The Wegner estimate and the integrated density of states for some random operators

Abstract

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Holder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on theL p-theory of the spectral shift function (SSF), forp ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace idealI p, for 0p ≤ 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local Holder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.