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.

A Matrix-Valued Point Interactions Model

“…This procedure allows us to construct from H(ω) an operatorĤ(ω) define on a bigger probability space which is R-ergodic.Ĥ(ω) is also constructed in a way such that its IDS and Lyapunov exponents exist if and only if those of H(ω) exist and in this case they are equal for both operators. Also, from the properties of the Floquet exponent w combined with previous results of Kotani (see [13]) one can repeat the discussions of Sections 4.1 and 4.2 in [4] to prove the following Thouless formula for H(ω).…”

“…Applying theorem 1 in [4] and proposition 2 we have proved theorem 3. Then applying theorem 3 on every compact interval I ⊂ R \ S where S is obtained in theorem 1, we get theorem 4.…”

“…In this section we prove theorem 3. We will see how to use general results of [4] for the operator H(ω). We can deduce theorem 3 from theorem 1, p.885 in [4] once we have proved the following estimates on the transfer matrices.…”

“…There already exists such formula for scalar-valued point interactions operators as presented in [7]. Adapting the Borel measure representation method of [7], using Lie-Trotter formula as it is done in [4] and noticing that the time-ordered exponential in [4] becomes now a usual exponential, one gets…”

Lifshitz Tails for Continuous Matrix-Valued Anderson Models

“…We have already proved in [1] that, for the general model H 0 (ω), for every E ∈ R, the limit (1.3) exists and is P-almost-surely independent of ω ∈ Ω (see [1,Corollary 1]). The question of the existence of (1.3) involves two problems to solve.…”