# On the continuity of the integrated density of states in the disorder

## TL;DR

This paper presents a new approach using Ky Fan inequalities to analyze the continuity of the integrated density of states in disordered quantum systems, improving existing estimates and extending results to continuous models.

## Contribution

It introduces an alternative method based on Ky Fan inequalities to establish sharper continuity estimates for the integrated density of states, including for continuous Schrödinger operators.

## Key findings

- Established a sharp continuity estimate for the integrated density of states
- Extended the analysis from discrete to continuous Schrödinger operators
- Provided an alternative proof approach using Ky Fan inequalities

## Abstract

Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schr\"odinger operators, and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schr\"odinger operators on $\R^d$.