Bounds on the spectral shift function and the density of states

We study spectra of Schr\"odinger operators on $\RR^d$. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values $\mu_n$ of the difference of the semigroups as $n\to \infty$ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schr\"odinger operators. The single site potential $u$ is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be H\"older continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies H\"older continuity of the integrated density of states.