 98391 Barbaroux J.M., Fischer W., M\"uller P.
 A Criterion for Dynamical Localization in Random Schrodinger Models
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May 28, 98

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Abstract. We study dynamical properties of random Schr\"odinger operators
$H^{(\omega)}$ defined on the Hilbert space $\ell^2(\bbZ^d)$ or
$L^2(\bbR^d)$. We give sufficient conditions on the decay of the
Green's function to obtain the dynamical localization property
$$
\bbE\left( \sup_{T>1} \, \la\la\vert X
\vert^2\ra\ra_{T,f_I(H^{(\omega)})\psi} \right) < {\rm \infty}\ ,
$$
where $\bbE$ is the expectation over randomness, $f_{I}$ is any smooth
characteristic function of a bounded energyinterval $I$ and $\psi$
is a state vector in the domain of $H^{(\omega)}$ with compact spatial
support. The quantity $\la\la X^2 \ra\ra_{T,\varphi}$ denotes the
Cesaro mean up to time $T$ of the second moment of position
$\la X^2\ra_{t,\varphi}$ at times $0\le t\le T$ of an initial
state vector $\varphi$. Under weaker assumptions, we also prove a theorem
on the absence of diffusion. The results are applied to a
simple Andersontype model in the lattice case and to a model with a
correlated random potential in the continuous case.
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