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...