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

Abstract

These lectures present some basic ideas and techniques in the spectral analysis of lattice Schrödinger 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 Fröhlich–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 Schrödinger cocycles assuming positive Lyapunov exponents.