  We prove a strictly positive, locally uniform lower bound on the density of
states (DOS) of continuum random Schr\"odinger operators on the entire
spectrum, i.e. we show that the DOS does not have a zero within the spectrum.
This follows from a lower Wegner estimate for finite-volume continuum random
Schr\"odinger operators. We assume throughout iid random variables and the
single-site distribution having a Lebesgue density bounded from below on its
support. The main mathematical novelty in this paper are pointwise-in-energy
bounds on the expectation of the spectral shift function at all energies for
these operators where we mainly focus on perturbations corresponding to a
change from Dirichlet to Neumann boundary conditions along the boundary of a
cube. We show that the bound scales with the area of the hypersurface where the
boundary conditions are changed. We also prove bounds on the averaged spectral
shift function for perturbations by bounded and compactly supported
multiplication operators.
