Lifshitz tails for matrix-valued Anderson models

TL;DR

This paper proves that the integrated density of states for a matrix-valued Anderson model exhibits Lifshitz tails with an exponent of -d/2 at the spectrum's bottom, independent of the matrix dimension.

Contribution

It establishes Lifshitz tail behavior for matrix-valued Anderson models and shows the Lifshitz exponent is independent of the matrix dimension D.

Findings

Lifshitz tails occur at the spectrum's bottom.

Lifshitz exponent is -d/2.

Behavior is independent of matrix dimension D.

Abstract

This paper is devoted to the study of Lifshitz tails for a continuous matrix-valued Anderson-type model $H_{\omega}$ acting on $L^2(\R^d)\otimes \C^{D}$, for arbitrary $d\geq 1$ and $D\geq 1$. We prove that the integrated density of states of $H_{\omega}$ has a Lifshitz behavior 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.