Contents -- Preface: A Description of Contents -- Acknowledgments -- 1. Introduction: A Summary of Mathematical and Physical Background for One-Particle Quantum Mechanics -- 2. Spreading and Asymptotic Decay of Free Wave Packets: The Method of Stationary Phase and van der Corput's Approach -- 3. The Relation Between Time-Like Decay and Spectral Properties -- 3.1 Decay on the Average Sense -- 3.1.1 Preliminaries: Wiener's, RAGE and Weyl theorems -- 3.1.2 Models of exotic spectra, quantum KAM theorems and Howland's theorem -- 3.1.3 UαH measures and decay on the average: Strichartz-Last theorem and Guarneri-Last-Combes theorem -- 3.2 Decay in the Lp-Sense -- 3.2.1 Relation between decay in the Lp-sense and decay on the average sense -- 3.2.2 Decay on the Lp-sense and absolute continuity -- 3.2.3 Sojourn time, Sinha's theorem and time-energy uncertainty relation -- 3.3 PointwiseDecay -- 3.3.1 Does decay in the Lp-sense and/or absolute continuity imply pointwise decay? -- 3.3.2 Rajchman measures, and the connection between ergodic theory, number theory and analysis -- 3.3.3 Fourier dimension, Salem sets and Salem's method -- 3.4 Quantum Dynamical Stability -- 4. Time Decay for a Class of Models with Sparse Potentials -- 4.1 Spectral Transition for Sparse Models in d = 1 -- 4.1.1 Existence of "mobility edges" -- 4.1.2 Uniform distribution of Prufer angles -- 4.1.3 Proof of Theorem 4.1 -- 4.2 Decay in the Average -- 4.2.1 Anderson-like transition for "separable" sparse models in d = 2 -- 4.2.2 Uniform α-Holder continuity of spectral measures -- 4.2.3 Formulation, proof and comments of the main result -- 4.3 PointwiseDecay -- 4.3.1 Pearson's fractal measures: Borderline time-decay for the least sparsemodel -- 4.3.2 Gevrey-type estimates -- 4.3.3 Proof of Theorem4.7 -- 5. Resonances and Quasi-exponential Decay -- 5.1 Introduction -- 5.2 The Model System.

5.3 Generalities on Laplace-Borel Transform and Asymptotic Expansions -- 5.4 Decay for a Class of Model Systems After Costin and Huang: Gamow Vectors and Dispersive Part -- 5.5 The Role of Gamow Vectors -- 5.6 A First Example of Quantum Instability: Ionization -- 5.7 Ionization: Study of a Simple Model -- 5.8 A Second Example of Multiphoton Ionization: The Work of M. Huang -- 5.9 The Reason for Enhanced Stability at Resonances: Connection with the Fermi Golden Rule -- 6. Aspects of the Connection Between Quantum Mechanics and Classical Mechanics: Quantum Systems with Infinite Number of Degrees of Freedom -- 6.1 Introduction -- 6.2 Exponential Decay and Quantum Anosov Systems -- 6.2.1 Generalities: Exponential decay in quantum and classical systems -- 6.2.2 QuantumAnosov systems -- 6.2.3 Examples of quantum Anosov systems and Weigert's configurational quantum cat map -- 6.3 Approach to Equilibrium -- 6.3.1 A brief introductory motivation -- 6.3.2 Approach to equilibrium in classical (statistical) mechanics 1: Ergodicity, mixing and the Anosov property. The Gibbs entropy -- 6.3.3 Approach to equilibrium in classical mechanics 2 -- The Ehrenfest model -- 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 -- 6.3.5 Approach to equilibrium in quantum systems: Analogies, differences, and open problems -- 6.4 Interlude: Systems with an Infinite Number of Degrees of Freedom -- 6.4.1 The Haag-Kastler framework -- 6.4.2 Quantum spin systems -- 6.5 Approach to Equilibrium and Related Problems in Quantum Systems with an Infinite Number of Degrees of Freedom -- 6.5.1 Quantum mixing, dynamical stability, return to equilibrium and weak asymptotic abelianness.

6.5.2 Examples of mixing and weak asymptotic abelianness: The vacuum and thermal states in rqft -- 6.5.3 Approach to equilibrium in quantum spin systems - the Emch-Radin model, rates of decay and stability -- Appendix A. A Survey of Classical Ergodic Theory -- Appendix B. Transfer Matrix, Prufer Variables and Spectral Analysis of Sparse Models -- Appendix C. Symmetric Cantor Sets and Related Subjects -- Bibliography -- Index.