98-391 Barbaroux J.M., Fischer W., M\"uller P.

A Criterion for Dynamical Localization in Random Schrodinger Models (74K, LATeX) 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 energy-interval $I$ 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 Anderson-type model in the lattice case and to a model with a correlated random potential in the continuous case.

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