HÖLDER CONTINUITY OF THE INTEGRATED DENSITY OF STATES FOR MATRIX-VALUED ANDERSON MODELS

Abstract

We study a class of continuous matrix-valued Anderson models acting on L2(ℝd) ⊗ ℂN. We prove the existence of their Integrated Density of States for any d ≥ 1 and N ≥ 1. Then, for d = 1 and for arbitrary N, we prove the Hölder continuity of the Integrated Density of States under some assumption on the group GμE generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group GμE is verified. Therefore, the general results developed here can be applied to this model.