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

Abstract

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 (a particular class of sets of positive measure). The proof exploits on one hand recently proven unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes and on the other hand the well developed machinery for proving Wegner estimates. We obtain a Wegner estimate with optimal volume dependence at all energies, and localization 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 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.