On the Integrated Density of States of Fractional Random Schrödinger Operators

We proof the existence of the Integrated Density of States (IDS) for fractional random Schrödinger operators with Gaussian potential based on the theory of isotropic $α$-stable Lévy processes. Further we show that the IDS exhibits Lifshitz tails and determine its asymptotics at the right end of the spectrum. Isotropic$α$-stable Lévy processes are self-similar but not continuous. Thus self-similarity is a fundamental concept for these quantum operators.