# Regularity of the density of states of Random Schr\"odinger Operators

## TL;DR

This paper proves that the density of states for a broad class of random Schr"odinger operators is differentiable up to a certain order in localized spectral regions, under specific conditions on the potentials and their distributions.

## Contribution

It extends regularity results of the density of states to more general continuous models with complete covering potentials, including magnetic fields, under new smoothness assumptions.

## Key findings

- Density of states is m-times differentiable in localized spectral regions.
- Regularity depends on the smoothness of the single site distribution.
- Results apply to models with complete covering condition, including some magnetic potentials.

## Abstract

In this paper we solve a long standing open problem for Random Schr\"odinger operators on $L^2(\mathbb{R}^d)$ with i.i.d single site random potentials. We allow a large class of free operators, including magnetic potential, however our method of proof works only for the case when the random potentials satisfy a complete covering condition. We require that the supports of the random potentials cover $\mathbb{R}^d$ and the bump functions that appear in the random potentials form a partition of unity. For such models, we show that the Density of States (DOS) is $m$ times differentiable in the part of the spectrum where exponential localization is valid, if the single site distribution has compact support and has H\"older continuous $m+1$ st derivative. The required H\"older continuity depends on the fractional moment bounds satisfied by appropriate operator kernels. Our proof of the Random Schr\"odinger operator case is an extensions of our proof for Anderson type models on $\ell^2(\mathbb{G})$, $\mathbb{G}$ a countable set, with the property that the cardinality of the set of points at distance $N$ from any fixed point grows at some rate in $N^\alpha, \alpha >0$. This condition rules out the Bethe lattice, where our method of proof works but the degree of smoothness also depends on the localization length, a result we do not present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition.