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

By Jean Michel Combes, Peter D. Hislop, Frédéric Klopp. 2006

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We prove that the integrated density of states (IDS) of random Schrodinger operators with Anderson-type potentials on $L^2 (\R^d)$, for $d \geq1$, is locally Holder continuous at all energies with the same Holder exponent $0<\alpha\leq1$ 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

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