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

Jean-Michel Combes·Peter Hislop·Frédéric Klopp
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Abstract

We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on $L^2 (\R^d)$, for $d \geq1$, is locally Hölder continuous at all energies with the same Hölder exponent $0<α\leq1$ 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 $u\in L\_0^\infty (\R^d)$ must be nonnegative and compactly-supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle. 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.

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Jean-Michel Combes, Peter Hislop, Frédéric Klopp. 2006-10-14. An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. https://arxiv.org/abs/math-ph/0605029

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