Local Wegner and Lifshitz tails estimates for the density of states for continuous random Schr\"odinger operators

TL;DR

This paper establishes local Wegner estimates for continuous Anderson Hamiltonians with independent, non-identically distributed variables, showing the density of states has Lifshitz tail behavior near the spectrum's bottom.

Contribution

It introduces new local Wegner estimates for non-i.i.d. variables and links the density of states' behavior to Lifshitz tails, extending previous results.

Findings

Wegner estimates with constants vanishing near the spectrum's bottom

Density of states exhibits Lifshitz tail upper bounds

Results apply to non-i.i.d. random variables

Abstract

We introduce and prove local Wegner estimates for continuous generalized Anderson Hamiltonians, where the single-site random variables are independent but not necessarily identically distributed. In particular, we get Wegner estimates with a constant that goes to zero as we approach the bottom of the spectrum. As an application, we show that the (differentiated) density of states exhibits the same Lifshitz tails upper bound as the integrated density of states.