An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

Abstract

We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on $L^2({\mathbb{R}}^d)$ for $d\ge 1$ is locally Hölder continuous at all energies with the same Hölder exponent $0<\alpha \le 1$ as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential $u\in L_0^{\infty }({\mathbb{R}}^d)$ must be nonnegative and compactly supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle (UCP). We also prove analogous continuity results for the IDS of random Anderson-type perturbations of the Landau Hamiltonian in two dimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures