# Dependence of the density of states on the probability distribution -- part II: Schr\"odinger operators on $\mathbb{R}^d$ and non-compactly supported probability measures

## TL;DR

This paper investigates how the density of states for random Schr"odinger operators varies with different probability distributions, extending previous results to non-compact supports and continuous models using advanced mathematical techniques.

## Contribution

It extends previous work on the density of states' continuity to non-compact probability measures and Schr"odinger operators on continuous spaces, employing the Combes-Thomas estimate and Helffer-Sj"ostrand formula.

## Key findings

- Established continuity properties for non-compact probability measures on lattice models.
- Proved analogous results for Schr"odinger operators on  with non-compact support.
- Utilized advanced mathematical tools to extend previous theoretical frameworks.

## Abstract

We extend our results in \cite{hislop_marx_1} on the quantitative continuity properties, with respect to the single-site probability measure, of the density of states measure and the integrated density of states for random Schr\"odinger operators. For lattice models on $\mathbb{Z}^d$, with $d \geq 1$, we treat the case of non-compactly supported probability measures with finite first moments. For random Schr\"odinger operators on $\mathbb{R}^d$, with $d \geq 1$, we prove results analogous to those in \cite{hislop_marx_1} for compactly supported probability measures. The method of proof makes use of the Combes-Thomas estimate and the Helffer-Sj\"ostrand formula.