Continuity with respect to Disorder of the Integrated Density of States

TL;DR

This paper proves that the integrated density of states (IDS) for a random Schrödinger operator varies continuously with respect to the disorder parameter, ensuring stability and convergence of the IDS as disorder diminishes.

Contribution

It establishes the local uniform Hölder continuity of the IDS with respect to the disorder parameter for the first time.

Findings

IDS is locally uniformly Hölder continuous in the disorder parameter

IDS converges to the unperturbed IDS as disorder tends to zero at energies where the unperturbed IDS is continuous

Provides a rigorous foundation for stability analysis of spectral properties under disorder variations

Abstract

We prove that the integrated density of states (IDS) associated to a random Schroedinger operator is locally uniformly Hoelder continuous as a function of the disorder parameter lambda. In particular, we obtain convergence of the IDS, as lambda tends to 0, to the IDS for the unperturbed operator at all energies for which the IDS for the unperturbed operator is continuous in energy.