Lifshitz Tails for Continuous Matrix-Valued Anderson Models

Abstract

This paper is devoted to the study of Lifshitz tails for a continuous matrix-valued Anderson-type model acting on $L^2(\mathbb {R}^d)\otimes \mathbb {C}^{D}$, for arbitrary $d\ge 1$ and $D\ge 1$. We prove that, under a hypothesis of non-degeneracy of the bottom of the spectrum, the integrated density of states of the model has a Lifshitz behaviour at the bottom of the spectrum. We obtain a Lifshitz exponent equal to $-d/2$ and this exponent is independent of $D$. It shows that the behaviour of the integrated density of states at the bottom of the spectrum of a quasi-d-dimensional Anderson model is the same as its behaviour for a d-dimensional Anderson model. For $d=1$, we prove that the bottom of the spectrum is always non-dege nerate, for any matrix-valued periodic background potential, and thus each quasi-one-dimensional Anderson model has a Lifshitz exponent equal to $-1/2$.