A lower Wegner estimate and bounds on the spectral shift function for continuum random Schr\"odinger operators

TL;DR

This paper establishes a positive lower bound on the density of states for continuum random Schrödinger operators and provides new bounds on the spectral shift function, enhancing understanding of spectral properties across the entire spectrum.

Contribution

It introduces a lower Wegner estimate for continuum operators and derives pointwise bounds on the spectral shift function for boundary condition perturbations.

Findings

Density of states is strictly positive across the spectrum.

Spectral shift function bounds scale with boundary surface area.

Bounds are obtained for perturbations by bounded, compactly supported operators.

Abstract

We prove a strictly positive, locally uniform lower bound on the density of states (DOS) of continuum random Schr\"odinger operators on the entire spectrum, i.e. we show that the DOS does not have a zero within the spectrum. This follows from a lower Wegner estimate for finite-volume continuum random Schr\"odinger operators. We assume throughout iid random variables and the single-site distribution having a Lebesgue density bounded from below on its support. The main mathematical novelty in this paper are pointwise-in-energy bounds on the expectation of the spectral shift function at all energies for these operators where we mainly focus on perturbations corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a cube. We show that the bound scales with the area of the hypersurface where the boundary conditions are changed. We also prove bounds on the averaged spectral shift function for perturbations by bounded and compactly supported multiplication operators.