On the integrated density of states for crystals with randomly distributed impurities

Abstract

In the present paper, we discuss spectral properties of a periodic Schrodinger operator which is perturbed by randomly distributed impurities; such operators occur as simple models for crystals (or semi-conductors) with impurities. While the spectrum itself is independent of the concentrationp of impurities, for 0<p<1, we focus our attention on the limiting behavior of the integrated density of states ρp of the random Schrodinger operator, inside a spectral gap of the periodic operator, asp→0. Denoting byU0 the set of eigenvalues (in the gap) of the reference problem having precisely one impurity (located at the origin, say), we show that the integrated density of states concentrates around the points ofU0, in the sense that ρp(Ue) is of orderp, for any fixed e-neighborhoodUe ofU0, while ρp(K)≦C·p2, for any compact subsetK of the gap which does not intersectUe.