Initial length scale estimate for the Schrödginer operator with a random fast oscillating potential in a multi-dimensional layer

Abstract

We consider the Dirichlet Laplacian in a multi-dimensional layer located between two parallel hyperplanes of codimension one. Such operator is perturbed by a fast oscillating random potential. Namely, the layer is partitioned into periodicity cells by a given periodic lattice and in each cell we consider a fast oscillating potential depending on a random variable multiplied by a global small parameters. All random variables associated with the periodicity cells are assumed to be independent and identically distributed. The fast oscillating potential introduced in the way standard for the homogenization theory. Namely, it depends on slow and fast variables, is compactly supported w.r.t. the slow variables and is periodic w.r.t. the fast ones. The main obtained result is the initial length scale estimate for the considered operator. Such estimate is the induction base for proving the spectral localization at the bottom of the spectrum by the multiscale analysis.