Wegner Estimate for Indefinite Anderson Potentials: Some Recent Results and Applications

TL;DR

This paper reviews recent and new results on spectral properties of random Schrödinger operators with indefinite alloy-type potentials, focusing on the Wegner estimate and its implications for spectral analysis.

Contribution

It provides new insights into the Wegner estimate for indefinite potentials and discusses its role in proving pure point spectrum and density of states existence.

Findings

Wegner estimate established for indefinite potentials

Pure point spectrum proven under certain conditions

Density of states existence demonstrated in specific models

Abstract

We review recent and give some new results on the spectral properties of Schroedinger operators with a random potential of alloy type. Our point of interest is the so called Wegner estimate in the case where the single site potentials change sign. The indefinitness of the single site potential poses certain difficulties for the proof of the Wegner estimate which are still not fully understood. The Wegner estimate is a key ingredient in an existence proof of pure point spectrum of the considered random Schroedinger operators. Under certain assumptions on the considered models additionally the existence of the density of states can be proven.