Integrated density of states and Wegner estimates for random Schr\"odinger Operators

TL;DR

This paper reviews recent advances in understanding the spectral properties of random Schrödinger operators, focusing on the integrated density of states (IDS), its existence, regularity, and implications for localization phenomena.

Contribution

It provides a general proof of the existence of a self-averaging IDS applicable to various operators and analyzes the regularity of IDS for alloy-type models in Euclidean space.

Findings

Existence of a self-averaging IDS for broad classes of operators

Regularity properties of IDS in alloy-type models

Connections between IDS regularity and localization phenomena

Abstract

We survey recent results on spectral properties of random Schr\"odinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a self-averaging IDS which is general enough to be applicable to random Schr\"odinger and Laplace-Beltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schr\"odinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.