Lifshitz tails for random acoustic operators

Abstract

This paper is devoted to the study of Lifshitz tails for random acoustic operators of the form Aω=−∇(1/ϱω)∇. We prove that the integrated density of states of Aω has a Lifshitz behavior at the edges of internal spectral gaps if and only if the integrated density of states of a well-chosen periodic operator is nondegenerate at the same edges.

Results dealing with the behavior of the integrated density of states of random divergence operators

“…It is quite close and follows the same steps used in [9,10] and [8] from which this work is inspired. Giving the main changes, we omit details and we refer the reader to the above references.…”

“…When d ν−d < κ + d 2 , using a similar result to Theorem 3.2 of [9] we get that for an energy E close to E + , N(E) − N(E + ) can be upper bounded by N E 0 (C · (E − E + ) + E + ), the IDS of the bounded random operator H 0 ω = Π 0 H ω Π 0 . Here Π 0 , is the spectral projection for H 1 on the band starting at E + .…”

“…The subtlety is in the good choice of the size of the cube. Using the same computation done in [8][9][10] we get that, in the case of the short range we have to estimate the double logarithm of:…”

“…In the case when (ν − d)κ < d, for ε being small, one computes the sum in (9) and gets the following estimation…”

“…This operator describes a vibrating membrane in the random medium as well as in the particular case when A ω = ω · I d (I d is the identity matrix and ω is a real function) we get the acoustic operator [2,9,10]. The interest of this operator both from the physical and the mathematical point of view is known [14].…”