An Invitation to Random Schroedinger operators

TL;DR

This paper provides an accessible overview of random Schrödinger operators, covering fundamental concepts, proofs of Lifshitz tails and Anderson localization, aimed at non-specialists with basic functional analysis and probability knowledge.

Contribution

It offers a comprehensive, beginner-friendly introduction with complete proofs of key phenomena in random Schrödinger operators, filling a gap in accessible literature.

Findings

Proof of Lifshitz tails

Proof of Anderson localization

Foundational overview of random Schrödinger operators

Abstract

This review is an extended version of my mini course at the Etats de la recherche: Operateurs de Schroedinger aleatoires at the Universite Paris 13 in June 2002, a summer school organized by Frederic Klopp. These lecture notes try to give some of the basics of random Schroedinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization.