Non-Selfadjoint Operators in Quantum Physics E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 697

Veröffentlichungsjahr: 2015

Beschreibung

A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses the recent emergence of unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis with potentially significant physical consequences. In addition to prompting a discussion on the role of mathematical methods in the contemporary development of quantum physics, the book features: * Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area * An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and perturbation theory * Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics and condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

CONTRIBUTORS

PREFACE

ACRONYMS

GLOSSARY

SYMBOLS

INTRODUCTION

References

CHAPTER 1: NON-SELF-ADJOINT OPERATORS IN QUANTUM PHYSICS: IDEAS, PEOPLE, AND TRENDS

1.1 THE CHALLENGE OF NON-HERMITICITY IN QUANTUM PHYSICS

1.2 A PERIODIZATION OF THE RECENT HISTORY OF STUDY OF NON-SELF-ADJOINT OPERATORS IN QUANTUM PHYSICS

1.3 MAIN MESSAGE: NEW CLASSES OF QUANTUM BOUND STATES

1.4 PROBABILISTIC INTERPRETATION OF THE NEW MODELS

1.5 INNOVATIONS IN MATHEMATICAL PHYSICS

1.6 SCYLLA OF NONLOCALITY OR CHARYBDIS OF NONUNITARITY?

1.7 TRENDS

REFERENCES

CHAPTER 2: OPERATORS OF THE QUANTUM HARMONIC OSCILLATOR AND ITS RELATIVES

2.1 INTRODUCING TO UNBOUNDED HILBERT SPACE OPERATORS

2.2 COMMUTATION RELATIONS

2.3 THE

q

–OSCILLATORS

2.4 BACK TO “HERMICITY”—A WAY TO SEE IT

CONCLUDING REMARKS

References

CHAPTER 3: DEFORMED CANONICAL (ANTI-)COMMUTATION RELATIONS AND NON-SELF-ADJOINT HAMILTONIANS

3.1 INTRODUCTION

3.2 THE MATHEMATICS OF -PBs

3.3 -PBs IN QUANTUM MECHANICS

3.4 OTHER APPEARANCES OF -PBs IN QUANTUM MECHANICS

3.5 A MUCH SIMPLER CASE: PSEUDO-FERMIONS

3.6 A POSSIBLE EXTENSION: NONLINEAR -PBs

3.7 CONCLUSIONS

3.8 ACKNOWLEDGMENTS

REFERENCES

CHAPTER 4: CRITERIA FOR THE REALITY OF THE SPECTRUM OF -SYMMETRIC SCHRÖDINGER OPERATORS AND FOR THE EXISTENCE OF -SYMMETRIC PHASE TRANSITIONS

4.1 INTRODUCTION

4.2 PERTURBATION THEORY AND GLOBAL CONTROL OF THE SPECTRUM

4.3 ONE-DIMENSIONAL -SYMMETRIC HAMILTONIANS: CRITERIA FOR THE REALITY OF THE SPECTRUM

4.4 -SYMMETRIC PERIODIC SCHRÖDINGER OPERATORS WITH REAL SPECTRUM

4.5 AN EXAMPLE OF -SYMMETRIC PHASE TRANSITION

4.6 THE METHOD OF THE QUANTUM NORMAL FORM

APPENDIX: MOYAL BRACKETS AND THE WEYL QUANTIZATION

REFERENCES

CHAPTER 5: ELEMENTS OF SPECTRAL THEORY WITHOUT THE SPECTRAL THEOREM

5.1 INTRODUCTION

5.2 CLOSED OPERATORS IN HILBERT SPACES

5.3 HOW TO WHIP UP A CLOSED OPERATOR

5.4 COMPACTNESS AND A SPECTRAL LIFE WITHOUT IT

5.5 SIMILARITY TO NORMAL OPERATORS

5.6 PSEUDOSPECTRA

REFERENCES

CHAPTER 6:

PT

-SYMMETRIC OPERATORS IN QUANTUM MECHANICS: KREIN SPACES METHODS

6.1 INTRODUCTION

6.2 ELEMENTS OF THE KREIN SPACES THEORY

6.3 SELF-ADJOINT OPERATORS IN KREIN SPACES

6.4 ELEMENTS OF -SYMMETRIC OPERATORS THEORY

REFERENCES

CHAPTER 7: METRIC OPERATORS, GENERALIZED HERMITICITY AND LATTICES OF HILBERT SPACES

7.1 INTRODUCTION

7.2 SOME TERMINOLOGY

7.3 SIMILAR AND QUASI-SIMILAR OPERATORS

7.4 THE LATTICE GENERATED BY A SINGLE METRIC OPERATOR

7.5 QUASI-HERMITIAN OPERATORS

7.6 THE LHS GENERATED BY METRIC OPERATORS

7.7 SIMILARITY FOR PIP-SPACE OPERATORS

7.8 THE CASE OF PSEUDO-HERMITIAN HAMILTONIANS

7.9 CONCLUSION

APPENDIX: PARTIAL INNER PRODUCT SPACES

REFERENCES

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

CHAPTER 1: NON-SELF-ADJOINT OPERATORS IN QUANTUM PHYSICS: IDEAS, PEOPLE, AND TRENDS

Figure 1.1 The -dependence (1.3.2) of the low-lying bound states in the exactly solvable -symmetric potential (1.3.1).

Figure 1.2 The spectrum of the non-self-adjoint harmonic-oscillator SUSY Hamiltonian of eqs (1.3.4) and (1.3.6).

CHAPTER 2: OPERATORS OF THE QUANTUM HARMONIC OSCILLATOR AND ITS RELATIVES

Figure 2.1 Positive definiteness (2.1.29):

too Little

.

Figure 2.3 “Half-plane” extended positive definiteness:

that's it!

Figure 2.2 Very extended positive definiteness:

too much

.

CHAPTER 5: ELEMENTS OF SPECTRAL THEORY WITHOUT THE SPECTRAL THEOREM

Figure 5.1 Spectrum (dark gray dots) and pseudospectra (enclosed by the light gray contour lines) of the imaginary cubic oscillator.

(Courtesy of Miloš Tater.)

CHAPTER 7: METRIC OPERATORS, GENERALIZED HERMITICITY AND LATTICES OF HILBERT SPACES

Figure 7.1 The lattice of Hilbert space s generated by a metric operator.

Figure 7.2 The lattice generated by a metric operator.

Figure 7.3 The lattice generated by a metric operator. Note that .

Figure 7.4 The semisimilarity scheme (from Ref. (11)).

List of Tables

Non-Selfadjoint Operators in Quantum Physics

Mathematical Aspects

Editors:

FABIO BAGARELLO Università di Palermo and INFN

JEAN PIERRE GAZEAU Université Paris Diderot, Sorbonne Paris Cité Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro

FRANCISZEK HUGON SZAFRANIEC Uniwersytet Jagielloński

MILOSLAV ZNOJIL Ústav jaderné fyziky AV ČR, Řež

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Library of Congress Cataloging-in-Publication Data:

Non-selfadjoint operators in quantum physics : mathematical aspects / editors: Fabio Bagarello, Jean Pierre Gazeau, Franciszek Hugon Szafraniec, Miloslav Znojil.

pages cm

Includes index.

ISBN 978-1-118-85528-7 (cloth)

1. Nonselfadjoint operators. 2. Spectral theory (Mathematics) 3. Quantum theory–Mathematics. 4. Hilbert space. I. Bagarello, Fabio, 1964- editor. II. Gazeau, Jean-Pierre, editor. III. Szafraniec, Franciszek Hugon, editor. IV. Znojil, M. (Miloslav), editor.

QA329.2.N67 2015

530.1201′515724–dc23

2014048325

To Charles Hermite with apologies

CONTRIBUTORS

SERGIO ALBEVERIO

Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany CERFIM, Locarno, Switzerland Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

JEAN-PIERRE ANTOINE

Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium

FABIO BAGARELLO

Università di Palermo and INFN, Torino, Italy

EMANUELA CALICETI

Dipartimento di Matematica, Università di Bologna, Bologna, Italy

SANDRO GRAFFI

Dipartimento di Matematica, Università di Bologna, Bologna, Italy

SERGII KUZHEL

AGH University of Science and Technology, Kraków, Poland

DAVID KREJČIŘĺK

Nuclear Physics Institute, ASCR, Řež, Czech Republic

PETR SIEGL

Mathematical Institute, University of Bern, Bern, Switzerland

FRANCISZEK HUGON SZAFRANIEC

Instytut Matematyki, Uniwersytet Jagielloński, Kraków, Poland

CAMILLO TRAPANI

Dipartimento di Matematica e Informatica, Università di Palermo, Palermo, Italy

MILOSLAV ZNOJIL

Nuclear Physics Institute, ASCR, Řež, Czech Republic

PREFACE

Although it is widely accepted common wisdom that the discussions between mathematicians and physicists are enormously rewarding and productive, it is usually much easier to select illustrative examples from the past than to convert such a nice-sounding observation into a concrete and constructive project or into a proposal of a collaboration between a mathematician and a physicist.

For people involved, the reasons are more than obvious: in contrast to physics in which one may always appeal to experiments, the mathematicians feel free to ask (and study) time-independent questions. Consequently, the communication among physicists is usually full of urgency and with emphasis on novelty, while the language used by mathematicians is perceivably different, more explicit and much less hasty. All of the statements in mathematics must be rigorous and made after precise definitions.

In this sense, the persuasive success of the mutual interaction between mathematics and physics appears slightly puzzling, because of the different goals and habits rather than the language itself. Obviously, it requires not only a lot of mutual tolerance and openness but also a change of the language. Fortunately, it is equally obvious that the necessary efforts almost always pay off. This is also one of the main reasons why we decided to collect a few members of the mathematical physics community and to compose an edited book in which an account of one of the most interesting developments in contemporary theoretical physics would be retold using mathematical style.

Our selection of the subject of the applicability and applications of non-self-adjoint operators in Hilbert spaces was dictated, first of all, by the current status of the development of the field in the context of physics and, in particular, of quantum physics. In parallel, many of the ideas involved in these recent developments may be identified as not so new in mathematics. For this reason, we believe that our current book could fill one of the increasingly visible gaps in the existing literature. We believe that the current emergence of multiple new ideas connected with the concepts of non-self-adjointness in physics will certainly profit from a less speedy return to the older knowledge and to the roots of at least some of these ideas in mathematics.

Naturally, the message delivered by our current book is far from being complete or exhaustive. We decided to prefer a selection of a few particular subjects, giving the authors more space for the presentation of their review-like summaries of the existing knowledge as well as of their own personal interpretation of the history of the field as well as of its expected further development in the nearest future.

F. BAGARELLO, J. P. GAZEAU, F. H. SZAFRANIEC, M. ZNOJILPalermo, Paris, Rio, Kraków,PragueSeptember, 2014

ACRONYMS

CCM

coupled clusters method

-PB(s)

pseudo-bosons(s) on a dense subspace

EEP

extreme exceptional point

EP

exceptional point

HO

harmonic oscillator

IB

interacting bosons

KLMN

Kato, Lax, Lions, Milgram, Nelson

LHS

lattice of Hilbert spaces

LBS

lattice of Banach spaces

MHD

magnetohydrodynamics

MP

Mathematical Physics/Mathematical Physicists

MPI

Max Planck Institute

NiTheP

National Institute of Theoretical Physics

PB(s)

pseudo-boson(s)

PF(s)

pseudo-fermion(s)

PIP space

partial inner product space

PHHQP

Pseudo-Hermitian Hamiltonians in Quantum Physics (conference series)

PT

see

in Symbols

PT symmetric

invariant under space reflection and complex conjugation

QM

quantum mechanics

RS

Rayleigh-Schrödinger

SIGMA

Symmetry, Integrability and Geometry: Methods and Applications

SSUSY

second-order supersymmetry

SUSY

supersymmetry

THS

three-Hilbert-space

WKB

Wentzel-Kramers-Brillouin

GLOSSARY

Hermitian

denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal diagonal are complex conjugates (from Oxford Dictionaries,

http://www.oxforddictionaries.com/

)

non-Hermitian

logical negation of “Hermitian”

Hermitesch

so heißt ein linearer Operator

R

selbstadjungiert oder Hermitesch…

J. v. Neumann 1930, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren,

Mathematische Annalen

102

49-131

nicht Hermitesch

logical negation of “self-adjoint”

self-adjoint

A densely defined operator

T

on a Hilbert space is called

symmetric

(or

Hermitian

) if

and

self-adjoint

if

(from M. Reed and B. Simon, Methods of Mathematical Physics, Vol 1: Functional Analysis Academic Press 1980)

non-Hermitian

logical negation of “symmetric”

non-selfadjoint

logical negation of “self-adjoint”

SYMBOLS

dense subspace

-quasi base (

Chapter 3

)

-quasi base (

Chapter 3

)

dense subspace (

Chapter 3

)

G(t)

non-Hermitian generator of quantum evolution in the time-dependent dynamical regime of Eqs. (

1.5.2

) and (

1.5.3

)

for operator

H

, one of the isospectral self-adjoint partners living in a “prohibited”, third Hilbert space

(

Chapter 1

)

H

Hamiltonian

Hilbert space

the adjoint of operator

H

in “the second”, physical Hilbert space

(Eq.

1.4.5

)

inner product

linear

in the

first

argument (

Chapter 2

)

inner product

linear

in the

second

argument (

Chapter 3

)

(u, v)

inner product

linear

in the

second

argument (

Chapter 5

)

[u, v]

indefinite inner product

linear

in the

first

argument (

Chapter 6

)

(u,v)

inner product

linear

in the

first

argument (

Chapters 6

,

7

)

inner product

linear

in the

first

argument (

Chapter 7

)

Dyson-map factor of metric

(Eq.

1.4.7

)

in “the first”, unphysical Hilbert space

, any biorthogonal partner of eigenket

of Hamiltonian

(

Chapter 1

)

parity times time-reversal

resolvent set of operator

A

spectrum of operator

A

point spectrum of operator

A

residual spectrum of operator

A

continuous spectrum of operator

A

INTRODUCTION

F. Bagarello, J.P. Gazeau, F. Szafraniec and M. Znojil

Palermo, Paris, Rio, Kraków, Prague

The overall conception of this multipurpose book found one of its sources of inspiration in a comparatively new series of international conferences “Pseudo-Hermitian Hamiltonians in Quantum Physics” (1). This series offered, from its very beginning in 2003, a very specific opportunity of confrontation of the mathematical and phenomenological approaches to the concepts of the non-self-adjointness of operators. At the same time, the recent meetings on this series (the conferences in Paris (2) and Istanbul (3)) seemed, to us at least, to convert this confrontation to a sort of just a polite coexistence.

We (i.e., our team of guest editors of this book) came to the conclusion that it is just time to complement the usual written outcome of these meetings (i.e., typically, the volumes of proceedings or special issues as published, more or less regularly, in certain physics-oriented journals) by a few more mathematically oriented texts, reviews, and/or studies.

The idea of collecting the contributions forming this volume came out from the workshop “Non-Hermitian operators in quantum physics,” held in Paris, in August 2012. It was the 11th meeting in the PHHQP series. Keeping track of the contemporary development of Quantum Physics, either monitoring the publications or attending conferences in diverse areas, we have realized that, in order to stimulate properly further progress as well as optimize the scientific efforts undertaken by researchers in the field, a résumé of mathematical methods used so far would surely be beneficial. As Mathematics is unquestionably a basic tool, people working in Quantum Physics should be aware of its applicability, deepening insight and widening perspectives. Therefore, we thought that any update in this direction should be welcome, particularly topics that refer to “non-self-adjoint operators,” primarily those involved in -symmetric Hamiltonians (4) and in their extensions. We are convinced that this relatively wide subject will attract the attention of many scientists, from mathematics to theoretical and applied physics, from functional analysis to operator algebras.

This mathematically oriented state of the art book is a result of these reflections and efforts. It includes a general survey of -symmetry, and invited chapters, reviewing, in a self-consistent way, various mathematical aspects of non-Hermitian or non-self-adjoint operators in mushrooming Quantum Physics. It is composed of contributions of several representative authors (or groups of authors) who accepted the challenge and who tried to promote the currently available physics - emphasizing accounts of the current status of the field to a level of more rigorous mathematical standards in the following areas:

Functional analytic methods for non-self-adjoint operators

Algebraic methods for non-self-adjoint lattices of Hilbert spaces

Perturbation theory

Spectral theory

Krein space theory

Metric operators and lattices of Hilbert spaces

The organization of the book follows more or less faithfully the aforementioned list of subjects. Each chapter can be read independently of the others and has its own references at the end.

Chapter 1 is thought as a comprehensive historical description of motivations and developments of those “non-hermitian” explorations and/or transgressions of self-adjointness, a crucial requirement for physical observability and dynamical evolution, lying at the heart of Von Neumann quantum paradigm. Its content reflects the selection of topics that are covered by the more mathematically oriented rest of the book. It intends, through the Hilbertian trilogy , , , to restrict the readership attention to a few moments at which a cross-fertilizing interaction between the phenomenological and formal aspects of the use of non-self-adjoint operators in physics proved particularly motivating and intensive.

Chapter 2 is intended to give those “operators” considered in mathematical physics a form of operators as mathematicians would like to see them. This in turn creates a need of having the commutation relations properly understood. As all this refers to the quantum harmonic oscillator and its relatives, the operators involved are rather nonsymmetric. The class of operators they belong to as well as their spatial properties are described in some detail. As a matter of fact, and besides isometries, there are only two classes of Hilbert space operators that are commonly recognizable in Quantum Mechanics: symmetric (essential self-adjoint, self-adjoint) and generators of different kinds of semigroups. Other important operators, for instance, those appearing in the quantum harmonic oscillator seem to be not categorized, at least unknown to the bystanders. One of the goals of this survey is to expose their role, enhancing the most distinctive features. The main “non-self-adjoint” object is the class of (unbounded) subnormal operators. This is compelling, and as such it determines our modus operandi: “spatial” approach rather than Lie group/algebra connections. A natural consequence is to refresh the meaning traditionally given to commutation relations.

Chapter 3 shows how a particular class of biorthogonal bases arises out of some deformations of the canonical commutation and anticommutation relations. The deformed raising and lowering operators define extended number operators, which are not self-adjoint but are related by a certain intertwining operator, which can also be used to introduce a new scalar product in the Hilbert space of the theory. The content of this chapter clarifies some of the questions raised by such deformations by making use of a rather general structure, with central ingredient being the so-called -pseudo-bosons (-PBs) or their fermionic counterparts, the pseudo-fermions (PFs). This structure is unifying as many examples introduced along the years in the literature on -quantum mechanics and its relatives can be rewritten in terms of -PBs or of PFs.

Chapter 4 is a review presenting some simple criteria, mainly of perturbative nature, entailing the reality or the complexity of the spectrum of various classes of -symmetric Schrödinger operators. These criteria deal with one-dimensional operators as well as multidimensional ones. Moreover, mathematical questions such as the diagonalizabilty of the -symmetric operators and their similarity with self-adjoint operators are also discussed, also through the technique of the convergent quantum normal form. A major mathematical problem in -symmetric quantum mechanics is to determine whether or not the spectrum of any given non-self-adjoint but -symmetric Schrödinger operator is real. Clearly, in this connection, an equally important issue is the spontaneous breakdown of the -symmetry, which might occur in a -symmetric operator family. The spontaneous violation of the -symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the -symmetric phase transition. This chapter is a review of the recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators.

Chapter 5 focuses on spectral theory. It is an extremely rich field, which has found applications in many areas of classical as well as modern physics and most notably in quantum mechanics. This chapter gives an overview of powerful spectral-theoretic methods suitable for a rigorous analysis of non-self-adjoint operators. It collects some classical results as well as recent developments in the field in one place, and it illustrates the abstract methods by concrete examples. Among other things, the notions of quasi-Hermiticity, pseudo-Hermiticity, similarity to normal and self-adjoint operators, Riesz-basicity, and so on, are recalled and treated in a unified manner. The presentation is accessible for a wide audience, including theoretical physicists interested in -symmetric models. It is a useful source of tools for dealing with physical problems involving non-self-adjoint operators.

Chapter 6 presents a variety of Krein-space methods in studying symmetric Hamiltonians and outlines possible developments. It bridges the gap between the growing community of physicists working with symmetry (4) with the community of mathematicians who study self-adjoint operators in Krein spaces for their own sake. The general mathematical properties of -symmetric operators are discussed within the Krein spaces framework, focusing on those aspects of the Krein spaces theory that may be more appealing to mathematical physicists. This supports the idea that every -symmetric operator corresponding to a quantum observable should be a self-adjoint operator in a suitably chosen Krein space and that a proper investigation of a -symmetric Hamiltonian involves the following stages: interpretation of as a self-adjoint operator in a Krein space ; construction of an operator for ; interpretation of as a self-adjoint operator in the Hilbert space .

Chapter 7 analyzes the possible role and structure of the generalized metric operators , which are allowed to be unbounded. As early as 1960, Dieudonné already tried to introduce and analyze such a concept. In the context of mathematics of Hilbert spaces he found, to his disappointment, that the properties of the operators A, which he suggested to be called quasi-Hermitian and which had to satisfy the generalized Hermiticity relation of the form , appeared not so attractive. Later, the class of the admissible 's has been narrowed by physicists. They found that once the 's are just bounded and strictly positive self-adjoint operators with bounded inverse, the quasi-Hermitian operators A reacquire virtually all of the properties that are needed in quantum mechanics. Unfortunately, in a number of examples including, in particular, many -symmetric models (4), the latter requirements proved too restrictive. Their moderate mathematical generalization appeared necessary. In Chapter 7, therefore, several generalizations of the notion of quasi-Hermiticity are introduced and the questions of the preservation of the spectral properties of operators are examined.

Canonical lattices of Hilbert spaces generated by unbounded metric operators are then considered. Such lattices constitute the simplest case of a partial inner product space (PIP space), and this justifies the employment of the technique of PIP space operators. Some of the previous results are applied to operators on a particular PIP space, namely, the scale of Hilbert spaces generated by a single metric operator. Finally, the notion of pseudo-Hermitian operators is reformulated in the preceding formalism.

As a concluding remark, the material presented in our book will certainly draw the attention of the reader to a well-known occurrence in the mutual irrigation of Mathematics and Physics, namely, the existence of basic, even trivial operations or properties leading to nontrivial developments in both disciplines. In the present case, there are two (very) discrete involutions in inner product complex vector spaces with countable basis , namely, antilinear complex conjugation of vectors