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 (R d ), 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 ∈ L ∞ 0 (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.

From Uncertainty Principles to Wegner Estimates

“…Combined with the analysis from [5], the latter estimate can be used to prove continuity of the IDS and Wegner estimates at all energies for Anderson models with quite irregular geometry of the set of impurities. We do not pursue this issue here.…”

“…The following result can be seen as a streamlined version of Theorem 3.1 in [5]. We fix a Hilbert space H and denote its inner product by (· | ·).…”

“…As discussed in [1], Appendix B and in [5], Section 3, the resolvent of a maximally accretive operator can always be written as the resolvent of a selfadjoint dilation.…”

“…The path described in the title has been laid by Combes, Hislop and Klopp, [5] to prove Wegner estimates with the correct volume factor for Anderson type models without a covering condition. Subsequently we showed in [3] how to use this approach to deduce localization for a great number of models that couldn't be treated before.…”

“…Actually, for low energies we derived an uncertainty principle by an extremely simple method. However, for the link from these uncertainty principles to Wegner estimates with the correct volume factor we still had to rely upon the analysis of [5] which is technically quite involved. Here, we will use the fact that the uncertainty principles in [3] apply to the spectral projections of the random operators and not just to to the spectral projections of the underlying periodic background, as was the case in [5].…”