Existence of the Density of States for One-dimensional Alloy-type Potentials with Small Support

We study spectral properties of Schrödinger operators with a random potential of alloy type on L2(R) and their restrictions to finite intervals. A Wegner estimates for non-negative single site potentials with small support is proven. It implies the existence and local uniform boundedness of the density of states. Our estimate is valid for all bounded energy intervals. Wegner estimates play a key role in an existence proof of pure point spectrum. 1. Model and results We study spectral properties of families of Schrödinger operators on L(R). The considered operators consist of a non-random periodic Schrödinger operator plus a random potential of Anderson or alloy type: Hω := H0 + Vω, H0 := −∆+ Vper. (1) Here ∆ is the Laplace operator on R and Vper ∈ L∞(R) is a Z-periodic potential. The random potential Vω is a stochastic process of the following form (2) Vω(x) = ∑