Universal regularity of higher-dimensional disorder and density of states under non-local interactions. I. Infinite smoothness and localization

It is shown that in a large class of disordered systems with singular alloy-type disorder and non-local media-particle interactions, the marginal measures of the induced random potential and the finite-volume integrated density of states (IDS) are infinitely differentiable in higher dimensions. The proposed approach complements the classical Wegner estimate which says that the IDS in the short-range models is at least as regular as the marginal distribution of the disorder. In the models with non-local interaction the finite-volume IDS is much more regular than the underlying disorder. In turn, smoothness of the finite-volume IDS is responsible for a mechanism complementing the Lifshitz tails phenomenon. The new eigenvalue concentration estimates give rise to relatively simple proofs of Anderson localization in several classes of discrete and continuous long-range models with arbitrarily singular disorder. The present paper addresses the model with power-law decay of the potential.