An introduction to multiscale techniques in the theory of Anderson localization, Part I. (July 2022)

• Record Type: Journal Article
• Authors: Schlag, Wilhelm
• Is Part Of: Nonlinear analysis. Volume 220(2022)
• Publication Date: 2022-07
• Subjects: Disordered models in quantum theory -- Anderson localization -- Multiscale techniques; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112869
• Languages: English
• ISSNs: 0362-546X

Abstract

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.