Lifshitz Tails for Continuous Matrix-Valued Anderson Models

摘要

This paper is devoted to the study of Lifshitz tails for a continuous matrix-valued Anderson-type model acting on , for arbitrary and . 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 and this exponent is independent of . 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 , 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 .