Wegner estimate and localisation for alloy type operators with minimal support assumptions on the single site potential

TL;DR

This paper establishes a Wegner estimate for alloy type models with minimal assumptions on the support of the single site potential, using new unique continuation principles and proving localization near the spectrum's minimum.

Contribution

It introduces a sharp condition on the support of the potential, showing that a lower bound by a characteristic function of a thick set is both necessary and sufficient for Wegner estimates.

Findings

Wegner estimate with optimal volume dependence at all energies.

Localization near the spectrum's minimum for certain non-stationary potentials.

Identification of a sharp support condition for the validity of Wegner estimates.

Abstract

We prove a Wegner estimate for alloy type models merely assuming that the single site potential is lower bounded by a characteristic function of a thick set, that is a particular set of positive measure. The proof is based on two ingredients: New unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes from [EV20] and established proofs for Wegner estimates, e.g.~from [CHK07]. We obtain a Wegner estimate with optimal volume dependence at all energies, and localisation near the minimum of the spectrum, even for some non-stationary random potentials. We complement the result by showing that a lower bound on the potential by the characteristic function of a thick set is necessary for a (translation uniform) Wegner estimate to hold. Hence, we have identified a sharp condition on the size for the support of random potentials that is sufficient and necessary for the validity of Wegner estimates.