Asymptotic time decay in quantum physics

Time decays form the basis of a multitude of important and interesting phenomena in quantum physics that range from spectral properties, resonances, return and approach to equilibrium, to quantum mixing, dynamical stability properties and irreversibility and the "arrow of time".This monograph is devoted to a clear and precise, yet pedagogical account of the associated concepts and methods.

Preface: A Description of Contents	p. vii
Acknowledgments	p. xiii
1 Introduction: A Summary of Mathematical and Physical Background for One-Particle Quantum Mechanics	p. 1
2 Spreading and Asymptotic Decay of Free Wave Packets: The Method of Stationary Phase and van der Corput's Approach	p. 21
3 The Relation Between Time-Like Decay and Spectral Properties	p. 39
3.1 Decay on the Average Sense	p. 40
3.1.1 Preliminaries: Wiener's, RAGE and Weyl theorems	p. 40
3.1.2 Models of exotic spectra, quantum KAM theorems and Howland's theorem	p. 44
3.1.3 U¿H measures and decay on the average: Strichartz-Last theorem and Guarneri-Last-Combes theorem	p. 57
3.2 Decay in the L p -Sense	p. 64
3.2.1 Relation between decay in the L p -sense and decay on the average sense	p. 64
3.2.2 Decay on the L p -sense and absolute continuity	p. 68
3.2.3 Sojourn time, Sinha's theorem and time-energy uncertainty relation	p. 76
3.3 Pointwise Decay	p. 85
3.3.1 Does decay in the L p -sense and/or absolute continuity imply pointwise decay?	p. 85
3.3.2 Rajchman measures, and the connection between ergodic theory, number theory and analysis	p. 88
3.3.3 Fourier dimension, Salem sets and Salem's method	p. 97
3.4 Quantum Dynamical Stability	p. 108
4 Time Decay for a Class of Models with Sparse Potentials	p. 119
4.1 Spectral Transition for Sparse Models in d = 1	p. 120
4.1.1 Existence of "mobility edges"	p. 121
4.1.2 Uniform distribution of Prüfer angles	p. 123
4.1.3 Proof of Theorem 4.1	p. 129
4.2 Decay in the Average	p. 133
4.2.1 Anderson-like transition for "separable" sparse models in d ≥ 2	p. 134
4.2.2 Uniform ¿-Hölder continuity of spectral measures	p. 136
4.2.3 Formulation, proof and comments of the main result	p. 138
4.3 Pointwise Decay	p. 142
4.3.1 Pearson's fractal measures: Borderline time-decay for the least sparse model	p. 142
4.3.2 Gevrey-type estimates	p. 153
4.3.3 Proof of Theorem 4.7	p. 161
5 Resonances and Quasi-exponential Decay	p. 167
5.1 Introduction	p. 167
5.2 The Model System	p. 169
5.3 Generalities on Laplace-Borel Transform and Asymptotic Expansions	p. 169
5.4 Decay for a Class of Model Systems After Costin and Huang: Gamow Vectors and Dispersive Part	p. 174
5.5 The Role of Gamow Vectors	p. 184
5.6 A First Example of Quantum Instability: Ionization	p. 189
5.7 Ionization: Study of a Simple Model	p. 191
5.8 A Second Example of Multiphoton Ionization: The Work of M. Huang	p. 194
5.9 The Reason for Enhanced Stability at Resonances: Connection with the Fermi Golden Rule	p. 200
6 Aspects of the Connection Between Quantum Mechanics and Classical Mechanics: Quantum Systems with Infinite Number of Degrees of Freedom	p. 203
6.1 Introduction	p. 203
6.2 Exponential Decay and Quantum Anosov Systems	p. 204
6.2.1 Generalities: Exponential decay in quantum and classical systems	p. 204
6.2.2 Quantum Anosov systems	p. 208
6.2.3 Examples of quantum Anosov systems and Weigert's configurational quantum cat map	p. 213
6.3 Approach to Equilibrium	p. 219
6.3.1 A brief introductory motivation	p. 219
6.3.2 Approach to equilibrium in classical (statistical) mechanics 1: Ergodicity, mixing and the Anosov property. The Gibbs entropy	p. 220
6.3.3 Approach to equilibrium in classical mechanics 2; The Ehrenfest model	p. 223
6.3.4 Approach to equilibrium in classical statistical mechanics 3: The initial sate, macroscopic states, Boltzmann versus Gibbs entropy. Examples: Reversible mixing systems and the evolution of densities	p. 228
6.3.5 Approach to equilibrium in quantum systems: Analogies, differences, and open problems	p. 241
6.4 Interlude: Systems with an Infinite Number of Degrees of Freedom	p. 242
6.4.1 The Haag-Kastler framework	p. 242
6.4.2 Quantum spin systems	p. 248
6.5 Approach to Equilibrium and Related Problems in Quantum Systems with an Infinite Number of Degrees of Freedom	p. 253
6.5.1 Quantum mixing, dynamical stability, return to equilibrium and weak asymptotic abelianness	p. 253
6.5.2 Examples of mixing and weak asymptotic abelianness: The vacuum and thermal states in rqft	p. 259
6.5.3 Approach to equilibrium in quantum spin systems - the Emch-Radin model, rates of decay and stability	p. 264
Appendix A A Survey of Classical Ergodic Theory	p. 277
Appendix B Transfer Matrix, Prüfer Variables and Spectral Analysis of Sparse Models	p. 303
Appendix C Symmetric Cantor Sets and Related Subjects	p. 323
Bibliography	p. 331
Index	p. 341

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