Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current

We study spectra of Schrödinger operators on R 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 µn of the difference of the semigroups as n →∞and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger 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 Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.

We survey recent results on spectral properties of random Schrodinger operators. The focus is set on the integrated density of states (IDS).

In this note we present a proof of the Wegner estimate for random Schrödinger operators of Anderson type that do not have necessarily overlapping single site potentials. We use the spectral averaging procedure via trace formula and spectral shift function.

We analyze the spectral shift function (SSF) of a Schrodinger operator due to a compactly supported potential. We give a bound on the integral of the SSF with respect to a bounded compactly supported function. It is based on the control of the singular values of the di#erence of two Schrodinger semigroups. As an application we improve some earlier results on the regularity of the integrated density of states. 1.

Abstract. We give a shortcut from simple uncertainty principles to Wegner estimates with correct volume term. 1.