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

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 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. 1. Model and results We prove a Wegner estimate for continuum random Schrödinger operators with a very weak assumption on the supports of the single site potentials, which turns out to be optimal. The random potential needs not be stationary. Together with an initial scale estimate we conclude localisation for such models. Until now, a fundamental assumption in this context has been that the potential either satisfies a covering condition, see e.g. [MH84, CH94, Kir96] or that it is uniformly positive on at least a non-empty open set, see e.g. [CHK03, CHK07, RMV13, Kle13, NTTV18]. We only ask the sum of potentials to be positive on a so-called thick set : Definition 1. Let γ ∈ (0, 1], a = (a1, . . . , ad) ∈ (0,∞)d, and set Aa := [0, a1] × · · · × [0, ad] ⊂ Rd. A measurable set S ⊂ Rd is called (γ, a)-thick if vol (S ∩ (x+Aa)) ≥ γ vol (Aa) for all x ∈ R (1) and simply thick if there exist γ ∈ (0, 1] and a ∈ (0,∞)d such that (1) holds. Thick sets are a generalization of periodic positive measure sets. Thickness is the minimal condition on the size of a characteristic function necessary for a uncertainty relation of spectral projectors to hold, cf. [EV20], reformulating a criterion of [Kov00]. We will study alloy type Hamiltonians on L2(Rd) satisfying the following assumption on the random variables: (U) Let [m−,m+] ⊂ R be an interval and Ω =×j∈Zd [m−,m+] a probability space equipped with a product measure P = ⊗ j∈Zd μj . Denote by πj : Ω → [m−,m+] the coordinate projections such that {πj}j∈Zd , are a family of independent and uniformly bounded random variables. We assume that all πj are non-trivial. While most of our results would hold also under more general conditions on the random variables πj, the spelled out condition is standard in the literature. In fact, we want to focus our attention on a different building block of the random potential and identify the