Lifshitz asymptotics for Hamiltonians monotone in the randomness

Abstract

In various aspects of the spectral analysis of random Schroedinger operators monotonicity with respect to the randomness plays a key role. In particular, both the continuity properties and the low energy behaviour of the integrated density of states (IDS) are much better understood if such a monotonicity is present in the model than if not. In this note we present Lifshitz-type bounds on the IDS for two classes of random potentials. One of them is a slight generalisation of a model for which a Lifshitz bound was derived in a recent joint paper with Werner Kirsch [KV]. The second one is a breather type potential which is a sum of characteristic functions of intervals. Although the second model is very simple, it seems that it cannot be treated by the methods of [KV]. The models and the proofs are motivated by well-established methods developed for so called alloy type potentials.