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Dietlein, Adrian; Gebert, Martin; Hislop, Peter D.; Klein, Abel and Müller, Peter (2018): A Bound on the Averaged Spectral Shift Function and a Lower Bound on the Density of States for Random Schrodinger Operators on R-d. In: International Mathematics Research Notices, No. 21: pp. 6673-6697

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Abstract

We obtain a bound on the expectation of the spectral shift function (SSF) for alloy-type random Schrodinger operators on R-d in the region of localisation, corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a finite volume. The bound scales with the area of the surface where the boundary conditions are changed. As an application of our bound on the SSF, we prove a reverse Wegner inequality for finite-volume Schrodinger operators in the region of localisation with a constant locally uniform in the energy. The application requires that the single-site distribution of the independent and identically distributed random variables has a Lebesgue density that is also bounded away from zero. The reverse Wegner inequality implies a strictly positive, locally uniform lower bound on the density of states for these continuum random Schrodinger operators.

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