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 D. Hislop, Frédéric Klopp

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71 Scopus citations

Abstract

We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on L 2(ℝd)for d ≥ 1 is locally Hölder continuous at all energies with the same Hölder exponent 0 < α ≤ 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 ∈ L0(ℝd) 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.

Original languageEnglish
Pages (from-to)469-498
Number of pages30
JournalDuke Mathematical Journal
Volume140
Issue number3
DOIs
StatePublished - Dec 1 2007

ASJC Scopus subject areas

  • Mathematics (all)

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