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Non-Selfadjoint Operators in Quantum Physics : Mathematical Aspects.

By: Contributor(s): Publisher: Hoboken : John Wiley & Sons, Incorporated, 2015Copyright date: ©2015Edition: 1st edDescription: 1 online resource (434 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781118855270
Subject(s): Genre/Form: Additional physical formats: Print version:: Non-Selfadjoint Operators in Quantum Physics : Mathematical AspectsDDC classification:
  • 530.1201/515724
LOC classification:
  • QA329.2 -- .N67 2015eb
Online resources:
Contents:
Cover -- Title Page -- Copyright -- Dedication -- Contributors -- Contents in Brief -- Contents -- Preface -- Acronyms -- Glossary -- Symbols -- Introduction -- References -- Chapter 1 Non-Self-Adjoint Operators in Quantum Physics: Ideas, People, and Trends -- 1.1 The Challenge of Non-Hermiticity in Quantum Physics -- 1.1.1 A Few Quantum Physics' Anniversaries, for Introduction -- 1.1.2 Dozen Years of Conferences Dedicated to Pseudo-Hermiticity -- 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 -- 1.2.2 The Periods of Growth -- 1.3 Main Message: New Classes of Quantum Bound States -- 1.3.1 Real Energies via Non-Hermitian Hamiltonians -- 1.3.2 Analytic and Algebraic Constructions -- 1.3.3 Qualitative Innovations of Phenomenological Quantum Models -- 1.4 Probabilistic Interpretation of the New Models -- 1.4.1 Variational Constructions -- 1.4.2 Non-Dirac Hilbert-Space Metrics Θ ≠ I -- 1.5 Innovations in Mathematical Physics -- 1.5.1 Simplified Schrödinger Equations -- 1.5.2 Nonconservative Systems and Time-Dependent Dyson Mappings -- 1.6 Scylla of Nonlocality or Charybdis of Nonunitarity? -- 1.6.1 Scattering Theory -- 1.6.2 Giving up the Locality of Interaction -- 1.6.3 The Threat of the Loss of Unitarity -- 1.7 Trends -- 1.7.1 Giving Up Metrics -- 1.7.2 Giving Up Unitarity -- 1.7.3 Giving Up Quantization -- References -- Chapter 2 Operators of the Quantum Harmonic Oscillator and Its Relatives -- 2.1 Introducing to Unbounded Hilbert Space Operators -- 2.1.1 How to Understand an Unbounded Operator -- 2.1.2 Very Basic Notions and Facts -- 2.1.3 Subnormal Operators -- 2.1.4 Operators in the Reproducing Kernel Hilbert Space -- 2.2 Commutation Relations -- 2.2.1 The Commutation Relation of the Quantum Harmonic Oscillator -- 2.2.2 Duality -- 2.3 The q-Oscillators.
2.3.1 Spatial Interpretation of (q-o) -- 2.3.2 Subnormality in the q-Oscillator -- 2.4 Back to "Hermicity"-A Way to See It -- Concluding Remarks -- References -- Chapter 3 Deformed Canonical (Anti-)Commutation Relations and Non-Self-Adjoint Hamiltonians -- 3.1 Introduction -- 3.2 The Mathematics of D-PBs -- 3.2.1 Some Preliminary Results on Bases and Complete Sets -- 3.2.2 Back to D-PBs -- 3.2.3 The Operators Sφ and Sψ -- 3.2.4 Θ-Conjugate Operators for D-Quasi Bases -- 3.2.5 D-PBs versus Bosons -- 3.3 D-PBs in Quantum Mechanics -- 3.3.1 The Harmonic Oscillator: Losing Self-adjointness -- 3.3.2 A Two-dimensional Model in a Flat noncommutative space -- 3.4 Other Appearances of D-PBs in Quantum Mechanics -- 3.4.1 The Extended Quantum Harmonic Oscillator -- 3.4.2 The Swanson Model -- 3.4.3 Generalized Landau Levels -- 3.4.4 An Example by Bender and Jones -- 3.4.5 A Perturbed Harmonic Oscillator in d=2 -- 3.4.6 A Last Perturbative Example -- 3.5 A Much Simpler Case: Pseudo-Fermions -- 3.5.1 A First Example from the Literature -- 3.5.2 More Examples from the Literature -- 3.6 A Possible Extension: Nonlinear D-PBs -- 3.7 Conclusions -- 3.8 Acknowledgments -- References -- Chapter 4 Criteria for the Reality of the Spectrum of PT-Symmetric Schrödinger Operators and for the Existence of PT-Symmetric Phase Transitions -- 4.1 Introduction -- 4.2 Perturbation Theory and Global Control of the Spectrum -- 4.3 One-Dimensional PT-Symmetric Hamiltonians: Criteria for the Reality of the Spectrum -- 4.4 PT-Symmetric Periodic Schrödinger Operators with Real Spectrum -- 4.5 An Example of PT-Symmetric Phase Transition -- 4.5.1 Holomorphy and Borel Summability at Infinity -- 4.5.2 Analytic Continuation of the Eigenvalues and Proof of the Theorem -- 4.6 The Method of the Quantum Normal Form -- 4.6.1 The Quantum Normal Form: the Formal Construction.
4.6.2 Reality of Bk: the Inductive Argument -- 4.6.3 Vanishing of the Odd Terms B2s+1 -- Appendix: Moyal Brackets and the Weyl Quantization -- A.1 Moyal Brackets -- A.2 The Weyl Quantization -- References -- Chapter 5 Elements of Spectral Theory without the Spectral Theorem -- 5.1 Introduction -- 5.2 Closed Operators in Hilbert Spaces -- 5.2.1 Basic Notions -- 5.2.2 Spectra -- 5.2.3 Numerical Range -- 5.2.4 Sectoriality and Accretivity -- 5.2.5 Symmetries -- 5.3 How to Whip Up a Closed Operator -- 5.3.1 Closed Sectorial Forms -- 5.3.2 Friedrichs' Extension -- 5.3.3 M-accretive Realizations of Schrödinger Operators -- 5.3.4 Small Perturbations -- 5.4 Compactness and a Spectral Life Without It -- 5.4.1 Compact Operators and Compact Resolvents -- 5.4.2 Essential Spectra -- 5.4.3 Stability of the Essential Spectra -- 5.5 Similarity to Normal Operators -- 5.5.1 Similarity Transforms -- 5.5.2 Quasi-Self-Adjoint Operators -- 5.5.3 Basis Properties of Eigensystems -- 5.6 Pseudospectra -- 5.6.1 Definition and Basic Properties -- 5.6.2 Main Tool from Microlocal Analysis -- References -- Chapter 6 PT-Symmetric Operators in Quantum Mechanics: Krein Spaces Methods -- 6.1 Introduction -- 6.2 Elements of the Krein Spaces Theory -- 6.2.1 Definition of the Krein Spaces -- 6.2.2 Bounded Operators C -- 6.2.3 Unbounded Operators C -- 6.3 Self-Adjoint Operators in Krein Spaces -- 6.3.1 Definitions and General Properties -- 6.3.2 Similarity to Self-adjoint Operators -- 6.3.3 The Property of Unbounded C-symmetry -- 6.4 Elements of PT-Symmetric Operators Theory -- 6.4.1 Definition of PT-Symmetric Operators and General Properties -- 6.4.2 Two-dimensional Case -- 6.4.3 Schrödinger Operator with PT-symmetric Zero-range Potentials -- References -- Chapter 7 Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces -- 7.1 Introduction -- 7.2 Some Terminology.
7.3 Similar and Quasi-Similar Operators -- 7.3.1 Similarity -- 7.3.2 Quasi-Similitarity and Spectra -- 7.3.3 Quasi-Similarity with an Unbounded Intertwining Operator -- 7.4 The Lattice Generated by a Single Metric Operator -- 7.4.1 Bounded Metric Operators -- 7.4.2 Unbounded Metric Operators -- 7.5 Quasi-Hermitian Operators -- 7.5.1 Changing the Norm: Two-Hilbert Space Formalism -- 7.5.2 Bounded Quasi-Hermitian Operators -- 7.5.3 Unbounded Quasi-Hermitian Operators -- 7.5.4 Quasi-Hermitian Operators with Unbounded Metric Operators -- 7.5.5 Example: Operators Defined from Riesz Bases -- 7.6 The LHS Generated by Metric Operators -- 7.7 Similarity for PIP-Space Operators -- 7.7.1 General PIP-Space Operators -- 7.7.2 The Case of Symmetric PIP-Space Operators -- 7.7.3 Semisimilarity -- 7.8 The Case of Pseudo-Hermitian Hamiltonians -- 7.8.1 An Example -- 7.9 Conclusion -- Appendix: Partial Inner Product Spaces -- A.1 PIP-Spaces and Indexed PIP-Spaces -- A.2 Operators on Indexed PIP-space S -- A.2.1 Symmetric Operators -- A.2.2 Regular Operators, Morphisms, and Projections -- References -- Index -- EULA.
Summary: 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-levelSummary: and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.
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Cover -- Title Page -- Copyright -- Dedication -- Contributors -- Contents in Brief -- Contents -- Preface -- Acronyms -- Glossary -- Symbols -- Introduction -- References -- Chapter 1 Non-Self-Adjoint Operators in Quantum Physics: Ideas, People, and Trends -- 1.1 The Challenge of Non-Hermiticity in Quantum Physics -- 1.1.1 A Few Quantum Physics' Anniversaries, for Introduction -- 1.1.2 Dozen Years of Conferences Dedicated to Pseudo-Hermiticity -- 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 -- 1.2.2 The Periods of Growth -- 1.3 Main Message: New Classes of Quantum Bound States -- 1.3.1 Real Energies via Non-Hermitian Hamiltonians -- 1.3.2 Analytic and Algebraic Constructions -- 1.3.3 Qualitative Innovations of Phenomenological Quantum Models -- 1.4 Probabilistic Interpretation of the New Models -- 1.4.1 Variational Constructions -- 1.4.2 Non-Dirac Hilbert-Space Metrics Θ ≠ I -- 1.5 Innovations in Mathematical Physics -- 1.5.1 Simplified Schrödinger Equations -- 1.5.2 Nonconservative Systems and Time-Dependent Dyson Mappings -- 1.6 Scylla of Nonlocality or Charybdis of Nonunitarity? -- 1.6.1 Scattering Theory -- 1.6.2 Giving up the Locality of Interaction -- 1.6.3 The Threat of the Loss of Unitarity -- 1.7 Trends -- 1.7.1 Giving Up Metrics -- 1.7.2 Giving Up Unitarity -- 1.7.3 Giving Up Quantization -- References -- Chapter 2 Operators of the Quantum Harmonic Oscillator and Its Relatives -- 2.1 Introducing to Unbounded Hilbert Space Operators -- 2.1.1 How to Understand an Unbounded Operator -- 2.1.2 Very Basic Notions and Facts -- 2.1.3 Subnormal Operators -- 2.1.4 Operators in the Reproducing Kernel Hilbert Space -- 2.2 Commutation Relations -- 2.2.1 The Commutation Relation of the Quantum Harmonic Oscillator -- 2.2.2 Duality -- 2.3 The q-Oscillators.

2.3.1 Spatial Interpretation of (q-o) -- 2.3.2 Subnormality in the q-Oscillator -- 2.4 Back to "Hermicity"-A Way to See It -- Concluding Remarks -- References -- Chapter 3 Deformed Canonical (Anti-)Commutation Relations and Non-Self-Adjoint Hamiltonians -- 3.1 Introduction -- 3.2 The Mathematics of D-PBs -- 3.2.1 Some Preliminary Results on Bases and Complete Sets -- 3.2.2 Back to D-PBs -- 3.2.3 The Operators Sφ and Sψ -- 3.2.4 Θ-Conjugate Operators for D-Quasi Bases -- 3.2.5 D-PBs versus Bosons -- 3.3 D-PBs in Quantum Mechanics -- 3.3.1 The Harmonic Oscillator: Losing Self-adjointness -- 3.3.2 A Two-dimensional Model in a Flat noncommutative space -- 3.4 Other Appearances of D-PBs in Quantum Mechanics -- 3.4.1 The Extended Quantum Harmonic Oscillator -- 3.4.2 The Swanson Model -- 3.4.3 Generalized Landau Levels -- 3.4.4 An Example by Bender and Jones -- 3.4.5 A Perturbed Harmonic Oscillator in d=2 -- 3.4.6 A Last Perturbative Example -- 3.5 A Much Simpler Case: Pseudo-Fermions -- 3.5.1 A First Example from the Literature -- 3.5.2 More Examples from the Literature -- 3.6 A Possible Extension: Nonlinear D-PBs -- 3.7 Conclusions -- 3.8 Acknowledgments -- References -- Chapter 4 Criteria for the Reality of the Spectrum of PT-Symmetric Schrödinger Operators and for the Existence of PT-Symmetric Phase Transitions -- 4.1 Introduction -- 4.2 Perturbation Theory and Global Control of the Spectrum -- 4.3 One-Dimensional PT-Symmetric Hamiltonians: Criteria for the Reality of the Spectrum -- 4.4 PT-Symmetric Periodic Schrödinger Operators with Real Spectrum -- 4.5 An Example of PT-Symmetric Phase Transition -- 4.5.1 Holomorphy and Borel Summability at Infinity -- 4.5.2 Analytic Continuation of the Eigenvalues and Proof of the Theorem -- 4.6 The Method of the Quantum Normal Form -- 4.6.1 The Quantum Normal Form: the Formal Construction.

4.6.2 Reality of Bk: the Inductive Argument -- 4.6.3 Vanishing of the Odd Terms B2s+1 -- Appendix: Moyal Brackets and the Weyl Quantization -- A.1 Moyal Brackets -- A.2 The Weyl Quantization -- References -- Chapter 5 Elements of Spectral Theory without the Spectral Theorem -- 5.1 Introduction -- 5.2 Closed Operators in Hilbert Spaces -- 5.2.1 Basic Notions -- 5.2.2 Spectra -- 5.2.3 Numerical Range -- 5.2.4 Sectoriality and Accretivity -- 5.2.5 Symmetries -- 5.3 How to Whip Up a Closed Operator -- 5.3.1 Closed Sectorial Forms -- 5.3.2 Friedrichs' Extension -- 5.3.3 M-accretive Realizations of Schrödinger Operators -- 5.3.4 Small Perturbations -- 5.4 Compactness and a Spectral Life Without It -- 5.4.1 Compact Operators and Compact Resolvents -- 5.4.2 Essential Spectra -- 5.4.3 Stability of the Essential Spectra -- 5.5 Similarity to Normal Operators -- 5.5.1 Similarity Transforms -- 5.5.2 Quasi-Self-Adjoint Operators -- 5.5.3 Basis Properties of Eigensystems -- 5.6 Pseudospectra -- 5.6.1 Definition and Basic Properties -- 5.6.2 Main Tool from Microlocal Analysis -- References -- Chapter 6 PT-Symmetric Operators in Quantum Mechanics: Krein Spaces Methods -- 6.1 Introduction -- 6.2 Elements of the Krein Spaces Theory -- 6.2.1 Definition of the Krein Spaces -- 6.2.2 Bounded Operators C -- 6.2.3 Unbounded Operators C -- 6.3 Self-Adjoint Operators in Krein Spaces -- 6.3.1 Definitions and General Properties -- 6.3.2 Similarity to Self-adjoint Operators -- 6.3.3 The Property of Unbounded C-symmetry -- 6.4 Elements of PT-Symmetric Operators Theory -- 6.4.1 Definition of PT-Symmetric Operators and General Properties -- 6.4.2 Two-dimensional Case -- 6.4.3 Schrödinger Operator with PT-symmetric Zero-range Potentials -- References -- Chapter 7 Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces -- 7.1 Introduction -- 7.2 Some Terminology.

7.3 Similar and Quasi-Similar Operators -- 7.3.1 Similarity -- 7.3.2 Quasi-Similitarity and Spectra -- 7.3.3 Quasi-Similarity with an Unbounded Intertwining Operator -- 7.4 The Lattice Generated by a Single Metric Operator -- 7.4.1 Bounded Metric Operators -- 7.4.2 Unbounded Metric Operators -- 7.5 Quasi-Hermitian Operators -- 7.5.1 Changing the Norm: Two-Hilbert Space Formalism -- 7.5.2 Bounded Quasi-Hermitian Operators -- 7.5.3 Unbounded Quasi-Hermitian Operators -- 7.5.4 Quasi-Hermitian Operators with Unbounded Metric Operators -- 7.5.5 Example: Operators Defined from Riesz Bases -- 7.6 The LHS Generated by Metric Operators -- 7.7 Similarity for PIP-Space Operators -- 7.7.1 General PIP-Space Operators -- 7.7.2 The Case of Symmetric PIP-Space Operators -- 7.7.3 Semisimilarity -- 7.8 The Case of Pseudo-Hermitian Hamiltonians -- 7.8.1 An Example -- 7.9 Conclusion -- Appendix: Partial Inner Product Spaces -- A.1 PIP-Spaces and Indexed PIP-Spaces -- A.2 Operators on Indexed PIP-space S -- A.2.1 Symmetric Operators -- A.2.2 Regular Operators, Morphisms, and Projections -- References -- Index -- EULA.

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.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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