Non selfadjoint operators in quantum physics mathematical aspects 1st Edition by Fabio Bagarello, Jean-Pierre Gazeau, Franciszek Hugon Szafraniec , Miloslav Znojil ISBN 978-1118855287 1118855287

Non selfadjoint operators in quantum physics mathematical aspects 1st Edition by Fabio Bagarello, Jean-Pierre Gazeau, Franciszek Hugon Szafraniec , Miloslav Znojil- Ebook PDF Instant Download/Delivery:  978-1118855287,  1118855287 
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Product details: 

ISBN 10:   1118855287 

ISBN 13:  978-1118855287

Author: Fabio Bagarello, Jean-Pierre Gazeau, Franciszek Hugon Szafraniec , Miloslav Znojil

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.

Table of contents: 

1 Non-Self-Adjoint Operators in Quantum Physics: Ideas, People, and Trends
Miloslav Znojil

• 1.1 The Challenge of Non-Hermiticity in Quantum Physics

• 1.2 A Periodization of the Recent History of Study of Non-Self-Adjoint Operators in Quantum Physics

• 1.3 Main Message: New Classes of Quantum Bound States

• 1.4 Probabilistic Interpretation of the New Models

• 1.5 Innovations in Mathematical Physics

• 1.6 Scylla of Nonlocality or Charybdis of Nonunitarity?

• 1.7 Trends

• References

2 Operators of the Quantum Harmonic Oscillator and Its Relatives
Franciszek Hugon Szafraniec

• 2.1 Introducing to Unbounded Hilbert Space Operators

• 2.2 Commutation Relations

• 2.3 The q–Oscillators

• 2.4 Back to “Hermicity”—A Way to See It

• Concluding Remarks

• References

3 Deformed Canonical (Anti-)Commutation Relations and Non-Self-Adjoint Hamiltonians
Fabio Bagarello

• 3.1 Introduction

• 3.2 The Mathematics of D-PBs

• 3.3 D-PBs in Quantum Mechanics

• 3.4 Other Appearances of D-PBs in Quantum Mechanics

• 3.5 A Much Simpler Case: Pseudo-Fermions

• 3.6 A Possible Extension: Nonlinear D-PBs

• 3.7 Conclusions

• 3.8 Acknowledgments

• References

4 Criteria for the Reality of the Spectrum of PT-Symmetric Schrödinger Operators and for the Existence of PT-Symmetric Phase Transitions
Emanuela Caliceti and Sandro Graffi

• 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

• 4.6 The Method of the Quantum Normal Form

• Appendix: Moyal Brackets and the Weyl Quantization

  • A.1 Moyal Brackets

  • A.2 The Weyl Quantization

• References

5 Elements of Spectral Theory without the Spectral Theorem
David Krejèiøík and Petr Siegl

• 5.1 Introduction

• 5.2 Closed Operators in Hilbert Spaces

• 5.3 How to Whip Up a Closed Operator

• 5.4 Compactness and a Spectral Life Without It

• 5.5 Similarity to Normal Operators

• 5.6 Pseudospectra

• References

6 PT-Symmetric Operators in Quantum Mechanics: Krein Spaces Methods
Sergio Albeverio and Sergii Kuzhel

• 6.1 Introduction

• 6.2 Elements of the Krein Spaces Theory

• 6.3 Self-Adjoint Operators in Krein Spaces

• 6.4 Elements of PT-Symmetric Operators Theory

• References

7 Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces
Jean-Pierre Antoine and Camillo Trapani

• 7.1 Introduction

• 7.2 Some Terminology

• 7.3 Similar and Quasi-Similar Operators

• 7.4 The Lattice Generated by a Single Metric Operator

• 7.5 Quasi-Hermitian Operators

• 7.6 The LHS Generated by Metric Operators

• 7.7 Similarity for PIP-Space Operators

• 7.8 The Case of Pseudo-Hermitian Hamiltonians

• 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

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Tags: Fabio Bagarello, Jean Pierre Gazeau, Franciszek Hugon Szafraniec, Miloslav Znojil, Non selfadjoint, quantum physics mathematical