Non-Selfadjoint Operators in Quantum Physics

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 recent emergence of the 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 of 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 pertubation 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, 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.

Preface xvii
Acronyms xix
Glossary xxi
Symbols xxiii
Introduction 1
F. Bagarello, J. P. Gazeau, F. Szafraniec, and M. Znojil
References 5
1 Non-Self-Adjoint Operators in Quantum Physics: Ideas, People, and Trends 7
Miloslav Znojil
1. 1 The Challenge of Non-Hermiticity in Quantum Physics 7
1. 2 A Periodization of the Recent History of Study of Non-Self-Adjoint Operators in Quantum Physics 11
1. 3 Main Message: New Classes of Quantum Bound States 18
1. 4 Probabilistic Interpretation of the New Models 29
1. 5 Innovations in Mathematical Physics 34
1. 6 Scylla of Nonlocality or Charybdis of Nonunitarity? 37
1. 7 Trends 45
References 50
2 Operators of the Quantum Harmonic Oscillator and Its Relatives 59
Franciszek Hugon Szafraniec
2. 1 Introducing to Unbounded Hilbert Space Operators 60
2. 2 Commutation Relations 88
2. 3 The q-Oscillators 106
2. 4 Back to "Hermicity"-A Way to See It 113
Concluding Remarks 115
References 115
3 Deformed Canonical (Anti-)Commutation Relations and Non-Self-Adjoint Hamiltonians 121
Fabio Bagarello
3. 1 Introduction 121
3. 2 The Mathematics of D-PBs 123
3. 3 D-PBs in Quantum Mechanics 145
3. 4 Other Appearances of D-PBs in Quantum Mechanics 158
3. 5 A Much Simpler Case: Pseudo-Fermions 174
3. 6 A Possible Extension: Nonlinear D-PBs 182
3. 7 Conclusions 184
3. 8 Acknowledgments 185
References 185
4 Criteria for the Reality of the Spectrum of PT -Symmetric Schrö dinger Operators and for the Existence of PT -Symmetric Phase Transitions 189
Emanuela Caliceti and Sandro Graffi
4. 1 Introduction 189
4. 2 Perturbation Theory and Global Control of the Spectrum 191
4. 3 One-Dimensional PT -Symmetric Hamiltonians: Criteria for the Reality of the Spectrum 194
4. 4 PT -Symmetric Periodic Schrö dinger Operators with Real Spectrum 200
4. 5 An Example of PT -Symmetric Phase Transition 206
4. 6 The Method of the Quantum Normal Form 219
Appendix: Moyal Brackets and theWeyl Quantization 232
A. 1 Moyal Brackets 232
A. 2 The Weyl Quantization 236
References 238
5 Elements of Spectral Theory without the Spectral Theorem 241
David Krejè iø í k and Petr Siegl
5. 1 Introduction 241
5. 2 Closed Operators in Hilbert Spaces 242
5. 3 How to Whip Up a Closed Operator 257
5. 4 Compactness and a Spectral Life Without It 266
5. 5 Similarity to Normal Operators 273
5. 6 Pseudospectra 281
References 288
6 PT -Symmetric Operators in Quantum Mechanics: Krein Spaces Methods 293
Sergio Albeverio and Sergii Kuzhel
6. 1 Introduction 293
6. 2 Elements of the Krein Spaces Theory 295
6. 3 Self-Adjoint Operators in Krein Spaces 304
6. 4 Elements of PT -Symmetric Operators Theory 320
References 340
7 Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces 345
Jean-Pierre Antoine and Camillo Trapani
7. 1 Introduction 345
7. 2 Some Terminology 347
7. 3 Similar and Quasi-Similar Operators 349
7. 4 The Lattice Generated by a Single Metric Operator 362
7. 5 Quasi-Hermitian Operators 367
7. 6 The LHS Generated by Metric Operators 380
7. 7 Similarity for PIP-Space Operators 382
7. 8 The Case of Pseudo-Hermitian Hamiltonians 389
7. 9 Conclusion 392
Appendix: Partial Inner Product Spaces 392
A. 1 PIP-Spaces and Indexed PIP-Spaces 392
A. 2 Operators on Indexed PIP-space S 395
A. 2. 1 Symmetric Operators 396
A. 2. 2 Regular Operators, Morphisms, and Projections 397
References 399
Index 403