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We obtain a bound on the expectation of the spectral shift function (SSF) for alloy-type random Schrödinger operators on   $\mathbb{R}^d$   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 Schrödinger 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 Schrödinger operators.

international mathematics research notices

A Bound on the Averaged Spectral Shift Function and a Lower Bound on the Density of States for Random Schrödinger Operators on   ${\mathbb{R}}^{\boldsymbol{d}}$