Existence of the Density of States for Some Alloy Type Models with Single Site Potentials of Changing Sign

We study spectral properties of ergodic random Schrödinger operators on L(R). The density of states is shown to exist for a certain class of alloy type potentials with single site potentials of changing sign. The Wegner estimate we prove implies Anderson localization under certain additional assumptions. For some examples we discuss briefly some properties of the common and conditional densities of the random coupling constants used in the proof of the Wegner estimate. Sažetak. Analiziraju se spektralna svojstva ergodičkih slučajnih Schrödingerovih operatora na L(R). Dokazuje se, da gustoća stanja postoji za odredjenu klasu potencijala tipa legure, kod kojih pojedinačni potencijal mijenja predznak. Uz odredjene dodatne uvjete Wegnerova ocjena koju dokazujemo implicira fenomen Andersonove lokalizacije. Na osnovu primjera promatramo neka svojstva zajedničke i uvjetne gustoće slučajnih konstanti veze. Te gustoće se koriste u dokazu Wegnerove ocjene. 1. Alloy type model and the integrated density of states We consider Schrödinger operators with a potential which is a stochastic process ergodic with respect to translations from Zd. Such operators model quantum mechanical Hamiltonians which govern the motion of single electrons in disordered solids. The spectral properties of the Schrödinger operator are related to the dynamical behaviour of the electron wave packets and thus to the charge transport properties of the described solid, cf. e.g. [4, 12, 29]. The random potential considered in this note is of alloy or continuous Anderson type: V : Ω× R → R, Vω(x) = ∑