F. Germinet, A. Klein
Bootstrap Multiscale Analysis and Localization in
Random Media (revised version)
(636K, .ps)
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.