, not to the Cauchy one. Remark 15.16. We remark that if (15.24) holds, then the generator of the corresponding contraction semigroup T(t) can be identified with the complex number α from the open upper half-plane. In particular, the (classical) dissipative system 1 d y = αy, i dt

y(0) = 1,

can be dilated (“quantized”) to the closed quantum system in L2 (R) initially prepared in the pure state ⎧ ⎨0, x>0 1 φ(x) =

2Im(α) ⎩e−iαx , x ≤ 0

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time evolution of which is given by the right shift operator V (t) in L2 (R), (V (t)φ)(x) = φ(x − t). d It is interesting to notice that the generator i dx of the shift group V (t) together with the multiplications operator by independent variable x solve the canonical commutation relations with which we began to present the results of our monograph.

We conclude this chapter with the following informal discussions. Perhaps the easiest informal way to understand the presence of exponentially decaying terms in the dynamics of open quantum system is as follows. Under certain conditions, the compressed resolvent of the Hamiltonian (onto the “unstable” subspace Ran(P )) admits an analytic continuation to the “unphysical sheet” where it may have poles type singularities (resonances). Representing the reduced evolution operator of the open quantum system as the Riesz type contour integral via the compressed resolvent P (H − λI)−1 P , one can deform the enclosing spectrum integration contour to the non-physical sheet. Evaluating the integral by the residue theorem yields exponentially decreasing terms in the description of the unitary reduced dynamics. In this schema, which is essentially due to Gamow, the analytic continuation ansatz necessarily requires the spectrum of the Hamiltonian to possess an absolutely continuous component. However, if the Hamiltonian of the system is a semi-bounded operator, the contour integration around the threshold of the continuous spectrum, the branching point of the resolvent kernel, gives rise to power law decay, which one again confirms that that there is no exponentially decay in the quantum systems with a semibounded Hamiltonian if one decides to start from the “first principles”. Under more additional restrictive assumptions it may happen that the compressed resolvent is the resolvent of a dissipative operator. Equivalently, the reduced evolution V (t) forms a strongly continuous semigroup of contractions, V (t + s) = V (t)V (s),

s, t ≥ 0,

and then a purely exponential decay in such systems is possible. However, in this case, the Hamiltonian of the system necessarily has absolutely continuous spectrum filling in the entire real axis. In particular, the Hamiltonian H is neither semi-bounded from below nor from above.

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135

If we assume that the quantum system is subject to continuous monitoring, the semigroup property for the reduced evolution V (t) is by no means necessary for exponential decay to occur. In some cases it is sufficient to require the existence of the Zeno limit evolution T (t) given by (15.22). Under these more relaxed assumptions one can get rid of the necessity for the Hamiltonian to have a portion of the absolutely continuous spectrum but nevertheless the energy spectrum of the exponentially decaying state must still be unbounded from both sides. It is worth mentioning that the sufficient conditions discussed above that guarantee the exponential decay of some states of the system i) are by no means necessary and ii) relate to quantities that are not observable (for instance, U (t), V (t) or T (t) are not self-adjoint operators) and therefore the study of the corresponding contractive semi-groups is of mathematical interest only: the natural physical requirement that the Hamiltonian of a quantum system has to a be a semi-bounded operator is violated. However, as we have shown in Example 15.14, the 1-stable limit theorem suggests a natural sufficient condition for the exponential decay of a state to occur. This condition does not require any of the above hypotheses: the Hamiltonian of the system is positive and has discrete spectrum. In particular, the Zeno limit evolution T (t) is ill defined in this case.

B1948

Governing Asia

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Chapter 16

THE QUANTUM ZENO VERSUS ANTI-ZENO EFFECT ALTERNATIVE

In this chapter we focus our attention on continuous monitoring of massive one-dimensional particles on a semi-axis. We assume that the Hamiltonian for a particle with one degree of freedom is the one-dimensional Schr¨odinger operator H =−

2 d2 2m dx2

in the Hilbert space L2 ((0, ∞)). In the system of units where  = 1 and mass m = 1/2 the Hamiltonian is given by the differential expression d2 dx2 with appropriate boundary conditions at the origin. It turns out that the results of frequent measurements for such quantum systems depend on the specific choice of the boundary conditions at the origin and they differ qualitatively. For instance, in the case of the Dirichlet Schr¨odinger operator, any smooth initial state with φ(0) = 0 is an Anti-Zeno state under the continuous monitoring. In contrast to this, if the quantum evolution is governed by any other self-adjoint realization of the second order differentiation operator, then all smooth initial states are Zeno states. The proper understanding of this phenomenon requires a more thorough analysis of the decay properties of quantum systems, which we will proceed below. H =−

137

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We start with a definition of a resonant state under continuous monitoring in a non-linear time-scale. Definition 16.1. Let t(t) be an increasing continuous function of t such that t(0) = 0. We say that the state φ is a resonant state under continuous monitoring of the quantum unitary evolution φ → eitH φ in the time-scale t(t) if lim [p(t(t/n))]n = e−σ|t| .

n→∞

where p(t) = |(eitH φ, φ)|2 is the survival probability. Remark 16.2. If φ ∈ Dom(H), then φ is a Zeno state by Proposition 15.3 in the standard (linear) time-scale t(t) = t, but φ is simultaneously a resonant state in the non-linear time-scale √ t(t) = t as it follows from (15.18) (see Subsection 15.4 where the concept of a prolonged frequent measurement is discussed). First, we treat the case of the Schr¨ odinger operator on the positive semi-axis with the Dirichlet boundary condition at the origin. Before formulating the corresponding result recall (see [33]) that if N denotes the normal distribution  λ 1 2 1 e− 2 y dy, N (λ) = √ 2π ∞ then

   σ Fσ (λ) = 2 1 − N , λ

λ > 0,

(16.1)

defines one-sided stable (L´evy) distribution with index of stability 12 the characteristic function f (t) of which is given by    t f (t) = exp −σ|t|1/2 1 − i . (16.2) |t|

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139

It is remarkable that along with the Gaussian and Cauchy distributions, the probability density function of the L´evy distribution is known in closed form [33]  σ 1/2 1 σ e− 2λ , λ > 0. ρ(λ) = 2π λ3/2 Our first result shows that for a typical initial state φ (φ ∈ W22 ((0, ∞)) such that φ(0) = 0) the spectral measure (EH (dλ)φ, φ) of the state φ has r-moments for all r < 1/2 but not for r = 1/2. Here H denotes the Schr¨ odinger operator with the Dirichlet boundary condition at the origin. In particular this means that such states are anti-Zeno states under continuous monitoring of the quantum evolution φ → eitH φ. However, in the time scale t(t) = t2 such states do exhibit exponential decay. Equivalently, short frequent measurements yield

2√n 1/2

it/nH

φ, φ)

= e−σ|t| lim (e n→∞

for some σ > 0. Theorem 16.3. Let H = L2 ((0, ∞)) and d2 (16.3) dx2 be the Schr¨ odinger operator with the Dirichlet boundary condition at the origin H =−

Dom(H) = {f ∈ W22 ((0, ∞)) | f (0) = 0}. Suppose that φ ∈ W22 ((0, ∞)) is such that φ(0) = 0 and φ = 1. Then the distribution function N (λ) of the spectral measure νφ (dλ) = (EH (dλ)φ, φ) of the element φ belongs to the domain of normal attraction of the one-sided 1 evy distribution Fσ (16.1) the characteristic function of which is 2 -stable L´ given by (16.2) with  2 σ= |φ(0)|2 . π In particular, the state φ is a resonant state in the time-scale t(t) = t2 . That is, lim [p(t(t/n))]n = exp (−2σ|t|) ,

n→∞

(16.4)

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where p(t) = |(eitH φ, φ)|2 is the survival probability. Proof. Let H˙ be the restriction of H on ˙ = {f ∈ Dom(H) | f (0) = f  (0) = 0}. Dom(H) It is known that the Weyl-Titchmarsh function M (z) associated with ˙ H) admits the representation [42] the pair (H, √ M (z) = i 2z + 1, z ∈ C+ . By the Stieltjes inversion formula, we have that   ∞ 1 λ M (z) = − 2 dμ(λ), λ−z λ +1 0 where μ(dλ) is an absolutely continuous measure supported by the positive semi-axis with the density √ 1 2√ dμ(λ) = Im(M (λ + i0)) = λ, λ > 0. (16.5) dλ π π ˙ ∗ ∓iI) Suppose that g± , g±  = 1, are deficiency elements g± ∈ Ker((H) such that g+ − g− ∈ Dom(H).

(16.6)

In fact, the deficiency elements g± of the symmetric operator H˙ can be chosen as (see, e.g., [42]) g+ (x) = 21/4 ei

√

2 2 x

e−

√

2 2 x

and g− (x) = g+ (x),

x ≥ 0.

(16.7)

In this case, g+ (0) − g− (0) = 0 which shows that (16.6) holds. One can apply Theorem C.1 in Appendix C to conclude that there is a unitary map U from L2 ((0, ∞)) onto L2 ((0, ∞); dμ) such that U HU −1 is the operator of multiplication by independent variable in L2 ((0, ∞); dμ). Since (16.6) holds, from Remark C.2 in Appendix C it follows that (U g± )(λ) = for some |Θ| = 1.

Θ , λ∓i

λ > 0,

(16.8)

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˙ ∗ ). Therefore, in accorBy the hypothesis, φ ∈ W22 ((0, ∞)) = Dom((H) dance with von Neumann’s formula the element φ admits the representation φ = αg+ + βg− + h

(16.9)

˙ ⊂ Dom(H). for some uniquely determined α, β ∈ C and h ∈ Dom(H) We claim that the distribution function N (λ) of the spectral measure νφ (dλ) = (EH (dλ)φ, φ) of the element φ admits the asymptotic representation √ 2 2 |α + β|2 √ + o(λ−5/4 ) as λ → ∞. 1 − N (λ) = π λ

(16.10)

Indeed, from (16.8) and (16.9) it follows that (U φ)(λ) =

a b + + (U h)(λ), λ−i λ+i

(16.11)

where a = Θα and b = Θβ. In particular, |a + b| = |α + β|.

(16.12)

Working out the computations in the model representation provided by Theorem C.1 in Appendix C, we obtain for the distribution function N (λ) the representation

2  ∞  ∞



a b dμ(s)

dμ(s) = |a + b|2

+ + (U h)(s) 1 − N (λ) =

s − i s + i s2 + 1 λ λ   ∞ 1 1 + 2Re ab − 2 dμ(s) 2 (s − i) s +1 λ   ∞ a b + (U h)(s)dμ(s) + 2Re s−i s+i λ  ∞ + |(U h)(s)|2 dμ(λ), λ ≥ 0. (16.13) λ

From (16.5) it follows √  ∞ √ √  ∞ 2 2 1 dμ(s) 2 s √ + O(λ−3/2 ) as λ → ∞. = ds = s2 + 1 π λ s2 + 1 π λ λ (16.14)

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Therefore, for the first term of the right hand side of (16.13) we have the asymptotic representation √  ∞ dμ(s) 2 22 2 1 √ + O(λ−3/2 ). = |α + β| |a + b| s2 + 1 π λ λ Here we have used (16.12). The remaining three terms in (16.13) can be estimated as follows

 ∞



1 3 ∞ dμ(s) 1



= O(λ−3/2 ), (16.15)

(s − i)2 − s2 + 1 dμ(s) ≤ λ 2+1 s λ λ



∞ (U h)(s)



dμ(s)



λ

s±i   ∞ ∞ 1 dμ(s) ≤ · (1 + s2 )|(U h)(s)|2 dμ(s) 2 λ s +1 λ λ = o(λ−5/4 ),

(16.16)

and 

∞

λ

|(U h)(s)|2 dμ(s) ≤

1 λ2



∞

λ

(1 + s2 )|(U h)(s)|2 dμ(s) = o(λ−2 ),

as λ → ∞.

(16.17)

˙ ⊂ Dom(H), Here, in (16.16) and (16.17) we have used that h ∈ Dom(H) so that U h ∈ L2 (R; (1 + λ2 )dμ(λ)). Combining (16.14) and the asymptotic estimates (16.15)-(16.17), from (16.13), we get √ 2 2 |α + β|2 √ + o(λ−5/4 ) as λ → ∞. (16.18) 1 − N (λ) = π λ Next, we evaluate |α + β|2 via the boundary data |φ(0)|2 . One observes (see (16.9)) that the boundary condition φ(0) = αg+ (0) + βg− (0) + h(0)

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˙ we have h(0) = 0, so that holds. Now, since h ∈ Dom(H), φ(0) = αg+ (0) + βg− (0) = 21/4 (α + β) as it follows from (16.7). Hence, |α + β|2 =

|φ(0)|2 √ . 2

Now, taking into account that φ(0) = 0, from (16.18) we get the asymptotic representation 2 |φ(0)|2 (1 + o(1)) as λ → ∞. π λ1/2 Moreover, since H is a non-negative operator, we obviously have N (λ) = 1 −

N (λ) = 0,

λ < 0.

By Theorem H.1 in Appendix H, the distribution N (λ) belongs to the domain of normal attraction of the one-sided stable L´evy distribution Fσ with the characteristic function    t 1/2 f (t) = exp −σ|t| 1−i , (16.19) |t| where



2 |φ(0)|2 . π Indeed, the distribution N (λ) satisfies the conditions (H.4) and (H.5) of Theorem H.1 in Appendix H with α = 12 , σ=

2 |φ(0)|2 and c2 = 0. π Therefore, N (λ) belongs to the domain of normal attraction of a stable law with the characteristic function     1 t 1/2 1 − i β ω t, f (t) = exp −σ|t| . |t| 2 c1 =

Here

    1 1 2 2 = |φ(0)| · d , σ = (c1 + c2 )d 2 π 2 c1 − c2 = 1, c1 + c2   π 1 = 1, ω t, = tan 2 4 β=

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and d

  1 1√ π 2π. = Γ(1/2) cos = 2 4 2

The main assertion of the theorem is now proven. To complete the proof of the theorem it remains to apply the 1/2-stable limit theorem to see that 



2n (16.20) lim exp in−2 tH φ, φ = exp −2σ|t|1/2 , n→∞



which justifies (16.4) by a change of variables.

The situation is quite different for any other self-adjoint realization of the free Schr¨ odinger operator H  on the semi-axis. In this case, the spectral measure (EH (dλ)φ, φ) of a typical state φ ∈ W22 ((0, ∞)) has r-moments for all r < 3/2. As a consequence, such states are Zeno states under continuous  monitoring of the quantum evolution φ → eitH φ. However, φ becomes a resonant state in the time scale t(t) = t2/3 . Equivalently, prolonged measurements “unfreeze” the quantum system and

2n√n  3/2



lim (eit/nH φ, φ)

= e−σ |t| n→∞



for some σ > 0. Theorem 16.4. Let H = L2 ((0, ∞)), γ ∈ R and H = −

d2 dx2

(16.21)

be the Schr¨ odinger operator with the mixed boundary condition at the origin Dom(H  ) = {f ∈ W22 ((0, ∞)) | f  (0) + γf (0) = 0}. Suppose that φ ∈ W22 ((0, ∞)), φ = 1, and assume, in addition, that φ (0) + γφ(0) = 0. Then the distribution function N (λ) of the spectral measure νφ (dλ) = (EH  (dλ)φ, φ) of the element φ belongs to the domain of normal attraction of the 3/2-stable law with the characteristic function  f (t) = exp −σ  |t|3/2 (1 + i sgn(t)) ,

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2  |φ (0) + γφ(0)|2 . π

In particular, the state φ is a resonant state in the time-scale t(t) = t2/3 . That is, lim [p(t(t/n))]n = exp (−2σ  |t|) ,

n→∞

(16.22)

where p(t) = |(eitH φ, φ)|2 is the survival probability. Proof. Let H˙ be the symmetric restriction of the operator H  on ˙ = {f ∈ Dom(H  ) | f (0) = f  (0) = 0}. Dom(H) Denote by g± , g±  = 1, the deficiency elements of the symmetric operator H˙ g+ (x) = 21/4 ei

√

2 2 x

e−

√

2 2 x

and g− (x) = g+ (x)Θ,

(16.23)

where Θ is chosen in such a way to ensure that g+ − g− ∈ Dom(H  ).

(16.24)

The parameter Θ can be determined as follows. From (16.24) it follows that g+ (x) − g− (x) should satisfy the boundary condition   (g+ (0) − g− (0)) + γ(g+ (0) − g− (0)) = 0

and hence ¯ + γ(1 − Θ) = 0, (ζ − ζΘ) where

√ 2 ζ= (1 − i). 2

Solving (16.25) for Θ yields ζ +γ Θ= ¯ . ζ +γ

(16.25)

(16.26)

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Since (16.24) holds, one can apply Theorem C.1 in Appendix C, and the same reasoning as the one in the proof of Theorem 16.3 shows that the leading term of the asymptotics of the distribution function N (λ) is given by 1 − N (λ) = |α + β|



2

∞ λ

dμ(s) + o(λ−7/4 ) s2 + 1

as λ → ∞.

(16.27)

Here μ(dλ) is the measure associated with the Herglotz-Nevanlinna decomposition for the Weyl-Titchmarsh function M (z) associated with the pair ˙ H ) (H,  M (z) =

spec(H  )



1 λ − 2 λ−z λ +1

 dμ(λ)

and α and β are determined by the von Neumann decomposition φ = αg+ + βg− + h,

˙ h ∈ Dom(H).

(16.28)

To justify the asymptotic representation (16.27) we argue as follows. First recall, that it is known that the Weyl-Titchmarsh function associ˙ H  ) has the form ated with the pair (H, √ cos α + sin α(i 2z + 1) √ , M (z) = sin α − cos α(i 2z + 1)

z ∈ C+ .

(16.29)

Here the boundary condition parameter γ and (the von Neumann extension) parameter α are related as [42] γ = 2−1/2 (1 − tan α),

α =

π . 2

(16.30)

From (16.29) it follows that the restriction of the measure μ(dλ) on the positive semi-axis is an absolutely continuous measure with the density given by 1 dμ(λ) = Im(M (λ + i0))dλ, dλ π

λ > 0.

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Explicit computations show that 1 Im(M (λ + i0))dλ π = = = =

√ 1 cos α + sin α(i 2λ + 1) √ Im π sin α − cos α(i 2λ + 1) √ √ 1 (cos α + sin α(i 2λ + 1))(sin α − cos α(−i 2λ + 1)) Im π (sin α − cos α)2 + 2λ cos2 α √ √ 1 (cos α + sin α + i 2λ sin α)(sin α − cos α + i 2λ cos α) Im π (sin α − cos α)2 + 2λ cos2 α √ 1 2λ , λ > 0. π (sin α − cos α)2 + 2λ cos2 α

Therefore, dμ(λ) =

√ 1 2λ dλ, π (sin α − cos α)2 + 2λ cos2 α

λ > 0.

(16.31)

To justify (16.27), in particular, to see that the error term is of the order of o(λ−7/4 ) as λ → ∞, we argue exactly as in the proof of Theorem 16.3. To do so, we need to estimate the following three integrals (we use the notation from the proof of Theorem 16.3)

 ∞

1 1



dμ(s), − I=

(s − i)2 s2 + 1

λ



∞ (U h)(s)



II =

dμ(s) ,

λ

s±i and  III =

∞

λ

|(U h)(s)|2 dμ(s).

We have (as λ → ∞) 3 I≤ λ



∞

λ

dμ(s) = O(λ−5/2 ). s2 + 1

(16.32)

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˙ ⊂ Dom(H), and therefore U h ∈ L2 (R; (1 + λ2 )dμ(λ)), Since h ∈ Dom(H) we also have the asymptotic estimates   ∞ ∞ dμ(s) 1 · (1 + s2 )|(U h)(s)|2 dμ(s) = o(λ−7/4 ) (16.33) II ≤ λ s2 + 1 λ λ and III ≤

1 λ2



∞

λ

(1 + s2 )|(U h)(s)|2 dμ(s) = o(λ−2 )

as λ → ∞.

(16.34)

Therefore, I + II + III = o(λ−7/4 )

as λ → ∞,

which completes the justification of the representation (16.27). Next, combining (16.27) and (16.31) we obtain √  ∞ ds 1 2s 2 1 − N (λ) = |α + β| + o(λ−7/4 ) π (sin α − cos α)2 + 2s cos2 α s2 + 1 λ √ 2 2 λ−3/2 (1 + o(1)) + o(λ−7/4 ) = |α + β| 3π cos2 α √ √ 2 2 (( 2γ − 1)2 + 1)λ−3/2 (1 + o(1)) + o(λ−7/4 ) = |α + β| 3π as λ → ∞.

(16.35)

Here we have used the relation √ 1 = (( 2γ − 1)2 + 1) 2 cos α that easily follows from (16.30). Recall that α = π2 and therefore cos α = 0. Our next claim is that 1 1 √ |φ (0) + γφ(0)|2 . (16.36) |α + β|2 = √ 2 (γ − 22 )2 + 12 From (16.28) it follows that φ(0) = αg+ (0) + βg− (0) = αg+ (0) + βΘg+ (0) = (α + βΘ)21/4

(16.37)

and   (0) + βg− (0) = (αζ + βζΘ)21/4 , φ (0) = αg+

where ζ is given by (16.26).

(16.38)

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Rewriting (16.37) and (16.38) as      1 Θ α φ(0) = 2−1/4 , ¯ ζ ζΘ β φ (0) and solving this system of algebraic equations one obtains       ¯ −Θ φ(0) α ζΘ 1 −1/4 . = 2 Θ(ζ¯ − ζ) −ζ 1 φ (0) β Therefore, 1 ¯ − ζ)φ(0) + (1 − Θ)φ (0)] [(ζΘ 21/4 Θ(ζ¯ − ζ)  ¯ ζΘ − ζ 1−Θ  φ(0) + φ (0) . = 1/4 ¯ 2 Θ(ζ − ζ) 1 − Θ

α+β =

From (16.25) it follows that γ=

¯ −ζ ζΘ , 1−Θ

so that α+β = One also observes that

1−Θ 21/4 Θ(ζ¯ −

ζ)

[γφ(0) + φ(0)].

1 1−Θ , = ¯ ζ +γ Θ(ζ − ζ)

which yields 1 1 |α + β|2 = √ |φ (0) + γφ(0)|2 2 |ζ + γ|2 1 1 √ = √ |φ (0) + γφ(0)|2 , 2 (γ − 22 )2 + 12 and the claim (16.36) follows. Combining (16.35) and (16.36) and taking into account that φ (0) + γφ(0) = 0 (by the hypothesis), we finally obtain the asymptotic representation √ 2 √ (( 2γ − 1)2 + 1)λ−3/2 (1 + o(1)) 1 − N (λ) = |α + β|2 3π √ √ 2 ( 2γ − 1)2 + 1  1 √ |φ (0) + γφ(0)|2 λ−3/2 (1 + o(1)) =√ · 2 3π (γ − 22 )2 + 12 =

2  |φ (0) + γφ(0)|2 λ−3/2 (1 + o(1)) as λ → ∞. 3π

(16.39)

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Notice that for γ < 0 the operator H  has a simple eigenvalue λ0 = −γ 2 and therefore N (λ) = 0 whenever λ < −γ 2 , and N (λ) = 0 for all λ < 0 if γ ≥ 0. Therefore, lim |λ|3/2 N (λ) = 0.

λ→−∞

(16.40)

By Theorem H.1 in Appendix H, the distribution N (λ) belongs to the domain of normal attraction of the one-sided 32 -stable distribution with the characteristic function    t  3/2 1+i f (t) = exp −σ |t| , (16.41) |t| where 2 σ = 3 



2  |φ (0) + γφ(0)|2 . π

Indeed, the distribution N (λ) satisfies the conditions (H.4) and (H.5) of Theorem H.1 in Appendix H with α = 32 , c1 =

2  |φ (0) + γφ(0)|2 3π

and c2 = 0.

By Theorem H.1, N (λ) belongs to the domain of normal attraction of a stable law with the characteristic function     3 t  3/2 f (t) = exp −σ |t| 1 − i β ω t, , |t| 2 where the parameters σ  and β are given by     3 3 2 σ  = (c1 + c2 )d = |φ (0) + γφ(0)|2 · d , 2 π 2 β=

c1 − c2 = 1, c1 + c2

with d and

Now (16.41) follows.

  3 1√ 3π = 2π, = Γ(−1/2) cos 2 4 2     3π 3 ω t, = tan = −1. 2 4

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To complete the proof of the theorem it remains to apply the 3/2-stable limit theorem to see that

  2n 



(16.42) lim exp in−2/3 tH φ, φ = exp −2σ  |t|3/2 , n→∞



which justifies (16.22) by a change of variables.

Remark 16.5. The right hand side of (16.42) is the characteristic functions of the Holtsmark distribution [47]. The Holtsmark distribution is a special case of a symmetric stable distribution with the index of stability α = 3/2 and skewness parameter β = 0 (see Appendix H, eqs. (H.1), (H.2) with α = 3/2 and β = γ = 0). Scholium. The 1/2- and 3/2-stable limit theorems, Theorems 16.3 and 16.4, respectively, show that the results of continuous monitoring of the quantum evolution of a smooth state φ are rather sensitive to the choice of a self-adjoint realization of the Hamiltonian, the Schr¨odinger operator (16.3) and(16.21), respectively. For instance, for the Schr¨odinger operator H with the Dirichlet boundary condition at the origin we have √

lim |(eit/nH φ, φ)|2

n

n→∞

where

 σ=

= e−2σ|t|

1/2

,

2 |φ(0)|2 . π

Therefore, if the probability density |φ(0)|2 to find a quantum particle at the origin does not vanish, then the state φ is an anti-Zeno state. That is, lim |(eit/nH φ, φ)|2n = 0.

n→∞

In the meanwhile, for the Schr¨ odinger operator H  with the mixed boundary condition f  (0) + γf (0) = 0, one obtains that 

√

lim |(eit/nH φ, φ)|2n

n→∞

where 2 σ = 3 



n



= e−2σ |t|

2  |φ (0) + γφ(0)|2 . π

3/2

,

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Hence, 

lim |(eit/nH φ, φ)|2n = 1.

n→∞

In other words, any smooth state φ is a Zeno state under the continuous  monitoring of the evolution φ → eit/nH φ where H  is any self-adjoint realizations of the second differentiation operator different form the Friedrichs ˙ extension H of H. We summarize the observations above in a more formal way. Corollary 16.6. Suppose that φ ∈ W22 ((0, ∞)), φ = 1. (i) Let H be the Schr¨ odinger operator with the Dirichlet boundary condition at the origin. Then φ is a Zeno state under the continuous monitoring of the unitary evolution φ → eitH φ if and only if φ(0) = 0. Otherwise, φ is an anti-Zeno state. (ii) If H  is any other self-adjoint realization of the differential expression τ =−

d2 dx2

different from its Friedrichs extension, then φ is a Zeno state under  the continuous monitoring of the unitary evolution φ → eitH φ. Proof. (i) If φ(0) = 0, then φ ∈ Dom(H) and therefore φ is a Zeno state under the continuous monitoring of the unitary evolution φ → eitH φ. If φ(0) = 0, by Theorem 16.3 the distribution function of the spectral measure of the element φ belongs to the domain of attraction of a 1/2-stable law, and therefore φ is an anti-Zeno state by Lemma 15.7. (ii) Notice that for any self-adjoint extension H  different from the Friedrichs extension H the domain of the quadratic form of H  coincides with the Sobolev class W21 ((0, ∞)). Since φ ∈ W22 ((0, ∞)) ⊂ W21 ((0, ∞)), we have that the distribution N (λ) of the spectral measure (EH  (dλ)φ, φ) of the state φ has the first moment and hence φ is necessarily a Zeno state.  Remark 16.7. (i) In the case of the Schr¨ odinger operator H with the Dirichlet boundary condition at the origin, one can slightly relax the smoothness requirement on the state φ that φ ∈ W22 ((0, ∞)): If φ ∈ W21 ((0, ∞)) only and φ(0) = 0, then the state φ belongs to the domain of the quadratic form of the Schr¨odinger operator H. In this case, φ is also

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a Zeno state under the continuous monitoring of the unitary evolution φ → eitH φ by Proposition 15.3. (ii) From Theorem 16.4 it follows that the spectral measure νφ (dλ) = (EH  (dλ)φ, φ) of the state φ has moments of order r for all r < 32 . In particular, the state φ belongs to the domain of the quadratic form of H  and therefore, φ is a Zeno state under the continuous monitoring of the unitary evolution  φ → eitH φ by Proposition 15.3. (iii). As far as the domain issues are concerned, we have the following inclusions ˙ ⊂ Dom(H) ⊂ Dom((H) ˙ ∗ ) = W22 ((0, ∞)) Dom(H) and ˙ ∗ ) ∩ Dom(H 1/2 ) = Dom(H) = {f ∈ W22 ((0, ∞)) | f (0+) = 0} Dom((H) ˙ H ≥ 0. For any self-adjoint extension for the Friedrichs extension H of H,  H different from the Friedrichs extension H we have ˙ ∗ ) ⊂ Dom(|H  |1/2 ) = W21 ((0, ∞)) ˙ ⊂ Dom(H  ) ⊂ Dom((H) Dom(H) and therefore ˙ ∗ ) = W22 ((0, ∞)). ˙ ∗ ) ∩ Dom(|H  |1/2 ) = Dom((H) Dom((H) Notice that for the Friedrichs extension H we have Dom(H 1/2 ) = {f ∈ W21 ((0, ∞)) | f (0+) = 0} = Dom(|H  |1/2 ) = W21 ((0, ∞)), which explains the peculiar “phase transition” in the relative geometry of domains (the Sobolev spaces) when replacing the Friedrichs extension H with any other self-adjoint extension H  .

B1948

Governing Asia

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Chapter 17

THE QUANTUM ZENO EFFECT VERSUS EXPONENTIAL DECAY ALTERNATIVE Throughout this chapter we assume that Y is a metric graph in one of the Cases (i)–(iii) (see the classification in the beginning of Chapter 4). Denote ˙ D,  D) the triple of differentiation operators on Y as introduced in by (D, Chapter 9. Recall that in Case (i), the metric graph has the form Y = (−∞, 0)  (0, ∞), in Case (ii), Y = (0, ), in Case (iii), Y = (−∞, 0)  (0, ∞)  (0, ). Also recall that the reference self-adjoint operator D is the differentiation operator on the graph Y defined on Dom(D) = {f∞ ∈ W21 (Y) | f∞ (0+) = −f∞ (0−)}, Dom(D) = {f ∈ W21 (Y) | f (0) = −f ()}, ⎫ ⎧ √ ⎧  ⎨f∞ (0+) = kf∞ (0−) + 1 − k 2 f () ⎬ ⎨  Dom(D) = f∞ ⊕ f ∈ W21 (Y) ,  ⎩f (0+) = √1 − k 2 f (0−) − kf () ⎭ ⎩  ∞  in Cases (i)–(iii), respectively. Here 0 < k < 1 is the parameter from the boundary condition (4.3) (the quantum gate coefficient) that determines the symmetric operator D˙ in Case (iii). More generally, see Theorem 17.4 below, we will also deal with the ˙ D,  DΘ ) where DΘ , |Θ| = 1 is the self-adjoint operator referred triples (D, to in Theorem 5.1.

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Our main concern is to study small-time asymptotic behavior of the quantum survival probability p(t) = |(eitH φ, φ)|2

as t → 0,

where H = D, or, more generally, H = DΘ , the magnetic Hamiltonian. We also assume that the state φ belongs to the test space ˙ ∗ ). L = Dom((D) We obviously have the inclusion L



W21 (e),

e⊂Y

where the sum is taken over all edges e of the graph Y. The main goal of this chapter is to show that the survival probability under continuous monitoring of the quantum evolution φ → eitH φ,

φ ∈ L,

on the metric graph either experiences an exponential decay or, alternatively, the quantum Zeno effect takes place. This justifies the complementarity of the Exponential Decay and the Quantum Zeno Effect scenarios for hyperbolic systems first indicated in [66]. We start our analysis with the observation that the normalized deficiency elements of the symmetric operator D˙ are resonant states under continuous monitoring of the unitary evolution φ → eitH φ where H = D. Lemma 17.1. Suppose that a metric graph Y is in one of the Cases (i)–(iii) and D˙ is the symmetric differentiation operator on Y with boundary condi˙ ∗ ∓iI), g± = 1, tions (4.1), (4.2) and (4.3), respectively. Let g± ∈ Ker((D) ˙ Then be normalized deficiency elements g± of the symmetric operator D. g± are equidistributed, that is, g± have the same spectral measure ν(dλ) = (EH (dλ)g+ , g+ ) = (EH (dλ)g− , g− ), where the Hamiltonian H is given by the differentiation operator D.

(17.1)

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Moreover, lim λν ((λ, ∞)) = lim λν ((−∞, −λ))

λ→+∞

λ→+∞

⎧ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎨ 1 coth  , = 2 π⎪ ⎪ ⎪  ⎪ ⎪ ⎩coth  +  , 2

in Case (i) in Case (ii)

(17.2)

in Case (iii).

Here, in Case (iii), 1 k and 0 < k < 1 is the quantum gate coefficient from the boundary condition (4.3) that determines the symmetric operator D˙ in Case (iii). In particular, the deficiency elements g± are resonant states with respect to the continuous monitoring of the unitary dynamics g± → eitH g± . In this case,  = log

lim |(eit/nH g± , g± )|2n = e−τ |t|,

n→∞

where the decay constant τ is given by ⎧ ⎪ 1, ⎪ ⎪ ⎪ ⎪  ⎨ τ = 2 coth 2 , ⎪ ⎪ ⎪  ⎪ ⎪ ⎩coth  +  , 2

(17.3)

in Case (i) in Case (ii) in Case (iii).

Proof. Let M (z) be the Weyl-Titchmarsh function associated with the ˙ D). By Corollary 7.2, pair (D,  1 λ M (z) = − dμ(λ), (17.4) λ − z 1 + λ2 R where the measure μ(dλ) is given by ⎧ ⎪ dλ, ⎪ ⎪ ⎪ ⎪ ⎪ 2π  coth δ (2k+1)π (dλ), 1⎨  μ(dλ) =  2 k∈Z π⎪ ⎪ ⎪ ⎪ ⎪  +  ⎪ ⎩coth Pe− (λ − π) dλ, 2

in Case (i) in Case (ii) in Case (iii).

(17.5)

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Here, in Case (iii), Pr (ϕ) =

1 − r2 1 + − 2r cos ϕ r2

denotes the Poisson kernel. Recall that by Lemma 4.4 the operator D˙ is a prime symmetric operator. Therefore, Theorem C.1 in Appendix C ensures the existence of a unitary map U from L2 (Y) onto the Hilbert space L2 (R, dμ), where μ(dλ) is given by (17.5), with the following properties: (i) UDU −1 coincides with the operator of multiplication by independent variable and (ii) the deficiency elements g± get mapped to simple fractions (Ug± )(λ) =

Θ± λ∓i

for some |Θ± | = 1.

In particular, for any Borel set δ ⊂ R we have (EH (δ)g+ , g+ ) = (EH (δ)g− , g− ) =

δ

dμ(s) , s2 + 1

which shows that g± are equidistributed and hence the spectral measure ν(dλ) in (17.1) is well defined. It follows that ∞ dμ(s) . λν((λ, ∞)) = λ s2 + 1 λ In Case (i), in view of (17.5) we have the following asymptotic representation ∞ 1 1 ds = (1 + o(1)) as λ → +∞, λν((λ, ∞)) = λ 2+1 π s π λ which proves that the first limit in (17.2) exists and coincides with the right hand side of (17.2). In Case (ii), by (17.5),  2 λν((λ, ∞)) = λ coth  2

=

 2 coth  2

 (2k+1)π 

 2π

 ≥λ

2 λ

1 (2k+1)π 

∞ λ 2π

2

+1

dk · (1 + o(1)) k2

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  2 coth · (1 + o(1))  2 2π 1  = coth (1 + o(1)) as λ → +∞, π 2

=

proving the first equality (17.2) in Case (ii). Finally, in Case (iii), using (17.5) we have ∞ 1  +  ds λν((λ, ∞)) = coth λ Pe− (s − π) 2 π 2 s +1 λ  + 1 (1 + o(1)) as λ → +∞, = coth π 2 which shows that the first limit in (17.2) exists and coincides with the right hand side of (17.2). Here we used that the Poisson kernel admits the representation Pr (s) =

1 − r2 = 1 + Gr (s), 1 + r2 − 2r cos s

where Gr (s) is a bounded 2π-periodic function with zero mean over the period such that ∞ ds lim λ = 0. Ge− (s − π) 2 λ→+∞ s +1 λ Notice that the equality above can be justified by integration by parts. In a completely similar way one shows that in all Cases (i)–(iii) the second limit in (17.2) exists and coincides with the right hand side of (17.2).  Remark 17.2. In Case (i), one can apply the residue theorem to see that the survival probability amplitude (eitD g± , g± ) itself is exponentially decaying as 1 ∞ iλt dλ = e−|t| , e (eitD g± , g± ) = π −∞ λ2 + 1 which in particular implies (17.3). In this case the result of continuous monitoring of the corresponding quantum system on the time interval [0, t] and a “one time observation” at the moment of time t are identical. That is, (eit/nD g± , g± )n = (eitD g± , g± )

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and therefore |(eit/nD g± , g± )|2n = |(eitD g± , g± )|2

for all t.

In this exceptional (resonant) case the continuous monitoring can neither stop nor modify the evolution. Notice that in this case the energy distribution of the states g± has a typical Cauchy-Lorentz (Breight-Wigner) shape which yields a purely exponential decay, see [29, Example 1.2.4], cf. [93, p. 759, a counterexample]. To understand better fine decay properties of a particular state from the test space L = Dom(D˙ ∗ ) we need a comprehensive information about the boundary functionals associated with the von Neumann decomposition of the test space ˙ ∗ − iI)+ ˙ ∗ + iI)+Dom( ˙ ˙ ∗ ) = Ker((D) ˙ Ker((D) ˙ D). L = Dom((D)

(17.6)

Lemma 17.3. Suppose that a metric graph Y is in one of the Cases (i)–(iii) and D˙ is the symmetric differentiation operator on Y with boundary conditions (4.1), (4.2) and (4.3), respectively. Denote by g± the deficiency elements of the symmetric operator D˙ referred to in Lemma 4.3. ˙ ∗ ) and let Assume that φ ∈ L = Dom((D) φ = αg+ + βg− + f,

˙ ∗ ), φ ∈ Dom((D)

be the decomposition associated with von Neumann’s ˙ α, β ∈ C and f ∈ Dom(D). Then ⎧ φ∞ (0−) + φ∞ (0+), ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ tanh (φ (0) + φ ()) , ⎨ 2 1  α+β = √  2⎪ ⎪ tanh  +  ⎪ ⎪ ⎪ 2 ⎪ 

⎪ ⎪ ⎪ φ∞ (0+) − kφ∞ (0−) ⎪ ⎩ √ , φ () − 1 − k2

(17.7)

formula (17.6), where

in Case (i) in Case (ii) (17.8)

in Case (iii),

where, in Case (iii),  = log

1 k

and 0 < k < 1 is the quantum gate coefficient from the boundary condition (4.3) that determines the symmetric operator D˙ in Case (iii).

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Proof. In Case (i), we have √ √ φ∞ (x) = α 2ex χ(−∞,0) (x) + β 2e−x χ(0,∞) (x) + f (x) ˙ for some f ∈ Dom(D). Since f (0−) = f (0+) = 0, we have √ φ∞ (0−) = α 2

√ and φ∞ (0+) = β 2.

Therefore α+β =

φ∞ (0+) + φ∞ (0+) √ , 2

proving (17.8) in that case. In Case (ii),  φ (x) = α

2 ex + β e2 − 1



2 e−x + f (x) e2 − 1

and therefore  

α+e β =

e2 − 1 φ (0) 2

and  

e α+β =

e2 − 1 φ (). 2

Hence  α+β =

e − 1 φ (0) + φ () √ = e + 1 2

 tanh

 φ (0) + φ () √ · , 2 2

proving (17.8) in Case (ii). In Case (iii), the elements φ and f from the von Neumann decomposition (17.7) are the two-component vector functions  φ=

φ∞ φ



 and f =

f∞ f

 .

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From (17.7) it follows that  φ∞ (x) = αξ 1 − k 2 ex χ(−∞,0) (x)  − βξ 1 − k 2 e−x χ(0,∞) (x) + f∞ (x),

x ∈ R,

and φ (x) = αξex + βξke−x + f (x),

x ∈ [0, ),

where the norming constant ξ is given by  2 ξ= . e2 − k 2 In particular,



1 − k 2 + f∞ (0−),  φ∞ (0+) = −βξ 1 − k 2 e + f∞ (0+), φ∞ (0−) = αξ

φ () = αξe + kβξ + f (). ˙ the boundary conditions Since f ∈ Dom(D), f∞ (0+) = kf∞ (0−),  f (0) = 1 − k 2 f∞ (0−), f () = 0 hold and hence



1 − k 2 + γ,  φ∞ (0+) = −βξ 1 − k 2 e + kγ,

φ∞ (0−) = αξ

φ () = αξe + kβξ, where we use the shorthand notation γ = f∞ (0−). Combining the obtained equations we arrive at the following system of equations ⎞⎛ ⎞ ⎛ ⎞ ⎛√ αξ x 1 − k2 0 1 √ ⎟⎜ ⎟ ⎜ ⎟ ⎜  2 0 −e 1 − k k ⎠ ⎝βξ ⎠ = ⎝y ⎠ , ⎝ e

k

0

γ

z

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where ⎛ ⎞ ⎛ ⎞ x φ∞ (0−) ⎜ ⎟ ⎜ ⎟ ⎝y ⎠ = ⎝φ∞ (0+)⎠ . z

φ ()

Taking into account that the inverse matrix of the system is of the form ⎛ 1 (e2

−

⎜ √ ⎝ 2 1−k

k2 )

−k 2

√ ⎞ e 1 − k 2 √ ⎟ −k 1 − k 2 ⎠

k

ke −e √ √ e2 1 − k 2 −k 1 − k 2 −e (1 − k 2 )

one easily obtains that αξ =

√ −k 2 x + ky + e 1 − k 2 z √ (e2 − k 2 ) 1 − k 2

and βξ =

√ ke x − e y − k 1 − k 2 z √ . (e2 − k 2 ) 1 − k 2

Therefore, α+β =ξ

−1 (ke



√ − k 2 )x + (k − e )y + (e − k) 1 − k 2 z √ (e2 − k 2 ) 1 − k 2

√ − k 2 )φ∞ (0−) + (k − e )φ∞ (0+) + (e − k) 1 − k 2 φ () √ =ξ (e2 − k 2 ) 1 − k 2 

1 φ∞ (0+) − kφ∞ (0−) √ = ξ −1  φ () − (e + k) 1 − k2  

 +  φ∞ (0+) − kφ∞ (0−) 1 √ tanh , = √ φ () − 2 2 1 − k2 −1 (ke



which completes the proof of (17.8) in Case (iii).



The main result of this chapter is the following Theorem 17.4 (Exponential Decay-Quantum Zeno Effect alternative). Suppose that a metric graph Y is in one of the Cases (i)–(iii). Let D˙ be the symmetric differentiation operator given by (4.1), (4.2) and ˙ ∗ ), φ = 1. (4.3), respectively. Assume, in addition, that φ ∈ Dom((D) Let H = DΘ , |Θ| = 1, be the (magnetic) Hamiltonian referred to in Theorem 5.1.

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Then lim |(eit/nH φ, φ)|2n = e−τ (Θ)|t|,

t ∈ R,

n→∞

where the decay constant τ (Θ) is given by ⎧ ⎪ |Θφ∞ (0−) + φ∞ (0+)|2 , ⎪ ⎪ ⎪ ⎨ 2 τ (Θ) = |Θφ () + φ (0)| ,  2 ⎪ ⎪  ⎪ φ∞ (0+) − kφ∞ (0−)  ⎪  ⎩Θφ () − √  , 1 − k2

(17.9)

in Case (i) in Case (ii)

(17.10)

in Case (iii).

Here 0 < k < 1 is the quantum gate coefficient from the boundary condition (4.3) that determines the symmetric operator D˙ in Case (iii). ˙ ∗ ) is a resonant state under In particular, the state φ ∈ L = Dom((D) continuous monitoring of the quantum unitary evolution φ → eitH φ if and only if φ∈ / Dom(H) = Dom(DΘ ). Otherwise, the state φ is a Zeno state. Proof. Part 1. First, we prove the assertion in the particular case of Θ = 1, where the Hamiltonian H is given by the differentiation operator D, i.e., H = D = DΘ |Θ=1 . ˙ ∗ ), by the von Neumann formula, the element φ Since φ ∈ Dom((D) admits a unique decomposition φ = αg+ + βg− + f,

(17.11)

˙ Here we choose the deficiency elements g± to where α, β ∈ C, f ∈ Dom(D). be given by (4.6), (4.7), and finally by (4.8) and (4.9) whenever the graph Y is in Cases (i), (ii), and (iii), respectively. Without loss of generality, we may assume that the operator H = D is already realized in its model representation in the Hilbert space L2 (R; dμ) as the operator of multiplication by independent variable with the measure μ(dλ) determined by (17.5). (z) associated with the Indeed, the Weyl-Titchmarsh function M(D,D) ˙ ˙ pair (D, D) and given by (7.1) admits the representation (17.4) with the measure μ(dλ) from (17.5). By Lemma 4.4, the symmetric differentiation

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operator D˙ is prime and therefore the Hamiltonian H = D is unitarily equivalent to its model representation in the Hilbert space L2 (R; dμ) by Theorem C.1 in Appendix C. By Lemma 6.1, g+ − g− ∈ Dom(H) = Dom(D). Therefore, one can also assume that the decomposition (17.11) takes place in the model Hilbert space L2 (R; dμ), where the deficiency elements g± are given by the partial fractions (see Remark C.2 in Appendix C) g± =

1 , λ∓i

λ ∈ R μ − a.e.,

and f ∈ L2 (R; (1 + λ2 )dμ(λ)).

(17.12)

The spectral measure (EH (dλ)φ, φ) of the element φ can be evaluated as follows 2    1 1  +β + f (λ) dμ(λ), (EH (δ)φ, φ) = α λ−i λ+i δ with δ ⊂ R a Borel set. Therefore,

dμ(λ) 2+1 λ δ  1 1 − + 2Re αβ dμ(λ) (λ − i)2 λ2 + 1 δ  1 1 +β f (λ)dμ(λ) α + 2Re λ−i λ+i δ 2 + |f (λ)| dμ(λ).

(EH (δ)φ, φ) = |α + β|

2

(17.13)

δ

It turns out that the first term in (17.13) determines the leading term of the asymptotics in the heavy-tailed distribution of the spectral measure (EH (dλ)φ, φ) whenever α + β = 0.

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Indeed, we have I =:

   1 2 dμ(s) 1   dμ(s) ≤ . −  (s − i)2  2 s +1 λ |s|>λ s2 + 1 |s|>λ

By Lemma 17.1, the following limit exists, dμ(s) lim λ λ and therefore I = O(λ−2 ) as λ → +∞. Next,     f (s)   II =:  dμ(s)  |s|>λ s ± i    ∞ 1 dμ(s) ≤ · (1 + s2 )|f (s)|2 dμ(s) 2 λ |s|>λ |s|>λ s + 1 = o(λ−3/2 )

as λ → +∞,

where we have used (17.12). Finally, III =: |f (s)|2 dμ(s) |s|>λ

1 ≤ 2 λ



|s|>λ

(1 + s2 )|f (s)|2 dμ(s) = o(λ−2 )

as λ → ∞.

Therefore, I + II + III = o(λ−3/2 ) as λ → +∞ and from (17.13) we obtain λ(EH ((λ, ∞))φ, φ) = |α + β|2 λ



∞

λ

dμ(s) + o(λ−1/2 ) as λ → ∞. s2 + 1

In a similar way one proves that λ(EH ((−∞, −λ))φ, φ) = |α + β|2 λ



−λ

−∞

dμ(s) + o(λ−1/2 ) as λ → ∞. s2 + 1

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By Lemma 17.1, lim λ

λ→+∞



∞

λ

λ→+∞

167

dμ(s) s2 + 1

= lim λ

page 167

−λ

−∞

⎧ ⎪ 1, in Case (i) ⎪ ⎪ ⎪ ⎪ ⎨  dμ(s) in Case (ii) = coth 2 , s2 + 1 ⎪ ⎪ ⎪ ⎪  +  ⎪ ⎩coth , in Case (iii). 2

Therefore, lim λ(EH ((λ, ∞))φ, φ)

λ→+∞

= lim λ(EH ((−∞, −λ))φ, φ) λ→+∞ ⎧ ⎪ 1, in Case (i) ⎪ ⎪ ⎪ ⎪ ⎨  1 in Case (ii) = |α + β|2 coth 2 , ⎪ π ⎪ ⎪ ⎪  +  ⎪ ⎩coth , in Case (iii). 2 On the other hand, from Lemma 17.3 it follows that ⎧ 2 |φ∞ (0−) + φ∞ (0+)| , in Case (i) ⎪ ⎪ ⎪ ⎪ ⎪  ⎨ 2 1 tanh |φ (0) + φ ()| , in Case (ii) |α + β|2 = 2 2⎪   ⎪ 2 ⎪ ⎪  +   φ∞ (0+) − kψ∞ (0−)  ⎪ ⎩tanh √ φ () −   , in Case (iii). 2  1 − k2 (17.14) Here, in Case (iii), 1  = log . k Hence lim λ(EH ((λ, ∞))φ, φ)

λ→+∞

= lim λ(EH ((−∞, −λ))φ, φ) λ→+∞ ⎧ 2 ⎪ |φ∞ (0−) + φ∞ (0+)| , in Case (i) ⎪ ⎪ ⎪ ⎪ 2 ⎨ in Case (ii) 1 |φ (0) + φ ()| , = 2π ⎪ 2  ⎪ ⎪  − kφ∞ (0−)  ⎪ ⎪φ () − φ∞ (0+) √ ⎩  , in Case (iii).  1 − k2

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To complete the proof of (17.10) in case Θ = 1 it remains to apply Theorem 15.12. Part 2. Now we can treat the general case of an arbitrary Θ, |Θ| = 1. Let H = DΘ be the magnetic Hamiltonian in Case (i). Denote by UΘ the unitary operator in L2 (Y) = L2 (R) defined as (UΘ f )(x) = Θχ(−∞,0) (x)f (x) + χ(0,∞) (x)f (x). One verifies that ∗ DΘ = UΘ DUΘ

and therefore ∗ ∗ lim |(eit/nH φ, φ)|2n = lim |(eit/nDΘ φ, φ)|2n = lim |(eit/nD UΘ φ, UΘ φ)|2n .

n→∞

n→∞

n→∞

By Part 1 of the proof,   ∗ ∗ ∗ ∗ lim |(eit/nD UΘ φ, UΘ φ)|2n = exp −|(UΘ φ∞ )(0+) + (UΘ φ∞ )(0−)|2 |t|

n→∞

  = exp −|φ∞ (0+) + Θφ∞ (0−)|2 |t| .

Therefore, τ (Θ) = |φ∞ (0+) + Θφ∞ (0−)|2 , which proves (17.10) in Case (i). In Cases (ii) and (iii), we have the commutation relation (cf. (5.6)) 

arg Θ ∗ I UΘ , DΘ = UΘ D +  where UΘ the unitary multiplication operator in L2 (Y) given by (UΘ φ(x)) = e−i

arg Θ x 

φ(x),

x ∈ Y.

Therefore, lim |(eit/nH φ, φ)|2n = lim |eit

n→∞

n→∞

arg Θ n

(eit/nD U ∗ φ, U ∗ φ)|2n

= lim |(eit/nD U ∗ φ, U ∗ φ)|2n . n→∞

Now the claim (17.10) follows by applying the result of Part 1 of the proof to the state U ∗ φ.

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To complete the proof it remains to show that under the hypothesis ˙ ∗ ) the decay constant τ (Θ) vanishes if and only if φ ∈ that φ ∈ Dom((D) Dom(DΘ ). Indeed, if φ ∈ Dom(DΘ ) ⊂ Dom(|DΘ |1/2 ), then φ is a Zeno state (by Proposition 15.3) and hence τ (Θ) = 0. One can also see right away that the boundary conditions (5.1), (5.2) and (5.3) imply τ (Θ) = 0. ˙ ∗ )) is also true. It The converse (under the hypothesis that φ ∈ Dom(D) is obvious in Cases (i) and (ii). In Case (iii) the equality τ (Θ) = 0 implies Θφ () =

φ∞ (0+) − kφ∞ (0−) √ 1 − k2

˙ ∗ ), from (4.4) it also follows that and since φ ∈ Dom((D)  φ∞ (0−) − k φ∞ (0+) − 1 − k 2 φ (0) = 0.

(17.15)

(17.16)

Multiplying (17.15) by k and using (17.16) we obtain kφ∞ (0+) − k 2 φ∞ (0−) √ 1 − k2 √ φ∞ (0−) − 1 − k 2 φ (0+) − k 2 φ∞ (0−) √ = 1 − k2  = 1 − k 2 φ∞ (0−) − φ (0),

kΘφ () =

which shows that φ (0) =

 1 − k 2 φ∞ (0−) − kΘφ ().

(17.17)

From (17.15) and (17.17) we get      √ φ∞ (0−) 1 − k2 Θ φ∞ (0+) k = √ φ (0) φ () 1 − k2 −kΘ and hence (5.3) holds proving that φ ∈ Dom(DΘ ).



 be the maximal dissipative differential operator defined by (8.1) Let D (with k = 0) whenever the graph Y is in Case (i) and by (8.4) and (8.5) whenever the graph Y is in Cases (ii) and (iii), respectively. Assume, in  ∪ Dom(D  ∗ ). addition, that the initial state φ is such that φ ∈ Dom(D) The following lemma shows that under these assumptions the decay rate of the state φ under continuous monitoring of the unitary evolution φ → eitH φ

(17.18)

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is determined by the state only and is, in fact, independent of the self˙ adjoint realization H = DΘ of the symmetric differentiation operator D.  be the maxLemma 17.5. Assume the hypothesis of Theorem 17.4. Let D imal dissipative differential operator defined by (8.1) (with k = 0) whenever the graph Y is in Case (i) and by (8.4) and (8.5) whenever the graph Y is in Cases (ii) and (iii), respectively. Then the decay constant τ = τ (Θ) given by (17.10) does not depend on Θ if and only if  ∪ Dom((D)  ∗ ). φ ∈ Dom(D) In this case, ⎧ |φ∞ (0−)|2 , ⎪ ⎪ ⎨ τ = |φ ()|2 , ⎪ ⎪ ⎩ 2 |φ ()| ,

in Case (i) in Case (ii)

 whenever φ ∈ Dom(D),

(17.19)

in Case (iii),

and

⎧ ⎪ |φ∞ (0+)|2 , ⎪ ⎪ ⎪ ⎪ ⎨ 2 τ = |φ (0)| , ⎪ ⎪ ⎪ ⎪ |φ (0+) − kφ∞ (0−)|2 ⎪ ⎩ ∞ , 1 − k2

in Case (i) in Case (ii)

 ∗ ). whenever φ ∈ Dom((D)

in Case (iii), (17.20)

Here, in Case (iii), k is the quantum gate coefficient. Proof. It is easily seen from (17.10) that the decay constant τ (Θ) does not depend on Θ if and only if either ⎧ φ∞ (0+) = 0, in Case (i) ⎪ ⎪ ⎨ (17.21) in Case (ii) φ (0) = 0, ⎪ ⎪ ⎩ φ∞ (0+) = kφ∞ (0−), in Case (iii), or

⎧ ⎪ ⎪φ∞ (0−) = 0, in Case (i) ⎨ ⎪ ⎪ ⎩

or both.

φ () = 0,

in Case (ii)

φ () = 0,

in Case (iii),

(17.22)

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Recall that the boundary conditions (8.1) (with k = 0), (8.4), and (8.5)  yield for the dissipative differentiation operator D

and

φ∞ (0+) = 0

(in Case (i))

φ (0) = 0

(in Case (ii))

(17.23)

⎧ ⎨φ∞ (0+) = kφ∞ (0−) ⎩φ (0) = √1 − k 2 φ∞ (0−)

(in Case (iii)).

(17.24)

Notice that in Case (iii) the first condition in (17.24) implies the second one ˙ ∗ ). Indeed, the membership φ ∈ Dom((D) ˙ ∗ ) means whenever φ ∈ Dom((D) that  φ∞ (0−) − kφ∞ (0+) − 1 − k 2 φ (0) = 0 by Lemma 4.2 and the claim follows by a simple computation. Now, it is straightforward to see that under the hypothesis that φ ∈ ˙ ∗ ) the boundary conditions (17.21) hold if and only if φ ∈ L = Dom((D)  Dom(D). In this case (17.19) follows from (17.10) in Theorem 17.4.  ∗ (9.13) (with Next, the boundary conditions for the adjoint operator (D) k = 0), (9.14), and (9.15) can be rewritten as

and

φ∞ (0−) = 0

(in Case (i))

φ () = 0

(in Case (ii))

(17.25)

⎧ √ ⎨φ∞ (0−) = kφ∞ (0+) + 1 − k 2 φ (0) ⎩φ ()

(in Case (iii)).

(17.26)

=0

By Lemma 4.2, the first condition in (17.26) simply means that φ ∈ ˙ ∗ ). Therefore, the boundary conditions (17.22) hold if and only Dom((D)  ∗ ), In this case (17.20) follows from (17.10) in Theoif and φ ∈ Dom(D rem 17.4.  If the initial state φ is taken from the somewhat narrower test space  ∪ Dom((D))  ∗ ⊂ Dom((D) ˙ ∗ ) = L, M = Dom(D) then Lemma 17.5 states that continuous monitoring of the unitary evolution (17.18) is universal (in the sense that the corresponding decay rate

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τ referred to in Lemma 17.5 is independent of the choice of the magnetic Hamiltonian H). In this case, i.e. if φ ∈ M, the universal exponent τ can also be recognized as the decay rate associated with continuous monitoring of the unitary evolution φ → eitH φ

(17.27)

in an extended Hilbert H containing L2 (Y) as a proper subspace. Here H = L2 (X), where X is the full metric graph: X = Y  Y if the metric graph Y in Cases (i) and (iii), and X can be identified with R if Y = (0, ) is in Case (ii), the Hamiltonian H is a self-adjoint dilation of the dissipative differentiation  and the new state φ ∈ L2 (X) of the extended quantum system operator D, is identified with the initial state φ being naturally imbedded to the space H = L2 (X). The precise statement is as follows. Corollary 17.6. Let H be a self-adjoint dilation in the Hilbert space H =  Assume that L2 (X) of the dissipative differentiation operator D.  ∪ Dom(D  ∗ ), φ ∈ M = Dom(D)

φ = 1.

Denote by φ a state in H such that φ = PL2 (Y) φ and

(I − PL2 (Y) )φ = 0,

where PL2 (Y) is the orthogonal projection from the Hilbert space L2 (X) onto its subspace L2 (Y). Then  φ)|  2n = e−τ |t| , lim |(eit/nH φ,

n→∞

t ∈ R,

(17.28)

 and by where the decay constant τ is given by (17.19) if φ ∈ Dom(D) ∗  (17.20) if φ ∈ Dom((D) ), respectively.  Proof. Suppose first that φ ∈ Dom(D). d on Denote by D the differentiation operator i dx Dom(D) =



W21 (e),

e⊂Y

where the sum is taken over all edges e of the graph Y.

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Integration by parts for φ ∈ Dom(D) yields ⎧ 2 2 ⎪ ⎪|φ∞ (0−)| − |φ∞ (0+)| ⎨

Im(Dφ, φ) =

1 |φ ()|2 − |φ (0)|2 2⎪ ⎪ ⎩ |φ∞ (0−)|2 − |φ∞ (0+)|2 + |φ ()|2 − |φ (0)|2

(17.29)

 taking into in Cases (i), (i) and (iii), respectively. Since φ ∈ Dom(D), account the boundary conditions (17.23) and (17.24), from (17.29) we obtain ⎧ |φ∞ (0−)|2 , in Case (i) ⎪ ⎪ ⎨ 1  φ) = Im(Dφ, φ) = Im(Dφ, in Case (ii) |φ ()|2 , 2⎪ ⎪ ⎩ |φ ()|2 , in Case (iii). Therefore,  φ) = τ, 2Im(Dφ, where τ is given by (17.19). By Lemma 15.15, 



lim |(eitD φ, φ)|2n = e−2Im(Dφ,φ)t ,

n→∞

t ≥ 0,

 φ ∈ Dom(D),

and therefore 

lim |(eit/nD φ, φ)|2n = e−τ t ,

n→∞

t ≥ 0,

 φ ∈ Dom(D),

(17.30)

where τ given by (17.19).  we have Since H dilates D,  φ)  = (eitD φ, φ), (eit/nH φ,

t ≥ 0,

(17.31)

and therefore 

 φ)|  2n = lim |(eit/nD φ, φ)|2n = e−τ t , lim |(eit/nH φ,

n→∞

n→∞

t ≥ 0.

(17.32)

 it remains to observe To complete the proof of (17.28) for φ ∈ Dom(D) it/nH   2 φ, φ)| is an even function in t. that the return probability p(t) = |(e  ∗ ). Next, suppose that φ ∈ Dom((D)

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As above, by Lemma 15.15, 



∗

lim |(e−it/n(D) φ, φ)|2n = e2Im((D)

∗

n→∞

φ,φ)t

,

t ≥ 0.

(17.33)

 ∗ ), one can use boundary conditions (17.25), (17.26) Since φ ∈ Dom((D) and the equality (17.29) to obtain that  ∗ φ, φ) = Im((−D)φ, φ) Im((−D) ⎧ 2 in Case (i) ⎪ ⎪|φ∞ (0+)| , 1⎨ 2 = in Case (ii) |φ (0)| , 2⎪ ⎪ ⎩ |φ∞ (0−)|2 − |φ∞ (0+)|2 − |φ (0)|2 , in Case (iii) . (In Case (iii), we took into account the second condition in (17.26) that φ () = 0). Now, using the first condition in (17.26), one computes    φ∞ (0+) − kφ∞ (0−) 2  ,  √ |φ∞ (0−)| − |φ∞ (0+)| − |φ (0)| =   1 − k2 2

2

2

which shows that ⎧ 2 ⎪ in Case (i) ⎪|φ∞ (0+)| , ⎪ ⎪ ⎪ ⎨ 2 in Case (ii) = τ,  ∗ φ, φ) = − |φ (0)| , 2Im((D) ⎪   ⎪ 2 ⎪ ⎪ φ (0+) − kφ∞ (0−)  ⎪ ⎩ ∞ √  , in Case (iii) 1 − k2 where τ is given by (17.20). Again, by Lemma 15.15, 



∗

lim |(e−it(D) φ, φ)|2n = e2Im((D)

n→∞

∗

φ,φ)t

,

t ≥ 0,

 ∗ ), φ ∈ Dom((D)

and therefore 

∗

lim |(e−it/n(D) φ, φ)|2n = e−τ t ,

n→∞

where τ given by (17.20).

t ≥ 0,

 ∗ ), φ ∈ Dom((D)

(17.34)

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 we obtain Since H dilates D, 

∗

lim |(e−it/n(D) φ, φ)|2n

n→∞





∗

= lim |(φ, (e−it/n(D) )∗ φ)|2n = lim |(eit/nD φ, φ)|2n n→∞

n→∞

= lim |(eit/nH φ, φ)|2n = e−τ t , n→∞

t > 0,

 ∗ ) with τ given by (17.20) and then, which proves (17.28) for φ ∈ Dom((D) by symmetry, for all t ∈ R.  ˙  ∩ Dom((D)  ∗ ) = Dom(D) Remark 17.7. We remark that if φ ∈ Dom(D) and therefore φ ∈ Dom(DΘ ) for all |Θ| = 1, then φ is a Zeno state under continuous monitoring of the dynamics φ → eitH φ for any self-adjoint realizations H = DΘ of the differentiation operator. Therefore, in this case the corresponding decay constant τ = τ (Θ) = 0 is Θ-independent for an obvious reason.  ∪ Also notice that under the requirement that φ ∈ M = Dom(D) ∗  Dom((D) ) the boundary data that determine the decay constant (17.19) and (17.20) can also be evaluated as follows. Assume, for instance, that the metric graph Y is in Case (iii) with its main vertex at the origin (μ = 0). Denote by X the full metric graph containing Y as its subgraph and let H be the self-adjoint dilation in the  in L2 (Y). extended Hilbert space L2 (X) of the dissipative operatorD  φ

Suppose that a two-component vector-function Ψ = φ↑↓ ∈ L2 (X) is a continuation of the function φ from the graph Y onto the full graph X, Ψ(x) = φ(x),

x ∈ Y,

(17.35)

such that Ψ ∈ Dom(H). Then the decay constant (17.19) can be evaluated via the second component of the vector-function Ψ as τ = |φ↓ ()|2 . Indeed, since Ψ ∈ Dom(H), the two-component vector-function Ψ(x) is continuous, so is its second component φ↓ (x). In particular, Ψ() = φ(), so that φ () = φ↓ ()

(17.36)

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and hence τ = |φ ()|2 = |φ↓ ()|2 . Moreover, for the decay constant (17.20) we have a similar expression τ = |φ↓ (0−)|2 .

(17.37)

Indeed, since Ψ ∈ Dom(H), by (14.5) we get      √ k − 1 − k2 φ↑ (0−) φ↑ (0+) = √ . 1 − k2 k φ↓ (0+) φ↓ (0−) In particular, φ↑ (0+) = kφ↑ (0−) −

 1 − k 2 φ↓ (0−).

Hence φ↓ (0−) =

kφ↑ (0−) − φ↑ (0+) kφ∞ (0−) − φ∞ (0+) √ √ = . 2 1−k 1 − k2

By (17.20),    kφ∞ (0−) − φ∞ (0+) 2   , √ τ =  1 − k2 which together with (17.38) proves (17.37).

(17.38)

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Chapter 18

PRELIMINARIES: PROBABILITIES VERSUS AMPLITUDES

To discuss applications of the continuous monitoring principle in connection with the exponential decay phenomenon in quantum mechanics, we need to warm up with some preliminaries. Recall that “when we deal with probabilities under ordinary circumstances, there are the following “rules of composition”: 1) if something can happen in alternative ways, we add the probabilities for each of the different ways: 2) if the event occurs as a succession of steps — or depends on a number of things happening —‘concomitantly’ (independently) — then we multiply the probabilities of each of the steps (or things)” [34, Ch. 3]. Apparently, under certain circumstances the probability P of an event that can be realized in two (at first glance mutually exclusive) alternative ways A1 and A2 is not necessarily equals the sum of probabilities P1 and P2 of the events A1 and A2 , that is, P = P1 + P2

(in general).

A more detailed analysis of the experimental data shows that the concept of an alternative should be analyzed more carefully and one has to distinguish between exclusive and interference alternatives. The latter occurs if there is no (experimental) evidence available to answer the question of how the final event has been realized, via the occurrence of A1 or A2 ? In other words, “when alternatives cannot possibly be resolved by any experiment, they always interfere” [35, page 14].

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If the alternative ways of a realization of the event are exclusive, one has the usual addition law of probabilities Pex = P1 + P2 .

(18.1)

In the case of an interference alternative, the rules of composition should be applied to the amplitudes of probability instead. Recall that there are complex numbers φ, φ1 , φ2 (the probability amplitudes), obtained, for example, by solving a kind of wave equation, such that Pint = |φ|2 ,

P1 = |φ1 |2

and P2 = |φ2 |2 .

In particular, the addition law for (probability) amplitudes φ = φ1 + φ2

(18.2)

yields Pint = |φ|2 = |φ1 + φ2 |2

(= Pex in general).

(18.3)

In this context it should be stressed that the (experimental) knowledge of the probabilities P1,2 (but not the amplitudes φ1,2 ) only gives the two sided-estimate for the probability Pint of the final event   P1 + P2 − 2 P1 P2 ≤ Pint ≤ min{1, P1 + P2 + 2 P1 P2 }. All of this is well known and has been extensively discussed in detail in connection with the two slit experiment (see, e.g., [34, 35, 44], also see [28, 70] for the concept of interaction-free measurements). Our goal is to provide a solid mathematical background for understanding the phenomenon on a simple one-dimensional example of a quantum system and develop a framework where the concepts of exclusive and interference alternatives can be rigorously discussed.

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Chapter 19

MASSLESS PARTICLES ON A RING Consider a quantum system the configuration space of which is a ring S obtained by identifying the end-points of a finite interval [0, ]. The dynamics of the system is described by the strongly continuous group of unitary operators U (t) = e−it/H , where the Hamiltonian H is given by the differentiation operator on the ring, or, equivalently, by the differentiation operator on the finite interval [0, ] with periodic boundary conditions. That is, d on Dom(H) = {f ∈ W21 ((0, )) | f (0) = f ()}. (19.1) dx To motivate the choice of the Hamiltonian we use the energy-momentum relation H = ic

E 2 = (cP )2 + (mc2 )2 and assume that we are dealing with a massless particle (m = 0) and then choose a square root brunch of (cP )2 to define the energy operator H as (cf. [130]) H = −cP. Here P denotes the momentum of a particle moving with no dispersion at the speed of light on the ring S in the direction from “x = 0 to x = .” In order to save ourselves from inventing new words such as “wavicles”, we have chosen to call these objects “‫גל‬-particles” (cf. [34, p. 85]). In our opinion, ‫גל‬-particles may, for instance, serve as a one-dimensional prototype of low energy electrons in the vicinity of an impurity in a zero-gap

179

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semiconductor. Recall that such electrons can formally be described by the two-dimensional Dirac-like Hamiltonian H = −ic σ · ∇ + V,

with

c = νF ,

where νF is the Fermi velocity, σ = (σx , σy ) are the 2×2 conventional Pauli matrices and V is a short range “defect” potential [20]. In this simplified model we will imagine these electrons as fake spin-zero electrons which, however, can carry the charge e. Suppose that φ ∈ L2 ((0, )), φ = 1, is a wave-function describing the initial state of the quantum system with the Hamiltonian H = ic

d dx

with periodic boundary conditions (19.1). If no observation is made whatsoever, the time evolution U (t)φ = e−it/H φ of the state φ is given by the family of unitary transformations  + ct), (U (t)φ)(x) = φ(x

x ∈ [0, ],

t ∈ R.

Here φ denotes the periodic extension of the function φ(x) from the interval [0, ] onto the full real axis. In other words, the wave packet U (t)φ is confined to move at the speed of light c without dispersion on the ring S of radius /2π, obtained from the interval [0, ] by identifying its end-points. In this case, the survival probability p(t) = |(e−it/H φ, φ)|2 to the initial state φ is a periodic function with the period T = /c. In the forthcoming chapters we will learn that under continuous monitoring the quantum system on the ring S becomes an open quantum system, the particles can be emitted and the whole system can be considered as a kind of quantum antenna.

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Chapter 20

CONTINUOUS MONITORING WITH INTERFERENCE Throughout this chapter we assume that the initial state φ is a W21 ((0, ))function that is allowed to have a discontinuity (jump) at the point of the observation x = 0 ≡ , that is, φ(0) = φ(),

in general.

Notice that although φ ∈ W21 ((0, )), the initial state φ is not required to belong to the domain of the Hamiltonian H in general. That is, it is not assumed that φ ∈ W21 (S), with S the ring obtained by identifying the end-points of the interval [0, ]. The decay properties of states with a unique jump-point on the ring under continuous monitoring are described by the following result. Theorem 20.1. Suppose that Y = (0, ) is a metric graph in Case (ii). In the Hilbert space H = L2 (Y) denote by H the differentiation operator H = ic

d dx

on

Dom(H) = {f ∈ W21 ((0, )) | f (0) = f ()}.

(20.1)

If φ ∈ W21 ((0, )) and φ = 1, then lim |(e−it/(n)H φ, φ)|2n = e−τ |t|,

n→∞

t ∈ R,

(20.2)

where τ = c|Δφ|2 = c|φ() − φ(0)|2 .

181

(20.3)

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182

d Proof. Let DΘ be the differentiation operator i dx on the finite interval [0, ] defined on

Dom(DΘ ) = {f ∈ W21 (0, ) | f (0) = −Θ()}. Since H = cDΘ , with Θ = −1, by Theorem 17.4 (see (17.10) in Case (ii) with Θ = −1) we have lim |(e−it/(n)H φ, φ)|2n = lim |(e−ict/nD−1 φ, φ)|2n

n→∞

n→∞

= e−|−φ()+φ(0)|

2

c|t|

= e−c|Δφ|

2

|t|

, 

which proves (20.2).

More generally, just repeating the proof presented above for an arbitrary Θ, |Θ| = 1, we have the following Corollary 20.2. Let H Φ , Φ ∈ R, be the self-adjoint realization of the differential expression H Φ = ic

d dx

on Dom(H Φ ) = {f ∈ W21 ((0, )) | f (0) = e−iΦ f ()}. Then, Φ

lim |(e−it/(n)H φ, φ)|2n = e−τΦ |t| ,

(20.4)

τΦ = c|ΔΦ φ|2 = c|e−iΦ φ() − φ(0)|2 .

(20.5)

n→∞

where

In particular, if e−iΦ φ() = φ(0), then φ is a resonant state under continuous monitoring of the unitary evolution Φ

φ → e−it/H φ governed by the Hamiltonian H Φ .

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Remark 20.3. Notice that the magnetic Hamiltonian H Φ is unitarily equivalent to the operator H + eA(x) with   e A(x)dx. Φ= c 0 Here e is the “charge” of the ‫גל‬-particle, A(x) is the magnetic potential (we   assume that A(x) is a piecewise real-valued continuous function), and 0 A(x)dx is the flux of the field through the ring. Indeed, denote by U the unitary multiplication operator    x e (U Ψ)(x) = exp i A(s)ds · Ψ(x), Ψ ∈ L2 ((0, )). c 0 Then U ∗ (H + eA(x))U = H Φ , which follows from the equality   d d + eA(x) [E(x) · Ψ(x)] = E(x) · ic Ψ(x), ic dx dx where

   x e E(x) = exp i A(s)ds , c 0

and the observation that

 Dom(H) = U (Dom(H Φ )) = U {f ∈ W21 ((0, )) | f (0) = e−iΦ f ()} .

Theorem 20.1 and Corollary 20.2 clearly suggest that continuous observation over a quantum system should rather be treated in the framework of open quantum systems theory. Below is a suitable model for that. Theorem 20.4. Given Φ ∈ [0, 2π), in the Hilbert space H = L2 ((0, )) ⊕ C

Φ defined on introduce the maximal dissipative operator H    f  Φ 1

)= Dom(H f ∈ W2 ((0, )), c = f (0) c  as

H

f Φ c

=i

d dx f (x) −iΦ

f (0) − e

f ()

Φ ), If φ ∈ H is a state such that φ ∈ Dom(H   |φ(x)|2 dx + |φ(0)|2 = 1, 0

.

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then Φ

lim |(e(it/n)H φ, φ)|2n = e−τΦ |t| ,

(20.6)

τΦ = |e−iΦ φ() − φ(0)|2 .

(20.7)

n→∞

where

Proof. We have  

Φ f, f ) = i (H f  (x)f (x)dx + i(f (0) − e−iΦ f ())f (0), 0

Φ ). f ∈ Dom(H

In particular,    

Φ f, f ) = 1 |f ()|2 − 1 |f (0)|2 + |f (0)|2 − Re e−iΦ f ()f (0) Im (H 2 2 1 −iΦ f () − f (0)|2 ≥ 0, = |e 2

Φ is a dissipative operator. To complete the proof it which shows that H remains to check that the lower half-plane belongs to the resolvent set of

Φ , so that H

Φ is a maximal dissipative operator, and then use the same H reasoning as the one in the proof of Lemma 15.15.



Remark 20.5. The idea to add to the original Hilbert space H a onedimensional “vacuum” subspace C is due to Schrader [123], also see [84], [103] and [126], where such extensions of a non-densely-defined symmetric operators found applications in modeling three-body systems with δ-like interactions that are free of the “fall to the center” phenomenon. For the general extension theory for non-densely-defined operators and its applications we also refer to [4, 61, 63, 102]. 20.1. Discussion The decay law (20.2) shows that continuous monitoring eventually triggers an exponential decay of the system. Meanwhile, the explicit expression (20.3) for the decay constant τ , τ = c|Δψ|2 , suggests that we are dealing with an interference alternative, which means that we cannot apply the laws of probabilities (18.1) and have to count on the composition laws of amplitudes (18.2). Indeed, a particle arriving at the junction point, the point of observation, has two options: a) either to stay on the track or b) be emitted. However, there is no way to “experimentally” confirm which option has been realized in reality. That is, we are not certain about what

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185

happened at the junction point x = 0 = , and consequently, the description of motion becomes an interference alternative. On the quantitative level, the reasoning presented above can be supported by the following considerations. The incoming φ(), outgoing φ(0) and emission amplitudes φem are to satisfy the “interference” relation φ() = φem + φ(0).

(20.8)

Since the quantity c|φem |2 t asymptotically describes the probability that the emitted particle can eventually be detected during the time interval [0, t],1 the probability P (t) of staying on the ring should fall off exponentially as P (t) = e−c|φem |

2

t

2

= e−c|Δφ| t ,

t ≥ 0,

(20.10)

1 This can be justified as follows. Suppose we put in an ideal detector that counts particles passing through the point x0 in the interval (0, ). If the initial state φ is a smooth function in a neighborhood of x0 , the probability px0 (t) that the detector will detect a particle during the time interval t is asymptotically given by

px0 (t) = c|φ(x0 )|2 t + o(t)

as

t → 0.

(20.9)

To justify the claim, recall that in accordance with the probabilistic interpretation of the wave-function, the probability to find the particle inside the interval δ ⊂ [0, ] is given by  Pr{“particle” ∈ δ} = In particular,

 Pr{“particle” ∈ [x0 − ε, x0 ]} =

δ

|φ(x)|2 dx.

x0

x0 −ε

|φ(x)|2 dx

= |φ(x0 )|2 ε + o(ε)

as

ε → 0.

If we repeat the experiment N∞ times, the quantity  x0 ΔN = N∞ |φ(x)|2 dx x0 −ε

gives the (average) number of outcomes when the (quantum) particle is accommodated by the interval [x0 − ε, x0 ]. One can change the point of view and assume that we are dealing with a beam of particles and that initially there were N∞ particles in the system. Therefore, ΔN would have the meaning of the averaged number of particles in the interval [x0 − ε, x0 ]. For the quantum system in question wave-particle duality is exact and hence we may assume that the particles are moving to the right with speed of light c. Therefore, in time ε t= c all the particles will leave the interval [x0 − ct, x0 ] and will eventually be counted by the detector.

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where |φem |2 = |Δφ|2 = |φ() − φ(0)|2 ,

(20.11)

which gives a heuristic justification of decay law (20.2) in Theorem 20.1. Notice that if φ ∈ Dom(H), then the wave function is continuous at the junction point, that is, φ() = φ(0).

(20.12)

Therefore, |φem |2 = 0 by (20.11) and hence there is no emission of particles. In this case, the dynamics is frozen by the continuous monitoring and we face the quantum Zeno effect. However, in the situation in question, in view of Corollary 20.2 and Remark 20.3 one can unfreeze the evolution (stopped by continuous monitoring) by switching on the magnetic field through the ring. Indeed, since the configuration space of the system (the ring S) is not a simply connected set, the effect of the magnetic potential will be to produce the phase shift of the wave function [2] at the junction point even if the magnetic field is absent in a neighborhood of the ring S (the Aharonov-Bohm effect) φ() → φ()e−iΦ , where Φ=

e c



 0

A(x)dx.

 Here e is the “charge” of the ‫גל‬-particle and 0 A(x)dx is the flux of the field through the ring. In this case, the interference relation (20.8) should be modified as e−iΦ φ() = φem + φ(0). Therefore, the decay properties of the state under continuous monitoring are determined by the quantity |ΔΦ φ|2 = |e−iΦ φ() − φ(0)|2 ,

(20.13)

Finally, the probability that the detector will go off within the time interval [0, t] is asymptotically given by  x0 ΔN = |φ(x)|2 dx = c|φ(x0 )|2 t + o(t) as t → 0, px0 (t) ∼ N∞ x0 −ct which justifies the claim (20.9).

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which finally leads to the decay law P (t) = e−|φem |

2

ct

= e−|ΔΦ φ|

2

ct

,

t ≥ 0,

for the probability P (t) to detect a particle remaining on the ring. In particular, under the assumption that |φ()| = |φ(0)|, it follows from (20.13) that the decrement |ΔΦ φ|2 experiences quite typical AharonovBohn oscillations which are periodic with respect to the flux of the field. That is, |ΔΦ φ|2 = |e−iΦ ei arg φ() − ei arg φ(0) |2 · |φ()|2   Φ − Δ arg φ 2 = 4 sin · |φ()|2 , 2

(20.14)

where Δ arg φ = arg φ() − arg φ(0). The above discussion provides a physically motivated example of a quantum system with the decay law (20.4), (20.5), see Corollary 20.2.

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Chapter 21

CONTINUOUS MONITORING WITH NO INTERFERENCE

Continuous monitoring of open quantum systems leads to a completely different understanding of decay processes. We will assume that the time evolution of the system is governed by a semi-group of contractive transformations generated by a dissipative differentiation operator such that the initial state belongs to the domain of the operator. Theorem 21.1. Suppose that Y = (0, ) is a metric graph in Case (ii).  κ the dissipative Given |κ| < 1, in the Hilbert space H = L2 (Y) denote by H differentiation operator  κ = ic d H dx on    κ ) = f ∈ W 1 ((0, )) | f (0) = κf () . Dom(H 2

(21.1)

 κ ) and φ = 1. Assume that φ ∈ Dom(H Then 

lim |(eit/(n)Hκ φ, φ)|2n = e−τ t ,

n→∞

t ≥ 0,

(21.2)

where τ = cΔ|φ|2 = c(|φ()|2 − |φ(0)|2 ).

189

(21.3)

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Proof. Integration by parts yields      2 2 φ (x)φ(x)dx = |φ()| − |φ(0)| − φ(x)φ (x)dx, 0

0

and therefore        0 φ (x)φ(x)dx + 0 φ (x)φ(x)dx Re φ (x)φ(x)dx = 2 0 =

|φ()|2 − |φ(0)|2 ≥0 2

 κ ). (for φ ∈ Dom(H

 κ ), as in the proof of Lemma 15.15 one obtains Since φ ∈ Dom(H 

lim |(eit/(n)Hκ φ, φ)|2n = e−2

−1

κ φ,φ)t Im(H

n→∞

with 2

−1

,

t ≥ 0,

      Im(Hκ φ, φ)t = 2c Im i φ (x)φ(x)dx 0

= c(|φ()|2 − |φ(0)|2 ) = τ.



One can go back from the reduced description of the open quantum system to the full one following the extended Hilbert space approach presented below. Consider an open quantum system prepared in the state φ ∈ L2 (S) = 2 L ((0, )) the time evolution of which generated by the dissipative operator  κ with the boundary condition parameter κ, |κ| < 1, referred to in H Theorem 21.1. Suppose that φ ∈ W21 ((0, )) is such that the radiation condition |φ()| > |φ(0)|

(21.4)

 κ ), that is, holds. Assume, in addition, that φ ∈ Dom(H κ=

φ(0) . φ()

Along with the open quantum system in the state space L2 ((0, )) introduce a new quantum system in an extended Hilbert space H containing L2 ((0, )) as a (proper) subspace. For the Hamiltonian H of the new system in the extended Hilbert space we choose a (minimal) self-adjoint dilation  κ and the new state of the system φ ∈ H is a of the dissipative operator H clone of φ considered as an element of the extended Hilbert space H.

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The results of continuous monitoring of the quantum evolution φ → e−it/H φ generated by the self-adjoint Hamiltonian H in the Hilbert space H can be described as follows. Corollary 21.2. Suppose that Y is the metric graph Y = (0, ) in Case (ii).  κ the dissipative Given |κ| < 1, in the Hilbert space H = L2 (Y) denote by H differentiation operator  κ = ic d H dx on    κ ) = f ∈ W 1 ((0, )) | f (0) = κf () . Dom(H 2

(21.5)

 κ in an Let H be a self-adjoint dilation of the dissipative operator H extended Hilbert H space containing the original Hilbert space H as a (proper) subspace H = L2 ((0, )) ⊂ H. Suppose that φ ∈ H,

φ = 1.

Then 

lim |(e−it/(n)H φ, φ)|2n = lim |(ei|t|/(n)Hκ φ, φ)|2n ,

n→∞

n→∞

t ∈ R,

provided that at least one (and therefore both) of the limits exist. In particular, if  κ ), φ ∈ Dom(H then lim |(e−it/(n)H φ, φ)|2n = e−τ t ,

n→∞

t > 0,

where the decay constant τ is given by τ = c(|φ()|2 − |φ(0)|2 ).

(21.6)

21.1. Discussion The decay law (21.2), (21.3) suggests that the interference effects are definitely absent. In order to get an adequate explanation for the phenomenon, instead of applying the composition law of amplitudes (20.8) one has to use the calculus of probabilities |φ()|2 = |φem |2 + |φ(0)|2 .

(21.7)

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Here is an argument supporting (21.7): a particle arriving at the junction point still has two options: a) either to stay on the track or b) be emitted. However, the transition from the open quantum system with state space H = L2 (S) referred to in Theorem (21.1) to the closed one in the extended Hilbert space H containing H as a proper subspace and discussed in Corollary 21.2 assumes that an additional scattering channel H  H is added and the emitted particles as well as the ones stayed on the track can eventually be counted. In other words, we are dealing with an exclusive alternative. From the experimental viewpoint in this case, the arrangement of the corresponding Gedankenexperiment involves the installation of two additional detectors D and D0 that count the particles that pass through the point x =  and x = 0. In other words, we accept the experimental condition that the emitted particles can eventually be counted (by combining the readings of the two detectors.) Given (21.7), arguing as in Chapter 19 we arrive to the exponential decay law P (t) = e−|φem |

2

ct

2

= e−cΔ|φ| t ,

t ≥ 0,

(21.8)

where P (t) stands for the probability to detect the particle on the ring at the moment of time t and |φem |2 = Δ|φ|2 = |φ()|2 − |φ(0)|2 .

(21.9)

Below we offer an heuristic explanation of the law (21.9) based on purely classical interpretation of the nature of a ‫גל‬-particle. When we watch the beam of ‫גל‬-particles by observing the readings of the two detectors, we indeed deal with an exclusive alternative. Denote by N∞ (t) the total amount of particles on the ring at the moment of time t and let N be the amount of particles that passed through the point x =  and arrived at the check point x = 0 during the time interval [0, t]. The arrived particles “have” the alternative: either to keep moving on the ring or to be emitted. By checking the readings of the detector D0 we know that N0 out of N∞ particles stayed traveling along the ring. Next, taking into account the readings of the detector D , we conclude that the remaining ΔN = N − N0 particles have beed emitted during the time interval [0, t]. (Here we implicitly assume that there is no other mechanism that causes the particles to radiate. Why this hypothesis is consistent with the way of the suggested reasoning will be explained later.)

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Given the wave-function probabilistic interpretation above, it is easy to see that N N0 = c|φ(−)|2 t + o(t) and = c|φ(0+)|2 t + o(t) as t → 0. N∞ N∞ Therefore,

ΔN = |φ()|2 − |φ(0)|2 ct + o(t). N∞ Repeating that monitoring over and over, in the limit t → 0, we arrive at the differential equation

dN = −c |φ()|2 − |φ(0)|2 N, dt N (0) = N∞ , that governs the counting process. Therefore, under continuous monitoring with detectors that are going off, the total number of particles N (t) as a function of time falls off exponentially as 2

N (t) = N∞ e−cΔ|φ| t ,

(21.10)

where the decrement Δ|φ|2 > 0 is given by (21.9) (provided that |φ| has a jump at the point x = 0). Notice that if the state φ is a continuous function on the ring, then no emission is observed and then the quantum Zeno effect takes place. Summarizing, we arrive at the conclusion that the computation of the emission probability for the quantum system (H, φ) referred to in Theorem 20.1 requires the application of the composition law of amplitudes (20.8) (the interference alternative scenario). Meanwhile, the decay rate  κ , φ) referred to in Theorem 21.1 or for for the open quantum system (H  the quantum system (H, φ) in the extended Hilbert space H (see Corollary 23.3) can be evaluated using the rules of the calculus of probabilities (21.7) (the exclusive alternative scenario).

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Chapter 22

THE SELF-ADJOINT DILATION  κ referred to in The self-adjoint dilation H of the dissipative operator H Corollary 21.2 can be described explicitly as the differentiation operator on the metric graph Y with appropriate boundary conditions. However, the geometry of the metric graph that determines the configuration space of the quantum system depends on whether or not the parameter κ in the boundary condition (21.1) vanishes. If κ = 0, as it follows from Theorem 5.7, the extended Hilbert space can be chosen to coincide with H = L2 (Y) = L2 (Y) where Y is the metric graph Y = (−∞, 0)  (0, )  (0, ∞) in Case (iii) and Y denotes its one-cycle completion, the configuration space of the extended quantum system, while the self-adjoint dilation H, the Hamiltonian, coincides with the differentiation operator on the graph H = ic

d dx

(22.1)

defined on the domain of functions satisfying the boundary conditions       κ f∞ (0+) f∞ (0−) |κ| − 1 − |κ|2 |κ| =  . (22.2) f (0) f () 1 − |κ|2 κ In the exceptional case κ = 0, the boundary condition (21.1) that deter 0 is local. Therefore, it is convenient to mines the dissipative operator H assume that the configuration space for the corresponding open quantum system (if κ = 0) is the finite interval (0, ) rather than a ring. Consequently, in this case the extended Hilbert space may be chosen as H = L2 (Y) 195

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where Y is the metric graph Y = (−∞, 0)  (0, )  (, ∞) in Case (i). The self-adjoint dilation (the Hamiltonian) H of the dissipative operator  0 can be chosen to be the differentiation operator on the graph H H = ic

d dx

(22.3)

with the self-adjoint boundary condition f (+) = Θf (−),

|Θ| = 1,

(22.4)

with an arbitrary choice of the unimodular extension parameter Θ. Notice that in the limit κ → 0 along the ray κ = −|κ|Θ the boundary conditions (22.2) split as f∞ (0+) = Θf () and f∞ (0−) = f (0).

(22.5)

In view of (22.5), the one-cycle graph Y “unwinds” to a straight line Y = (−∞, 0)  (0, )  (, ∞) which can naturally be identified with the real axis. One can show that  −|κ|Θ the corresponding self-adjoint dilations of the dissipative operators H approach (in the strong resolvent sense) the operator (22.3) with the boundary condition (22.4). Notice that emission amplitude φem from (21.9) can be evaluated by directly solving the Schr¨odinger equation i

∂ Ψ = HΨ ∂t

(22.6)

on the one-cycle graph Y. Indeed, representing the initial state φ in the extended Hilbert space H = L2 (Y) as the two-component vector function     φ∞ 0 φ = = , φ ∈ L2 (Y),  φ φ

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denoted by Ψ(t, x), x ∈ Y the solution of the Schr¨ odinger equation (22.6) with the initial condition   0 Ψ|t=0 = φ = . φ We claim that emission amplitude φem can be evaluated as the limiting value of the first component Ψ∞ (t, 0+) of the solution Ψ as t → 0. Indeed, since       κ f∞ (0+) 0 |κ| − 1 − |κ|2 |κ| lim lim Ψ(t, ε) = =  , t↓0 ε↓0 f (0) φ() 1 − |κ|2 κ we have  κ φ(). φem = Ψ∞ (0+, 0+) = − 1 − |κ|2 |κ| Therefore, the probability density of the event of emission of a particle is given by |φem |2 = (1 − |κ|2 )|φ()|2 = |φ()|2 − |φ(0)|2 = Δ|φ|2 , which agrees with (21.2), (21.3) (cf. Remark 17.7). Here we have used that the boundary condition φ(0) = κφ() holds, that is,  κ ). φ ∈ Dom(H Notice, that in the exceptional case, that is, if the initial state is such  0 associated with the correspondthat φ(0) = 0, the dissipative operator H ing open quantum system has the domain  0 ) = {f ∈ W21 ((0, )) | f (0) = 0}. Dom(H In this case, the differentiation operator H on the whole real axis, H = ic

d dx

on Dom(H) = W21 ((−∞, ∞)),

 0 . Therefore, the configuration space of the extended quantum dilates H system is just the real axis Y = R, not the one-cycle graph Y. The corresponding solution Ψ(x, t) of the Schr¨ odinger equation (22.6) with the initial

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data Ψ|t=0 = φ (here we do not distinguish the function φ on [0, ] and its extension by zero on the whole real axis R) is a one-component function given by Ψ(x, t) = φ(x − ct). In this case the emission amplitude φem can be evaluated as φem = Φ(x − ct)|x=, t=0 = φ(), so that again |φem |2 = |φ()|2 = |φ()|2 − |φ(0)|2 = Δ|φ|2 . It is also worth mentioning that the decrement Δ|φ|2 given by (21.9) and referred to in Theorem 21.1 is gauge invariant while |Δφ|2 defined in (20.11) is not. Indeed, if (V f )(x) = eiλ(x) f (x),

x ∈ (0, ),

(22.7)

is a (unitary) gauge transformation, where λ(x) is a differentiable function on [0, ], then Δ|V φ|2 = Δ|φ|2 and |ΔV φ|2 = |eiΦ φ() − φ(0)|2 = |Δφ|2

(in general).

Here,  Φ = λ() − λ(0) =



0

d λ(x)dx. dx

is a shift of the relative phase of the wave function. As we have already pointed out, if the wave function is continuous at the junction point, that is φ(0) = φ(), then Δ|φ|2 = |Δφ|2 = 0.

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In this case the quantum Zeno effect takes place regardless of whether both of the detectors D0 and D go off or only one of them does. Moreover, if V is a gauge transformation, we also have that Δ|V φ|2 = Δ|φ|2 = 0 that shows that the Zeno effect is stable with respect to the gauge transformations when both of the detectors D0 and D go off. The situation is quite different in the experiment when only one of the detectors goes off. A tiny variation of the phase of the wave functions can easily transfer the system from the quantum Zeno mode to the exponential decay regime with the decay rate given by the “magnetic” decrement |ΔΦ φ|2 = |e−iΦ φ() − φ(0)|2 . We remark that (|φ()| − |φ(0)|)2 ≤ |ΔΦ φ|2 ≤ (|φ()| + |φ(0)|)2 . Moreover, by changing the “flux” Φ, the upper and as well as the lower bound for the magnetic decrement can easily be attained. In particular, 0 ≤ |ΔΦ φ|2 ≤ 4|φ(0)|2 , whenever the continuity condition (20.12) at the junction point holds.

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Chapter 23

GENERAL OPEN QUANTUM SYSTEMS ON A RING  κ , φ) on a ring Notice that while considering open quantum systems (H referred to in Theorem 21.1, we assumed  κ ). φ ∈ Dom(H This requirement can be relaxed and we arrive at the following more general result. Theorem 23.1. Suppose that φ ∈ W21 ((0, )) is a state, φ = 1. Given |κ| ≤ 1, in the Hilbert space H = L2 (Y), Y = (0, ), denote by  κ = ic d H dx the differentiation operator with the boundary condition f (0) = κf (), that is,    κ ) = f ∈ W21 ((0, )) | f (0) = κf () . Dom(H Then 

lim |(eit/nHκ φ, φ)|2n = e−τ t ,

t > 0,

(23.1)

τ = |φ(0) − κφ()|2 + (1 − |κ|2 )|φ()|2 .

(23.2)

n→∞

where

201

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Proof. Without loss we will assume that we work in the system of units where c = 1 and  = 1. It is sufficient to prove the asymtotic representation 1  Re(eitHκ φ, φ) = 1 − τ t + o(t) 2

as

t ↓ 0.

(23.3)



Denote by W (t) = eitHκ the contractive semi-group generated by the  κ . Notice that operator H W (t + ) = κW (t), with

t ≥ 0,

⎧ ⎨κφ( + x − t), 0 < x < t

(W (t)φ(x) =

⎩φ(x − t),

x 0 and P is a rank-one orthogonal projection, P = (·, g)g,

g = 1.

 Denote by A the real part of A,  = 1 (A  + (A)  ∗ ), A = Re(A) 2

 = Dom((A)  ∗ ), Dom(A) = Dom(A)

so that  = A + itP. A  is described in the following lemma. The resolvent of A 221

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 be a maximal dissipative operator with a Lemma A.1 (cf. [124]). Let A rank-one imaginary part,  = A + itP, A A = A∗ , P = (·, g)g, g = 1, and t > 0. Denote by M (z) = ((A − zI)−1 g, g),

z ∈ ρ(A),

 of the operator A  and the M -function associated with the real part Re(A) the unit vector g. Then the following resolvent formula  − zI)−1 = (A − zI)−1 − p(z)(A − zI)−1 P (A − zI)−1 , (A  ∩ ρ(A), z ∈ ρ(A)

(A.2)

holds, where p(z) =

1 M (z) +

1 it

.

In particular, M (z) . 1 + itM (z)

(A.3)

 ⊂ {z : 0 ≤ Im(z) ≤ t}. spec(A)

(A.4)

 − zI)−1 g, g) = ((A Moreover,

Proof. To prove the resolvent formula (A.2), one observes that  − zI)−1 = (A − zI)−1 − it(A  − zI)−1 P (A − zI)−1 , (A and hence  − zI)−1 g = (A − zI)−1 g − it((A − zI)−1 g, g)(A  − zI)−1 g, (A which yields the representation  − zI)−1 g = (1 + itM (z))−1 (A − zI)−1 g. (A

(A.5)

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Substituting this equality back to (A.5), one obtains  − zI)−1 = (A − zI)−1 − (A

it (A − zI)−1 P (A − zI)−1 , 1 + itM (z)

which proves (A.2).  coincides with Now, it is easy to see that the non-real spectrum of A those z that satisfy the equation 1 + M (z) = 0, Im(z) = 0. it To complete the proof, we use the inequality |M (z)| ≤

1 , Im(z)

z ∈ C+ .

 which proves (A.4). Therefore, {z : Im(z) > t} ⊂ ρ(A),



 be a maximal dissipative operator with a rank-one Theorem A.2. Let A imaginary part  = A + itP, A  is completely nonA = A∗ , P = (·, g)g, g = 1, and t > 0. Then A self-adjoint if and only if the element g is generating for the self-adjoint  operator A = Re(A). Proof. Introduce the subspace, Hg = spanδ∈B(R) {EA (δ)g},

with B(R) the Borel σ-algebra.

 is completely non-self-adjoint. Assume Only If Part. Suppose that A  that g is not a generating element for the self-adjoint operator A = Re(A). Then the orthogonal complement Hg reduces the real part A and since the element g is orthogonal to Hg⊥ of H, one obtains that  = Ah Ah

for

h ∈ Hg⊥ .

 as well. Furthermore, the Since Hg reduces A, the subspace Hg⊥ reduces A  on Hg⊥ is a self-adjoint operator. Therefore, A  is not completely part of A non-self-adjoint. We get a contradiction. If Part. Suppose that g is a generating element for the self-adjoint oper Assume that A  is not completely non-self-adjoint and let ator A = Re(A).  in this subspace is a H0 be its reducing subspace such that the part of A self-adjoint operator.

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Given 0 = h ∈ H0 , using the resolvent formula (A.2), one obtains that  − iyI)−1 h, h) = ((A − ityI)−1 h, h) ((A − |y| > t,

((A − ityI)−1 h, g)((−ityI)−1 g, h) , 1 it + M (iy) y ∈ R.

Therefore, f1 (y) = f2 (y) −

f3 (y) , f4 (y)

|y| > t,

(A.6)

where the functions fk , k = 1, 2, 3, 4 are given by  − iyI)−1 h, h), f1 (y) = ((A f2 (y) = ((A − ityI)−1 h, h), f3 (y) = ((A − ityI)−1 h, g)((A − ityI)−1 g, h), f4 (y) =

1 + M (iy). it

One observes that fk (y) = fk (−y),

y ∈ R,

|y| > t,

k = 1, 2, 3.

(A.7)

However, f4 (y) = f4 (−y)

(A.8)

for f4 (−y) =

(−1) 1 1 + M (−iy) = + M (iy) = + M (iy) = f4 (y). it it it

Inequality (A.8) together with (A.7) is inconsistent with (A.6), provided that f3 (y) = 0. Finally, the fact that the function f3 (y) is not identically zero easily follows from the assumption that the element g is generating for A. The obtained contradiction shows that there is no reducing subspace  in this subspace is self-adjoint. Therefore, A  is such that the part of A completely non-self-adjoint.   is a maximal dissipative operator Theorem A.3 (cf. [14]). Assume that A  with a rank-one imaginary part, A = A + itP, A = A∗ , P = (·, g)g, g = 1,  is completely non-self-adjoint. and t > 0. Assume, in addition, that A

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225

Let μ(dλ) be the probability measure from the representation  dμ(λ) −1 , z ∈ C+ . ((A − zI) g, g) = R λ−z  is unitarily equivalent to the operator B of the form Then the operator A  )(λ) = λf (λ) + it(f, 1)1(λ), (Bf     2  Dom(B) = f ∈ L (R; dμ)  λ2 |f (λ)|2 dμ(λ) < ∞ , R

in the Hilbert space L2 (R; dμ), where 1(λ) = 1

for μ-a.e. λ ∈ R.

(A.9)

 is completely non-self-adjoint and therefore, by Proof. By hypothesis A Theorem A.2, the real part A has simple spectrum and the vector g is generating for A. It follows that the spectral measures ν(dλ) = (EA (dλ)g, g) and μ(dλ) = (EB (dλ)1, 1), where B is the self-adjoint operator of multiplication by independent variable in L2 (R; dμ) and 1(λ) is given by (A.9), coincide. Since B has simple spectrum and 1(λ) is a generating vector for the self-adjoint operator B, the Spectral Theorem for self-adjoint operators yields the existence of a unitary operator U : H → L2 (R; dμ) such that UAU −1 = B

and Ug = 1.

Hence  −1 = B,  U AU which completes the proof.



 with a rank-one imagiGiven a non-self-adjoint dissipative operator A ∗  = A + itP, A = A , P = (·, g)g, g = 1, and t > 0, denote nary part, A  following [77] by S(z) the characteristic function of operator A    ∗ − zI)−1 g, g , z ∈ C+ . (A.10) S(z) = 1 + 2it ((A)

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 be a maximal dissipative operator with a rank-one Lemma A.4. Let A  = A + itP, A = A∗ , P = (·, g)g, g = 1, and t > 0. imaginary part, A  admits the representation Then the characteristic function S(z) of A S(z) =

1 + itM (z) , 1 − itM (z)

z ∈ C+ ,

 = A given by where M (z) is the M -function of Re(A) M (z) = ((A − zI)−1 g, g),

z ∈ ρ(A).

Proof. Introduce the function  − zI)−1 g, g) = ((A + itP − zI)−1 g, g), Mt (z) = ((A

z∈

⎧ ⎨C− , if t > 0 ⎩C+ ,

if t < 0

.

From the resolvent formula (A.2) one gets that M (z) , Mt (z) = 1 + itM (z)

z∈

⎧ ⎨C− , if t > 0 ⎩C+ ,

if t < 0

,

and hence S(z) = 1 + 2itM−t (z) =

1 + itM (z) , 1 − itM (z)

z ∈ C+ , 

completing the proof.

Theorem A.5 ([77]). The characteristic function of a completely nonself-adjoint operator with a rank-one imaginary part uniquely determines the operator up to unitary equivalence.  is a completely non-self-adjoint operator with a Proof. Suppose that A rank-one imaginary part so that  = A + it(·, g)g A for some g = 1, A = A∗ and t > 0. In view of Theorem A.3, it suffices to show that the characteristic function uniquely determines the parameter t and the (probability) measure

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μ from the representation for the M -function (in the case in question, μ(R) = g2 = 1)  dμ(λ) , z ∈ C+ . (A.11) M (z) = ((A − zI)−1 g, g) = R λ−z Indeed, by Lemma A.4, S(z) =

1 − itz −1 + o(z −1 ) 2it 1 + itM (z) = =1− + o(z −1 ), 1 − itM (z) 1 + itz −1 + o(z −1 ) z

z → ∞,

and hence i lim z(S(z) − 1) = t, 2 z→∞ which uniquely determines the perturbation parameter t. Since M (z) =

1 S(z) − 1 · , it S(z) + 1

the knowledge of the characteristic function S(z) also uniquely determines the probability measure μ(dλ) in (A.11) by the Stieltjes inversion formula. 

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Appendix B

PRIME SYMMETRIC OPERATORS Recall that a densely defined linear operator A˙ in a Hilbert space H is called symmetric if ˙ y) = (x, Ay) ˙ ˙ (Ax, for all x, y ∈ Dom(A). Definition B.1. A symmetric operator A˙ is called a prime operator if there does not exist a (non-trivial) subspace invariant under A˙ such that the restriction of A˙ on this subspace is self-adjoint. If a symmetric operator is not prime, it is useful to separate its selfadjoint part from its prime part. Theorem B.2. Let A˙ be a closed symmetric operator with equal deficiency indices in a Hilbert space H. Then the Hilbert space splits into the orthogonal sum of two subspaces H = K ⊕ L,

(B.1)

where ˙ ∗ − zI) K = spanIm(z)=0 Ker((A) and L=



Ran(A˙ − zI).

Im(z)=0

˙ Both of the subspaces K and L reduce the symmetric operator A. ˙ ˙ Moreover, the part A|L of A in L is a self-adjoint operator and the part ˙ K of A˙ in K is a prime symmetric operator. A| 229

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Proof. Since by hypothesis the operator A˙ is a closed symmetric operator, Ran(A˙ − zI), Im(z) = 0, is a closed subspace and hence L being the intersection of closed subspaces is a closed subspace itself. Assume that h ∈ L and hence h ∈ Ran(A˙ − zI) for all z ∈ C \ R.

(B.2)

Since ˙ ∗ − zI) ⊕ Ran(A˙ − zI), H = Ker((A)

z ∈ C \ R,

(B.3)

from (B.2) and (B.3) one concludes that h is orthogonal to the subspace ˙ ∗ − zI) for any z ∈ C \ R. Hence h is orthogonal to K, which means Ker((A) that L ⊂ K⊥ .

(B.4)

Now, assume that an element h is orthogonal to K. ˙ ∗ − zI) is a dense subset Since the linear set D = spanz∈C\R Ker((A) ˙ ∗ − zI) for any z ∈ C \ R. in K, the element h is orthogonal to Ker((A) Therefore, by (B.3), h ∈ Ran(A˙ − zI) for all z ∈ C \ R and hence h ∈ L which means that K⊥ ⊂ L.

(B.5)

Combining (B.4) and (B.5) completes the proof of (B.1). To prove the remaining assertion of the theorem, we show first that the ˙ subspace L is A-invariant. ˙ and hence Indeed, assume that f ∈ L ∩ Dom(A) f ∈ Ran(A˙ − zI) for all z ∈ C \ R.

(B.6)

Then from (B.6) follows that ˙ = (A˙ − zI)f + zf ∈ Ran(A˙ − zI) for all z ∈ C \ R, Af ˙ ∈ L by the definition of the space L. and hence Af Next, we will show that ˙ = L for any z ∈ C \ R. (A˙ − zI)(L ∩ Dom(A))

(B.7)

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To see this, take an f ∈ L. Then f ∈ Ran(A˙ − zI) for any z ∈ C \ R, ˙ such and therefore, for any z ∈ C \ R there exists an element gz ∈ Dom(A) that f = (A˙ − zI)gz . We claim that ˙ ∗ − zI) for all z ∈ C \ R. gz ⊥ Ker((A) Indeed, let A be a(ny) self-adjoint extension of A˙ (recall that A˙ has equal deficiency indices and therefore A˙ admits self-adjoint extensions). Then f = (A˙ − zI)gz = (A − zI)gz and hence gz = (A − zI)−1 f,

z ∈ C \ R.

(B.8)

Fix a z with Im(z) = 0. For any element fζ such that ˙ ∗ − ζI), fζ ∈ Ker((A)

ζ ∈ C \ R,

with ζ = z, one gets that (gz , fζ ) = ((A − zI)−1 f, fζ ) = (f, (A − zI)−1 fζ ) =

 1  (f, (A − ζI)(A − zI)−1 fζ ) − (f, fζ ) . z−ζ

(B.9)

By assumption f ∈ L and hence f ⊥ K by the first part of the proof. Since ˙ ∗ − zI) ⊂ K, (A − ζI)(A − zI)−1 fζ ∈ Ker((A) ˙ ∗ − ζI) ⊂ K, fζ ∈ Ker((A) and f ⊥ K, from (B.9) follows that (gz , fζ ) = 0, i.e., ˙ ∗ − ζI), gz ⊥ Ker((A)

ζ = z,

Im(ζ) = 0.

(B.10)

It remains to show that ˙ ∗ − zI). gz ⊥ Ker((A) ˙ ∗ − zI) we have that Indeed, for any fz ∈ Ker((A) ˙ ∗ − ζI). (A − zI)(A − ζI)−1 fz ∈ Ker((A)

(B.11)

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Therefore, by (B.10), (gz , (A − zI)(A − ζI)−1 fz ) = 0,

ζ = z,

Im(ζ) = 0.

Hence, (gz , fz ) = lim (gz , (A − zI)(A − ζI)−1 fz ) = 0, ζ→z

˙ ∗ − zI), for all fz ∈ Ker((A)

which proves (B.11) From (B.1) follows that gz ∈ L which justifies (B.7) for gz was chosen ˙ to be an element of Dom(A). Denote by B the restriction of A˙ on ˙ Dom(B) = L ∩ Dom(A). Our next claim is that Dom(B) is dense in L. Indeed, let f ∈ L and f ⊥ Dom(B), that is, (f, g) = 0

for all g ∈ Dom(B).

From (B.7) follows that f ∈ Ran(B − iI) and hence f = (B − iI)h for some h ∈ Dom(B). Thus, for all g ∈ Dom(B) one obtains that (f, g) = ((B − iI)h, g) = (h, (B + iI)g) = 0.

(B.12)

On the other hand, (B.7) yields Ran(B + iI) = L and therefore from (B.12) follows that h = 0 and hence f = (B − iI)h = 0. So, we have shown that the operator B is a densely defined symmetric operator in the Hilbert space L such that Ran(B ± iI) = L which means that B is self-adjoint. ˙ K0 is Now suppose that a subspace K0 reduces A˙ and that the part A| self-adjoint. Then ˙ K0 − zIK0 ) = K0 , Ran(A|

z ∈ C \ R,

which means that K0 ⊂ L, proving that the part A˙ in the subspace K is a prime symmetric operator. The proof is complete.  ˙ K is called Remark B.3. In the situation of Theorem B.2 the operator A| ˙ the prime part of the symmetric operator A.

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Recall that an element h ∈ H is said to be a generating element for a self-adjoint operator H in the Hilbert space H if spanIm(z)=0 {(H − zI)−1 h} = H. We also say that a self-adjoint operator has simple spectrum if it has a generating element. Corollary B.4 (cf. [135]). Let A˙ be a closed symmetric operator with equal deficiency indices in a Hilbert space H. Then A˙ is a prime operator if and only if ˙ ∗ − zI). H = spanIm(z)=0 Ker((A) If, in addition, A˙ has deficiency indices (1, 1), then A˙ is a prime operator if and only if for any self-adjoint extension A of A˙ a deficiency element ˙ ∗ − iI) is generating, that is, 0 = g+ ∈ Ker((A) H = spanIm(z)=0 (A − zI)−1 g+ .

(B.13)

In particular, in this case, any self-adjoint extension of A˙ has simple spectrum. Proof. The first assertion has already been proven in Theorem B.2. To prove the remaining statement, one proceeds as follows. Suppose that (B.13) fails to hold and therefore the orthogonality condition ((A − zI)−1 g+ , h) = 0

for all z ∈ C \ R

(B.14)

holds for some element h ∈ H, h = 0. Since ((A − zI)−1 g+ , h) = ((A − zI)−1 g+ , (A + iI)(A + iI)−1 h) = ((A − iI)(A − zI)−1 g+ , (A + iI)−1 h), one concludes that ((A − iI)(A − zI)−1 g+ , g) = 0

for all z ∈ C \ R,

where g = (A + iI)−1 h. Observing that ˙ ∗ − zI), (A − iI)(A − zI)−1 g+ ∈ Ker((A)

(B.15)

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the hypothesis that A˙ has deficiency indices (1, 1) yields the orthogonality condition ˙ ∗ − zI), g ⊥ spanIm(z)=0 Ker((A) and therefore A˙ is not a prime operator by Theorem B.2. Conversely, suppose that A˙ is not a prime operator and therefore (B.15) holds for some 0 = g ∈ H. In particular, (g+ , g) = 0.

(B.16)

Using the first resolvent identity, (A − zI)−1 − (A − iI)−1 = (z − i)(A − iI)−1 (A − zI)−1 , one obtains (A − iI)(A − zI)−1 = I + (z − i)(A − zI)−1 . Therefore, ((A − iI)(A − zI)−1 g+ , g) = (g+ , g) + ((z − i)(A − zI)−1 g+ , g) for all z ∈ C \ R. Using (B.15) and (B.16), this equality yields ((A − zI)−1 g+ , g) = 0

for all z ∈ C \ R,

z = i,

and therefore, by continuity, ((A − zI)−1 g+ , g) = 0

for all z ∈ C \ R,

which shows that H = spanIm(z)=0 (A − zI)−1 g+ . So, we have shown that A˙ is a prime operator if and only if (B.13) holds. The proof is complete.  We will also need a variant of the first part of this corollary in case when A˙ has deficiency indices (0, 1) or (1, 0). Lemma B.5. Let A˙ be a symmetric operator with deficiency indices (0, 1) in a Hilbert space H. Then A˙ is a prime operator if and only if ˙ ∗ − zI). H = spanIm(z) 0,

and hence f = 0 by the uniqueness theorem for the Laplace transform (see, e.g., [22, Th. 5.5]. Therefore, (B.17) holds which completes the proof. “If ” Part. Suppose that A˙ is not a prime operator. Therefore, by Theorem B.2, ˙ ∗ − zI). H = spanIm(z)=0 Ker((A) Since A˙ has deficiency indices (0, 1), ˙ ∗ − zI) = spanIm(z) 0.

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Lemma E.3. Let A˙ be a symmetric operator with deficiency indices (1, 1) and A its self-adjoint extension. Suppose that f (z) = az + b with a, b ∈ R, a > 0 is an affine transformation. ˙ f (A)) admits Then the Livˇsic function associated with the pair (f (A), the representation s(f (A),f ˙ (A)) (z) =

m(z) − m(i) m(z) − m(i)

,

(E.6)

where m(z) = M (f −1 (z)) (z) is the Weyl-Titchmarsh function associated with and M (z) = M(A,A) ˙ ˙ the pair (A, A). Proof. By Remark 2.5, without loss of generality one may assume that A˙ is a prime symmetric operator. Let M (z) = M(A,A) (z) be the Weyl˙ ˙ A). Next, we may assume Titchmarsh function associated with the pair (A, that A is the multiplication operator by independent variable in L2 (R, dμ) and A˙ is its restriction on      ˙ Dom(A) = f ∈ Dom(A)  f (λ)dμ(λ) = 0 , R

where μ(dλ) is the representing measure for M (z) (see Theorem C.1). Introduce the family of functions Gz (λ) =

1 , λ − f −1 (z)

Im(z) = 0.

Clearly, ˙ ∗ ) − zI), Gz ∈ Ker((f (A)

Im(z) = 0.

Set G+ = Gf −1 (i)

and G− = Gf −1 (−i) .

One easily checks that G+  = G− ,

˙ ∓ iI), G± ∈ Ker(f (A)

and that G+ − G− ∈ Dom(A) = Dom(f (A)).

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253

Therefore, the Livˇsic function s(f (A),f ˙ (A)) (z) has the representation s(f (A),f ˙ (A)) (z) =

z − i (Gz , G− ) . z + i (Gz , G+ )

(E.7)

We have  (Gz , G− ) =

dμ(λ)

R

(λ −

f −1 (z))(λ

− f −1 (−i))

and  (Gz , G+ ) =

R

dμ(λ) (λ − f −1 (z))(λ − f −1 (i))

,

where μ(dλ) is the measure from the representation    1 λ M (z) = − dμ(λ), Im(z) = 0. λ − z 1 + λ2 R Therefore, 

dμ(λ) −1 (z))(λ − f −1 (i)) (λ − f R

z−i = −1 M (f −1 (z)) − M (f −1 (i)) −1 f (i) − f (z)

= −a M (f −1 (z)) − M (f −1 (i)) (E.8)

(z − i)(Gz , G− ) = (z − i)

and 

dμ(λ) −1 (z))(λ − f −1 (−i)) (λ − f R

z+i = −1 M (f −1 (z)) − M (f −1 (−i)) −1 f (−i) − f (z)

(E.9) = −a M (f −1 (z)) − M (f −1 (−i)) .

(z + i)(Gz , G+ ) = (z + i)

Now (E.7), (E.8) and (E.9) yield s(f (A),f ˙ (A)) (z) =

m(z) − m(i) M (f −1 (z)) − M (f −1 (i)) = , M (f −1 (z)) − M (f −1 (−i)) m(z) − m(i)

which completes the proof.



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Next, we discuss the transformation law under the affine transformation ˙ A) given by of the pair (A, ˙ A) −→ (−A, ˙ −A). (A, Lemma E.4. If A˙ is a closed symmetric operator with deficiency indices (1, 1) and A its self-adjoint extension, then the Weyl-Titchmarsh functions ˙ ±A) are related as follows M± (z) associated with the pairs (±A, M− (z) = −M+ (−z),

z ∈ ρ(A).

(E.10)

˙ A) In particular, for the Liˇsic functions associated with the pairs (A, ˙ −A) we have and (−A, s(−A,−A) (z) = s(A,A) (−z), ˙ ˙

z ∈ C+ .

(E.11)

˙ ∗ − iI) and Proof. Let n be a unit vector in Ker((A) m = (A − iI)(A + iI)−1 n.

(E.12)

˙ ∗ − iI). By the definition of the ˙ ∗ + iI) = Ker((−A) Then m ∈ Ker((A) Weyl-Titchmarsh function one obtains that

M− (z) = (−Az + I)(−A − zI)−1 m, m . Therefore, −M− (−z) = ((Az + I)(A − zI)−1 m, m) = ((Az + I)(A − zI)−1 (A − iI)(A + iI)−1 n, (A − iI)(A + iI)−1 n) = ((A − iI)(A + iI)−1 (Az + I)(A − zI)−1 n, (A − iI)(A + iI)−1 n) = ((Az + I)(A − zI)−1 n, n) = M+ (z) = M+ (z).

(E.13)

Here we have used the Schwarz symmetry principle for the Weyl-Titchmarsh function M+ (z) = M+ (z),

z ∈ ρ(A),

and the observation that the Cayley transform (A − iI)(A + iI)−1 is a unitary operator commuting with the operator A. Finally, (E.10) follows from (E.13) by the substitution z → −z.

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To prove the last assertion, we use the relation (2.4) to conclude that s(−A,−A) (z) = ˙ completing the proof.

M− (z) − i −M+ (−z) − i (−z), = s(A,A) = ˙ M− (z) + i −M+ (−z) + i 

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Appendix F

THE INVARIANCE PRINCIPLE

The main goal of this chapter is to establish an invariance principle for the characteristic function of a triple of operators under linear transformations of the operators from the triple. Introduce the class D(H) of maximal dissipative unbounded densely  (A  = (A)  ∗ ), in the Hilbert space H such that defined operators A,  A˙ = A|  ∗ ) Dom(A)∩Dom( A is a densely defined symmetric operator with deficiency indices (1, 1). In this case,  ⊂ (A) ˙ ∗ A˙ ⊂ A  is automatically a quasi-selfadjoint extension of A˙ (see, and therefore A e.g., [86]). If f (z) is the affine transformation f (z) = az + b, introduce the triple f (A) as ˙ f (A),  f (A)). f (A) = (f (A), ˙ A,  A) be a triple such that A  ∈ D(H). Suppose Theorem F.1. Let A = (A, that f (z) = az + b with a, b ∈ R, a > 0, is an affine transformation. Let M (z) be the Weyl-Titchmarsh function associated with the pair ˙ A). Then the von Neumann parameters κ and κ  of the triples A and (A, f (A) are related as 1+κ m − κ m = , 1−κ i(1 − κ  ) where m = M (f −1 (i)). 257

(F.1)

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Moreover, the characteristic functions Sf (A) (z) and SA (z) are related as Sf (A) (f (z)) = Θf SA (z),

z ∈ C+ ,

(F.2)

where  Θf =

1−κ 1−κ

−1

1 − κ . 1 − κ

·

is a unimodular factor. In particular, Θf continuously depends on f . Proof. As in the proof of Lemma E.3, from the very beginning one can assume that A˙ is a prime symmetric operator. Let μ(dλ) denote the representing measure for the Weyl-Titchmarsh function M (z). Without loss of generality (see Theorem C.1) one may assume that A is the multiplication operator by independent variable in L2 (R, dμ) and A˙ coincides with its restriction on     ˙ Dom(A) = h ∈ Dom(A)  h(λ)dμ(λ) = 0 . R

In this case, from Theorem D.2 (see (D.4)), we know that the functions g+ (λ) =

1 λ−i

and

g− (λ) =

1 λ+i

form a basis in the deficiency subspace, ˙ ∗ ∓ iI), g± ∈ Ker((A)

g±  = 1.

From (C.9) is also follows that g+ − g− ∈ Dom(A).

(F.3)

Clearly, the functions G± (λ) =

1 λ − f −1 (±i)

have the properties ˙ ∗ ) ∓ I), G± ∈ Ker((f (A)

G+  = G− ,

and G+ − G− ∈ Dom(A) = Dom(f (A)).

(F.4)

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From the definition of the von Neumann parameters κ, κ  ∈ D for the triples A and f (A) it follows that  g+ − κg− ∈ Dom(A)

(F.5)

 = Dom(A).  G+ − κ  G− ∈ Dom(f (A))

(F.6)

and

Introduce the function m(z) = M (f −1 (z)),

Im(z) = 0.

(F.7)

In order to establish the relationship (F.1) between the von Neumann parameters, notice that

g+ + g− g+ − g− ˙ + ∈ Dom(f (A)), (F.8) G± − m(±i) 2i 2 that is,

   g+ (λ) − g− (λ) g+ (λ) + g− (λ) + G± (λ) − m(±i) dμ(λ) = 0. 2i 2 R Indeed, since 1 g+ (λ) − g− (λ) = 2 , 2i λ +1

g+ (λ) + g− (λ) λ = 2 , 2 λ +1

and G± (λ) =

1 λ−

f −1 (±i)

,

one needs to verify the equality

   M (f −1 (±i)) 1 λ − + dμ(λ) = 0, λ − f −1 (±i) λ2 + 1 λ2 + 1 R which simply follows from the observations that  M (f −1 (±i)) dμ(λ) = M (f −1 (±i)) λ2 + 1 R and

  R

1 λ − f −1 (±i)

−

λ λ2 + 1



dμ(λ) = M (f −1 (±i)).

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Combining (F.6) and (F.8) we get that     g+ + g− g+ + g− g+ − g− g+ − g−  + + h= m −κ m 2i 2 2i 2 =

i − iκ  − m + κ  m m − κ  m + i − iκ   g+ + g− ∈ Dom(A). 2i 2i

Therefore, in view of (F.5), κ=

m − i − κ  (m − i) m − κ  m − i + iκ  =   m − κ m + i − iκ m + i − κ  (m + i)

(F.9)

and (F.1) follows. From (F.5) and (F.6) it follows that the characteristic function SA (z) ˙ A,  A) (see (2.6)) can be evaluated as associated with the triple A = (A, SA (z) =

(z) − κ s(A,A) ˙ κs(A,A) (z) − 1 ˙

.

Representing s(A,A) (z) via the Weyl-Titchmarsh function M (z), ˙ s(A,A) (z) = ˙

M (z) − i , M (z) + i

one concludes that SA (z) =

M(z)−i M(z)+i − κ . κ M(z)−i − 1 M(z)+i

Therefore, taking into account (F.7), one obtains SA (f −1 (z)) =

m(z)−i m(z)+i − κ . κ m(z)−i m(z)+i − 1

(F.10)

In a similar way, using that G+ − G− ∈ Dom(f (A))

 and G+ − κG+ ∈ Dom(f (A)),

one also gets Sf (A) (z) =

 s(f (A),f ˙ (A)) (z) − κ

κ  s(f (A),f ˙ (A)) (z) − 1

By Lemma E.3, s(f (A),f ˙ (A)) (z) =

m(z) − m , m(z) − m

.

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so that Sf (A) (z) =

m(z)−m  m(z)−m − κ . κ  m(z)−m m(z)−m − 1

(F.11)

From (F.10) one gets that SA (f

−1

(z)) =

=

m(z)−i m(z)+i − κ κ m(z)−i m(z)+i − 1

=

m(z) − i − κ(m(z) + i) κ(m(z) − i) − (m(z) + i)

1+κ (1 − κ)m(z) − i(1 + κ) 1 − κ m(z) − i 1−κ . = · 1+κ (κ − 1)m(z) − i(1 + κ) κ − 1 m(z) + i 1−κ

(F.12)

A similar computation for the right hand side of (F.11) yields Sf (A) (z) =

m(z)−m  m(z)−m − κ κ  m(z)−m m(z)−m − 1

=

m(z) − m − κ  (m(z) − m) κ  (m(z) − m) − (m(z) − m)

(1 − κ  )m(z) − (m − κ  m) 1 − κ  m(z) − =  =  ·  (κ − 1)m(z) + (m − κ m) κ − 1 m(z) + =

1+κ 1 − κ  m(z) − i 1−κ · , 1+κ κ  − 1 m(z) + i 1−κ

m−κ  m 1−κ  m−κ  m κ  −1

(F.13)

where we used the relation (F.1) on the last step. Comparing (F.12) and (F.13), we obtain  −1 1−κ 1 − κ Sf (A) (f (z)) = · · SA (z), 1−κ 1 − κ 

which proves (F.2).

We conclude this chapter by establishing an invariance principle under the anti-holomorphic transformation (involution) of the triple ˙ −(A)  ∗ , −A). ˙ A,  A) −→ −A∗ = (−A, A = (A, Theorem F.2. Let A˙ be a densely defined, closed symmetric operator with  quasi-selfadjoint deficiency indices (1, 1), A its self-adjoint extension and A ˙ dissipative extension of A. Then the characteristic functions associated with the triples A = ˙ −(A)  ∗ , −A) are related as follows ˙ A,  A) and −A∗ = (−A, (A, S−A∗ (z) = SA (−z).

(F.14)

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˙ Proof. Let g± be normalized deficiency elements of A, ˙ ∗ ∓ I), g± ∈ Ker((A) such that g+ − g− ∈ Dom(A)

 and g+ − κg− ∈ Dom(A).

Clearly, ˙ ∗ ± I), g± ∈ Ker((−A) g− − g+ ∈ Dom(A) = Dom(−A), and  ∗ ) = Dom((−A)  ∗ ). g− − κg+ ∈ Dom((A) Hence, using Lemma E.4, one obtains S−A∗ (z) =

(z) − κ s(−A,−A) ˙ κs(−A,−A) (z) − 1 ˙

=

 A)(−z) − κ s(A, κs(A,A) (−z) − 1 ˙

= SA (−z). The proof is complete.



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Appendix G

THE OPERATOR COUPLING AND THE MULTIPLICATION THEOREM Introduce the class D0 (H) of maximal dissipative densely defined operators  in the Hilbert space H of the form A  = A + itP, A  is a self-adjoint operator, t > 0, and P is a rank-one where A = Re(A) orthogonal projection [4, 15, 77]. Introduce the concept of an operator coupling of two operators from the classes D0 (H1 ) and D0 (H2 ). 1 ∈ D0 (H1 ) and A 2 ∈ D0 (H2 ) are maxiDefinition G.1. Suppose that A mal dissipative operators acting in the Hilbert spaces H1 and H2 , respectively.  from the class D0 (H1 ⊕ We say that a maximal dissipative operator A   H2 ) is an operator coupling of A1 and A2 , in writing, =A 1  A 2 , A  (A 1 ⊕ A 2 ) is a rank-one operator, the Hilbert space H1 is invariant for if A−  and the restriction of A  on H1 coincides with the dissipative operator A,  A1 . That is, 1 )  ∩ H1 = Dom(A Dom(A) and   A| 1 ) = A1 . H1 ∩Dom(A 263

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= A 1  A 2 be an operator Theorem G.2 (cf. [4, 14, 15, 82]). Let A k ∈ D0 (Hk ), k = 1, 2. coupling of two maximal dissipative operators A 2 coincides 1  A Then the characteristic function of an operator coupling A   with the product of the ones of A1 and A2 , SA1 A2 (z) = SA1 (z) · SA2 (z),

z ∈ C+ .

(G.1)

Proof. Suppose that k = Ak + i(·, gk )gk , A

k = 1, 2,

A∗k ,

where Ak = gk ∈ Hk . Denote by Pk (k = 1, 2) the orthogonal projections onto the subspaces Hk , respectively. From the definition of an operator coupling it follows that =A 1 P1 + A 2 P2 + (·, g)g A for some g,  g ∈ H1 ⊕ H2 and that  1=A 1 P1 . AP In particular, 2 )∗ P2  ∗ P2 = (A (A) and therefore g ∈ H2

and g ∈ H1 .

(G.2)

First we show that  = (·, φ)φ, Im(A) where φ = (Θ1 g1 ) ⊕ (Θ2 g2 )

(G.3)

for some |Θk | = 1, k = 1, 2. Indeed, we have (·, g1 )g1 + (·, g2 )g2 +

1 ((·, g)g − (·, g) g ) = (·, φ)φ. 2i

Introducing φk = Pk φ,

k = 1, 2,

from (G.2) and (G.4) it follows that |(gk , gk )| = |(gk , φ)| = |(gk , φk )|, and then we get (G.3).

k = 1, 2,

(G.4)

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265

Rewrite the equality (G.4) one more time (·, g1 )g1 + (·, g2 )g2 +

1 ((·, g)g − (·, g) g) 2i

= (·, (Θ1 g1 ) ⊕ (Θ2 g2 ))(Θ1 g1 ) ⊕ (Θ2 g2 ). We get 1 ((·, g)g − (·, g) g ) = Θ1 Θ2 (·, g1 )g2 + Θ1 Θ2 (·, g2 )g1 2i and therefore (·,  g)g = 2i(·, φ2 )φ1 . In particular, we have that  = (A1 + i(·, φ1 )φ1 )P1 + (A2 + i(·, φ2 )φ2 )P2 + 2i(·, φ2 )φ1 , A  = (·, (φ1 + φ2 ))(φ1 + φ2 ) Im(A) and we arrive at the definition of an operator coupling as presented in [16, eq. (2.1)]. Literally repeating step by step the proof of the Multiplication Theorem [15, Theorem 2.1] one justifies (G.1).  Remark G.3. We remark that an operator coupling of two dissipative operators from the classes D0 (H1 ) and D0 (H2 ) is not unique. In fact, we 2 of two dissipative operators 1  A have shown that an operator coupling A k = Ak + i(·, gk )gk , A

k = 1, 2,

is necessarily of the form 1  A 2 = (A1 + i(·, g1 )g1 )P1 + (A2 + i(·, g2 )g2 )P2 + 2iΘ(·, g2)g1 , (G.5) A for some |Θ|=1. Moreover, for any choice of Θ such that |Θ| = 1 the right hand side of (G.5) meets the requirements to be an operator coupling of 1 and A 2 . A Recall that the class D(H) consists of all maximal dissipative unbounded  (A  = (A)  ∗ ), in the Hilbert space H such that densely defined operators A,  A˙ = A|   ∗) Dom(A)∩Dom(( A) is a densely defined symmetric operator with deficiency indices (1, 1) (see Appendix F).

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1 ∈ D(H1 ) and A 2 ∈ D(H2 ). We Definition G.4 ([87]). Suppose that A 1 and A 2 , in writing,  ∈ D(H1 ⊕H2 ) is an operator coupling of A say that a A =A 1  A 2 , A if  and the restriction of A  on H1 (i) the Hilbert space H1 is invariant for A  coincides with the dissipative operator A1 , that is,  ∩ H1 = Dom(A 1 ), Dom(A)   A| 1 ) = A1 , H1 ∩Dom(A and  (ii) the symmetric operator A˙ = A|   ∗ ) has the property Dom(A)∩Dom(( A) 1 ⊕ (A 2 )∗ . A˙ ⊂ A The corresponding multiplication theorem for the class D(H) can be formulated as follows (see [87, Theorem 6.1, cf. Theorem 5.4]). 2 is an operator coupling of two =A 1  A Theorem G.5. Suppose that A ˙ A˙ 1 and  maximal dissipative operators Ak ∈ D(Hk ), k = 1, 2. Denote by A, ˙ A2 the corresponding symmetric operators with deficiency indices (1, 1), respectively. That is,  A˙ = A|   ∗) Dom(A)∩Dom(( A) and k | A˙ k = A k )∩Dom((A k )∗ ) , Dom(A

k = 1, 2.

Then there exist self-adjoint reference operators A, A1 , and A2 , extend˙ A˙ 1 and A˙ 2 , respectively, such that ing A, S(A, ˙ A 1 A 2 ,A) (z) = S(A˙ 1 ,A 1 ,A1 ) (z) · S(A˙ 2 ,A 2 ,A2 ) (z),

z ∈ C+ .

(G.6)

 of A 1 and A 2 , the multiplication Moreover, for any operator coupling A rule  =κ 1 ) · κ 2 ) κ (A) (A (A

(G.7)

holds. Here κ (·) stands for the absolute value of the von Neumann parameter of a dissipative operator defined by (2.16).

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267

Corollary G.6. Assume the hypotheses of Theorem G.5. Then the von Neumann logarithmic potential ΓA(z) (see Definition 2.4) is an additive functional in the sense that ΓA1 A2 (z) = ΓA1 (z) + ΓA2 (z),

z ∈ ρA1 ∩ ρA2 ∩ ρA1 A2 ∩ C+ .

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Appendix H

STABLE LAWS Recall (see, e.g., [33, 48, 141]) that a distribution G (of a random variable) is said to be stable, if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. That is, for any b1 , b2 > 0, there exist a b > 0 and a ∈ R such that       x x x−a G G =G , b1 b2 b where  denotes the convolution of distributions (see [33, Ch. V.4]). It turns out that a (non-degenerated) law G is stable if and only if the logarithm of its characteristic function has the representation [141, Theorem B.2]    t (H.1) log g(t) = σ itγ − |t|α 1 − iβ ω(t, α) |t| for some σ > 0, −∞ < γ < ∞, 0 < α ≤ 2, −1 ≤ β ≤ 1,

(the index of stability) (the skew parameter).

(H.2)

Here

⎧ π ⎪ α , α = 1, ⎨tan 2 ω(t, α) = ⎪ ⎩− 2 log |t|, α = 1. π The skew parameter β is irrelevant when α = 2.

269

(H.3)

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Recall (see, e.g., [141]) that a distribution F is said to belong to the domain of attraction of a law if there are constants An and Bn > 0 such that the following non-zero limit n

lim log [f (t/Bn )] eiAn t

n→∞

exists, where f (t) is the characteristic function of the distribution F ,

f (t) = eitx dF (x). R

In this case the limit coincides with the logarithm of a stable law (H.1) for an appropriate choice of the parameters α, β, γ and σ. Recall that a positive function h(x), defined for x ≥ 0, is said to be slowly varying if, for all t > 0, lim

x→∞

h(tx) = 1. h(x)

Also, by the Karamata theorem (see, e.g., [48, Appendix 1] for an exposition of the Karamata theory), a slowly varying function h which is integrable on any finite interval can be represented in the form  x ε(t) dt , x0 > 0, h(x) = c(x) exp t x0 where lim c(x) = c = 0

x→∞

and

lim ε(x) = 0.

x→∞

A key result in this area is the following Gnedenko-Kolmogorov limit theorem. Theorem H.1 ([48, Theorem 2.6.1]). A distribution F belongs to the domain of attraction of a stable law (H.1) with exponent α, 0 < α ≤ 2, and parameters σ, β and γ if and only if c1 + o(1) h(x), x > 0 xα c2 + o(1) F (x) = h(−x), x < 0 (−x)α

1 − F (x) =

(H.4) (H.5)

as |x| → ∞, where c1 , c2 ≥ 0, c1 + c2 > 0 and h is slowly varying in the sense of Karamata.

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271

In this case, σ = (c1 + c2 )d(α), where

  ⎧ ⎪ ⎨Γ(1 − α) cos 1 πα , 2 d(α) = ⎪ ⎩π , 2

(H.6)

α = 1

,

(H.7)

α=1

and β=

c1 − c2 . c1 + c2

(H.8)

Remark H.2. The (tauberian type) relationship between the set of data (c1 , c2 , h) and (α, β, γ, σ) referred to in Theorem H.1 (also see (H.2)) can be described as follows: if a distribution F belongs to the domain of attraction of the stable law (H.1), that is,   t n iAn t α = σ itγ − |t| + iβ ω(t, α) lim log [f (t/Bn )] e n→∞ |t| for some constants An and Bn > 0, then (see, e.g., [48, Theorem 2.6.5])   t α log f (t) = i˜ γ t − σ|t| h(1/t) 1 − iβ ω(t, α) (1 + o(1)) as t → 0, |t| where γ˜ is in general not necessarily the same as γ. Recall that in this case the norming constants Bn necessarily satisfy the relation lim nBn−α h(Bn ) = 1.

n→∞

If in the hypothesis of Theorem H.1 the slowly varying function h(x) has the property that lim h(x) = 1, then the scaling factors Bn can be x→∞ given by Bn = n1/α . Under this hypothesis, the probability distribution F is said to belong to the domain of normal attraction of a stable law. In particular, every stable law belongs to the normal of its own normal attraction.

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page 283

INDEX

‫גל‬-particles, 179

contractive semi-group, 202 convergence in distribution, 212 convolution of distributions, 269

affine transformation of the pair, 254 Aharonov-Bohm effect, 186 Aharonov-Bohn oscillations, 187 anti-Zeno state, 124

decay constant, 164 deficiency elements, 13 Dirac-like Hamiltonian, 180 directed metric graph, 29 dissipative canonical pair, 100 domain of attraction of a law, 270 domain of attraction of an α-stable law, 124 domain of normal attraction of a stable law, 271

Blaschke product, 25 bond scattering matrix, 50 Breit-Wigner-Cauchy-Lorentz shape, 133 canonical commutation relations (CCR), 3 canonical commutation relations in the Weyl form, 3 canonical dissipative triple, 100 Cauchy distribution, 139 Cayley transform, 254 character-automorphic function, 25 characteristic function, 14 coincide in distribution, 211 collapse in a three-body system, 41 completely non-self-adjoint operator, 211 composition law of amplitudes, 191 compression of the dissipative dynamics, 113 configuration space, 207

emission amplitude, 198 Euler characteristic, 19 Euler characteristic χ(Y) of the graph Y, 30 exclusive alternative, 177 Exponential Decay, 121 extension of the Stone-von Neumann uniqueness theorem, 109 Fermi velocity, 180 first Betti number, 30 flux of the magnetic field, 42 Friedrichs extension, 152 full metric graph, 115

283

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functional model of a prime dissipative triple, 237 Gaussian distribution, 139 Gnedenko-Kolmogorov limit theorem, 205, 270 Herglotz-Nevanlinna function, 25 Herglotz-Nevanlinna representation, 246 Hermite-Tchebyscheff polynomials, 128 Holtsmark distribution, 151 hyperbolic system, 115 incoming bonds, 29 index of stability, 269 inner-outer factorization, 55 interference alternative, 177 invariance principle for the characteristic function, 257 Jørgensen-Muhly problem, 4 Karamata theory, 270 Krein formula for resolvents, 245 Landau distribution, 129 Laplace transform, 235 L´evy distribution, 139 light tails requirement, 123 limit distribution universality, 218 Livˇsic function, 13 local gauge transformation, 35 “magnetic” decrement, 199 magnetic differentiation operator, 41 magnetic Hamiltonian, 42 (magnetic) potential, 36 Mandelstam-Tamm time-energy uncertainty relation, 131 M -function, 222 minimal self-adjoint dilation, 46 model triple, 16 model triple on Y, 71 momentum operator, 99

multiplication rule, 266 multiplication theorem, 266 mutually unitary equivalence, 239 non-linear time-scale, 138 one-cycle graph, 196 one-sided stable (L´evy) distribution, 138 operator coupling, 263 operator stable distribution, 215 outer function, 55 outgoing bonds, 29 Pauli matrices, 180 Poisson kernel, 56 position operator, 99 prime part of the symmetric operator, 232 prime symmetric operator, 33, 229 probability amplitudes, 178 prolonged frequent quantum measurements, 130 quantum Anti-Zeno effect, 121 quantum gate coefficient, 65 quantum Kirchhoff rule, 31, 67 quantum oscillator (pendulum), 129 quantum Zeno effect, 121 quasi-selfadjoint extension, 43 radiation condition, 190 rank-one self-adjoint operator, 248 resonant state under continuous monitoring, 125 restricted Weyl commutation relations, 97 Riemann zeta-function, 128 Schwarz reflection principle, 246 self-adjoint dilation, 195 self-adjoint momentum operator, 116 semi-Weyl commutation relations, 35 singular inner function, 55 skew parameter, 269 slowly varying function, 270

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Index

stable distribution, 215, 269 Stieltjes transform, 220 strong law of large numbers, 123 survival probability, 121 transmission coefficient, 84 Volterra operator, 218 von Neumann’s (logarithmic) potential, 18

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von Neumann’s parameter of the triple, 14 wave packet, 180 wave-function, 180 Weyl canonical triple, 101 Weyl-Titchmarsh function, 13 Zeno state, 122