 99254 JeanMarie Barbaroux, Werner Fischer, Peter M\"uller
 Dynamical properties of random Schr\"odinger operators
(335K, Postscript)
Jul 1, 99

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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
$L^2(\bbR^d)$. Building on results from existing multiscale
analyses, we give sufficient conditions on $H^{(\omega)}$ to obtain
the vanishing of the diffusion exponent
$$
\sigma_{\rm diff}^+ := \limsup_{T\rightarrow\infty } \frac{\log
\bbE \left(\la\la\vert X
\vert^2\ra\ra_{T,f_I(H^{(\omega)})\psi}\right) }{\log T}=0.
$$
Here $\bbE$ 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 $\varphi$.
If the Hilbert space is $\ell^2(\bbZ^d)$, the method of proof can be
strengthened to yield dynamical localization.
Under weaker assumptions, we also prove a theorem on the absence of
diffusion. The results are applied to a randomly perturbed periodic
Schr\"odinger operator on $L^2(\bbR^d)$, to a simple Andersontype
model on the lattice and to a model with a correlated random
potential in continuous space.
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