A bound on the averaged spectral shift function and a lower bound on the density of states for random Schr\"odinger operators on $\mathbb{R}^d$

TL;DR

This paper establishes a bound on the spectral shift function for random Schrödinger operators, leading to a lower bound on the density of states, with implications for understanding localization phenomena in quantum systems.

Contribution

It introduces a novel bound on the spectral shift function in the localization region, enabling a lower bound on the density of states for alloy-type random Schrödinger operators.

Findings

Bound scales with the surface area of boundary condition change

Proves a reverse Wegner inequality with a uniform energy constant

Establishes a positive lower bound on the density of states

Abstract

We obtain a bound on the expectation of the spectral shift function for alloy-type random Schr\"odinger operators on $\mathbb{R}^d$ in the region of localisation, corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a finite volume. The bound scales with the area of the surface where the boundary conditions are changed. As an application of our bound on the spectral shift function, we prove a reverse Wegner inequality for finite-volume Schr\"odinger operators in the region of localisation with a constant locally uniform in the energy. The application requires that the single-site distribution of the independent and identically distributed random variables has a Lebesgue density that is also bounded away from zero. The reverse Wegner inequality implies a strictly positive, locally uniform lower bound on the density of states for these continuum random Schr\"odinger operators.