A bound on the averaged spectral shift function and a lower bound on the density of states for random Schr\"odinger operators on $\mathbb{R}^d$

We obtain a bound on the expectation of the spectral shift function for alloy-type random Schr\"odinger 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 spectral shift function, we prove a reverse Wegner inequality for finite-volume Schr\"odinger 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 Schr\"odinger operators.