Non-Selfadjoint Operators in Quantum Physics

Fabio Bagarello, PhD, is Professor in the Department of Energy, Information Engineering, and Mathematical Models at the University of Palermo, Italy. Dr. Bagarello is the author of over 160 journal articles and Quantum Dynamics for Classical Systems: With Applications of the Number Operator, also published by Wiley.Jean Pierre Gazeau, PhD, is Emeritus Professor of Physics in the Laboratory of Astroparticles and Cosmology at the University Paris Diderot, France.  Dr. Gazeau is the author of over 200 journal articles and two books, including Coherent States in Quantum Physics, also published by Wiley.Franciszek Hugon Szafraniec, Prof. Dr hab., is a retired professor from the Jagiellonian University in Kraków, Poland. He remains an active researcher in functional analysis, and his research interests include operator theory, harmonic analysis, complex function theory, and mathematical foundations of quantum physics. Dr. Szafraniec is also an authority on reproducing kernel Hilbert spaces.Miloslav Znojil, DrSc, is Leading Research Worker at the Nuclear Physics Institute of the Czech Academy of Sciences and the Deputy Director of the Doppler Institute for Mathematical Physics and Applied Mathematics in the Czech Republic. Dr. Znojil is the author of over 300 journal articles and a member of the Czech Union of Mathematicians and Physicists.

• Preface xviiAcronyms xixGlossary xxiSymbols xxiiiIntroduction 1F. Bagarello, J.P. Gazeau, F. Szafraniec, and M. ZnojilReferences 51 Non-Self-Adjoint Operators in Quantum Physics: Ideas, People, and Trends 7Miloslav Znojil1.1 The Challenge of Non-Hermiticity in Quantum Physics 71.2 A Periodization of the Recent History of Study of Non-Self-Adjoint Operators in Quantum Physics 111.3 Main Message: New Classes of Quantum Bound States 181.4 Probabilistic Interpretation of the New Models 291.5 Innovations in Mathematical Physics 341.6 Scylla of Nonlocality or Charybdis of Nonunitarity? 371.7 Trends 45References 502 Operators of the Quantum Harmonic Oscillator and Its Relatives 59Franciszek Hugon Szafraniec2.1 Introducing to Unbounded Hilbert Space Operators 602.2 Commutation Relations 882.3 The q–Oscillators 1062.4 Back to “Hermicity”—A Way to See It 113Concluding Remarks 115References 1153 Deformed Canonical (Anti-)Commutation Relations and Non-Self-Adjoint Hamiltonians 121Fabio Bagarello3.1 Introduction 1213.2 The Mathematics of D-PBs 1233.3 D-PBs in Quantum Mechanics 1453.4 Other Appearances of D-PBs in Quantum Mechanics 1583.5 A Much Simpler Case: Pseudo-Fermions 1743.6 A Possible Extension: Nonlinear D-PBs 1823.7 Conclusions 1843.8 Acknowledgments 185References 1854 Criteria for the Reality of the Spectrum of PT -Symmetric Schrödinger Operators and for the Existence of PT -Symmetric Phase Transitions 189Emanuela Caliceti and Sandro Graffi4.1 Introduction 1894.2 Perturbation Theory and Global Control of the Spectrum 1914.3 One-Dimensional PT -Symmetric Hamiltonians: Criteria for the Reality of the Spectrum 1944.4 PT -Symmetric Periodic Schrödinger Operators with Real Spectrum 2004.5 An Example of PT -Symmetric Phase Transition 2064.6 The Method of the Quantum Normal Form 219Appendix: Moyal Brackets and theWeyl Quantization 232A.1 Moyal Brackets 232A.2 The Weyl Quantization 236References 2385 Elements of Spectral Theory without the Spectral Theorem 241David Krejèiøík and Petr Siegl5.1 Introduction 2415.2 Closed Operators in Hilbert Spaces 2425.3 How to Whip Up a Closed Operator 2575.4 Compactness and a Spectral Life Without It 2665.5 Similarity to Normal Operators 2735.6 Pseudospectra 281References 2886 PT -Symmetric Operators in Quantum Mechanics: Krein Spaces Methods 293Sergio Albeverio and Sergii Kuzhel6.1 Introduction 2936.2 Elements of the Krein Spaces Theory 2956.3 Self-Adjoint Operators in Krein Spaces 3046.4 Elements of PT -Symmetric Operators Theory 320References 3407 Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces 345Jean-Pierre Antoine and Camillo Trapani7.1 Introduction 3457.2 Some Terminology 3477.3 Similar and Quasi-Similar Operators 3497.4 The Lattice Generated by a Single Metric Operator 3627.5 Quasi-Hermitian Operators 3677.6 The LHS Generated by Metric Operators 3807.7 Similarity for PIP-Space Operators 3827.8 The Case of Pseudo-Hermitian Hamiltonians 3897.9 Conclusion 392Appendix: Partial Inner Product Spaces 392A.1 PIP-Spaces and Indexed PIP-Spaces 392A.2 Operators on Indexed PIP-space S 395A.2.1 Symmetric Operators 396A.2.2 Regular Operators, Morphisms, and Projections 397References 399Index 403