Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects

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.  INDICE: Preface xvii.Acronyms xix.Glossary xxi.Symbols xxiii.Introduction 1F. Bagarello, J.P. Gazeau, F. Szafraniec, and M. ZnojilReferences, 5.1 Non–Self–Adjoint Operators in Quantum Physics: Ideas, People, and Trends 7Miloslav Znojil.1.1 The Challenge of Non–Hermiticity in Quantum Physics 7.1.1.1 A Few Quantum Physics Anniversaries, for Introduction 7.1.1.2 Dozen Years of Conferences Dedicated to Pseudo–Hermiticity 10.1.2 A Periodization of the Recent History of Study of Non–Self–Adjoint Operators in Quantum Physics 11.1.2.1 The Years of Crises 11.1.2.2 The Periods of Growth 15.1.3 Main Message: New Classes of Quantum Bound States 18.1.3.1 Real Energies via Non–Hermitian Hamiltonians 18.1.3.2 Analytic and Algebraic Constructions 21.1.3.3 Qualitative Innovations of Phenomenological Quantum Models 24.1.4 Probabilistic Interpretation of the New Models 29.1.4.1 Variational Constructions 29.1.4.2 Non–Dirac Hilbert–Space Metrics I 32.1.5 Innovations in Mathematical Physics 34.1.5.1 Simplified Schrödinger Equations 34.1.5.2 Nonconservative Systems and Time–Dependent Dyson Mappings 36.1.6 Scylla of Nonlocality or Charybdis of Nonunitarity? 37.1.6.1 Scattering Theory 38.1.6.2 Giving up the Locality of Interaction 39.1.6.3 The Threat of the Loss of Unitarity 43.1.7 Trends 45.1.7.1 Giving Up Metrics 45.1.7.2 Giving Up Unitarity 46.1.7.3 Giving Up Quantization 47.References 50.2 Operators of the Quantum Harmonic Oscillator and Its Relatives 59Franciszek Hugon Szafraniec.2.1 Introducing to Unbounded Hilbert Space Operators 60.2.1.1 How to Understand an Unbounded Operator 60.2.1.2 Very Basic Notions and Facts 60.2.1.3 Subnormal Operators 73.2.1.4 Operators in the Reproducing Kernel Hilbert Space 78.2.2 Commutation Relations 88.2.2.1 The Commutation Relation of the Quantum Harmonic Oscillator 90.2.2.2 Duality 101.2.3 The q Oscillators 106.2.3.1 Spatial Interpretation of (q–o) 107.2.3.2 Subnormality in the q–Oscillator 110.2.4 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 121Fabio Bagarello.3.1 Introduction 121.3.2 The Mathematics of D–PBs 123.3.2.1 Some Preliminary Results on Bases and Complete Sets 123.3.2.2 Back to D–PBs 126.3.2.3 The Operators S and S 129.3.2.4 –Conjugate Operators for D–Quasi Bases 134.3.2.5 D–PBs versus Bosons 140.3.3 D–PBs in Quantum Mechanics 145.3.3.1 The Harmonic Oscillator: Losing Self–adjointness 145.3.3.2 A Two–dimensional Model in a Flat noncommutative space 152.3.4 Other Appearances of D–PBs in Quantum Mechanics 158.3.4.1 The Extended Quantum Harmonic Oscillator 158.3.4.2 The Swanson Model 159.3.4.3 Generalized Landau Levels 163.3.4.4 An Example by Bender and Jones 165.3.4.5 A Perturbed Harmonic Oscillator in d = 2 169.3.4.6 A Last Perturbative Example 171.3.5 A Much Simpler Case: Pseudo–Fermions 174.3.5.1 A First Example from the Literature 178.3.5.2 More Examples from the Literature 180.3.6 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 Schrödinger Operators and for the Existence of PT –Symmetric Phase Transitions 189Emanuela 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 Schrödinger Operators with Real Spectrum 200.4.5 An Example of PT –Symmetric Phase Transition 206.4.5.1 Holomorphy and Borel Summability at Infinity 213.4.5.2 Analytic Continuation of the Eigenvalues and Proof of the Theorem 217.4.6 The Method of the Quantum Normal Form 219.4.6.1 The Quantum Normal Form: the Formal Construction 226.4.6.2 Reality of Bk: the Inductive Argument 229.4.6.3 Vanishing of the Odd Terms 2s+1 230.Appendix: Moyal Brackets and the Weyl Quantization 232.A.1 Moyal Brackets 232.A.2 The Weyl Quantization 236.References 238.5 Elements of Spectral Theory without the Spectral Theorem 241David Krej i ík and Petr Siegl.5.1 Introduction 241.5.2 Closed Operators in Hilbert Spaces 242.5.2.1 Basic Notions 243.5.2.2 Spectra 246.5.2.3 Numerical Range 249.5.2.4 Sectoriality and Accretivity 250.5.2.5 Symmetries 253.5.3 How to Whip Up a Closed Operator 257.5.3.1 Closed Sectorial Forms 258.5.3.2 Friedrichs Extension 259.5.3.3 M–accretive Realizations of Schrödinger Operators 262.5.3.4 Small Perturbations 263.5.4 Compactness and a Spectral Life Without It 266.5.4.1 Compact Operators and Compact Resolvents 266.5.4.2 Essential Spectra 269.5.4.3 Stability of the Essential Spectra 272.5.5 Similarity to Normal Operators 273.5.5.1 Similarity Transforms 274.5.5.2 Quasi–Self–Adjoint Operators 276.5.5.3 Basis Properties of Eigensystems 278.5.6 Pseudospectra 281.5.6.1 Definition and Basic Properties 281.5.6.2 Main Tool from Microlocal Analysis 283.References 288.6 PT–Symmetric Operators in Quantum Mechanics: Krein Spaces Methods 293Sergio Albeverio and Sergii Kuzhel.6.1 Introduction 293.6.2 Elements of the Krein Spaces Theory 295.6.2.1 Definition of the Krein Spaces 295.6.2.2 Bounded Operators C, 297.6.2.3 Unbounded Operators C, 300.6.3 Self–Adjoint Operators in Krein Spaces 304.6.3.1 Definitions and General Properties 304.6.3.2 Similarity to Self–adjoint Operators 307.6.3.3 The Property of Unbounded C–symmetry 313.6.4 Elements of PT–Symmetric Operators Theory 320.6.4.1 Definition of PT–Symmetric Operators and General Properties 320.6.4.2 Two–dimensional Case 325.6.4.3 Schrödinger Operator with C–symmetric Zero–range Potentials 333.References 340.7 Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces 345Jean–Pierre Antoine and Camillo Trapani.7.1 Introduction 345.7.2 Some Terminology 347.7.3 Similar and Quasi–Similar Operators 349.7.3.1 Similarity 349.7.3.2 Quasi–Similitarity and Spectra 353.7.3.3 Quasi–Similarity with an Unbounded Intertwining Operator 359.7.4 The Lattice Generated by a Single Metric Operator 362.7.4.1 Bounded Metric Operators 365.7.4.2 Unbounded Metric Operators 365.7.5 Quasi–Hermitian Operators 367.7.5.1 Changing the Norm: Two–Hilbert Space Formalism 367.7.5.2 Bounded Quasi–Hermitian Operators 371.7.5.3 Unbounded Quasi–Hermitian Operators 371.7.5.4 Quasi–Hermitian Operators with Unbounded Metric Operators 377.7.5.5 Example: Operators Defined from Riesz Bases 377.7.6 The LHS Generated by Metric Operators 380.7.7 Similarity for PIP–Space Operators 382.7.7.1 General PIP–Space Operators 382.7.7.2 The Case of Symmetric PIP–Space Operators 384.7.7.3 Semisimilarity 387.7.8 The Case of Pseudo–Hermitian Hamiltonians 389.7.8.1 An Example 391.7.9 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

• ISBN: 978-1-118-85528-7
• Editorial: Wiley–Blackwell
• Encuadernacion: Cartoné
• Páginas: 432
• Fecha Publicación: 29/05/2015
• Nº Volúmenes: 1
• Idioma: Inglés