Bootstrap Multiscale Analysis and Localization¶in Random Media

We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e −Lζ, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 −\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions.