Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects

By Fabio Bagarello and Jean-Pierre Gazeau

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.

• Language: English
• Publisher: Wiley
• Release date: Sep 9, 2015
• ISBN: 9781118855270

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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. ZNOJIL

Palermo, Paris, Rio, Kraków,

PragueSeptember, 2014

ACRONYMS

GLOSSARY

SYMBOLS

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 f01-math-0001 -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 f01-math-0003 -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 f01-math-0004 , f01-math-0005 , f01-math-0006 , 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 f01-math-0007 -pseudo-bosons ( f01-math-0008 -PBs) or their fermionic counterparts, the pseudo-fermions (PFs). This structure is unifying as many examples introduced along the years in the literature on f01-math-0009 -quantum mechanics and its relatives can be rewritten in terms of f01-math-0010 -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 f01-math-0011 -symmetric Schrödinger operators. These criteria deal with one-dimensional operators as well as multidimensional ones. Moreover, mathematical questions such as the diagonalizabilty of the f01-math-0012 -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 f01-math-0013 -symmetric quantum mechanics is to determine whether or not the spectrum of any given non-self-adjoint but f01-math-0014 -symmetric Schrödinger operator is real. Clearly, in this connection, an equally important issue is the spontaneous breakdown of the f01-math-0015 -symmetry, which might occur in a f01-math-0016 -symmetric operator family. The spontaneous violation of the f01-math-0017 -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 f01-math-0018 -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 f01-math-0019 -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 f01-math-0020 symmetric Hamiltonians and outlines possible developments. It bridges the gap between the growing community of physicists working with f01-math-0021 symmetry (4) with the community of mathematicians who study self-adjoint operators in Krein spaces for their own sake. The general mathematical properties of f01-math-0022 -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 f01-math-0023 -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 f01-math-0024 -symmetric Hamiltonian f01-math-0025 involves the following stages: interpretation of f01-math-0026 as a self-adjoint operator in a Krein space f01-math-0027 ; construction of an operator f01-math-0028 for f01-math-0029 ; interpretation of f01-math-0030 as a self-adjoint operator in the Hilbert space f01-math-0031 .

Chapter 7 analyzes the possible role and structure of the generalized metric operators f01-math-0032 , 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 f01-math-0033 , appeared not so attractive. Later, the class of the admissible f01-math-0034 's has been narrowed by physicists. They found that once the f01-math-0035 '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 f01-math-0036 -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 f01-math-0037 , namely, antilinear complex conjugation of vectors f01-math-0038 and linear parity f01-math-0039 . The next 400 pages are recurrent symphonic variations around that f01-math-0040 e f01-math-0041 ite phrase musicale (5) pervading our lost f01-math-0042 roustian f01-math-0043 ime.

References

1. Available at http://gemma.ujf.cas.cz/%7Eznojil/conf/index.html

2. Available at http://phhqp11.in2p3.fr/Home.html Accessed 2014 Nov 13.

3. Available at http://home.ku.edu.tr/%7Eamostafazadeh/workshop_2012/index.html

4. Bender CM, Boetcher S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys Rev Lett 1998;80:5243–55246–.

5. f01-math-0044 roust M. Vinteuil Sonate in Un amour de Swann, A la recherche du f01-math-0045 f01-math-0046 erdu. Grasset and Gallimard, Paris, 1913, p 1871–1922–.

CHAPTER 1

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

Miloslav Znojil

Nuclear Physics Institute, ASCR, Řež, Czech Republic

1.1 THE CHALLENGE OF NON-HERMITICITY IN QUANTUM PHYSICS

1.1.1 A Few Quantum Physics' Anniversaries, for Introduction

The year of writing this history-oriented chapter on the appeal of non-Hermiticity was also the year of several minor but interesting anniversaries in quantum physics. So let us start by recalling some of these dates.

1.1.1.1 Hundred Years of the Bohr's Model

In 2013, on occasion of the centenary of the Bohr's model of atom (1) (marking, in a way, the birth of quantum theory), one should appreciate the multitude of results of the first hundred years of our study of quantum world. One of typical characteristics of these developments may be seen in an incessant emergence of dramatic innovations and changes in our perception of what is measurable. The process still remains unfinished. Even the Nobel Prize in Physics for year 2012 was not awarded for the fresh, expensive, and long expected discovery of the Higgs boson in particle physics (which had to wait for one more year) but rather for the invention of ground-breaking methods of quantum measurements (2).

One must underline that during the century, the fundamental quantum physics remained a vivid discipline and that its experimental side never ceased to be a topical subject. What should be appreciated, in parallel, is the fact that none of these innovations ever disproved any of the apparently counterintuitive basic principles of the theory. One must admire the robust nature of the basic mathematical ideas.

In particular, it was not necessary to change the theory after Herman Feshbach (3) succeeded in describing the usual processes of quantum scattering and reactions in atomic nuclei (including the elastic ones) by means of a complex effective potential. On this occasion, the exotic non-self-adjoint (alias, in the physicist's language, non-Hermitian) operators seem to have entered the scene.

1.1.1.2 Fifty-five Years of the Feshbach's Non-self-adjoint Hamiltonians

For stable quantum systems, the evolution in time is usually assumed generated by a physical Hamiltonian c01-math-0001 which is defined as acting in a suitable representation c01-math-0002 of the physical Hilbert space of states. It is very important that the popular principle of correspondence, albeit vaguely defined, often enables us to choose, in realistic models, constructively tractable versions of spaces c01-math-0003 and Hamiltonians c01-math-0004 .

The practical feasibility of calculations quickly decreases during transition to more complicated systems. One may recall multiple examples, say, in nuclear physics where the computer-assisted numerical determination of the bound-state energies hardly remains sufficiently routine even in the lightest nuclei. For the heavier nuclei, the growth of complexity of calculations may be perceived as one of the fundamental methodical challenges in nuclear physics.

As we already mentioned, one of the productive tools of an amendment of the algorithms has been proposed by Feshbach (3). In his considerations, he admits that even if one knows Hamiltonians c01-math-0005 , many time-independent Schrödinger equations c01-math-0006 describing bound states (with E = real) or resonant states (with E = complex) prove prohibitively difficult to solve in practice. He recalled that in the majority of applications just the knowledge of the low-lying spectrum of energies is asked for. This led him to the conclusion that a judicious restriction of physical space c01-math-0009 to a suitable subspace c01-math-0010 should be performed in such a way that the reduction of Hamiltonian c01-math-0011 remains compatible with the requirement of an at least partial isospectrality of the two operators.

Ambitious as the project might have seemed, its analysis resulted into a recipe which proved enormously popular and successful in practice (4). In fact, its basic idea is fairly elementary. One simply partitions the big Hilbert space c01-math-0012 into two subspaces via projectors Q (on an irrelevant part of the bigger Hilbert space c01-math-0014 ) and c01-math-0015 (in our present notation, the projector on the model space c01-math-0016 ). This yields the partitioned Schrödinger equation

equation

and formula

equation

for the Q-projection of the exact wave function, which is, by assumption, less relevant. Its elimination provides the ultimate compactified, nonlinear, effective Schrödinger eigenvalue problem

1.1.1 equation

defined inside the subspace c01-math-0021 . The action of the effective Hamiltonian

equation

is energy dependent but it remains restricted just to the relevant, R-projected subspace c01-math-0024 .

One must emphasize that the required strict isospectrality between H and c01-math-0026 is guaranteed. Unfortunately, owing to the manifest energy dependence of c01-math-0027 , the costs grew high. They became even higher in the original nuclear-reaction context in which the physical values of energies lied in the essential part of the spectrum. One must then generalize the definition of the operator pencil c01-math-0028 and start working with the complex values of the variable parameter c01-math-0029 .

In the energy range of interest, the spectral shift caused by the presence of z or E in denominators is often being ignored as not too relevant in practice. Still, it must be reemphasized that the simplified effective-Hamiltonian operator c01-math-0032 is manifestly non-self-adjoint in general.

With such an observation, the projection-operator studies of quantum systems rarely remain restricted to the mere stable dynamical regime with real spectra and unitary evolution in time. The loss of the self-adjoint nature of the effective Hamiltonians is usually interpreted as implying a necessary loss of the reality of the energies. Strictly speaking, such a deduction is not always correct. As a counterexample, one may recall, for example, the so-called c01-math-0033 -symmetric systems and Hamiltonians.

1.1.1.3 Fifteen Years of c01-math-0034 -symmetry alias Pseudo-Hermiticity

In the broader context of preceding paragraph, the abstract formalism of quantum theory encountered an unexpected challenge circa 15 years ago, after several parallel innovative proposals of inclusion, in the mainstream formalism, of certain manifestly non-self-adjoint Hamiltonian-like operators c01-math-0035 possessing strictly real, bound-state-like spectra. Fortunately, during the subsequent years, the acceptance of such a class of models proved fully compatible with the first principles of quantum theory. Moreover, an intensified study of mathematics of manifestly non-self-adjoint candidates for observables became an inseparable part of quantum theory.

An official start of studies of the possibility of having non-self-adjoint operators in a unitary theory may be dated back to 1998 when Bender and Boettcher published their letter (5). Its title Real spectra in non-Hermitian Hamiltonians having c01-math-0036 -symmetry sounded truly provocative at that time because, according to conventional wisdom, the spectra of non-Hermitian operators can hardly be purely real. The explicit construction of a non-Hermitian quantum Hamiltonian c01-math-0037 with real spectrum sounded, therefore, like a joke or paradox rather than like a serious scientific proposal¹.

Later on, the situation and attitudes have changed. A sample of the progress is to be reported in this book. As long as the main emphasis will be laid, in the forthcoming chapters, upon the mathematical aspects of the theory, the collected material will be preceded and, in some sense, interconnected by this chapter offering a compact outline of the field, with particular emphasis upon historical and phenomenological context.

Our considerations will reflect the selection of topics to be covered by the more mathematically oriented rest of the book. In a sketchy and incomplete, selective outline of key ideas, we intend to restrict our 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.

1.1.2 Dozen Years of Conferences Dedicated to Pseudo-Hermiticity

Letter (5) inspired a lot of research activities. In the literature, a number of emerging paradoxes was spotted, exposed to a thorough scrutiny—and shown to disappear. At present, one can say that from the point of view of the recent history of quantum physics, the date of publication of this remarkable letter may be perceived as one of the most important turning points. Not only within the theory itself (in which, later, the concepts of stability and evolution were thoroughly revisited and clarified under its influence) but also in experiments.

At present, the progress and publications of the related results in prestigious physics Journals may be followed online, via dedicated bookkeeping webpage as maintained by Daniel Hook (8). Through this page, one can trace the recent history of the field. One can download a lot of papers that caused the change (or rather a complete reversal) of attitude of the international scientific community toward the real spectra in non-Hermitian Hamiltonians.

The change of the paradigm was due to the work by multiple active authors, so the results cannot be summarized easily (cf. review papers (9, 11) for many details and references). Among these results, one should recollect, first of all, the introduction of the influential concepts of c01-math-0038 -symmetry of a quantum Hamiltonian H (i.e., of the rule c01-math-0040 where symbols c01-math-0041 and c01-math-0042 denote parity and time reversal, respectively) or, after a slight generalization, of the c01-math-0043 -pseudo-Hermiticity (or, briefly, pseudo-Hermiticity) of an observable c01-math-0044 (i.e., of the rule c01-math-0045 written in terms of a suitable generalization c01-math-0046 of the parity operator).

Although the notion of c01-math-0047 -symmetry may be already found in 1993 paper (12), it only played a marginal role there. Probably, the concept was in current use even earlier (13). Anyhow, its heuristic relevance and productivity remained practically unknown before 1998.

Several years after 1998, symbol c01-math-0048 entering the c01-math-0049 -symmetry relation was mostly perceived as the mere parity in one-dimensional bound-state Schrödinger equations

1.1.2

equation

while symbol c01-math-0051 was strictly identified with an antilinear operator of time reversal. Even under these constraints, the appeal of the innovative notion grew quickly. Its use led to conjectures of multiple toy models (1.1.2) with quantum Hamiltonians of the usual form c01-math-0052 (and real spectra) but with an unusual non-self-adjointness property c01-math-0053 in the underlying friendly Hilbert space c01-math-0054 .

In the contemporary context of quantum model-building practice, the majority of innovations sounded strangely. One of their presentations to a larger audience during an international scientific conference took place in Paris in 2002. On this occasion, the invited speaker (naturally, Carl Bender) plus four other authors (cf. the written form of the talks in proceedings (14) discussed the response and concluded that the subject might deserve a separate series of dedicated conferences.

Supported by several other enthusiasts, the dedicated series really started, a year later, by the meeting of 27 participants from as many as 13 different countries in Prague (cf. (15). The next, similarly compact international workshop followed within a year. Subsequently, the number of participants jumped up (i.e., close to one hundred) in 2005, after the transfer of the meeting from Villa Lanna in Prague to the Universities in Istanbul (Turkey, June 2005) and Stellenbosch (South Africa, November 2005), etc².

The proceedings of the PHHQP series of conferences were mostly published in the form of a dedicated and refereed special issue (cf. (17, 19), etc.). These materials may be recalled as offering a compact (i.e., introductory, history-oriented, and time-ordered) sample of a few characteristic results, mainly in the field of quantum physics.

Even such a restricted inspection of the history of acceptance of non-self-adjoint operators in quantum physics reveals a sequence of ups (i.e., of the periods of a more or less uninterrupted growth), which were followed by downs, characterized by a sudden emergence of serious obstacles and crises.

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

1.2.1 The Years of Crises

In retrospective, one of the least expected observations resulting from the recollection of history of c01-math-0055 -symmetry and pseudo-Hermiticity shows an amazing regularity in the occurrence of crises. Several less regular precursors of these crises may even be dated before the above-mentioned year 1998. One may recollect, for example, that on the basis of multiple numerical experiments with the purely imaginary cubic interaction

1.2.1 equation

Daniel Bessis with Zinn-Justin already believed, in 1992 at the latest (20), in the strict reality of the bound-state spectrum. Anyhow (was it a crisis?), up to the present author' knowledge, they never published anything on their observations before the years when Carl Bender did.

1.2.1.1 The Year 2001: the First, Spectral-reality-proof Crisis

The first regular crisis came after circa 3 years of intensified studies of various c01-math-0057 -symmetric potentials and, in particular, of the Bender's and Milton's (21) (or, if you wish, of the Bender's and Boettcher's (5) extremely popular model

1.2.2 equation

In this potential, the choice of the general power-law c01-math-0059 dependence appeared to support an extension of the above-mentioned c01-math-0060 hypothesis of the reality of the spectrum. During the years 2000 and 2001, nevertheless, people started feeling more and more aware of the lasting absence of a reliable, rigorous proof.

The first proof applicable to the important family of c01-math-0061 -symmetric models (1.2.2) already appeared in 2001 (22). Just in time. What followed was a quick acceptance of the promising perspective of a noncontradictory existence of the real bound-state energies even when obtained from non-Hermitian Hamiltonians.

The much-required clarification of this point encouraged the community to make the next step and to return, i.a., to the Streater's criticism (6), (7) and to the related doubts about the possible physics behind the new and highly nonstandard models as exemplified by eqs (1.2.1) or (1.2.2). During the first years after the first crisis, the decisive suppression of these doubts has been achieved, basically, via a discovery (or rather rediscovery) of the possibility of using an ad hoc inner product and of defining a different Hilbert space of states in which a correct probabilistic interpretation of the models can be provided. In this text, we denote such a second or standard Hilbert space by dedicated symbol c01-math-0062 .

1.2.1.2 The Year 2004: the Metric-ambiguity Crisis

In the amended Dirac's notation of Ref. (23), the generalized inner products may be defined as overlaps c01-math-0063 where operator c01-math-0064 may be called Hilbert-space metric. During a year or two, the discovery has been slightly reformulated and found equivalent to a resuscitation of the three-Hilbert-space (THS) quantum-system representation as known and used, in nuclear physics, as early as in 1992 (24)³. In this manner, an overall, sketchy formulation of the theory was more or less completed.

The ultimate moment of acceptance of c01-math-0065 -symmetric Hamiltonians by physicists may be identified with the year 2004 of publication of erratum (26). During this year, any nontrivial metric c01-math-0066 (known, in the conventional physical terminology, as non-Dirac metric) became perceived as a fundamental ingredient in the description of quantum system, in principle at least.

The nontrivial-metric-dependent THS background of the theory was accepted, redirecting attention to the next open problem, namely, to the immanent ambiguity of the THS recipe. People imagined that the assignment of the desirable Hilbert-space metric c01-math-0067 to a preselected (and, say, c01-math-0068 -symmetric) Hamiltonian c01-math-0069 is far from unique. Deep crisis number two followed almost immediately.

At a comparable speed, the crisis was suppressed, mainly due to an overall acceptance of an additional postulate of observability of a new quantity c01-math-0070 called quasi-parity (27) or charge (28). It has been clarified that the requirement of the observability of charge c01-math-0071 makes the metric unique. A new wave of optimism followed.

1.2.1.3 The Year 2007: the Nonlocality Crisis

The third crisis emerged 3 years later, in 2007, when Jones noticed several counterintuitive features and obstructions to realization of a quantum-scattering-type experiment in c01-math-0072 symmetrized quantum mechanics arrangement (29). This discovery forced people to reanalyze the concepts of c01-math-0073 -symmetry and of the c01-math-0074 -symmetry, with the latter name being, for physicists, just the most common alias to the above-mentioned compatibility of the THS representation of quantum systems (using a special metric c01-math-0075 ) with the standard textbooks on quantum theory where only too often, just for the sake of simplicity, the metric is being set equal to the identity operator.

The simplest way out of the Jones' trap was proposed by Jones himself. He conjectured that the local complex potentials c01-math-0076 should be perceived as effective, say, in the Feshbach's subspace projection sense. In such an approach, the key scattering-unitarity assumption is declared redundant. Although this implies that, say, the flow of mass need not be conserved, that is, certain deus ex machina of sinks and sources of particles is admitted, one simply accepts an overdefensive argument that the evolution processes and, in particular, the scattering in a given quantum system is in fact just partially controlled and described by our effective local potentials c01-math-0077 . In other words, one admits that our information about the quantum dynamics is incomplete.

During the subsequent growth of theoretical efforts, one of the most consequent resolutions of the Jones' apparent paradoxes was offered in Ref. (30). The main source of misunderstanding has been identified as lying in an inconsistence of our assumptions. In the context of a conventional unitary quantum scattering theory, one simply asks for too much when demanding that the scattering potential may be chosen complex and local. In a way based on the construction of several explicit models, a transition to nonlocal c01-math-0078 -symmetric complex forces V was recommended (see more details in the following sections).

1.2.1.4 The Year 2010: the Construction-difficulty Crisis

The fourth crisis may be localized, roughly, to the year 2010 when it became clear that the emergent necessity of working with nonlocal potentials might lead to an enormous increase in technical difficulties during the applications of the unitary THS recipe. The same danger of encountering obstacles was found connected with the need of working with very complicated metrics c01-math-0080 even for originally not too complicated Hamiltonians.

A better profit has been found provided by a transfer of technologies beyond quantum physics. Around the year 2010, the active research in the area of quantum theory dropped perceivably down, therefore. This tendency was clearly reflected by a drastic decrease in the number of foreign participants in the meeting PHHQP IX in 2010 in China (31).

During the crisis, it became clear that the transfer of the concept of c01-math-0081 -symmetry out of quantum theory may in fact prove not only necessary but also unexpectedly rewarding. A real boom followed, for example, in experimenting with simulations of c01-math-0082 -symmetry in various gain-or-loss media in nonquantum settings (pars pro toto, let us mention here just the quick growth of popularity of c01-math-0083 -symmetry in nonquantum optics (32).

Within quantum physics, the boom was paralleled by a reenhancement of interest in the traditional studies of open quantum systems and also of unstable complex systems, with the dynamics controlled by the presence of resonances. Symptomatically, Nimrod Moiseyev's monograph Non-Hermitian Quantum Mechanics (33) dealing with these topics appeared published in 2011.

During the same, first-after-the-crisis year, the place of the jubilee tenth meeting PHHQP (34) (viz., MPI in Dresden) was packed by participants up to the roof again. The conference lasted much longer than usual (viz., full two weeks) and marked the onset of a new period of growth. The meeting was truly successful in putting emphasis on the closeness of connections between the quantum and nonquantum worlds. In addition, the not-entirely-expected influx of a lot of specialists from open-system quantum phenomenology contributed, by feedback, to the subsequent new growth of activities in all of the neighboring fields.

For mathematical physicists, attention has been redirected to the formal aspects of energy-dependent (i.e., nonlinear, effective, and subspace projected) non-self-adjoint Hamiltonians. Last but not least one should mention that as many as two separate special issues of Journals were needed to play the role of proceedings of the remarkable, direction-changing jubilee conference in 2011.

1.2.1.5 At Present: the Ill-defined-metrics Mathematical Crisis

Toward the end of 2013, we already know that we are just in the middle of another, fifth serious crisis. Its roots may be traced back to several, not always noticed critical comments on the THS representation formalism as made, in the recent past, by mathematicians (35). The essence of the problem is connected with several failures of the physicists to check the validity of certain necessary mathematical assumptions in their models.

The problem is serious—even for the most popular examples, unexpected no-go theorems were proved in the second half of the year 2012 (36). Several parallel attempts at circumventing the obstacles followed (37, 38). Still, the essence of the conflict remains unresolved at present. A final outcome of the last crisis is not yet known. We only have to stay optimistic, recollecting that