01-228 F. Germinet, A. Klein

Bootstrap Multiscale Analysis and Localization in Random Media (revised version) (636K, .ps) Jun 25, 01

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Abstract. We introduce an enhanced multiscale analysis that yields sub-exponentially decaying probabilities for \emph{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 $1-\mathrm{e}^ {-L^\zeta}$, for any $0<\zeta<1$. 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 $1 - \frac 1 {841^d}$ ($d$ is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any $0<\zeta<1$. 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.

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