## Duke Mathematical Journal

### 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 $d \geq 1$ is locally Hölder continuous at all energies with the same Hölder exponent $0 \lt \alpha \leq 1$ 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

#### Article information

Source
Duke Math. J., Volume 140, Number 3 (2007), 469-498.

Dates
First available in Project Euclid: 8 November 2007

https://projecteuclid.org/euclid.dmj/1194547696

Digital Object Identifier
doi:10.1215/S0012-7094-07-14032-8

Mathematical Reviews number (MathSciNet)
MR2362242

Zentralblatt MATH identifier
1134.81022

#### Citation

Combes, Jean-Michel; Hislop, Peter D.; Klopp, Frédéric. An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140 (2007), no. 3, 469--498. doi:10.1215/S0012-7094-07-14032-8. https://projecteuclid.org/euclid.dmj/1194547696

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