 0045 Francois Germinet
 Dynamical Localization II with an Application to the Almost Mathieu Operator  Published version
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Abstract. Several recent works have established dynamical localization for Schr\"o\dinger
operators, starting from control on the localization length of their eigenfunctions,
in terms of their centers of localization. We provide an alternative way to obtain
dynamical localization, without resorting to such a strong condition on
the exponential decay of the eigenfunctions. Furthermore, we illustrate our
purpose with the almost Mathieu operator, $H_{\theta,\lambda,\omega} =
\Delta + \lambda \cos(2\pi(\theta+x\omega))$, $\lambda\geq 15$ and $\omega$
with good Diophantine properties. More precisely, for almost all $\theta$, for
all $q>0$ and for all functions $\psi\in\ell^2(\Z)$ of compact support, we show that
$$
\sup_t \la e^{iH_{\theta,\lambda,\omega} t}\psi, X^q
e^{iH_{\theta,\lambda,\omega} t}\psi \ra \ < C_\psi.
$$
The proof applies equally well to discrete and continuous random
Hamiltonians. In all cases, it uses as input a repulsion principle of singular boxes,
supplied in the random case by the multiscale analysis.
Appeared in J. Stat. Phys., vol 95, 273286 (1999)
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