Random Schr\"odinger Operators on discrete structures

TL;DR

This paper reviews the spectral and dynamical properties of the Anderson model on discrete structures, focusing on localization phenomena and the methods used to establish these properties in quantum disordered systems.

Contribution

It provides a comprehensive overview of localization results and proof techniques for the Anderson model on lattices and Bethe trees, based on a course from 2016.

Findings

Review of spectral properties of the Anderson model

Discussion of localization proofs on discrete structures

Summary of methods used in quantum disordered systems

Abstract

The Anderson model serves to study the absence of wave propagation in a medium in the presence of impurities, and is one of the most studied examples in the theory of quantum disordered systems. In these notes we give a review of the spectral and dynamical properties of the Anderson Model on discrete structures, like the $d$-dimensional square lattice and the Bethe lattice, and the methods used to prove localization. These notes are based on a course given at the CIMPA School "Spectral Theory of Graphs and Manifolds" in Kairouan, 2016.