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.