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Comment: This is a note for the report on the Oberwolfach Mini-Workshop: Multiscale and Variational Methods in Material Science and Quantum Theory of Solids
We study Anderson and alloy type random Schr\"odinger operators on $\ell^2(\ZZ^d)$ and $L^2(\RR^d)$. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. For a certain class of models we prove a Wegner estimate which is linear in the volume of the box and the length of the considered energy interval. The single site potential of the Anderson/alloy type model does not need to have fixed sign, but it needs be of a generalised step function form. The result implies the Lipschitz continuity of the integrated density of states.
The integrated density of states of a Schroedinger operator with random potential given by a homogeneous Gaussian field whose covariance function is continuous, compactly supported and has positive mean, is locally uniformly Lipschitz-continuous. This is proven using a Wegner estimate.
Comment: 9 pages, a different version to appear in proceedings of the Conference on Applied Mathematics and Scientific Computing, Dubrovnik, Croatia, 2001
Comment: 87 pages; corrected and extended version
Comment: 10 pages, LaTeX 2e, to appear in "Applied Mathematics and Scientific Computing", Brijuni, June 23-27, 2003. by Kluwer publishers
Comment: 19 pages, LaTeX 2e. See also preprint 04-326 on mp_arc. To appear in a slightly different version in "Mathematische Annalen", see the DOI
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