Asymptotic Time Decay in Quantum Physics

The following sections are included:

• Dedication

• Preface: A Description of Contents

• Acknowledgments

• Contents

In this chapter, we pose some problems which will be discussed throughout the book, at the same time, fixing some important notations and levels of knowledge assumed from the reader.

As remarked in the preface, we shall be concerned with the decay of various quantities in time such as the return or survival probability which lie at the basis of various phenomena in quantum physics. We shall be dealing with one-particle systems subject to certain potentials which may or may not depend on time (in the positive case, we speak of external potentials)…

In this chapter, we shall be almost exclusively concerned with the propagation of free wave packets and their decay in time (pointwise or in L^p, see Chap. 3 for much more about this distinction). This means that for most of the time our Hamiltonian will be , where is the Laplacian on , or H_0l = −△_l which is the difference Laplacian (1.2d) and (1.2e) on , with n denoting the dimension as before. The difference Laplacian describes the tight-binding model whose physical significance was discussed in Chap. 1, for E₀ = 2A in (1.3f); in this case, the energy E = E(k) is, for sufficiently small modulus of the wave vector |k|, of the form 2Ak²…

The following sections are included:

• Decay on the Average Sense
  • Preliminaries: Wiener's, RAGE and Weyl theorems
  • Models of exotic spectra, quantum KAM theorems and Howland's theorem
  • UαH measures and decay on the average: Strichartz–Last theorem and Guarneri–Last–Combes theorem

• Decay in the L^p-Sense
  • Relation between decay in the L^p-sense and decay on the average sense
  • Decay on the L^p-sense and absolute continuity
  • Sojourn time, Sinha's theorem and time–energy uncertainty relation

• Pointwise Decay
  • Does decay in the L^p-sense and/or absolute continuity imply pointwise decay?
  • Rajchman measures, and the connection between ergodic theory, number theory and analysis
  • Fourier dimension, Salem sets and Salem's method

• Quantum Dynamical Stability

The following sections are included:

• Spectral Transition for Sparse Models in d = 1
  • Existence of “mobility edges”
  • Uniform distribution of Prüfer angles
  • Proof of Theorem 4.1

• Decay in the Average
  • Anderson-like transition for “separable” sparse models in d ≥ 2
  • Uniform α-Hölder continuity of spectral measures
  • Formulation, proof and comments of the main result

• Pointwise Decay
  • Pearson's fractal measures: Borderline time-decay for the least sparse model
  • Gevrey-type estimates
  • Proof of Theorem 4.7

The following sections are included:

• Introduction

• The Model System

• Generalities on Laplace–Borel Transform and Asymptotic Expansions

• Decay for a Class of Model Systems After Costin and Huang: Gamow Vectors and Dispersive Part

• The Role of Gamow Vectors

• A First Example of Quantum Instability: Ionization

• Ionization: Study of a Simple Model

• A Second Example of Multiphoton Ionization: The Work of M. Huang

• The Reason for Enhanced Stability at Resonances: Connection with the Fermi Golden Rule

The following sections are included:

• Introduction

• Exponential Decay and Quantum Anosov Systems
  • Generalities: Exponential decay in quantum and classical systems
  • Quantum Anosov systems
  • Examples of quantum Anosov systems and Weigert's configurational quantum cat map

• Approach to Equilibrium
  • A brief introductory motivation
  • Approach to equilibrium in classical (statistical) mechanics 1: Ergodicity, mixing and the Anosov property. The Gibbs entropy
  • Approach to equilibrium in classical mechanics 2; The Ehrenfest model
  • 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
  • Approach to equilibrium in quantum systems: Analogies, differences, and open problems

• Interlude: Systems with an Infinite Number of Degrees of Freedom
  • The Haag–Kastler framework
  • Quantum spin systems

• Approach to Equilibrium and Related Problems in Quantum Systems with an Infinite Number of Degrees of Freedom
  • Quantum mixing, dynamical stability, return to equilibrium and weak asymptotic abelianness
  • Examples of mixing and weak asymptotic abelianness: The vacuum and thermal states in rqft
  • Approach to equilibrium in quantum spin systems — the Emch–Radin model, rates of decay and stability

The following sections are included:

• Appendix A: A Survey of Classical Ergodic Theory

• Appendix B: Transfer Matrix, Prüfer Variables and Spectral Analysis of Sparse Models

• Appendix C: Symmetric Cantor Sets and Related Subjects

• Bibliography

• Index