TY - JOUR

T1 - An optimal wegner estimate and its application to the global continuity of the integrated density of states for random schrödinger operators

AU - Combes, Jean Michel

AU - Hislop, Peter D.

AU - Klopp, Frédéric

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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.

AB - 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.

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U2 - 10.1215/S0012-7094-07-14032-8

DO - 10.1215/S0012-7094-07-14032-8

M3 - Article

AN - SCOPUS:36749088515

SN - 0012-7094

VL - 140

SP - 469

EP - 498

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 3

ER -