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Full Description
This book provides an introduction to the mathematical theory of disorder effects on quantum spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics of self-adjoint operators through Anderson localization-presented here via the fractional moment method, up to recent results on resonant delocalization.
The subject's multifaceted presentation is organized into seventeen chapters, each focused on either a specific mathematical topic or on a demonstration of the theory's relevance to physics, e.g., its implications for the quantum Hall effect. The mathematical chapters include general relations of quantum spectra and dynamics, ergodicity and its implications, methods for establishing spectral and dynamical localization regimes, applications and properties of the Green function, its relation to the eigenfunction correlator, fractional moments of Herglotz-Pick functions, the phase diagram for tree graph operators, resonant delocalization, the spectral statistics conjecture, and related results.
The text incorporates notes from courses that were presented at the authors' respective institutions and attended by graduate students and postdoctoral researchers.
Contents
Introduction
General relations between spectra and dynamics
Ergodic operators and their self-averaging properties
Density of states bounds: Wegner estimate and Lifshitz tails
The relation of Green functions to eigenfunctions
Anderson localization through path expansions
Dynamical localization and fractional moment criteria
Fractional moments from an analytical perspective
Strategies for mapping exponential decay
Localization at high disorder and at extreme energies
Constructive criteria for Anderson localization
Complete localization in one dimension
Diffusion hypothesis and the Green-Kubo-Streda formula
Integer quantum Hall effect
Resonant delocalization
Phase diagrams for regular tree graphs
The eigenvalue point process and a conjectured dichotomy
Elements of spectral theory
Herglotz-Pick functions and their spectra
Bibliography
Index
