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Pt Symmetry: In Quantum And Classical Physics [Minkštas viršelis]

(Washington Univ In St Louis, Usa), Contributions by (Hungarian Academy Of S), Contributions by (Imperial College London, Uk), Contributions by (Imperial College London, Uk), Contributions by (City, Univ Of London, Uk), Contributions by (Agh Univ Of Science & Technology, Poland), Contributions by (Durham Univ, Uk), Contributions by , Contributions by (Univ Of Kent, Uk)
  • Formatas: Paperback / softback, 468 pages
  • Išleidimo metai: 24-Jan-2019
  • Leidėjas: World Scientific Europe Ltd
  • ISBN-10: 1786346680
  • ISBN-13: 9781786346681
Kitos knygos pagal šią temą:
Pt Symmetry: In Quantum And Classical PhysicsDidesnis paveikslėlis
  • Formatas: Paperback / softback, 468 pages
  • Išleidimo metai: 24-Jan-2019
  • Leidėjas: World Scientific Europe Ltd
  • ISBN-10: 1786346680
  • ISBN-13: 9781786346681
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781786346681.html
Raktažodžiai:
'The text is easy to read because the matter is clearly explained. Symmetries are a central component of physical laws, and the PT-symmetry proves to be very interesting and fruitful. The discussion of the matter is up-to-date and self-contained. The book is recommended to students of higher courses, PhD and researchers. It is also a basic read to those who wish to have an insight into this field.'Contemporary PhysicsOriginated by the author in 1998, the field of PT (parity-time) symmetry has become an extremely active and exciting area of research. PT-symmetric quantum and classical systems have theoretical, experimental, and commercial applications, and have been the subject of many journal articles, PhD theses, conferences, and symposia. Carl Bender's work has influenced major advances in physics and generations of students.This book is an accessible entry point to PT symmetry, ideal for students and scientists looking to begin their own research projects in this field.
Preface vii
About the Authors xi
Acknowledgment xv
Part I Introduction to PT Symmetry
1(172)
1 Basics of PT Symmetry
3(36)
1.1 Open, Closed, and PT-Symmetric Systems
3(4)
1.2 Simple PT-Symmetric Matrix Hamiltonians
7(4)
1.3 Real Secular Equation for PT-Symmetric Hamiltonians
11(1)
1.4 Classical PT-Symmetric Coupled Oscillators
11(6)
1.5 Complex Deformation of Real Physical Theories
17(8)
1.6 Classical Mechanics in the Complex Domain
25(5)
1.7 Complex Deformed Classical Harmonic Oscillator
30(9)
2 PT-Symmetric Eigenvalue Problems
39(44)
2.1 Examples of PT-Deformed Eigenvalue Problems
41(6)
2.2 Deformed Eigenvalue Problems and Stokes Sectors
47(8)
2.2.1 Resolution of puzzles in examples of Sec. 2.1
47(3)
2.2.2 Analytic deformation of eigenvalue problems
50(5)
2.3 Proof of Spectral Reality for -- x4 Potential
55(4)
2.4 Additional PT-Deformed Eigenvalue Problems
59(12)
2.5 Numerical Calculation of Eigenvalues
71(3)
2.5.1 Shooting method
72(1)
2.5.2 Variational techniques
73(1)
2.6 Approximate Analytical Calculation of Eigenvalues
74(1)
2.7 Eigenvalues in the Broken-PT-Symmetric Region
75(8)
3 PT-Symmetric Quantum Mechanics
83(24)
3.1 Hermitian Quantum Mechanics
84(2)
3.2 PT-Symmetric Quantum Mechanics
86(5)
3.3 Comparison of Hermitian and PT-Symmetric Theories
91(1)
3.4 Observables
92(1)
3.5 Pseudo-Hermiticity and Quasi-Hermiticity
93(1)
3.6 Model 2 × 2 PT-Symmetric Matrix Hamiltonian
94(2)
3.7 Calculating the C Operator
96(2)
3.8 Algebraic Equations Satisfied by C
98(6)
3.8.1 Perturbative calculation of C
99(2)
3.8.2 Calculation of C for other Hamiltonians
101(3)
3.9 Mapping PT-Symmetric to Hermitian Hamiltonians
104(3)
4 PT-Symmetric Classical Mechanics
107(26)
4.1 Classical Trajectories for Noninteger ε
108(5)
4.2 Some PT-Symmetric Classical Dynamical Systems
113(11)
4.2.1 Lotka--Volterra equations for predator-prey models
113(3)
4.2.2 Euler's equation for a rotating rigid body
116(2)
4.2.3 Simple pendulum
118(3)
4.2.4 Classical trajectories for isospectral Hamiltonians
121(2)
4.2.5 A more complicated oscillatory system
123(1)
4.3 Complex Probability
124(8)
4.3.1 PT-symmetric classical random walk
125(3)
4.3.2 Probability density for PT quantum mechanics
128(4)
4.4 PT-Symmetric Classical Field Theory
132(1)
5 PT-Symmetric Quantum Field Theory
133(40)
5.1 Introduction to PT-Symmetric Quantum Field Theory
134(2)
5.2 Perturbative and Nonperturbative Behavior
136(10)
5.2.1 Cubic PT-symmetric quantum field theory
136(2)
5.2.2 Quartic PT-symmetric quantum field theory
138(2)
5.2.3 Zero-dimensional PT-symmetric field theory
140(2)
5.2.4 Saddle-point analysis of PT-symmetric theories
142(4)
5.3 Nonvanishing One-Point Green's Function
146(5)
5.3.1 Derivation of the Dyson-Schwinger equations
147(3)
5.3.2 Truncation of the Dyson-Schwinger equations
150(1)
5.4 C Operator for Cubic PT-Symmetric Field Theory
151(4)
5.4.1 iφ3 quantum field theory
152(1)
5.4.2 Other cubic quantum field theories
153(2)
5.5 Bound States in a PT-Symmetric Quartic Potential
155(2)
5.6 Lee Model
157(7)
5.7 Other PT-Symmetric Quantum Field Theories
164(9)
5.7.1 Higgs sector of the Standard Model
165(1)
5.7.2 PT-symmetric quantum electrodynamics
166(1)
5.7.3 Dual PT-symmetric quantum field theories
167(4)
5.7.4 PT-symmetric theories of gravity and cosmology
171(1)
5.7.5 Double-scaling limit
172(1)
5.7.6 Fundamental properties of fermionic theories
172(1)
Part II Advanced Topics in PT Symmetry
173(220)
6 Proof of Reality for Some Simple Examples
175(46)
6.1 Stokes Phenomenon
176(4)
6.2 Functional Relations
180(5)
6.3 Proof of Reality
185(3)
6.4 General Cubic Oscillators
188(8)
6.5 Generalized Bender--Boettcher Hamiltonian
196(2)
6.6 Reality Domains for the Generalized Problem
198(9)
6.7 Quasi-Exactly-Solvable Models
207(12)
6.8 Concluding Remarks
219(2)
7 Exactly Solvable PT-Symmetric Models
221(40)
7.1 Exactly Solvable Potentials
221(1)
7.2 Generating Real Exactly Solvable Potentials
222(4)
7.2.1 Method 1: Variable transformations
223(1)
7.2.2 Method 2: Supersymmetric quantum mechanics
224(2)
7.3 Types of Exactly Solvable Potentials
226(9)
7.3.1 Natanzon and Natanzon-confluent potentials
227(2)
7.3.2 Shape-invariant potentials
229(2)
7.3.3 Beyond Natanzon class: More general ψ(x)
231(1)
7.3.4 Beyond Natanzon class: Other functions F(z)
232(2)
7.3.5 Further types of solvable potentials
234(1)
7.4 Aspects of PT-Symmetric Potentials
235(4)
7.4.1 Constructing PT-symmetric potentials
235(1)
7.4.2 Energy spectrum and breaking of PT symmetry
236(1)
7.4.3 Inner product, pseudonorm, and the C operator
237(1)
7.4.4 SUSYQM and PT symmetry
238(1)
7.4.5 Scattering in PT-symmetric potentials
239(1)
7.5 Examples of Solvable PT-Symmetric Potentials
239(21)
7.5.1 Shape-invariant potentials
239(9)
7.5.2 Natanzon potentials --- An example
248(4)
7.5.3 Potentials generated by SUSY transformations
252(2)
7.5.4 Potentials solved by other functions
254(4)
7.5.5 Further solvable potentials and generalizations
258(2)
Acknowledgment
260(1)
8 Krein-Space Theory and PTQM
261(44)
8.1 Introduction
261(5)
8.2 Terminology and Notation
266(4)
8.3 Elements of Krein-Space Theory
270(18)
8.3.1 Definition and elementary properties
270(5)
8.3.2 Definition of C operators
275(3)
8.3.3 Bounded and unbounded C operators
278(1)
8.3.4 Linear operators having C-symmetry
279(1)
8.3.5 P-symmetric and P-Hermitian operators
280(4)
8.3.6 C-symmetries for P-Hermitian operators
284(2)
8.3.7 Bounded C operators and Riccati equation
286(2)
8.4 P-Symmetric Operators with Complete Set of Eigenvectors
288(8)
8.4.1 Preliminaries: Best-choice problem
288(2)
8.4.2 Riesz basis of eigenvectors
290(1)
8.4.3 Schauder basis of eigenvectors
291(1)
8.4.4 Complete set of eigenvectors and quasi basis
292(4)
8.5 Property of PT Symmetry
296(9)
8.5.1 "PT-symmetric operators
296(6)
8.5.2 PT-symmetric operators having C-symmetry
302(3)
9 PT-Symmetric Deformations of Nonlinear Integrable Systems
305(46)
9.1 Basics of Classical Integrable Systems
306(11)
9.1.1 Isospectral deformation method (Lax pairs)
307(5)
9.1.2 Painleve test
312(1)
9.1.3 Transformation methods
313(4)
9.2 PT Deformation of Nonlinear Wave Equations
317(7)
9.2.1 PT-deformed supersymmetric equations
319(2)
9.2.2 PT-deformed Burgers equation
321(1)
9.2.3 PT-deformed KdV equation
322(1)
9.2.4 PT-deformed compacton equations
323(1)
9.2.5 PT-deformed supersymmetric equations (revisited)
323(1)
9.3 Properties of PT-Deformed Nonlinear Wave Equations
324(17)
9.3.1 Painleve tests of PT-deformed Burgers equations
324(4)
9.3.2 Painleve procedure for deformed KdV equation
328(2)
9.3.3 Conserved quantities
330(1)
9.3.4 Solutions of PT-deformed nonlinear equations
331(10)
9.3.5 From wave equations to quantum mechanics
341(1)
9.4 PT-Deformed Calogero-Moser-Sutherland Models
341(9)
9.4.1 Extended Calogero-Moser-Sutherland models
342(2)
9.4.2 From fields to particles
344(2)
9.4.3 Deformed Calogero-Moser-Sutherland models
346(4)
Acknowledgment
350(1)
10 PT Symmetry in Optics
351(42)
10.1 Paraxial Approximation
352(2)
10.2 First Applications
354(4)
10.3 A Simpler System: Coupled Waveguides
358(3)
10.4 Unidirectional Invisibility
361(11)
10.4.1 Coupled-mode approximation
363(1)
10.4.2 Analytic solution for the scattering coefficients
363(2)
10.4.3 Wronskians and pseudo-unitarity
365(3)
10.4.4 Transfer matrix
368(4)
10.5 PT Lasers
372(5)
10.6 Supersymmetry in Quantum Mechanics and Optics
377(3)
10.7 Wave Propagation in Discrete PT Systems
380(5)
10.7.1 Propagation in infinite systems
380(3)
10.7.2 Finite systems: Dimers, trimers, quadrimers
383(2)
10.8 Optical Solitons
385(2)
10.9 Cloaking, Metamaterials, and Metasurfaces
387(5)
10.9.1 One-way invisibility cloak
388(1)
10.9.2 Cloaking by metasurface
389(3)
10.10 Conclusion
392(1)
Bibliography 393(44)
Index 437