An introduction to multiscale techniques in the theory of Anderson localization. Part I

TL;DR

This paper introduces multiscale techniques for analyzing Anderson localization in disordered lattice Schrödinger operators, covering classical and quasiperiodic potentials, with a focus on spectral analysis and localization theorems.

Contribution

It extends classical localization results to potentials with finitely many degrees of freedom, including quasiperiodic cases, and discusses advanced spectral analysis methods.

Findings

Classical localization theorems for large disorders in all dimensions

Localization results for quasiperiodic potentials on the line

Application of Bourgain-Goldstein theorem assuming positive Lyapunov exponents

Abstract

These lectures present some basic ideas and techniques in the spectral analysis of lattice Schrodinger operators with disordered potentials. In contrast to the classical Anderson tight binding model, the randomness is also allowed to possess only finitely many degrees of freedom. This refers to dynamically defined potentials, i.e., those given by evaluating a function along an orbit of some ergodic transformation (or of several commuting such transformations on higher-dimensional lattices). Classical localization theorems by Frohlich--Spencer for large disorders are presented, both for random potentials in all dimensions, as well as even quasi-periodic ones on the line. After providing the needed background on subharmonic functions, we then discuss the Bourgain-Goldstein theorem on localization for quasiperiodic Schrodinger cocycles assuming positive Lyapunov exponents.