 97430 F. Germinet, S. De Bi\`evre
 Dynamical Localization for Discrete and Continuous Random Schr\"odinger Operators
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Aug 4, 97

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Abstract. We show for a large class of random Schr\"odinger operators
$H(\omega)$ on $\ell^2(\Z^\nu)$ and on $L^2(\R^\nu)$ that
dynamical localization holds, {\em i.e.} that, with probability
one, for a suitable energy interval $I$ and for $q$ a positive
real,
$$
\sup_t <P_I(H_\omega)\psi_t, \ X^q P_I(H_\omega)\psi_t>\ <\infty.
$$
Here $\psi$ is a function of sufficiently rapid decrease,
$\psi_t=e^{iH_\omega t} \psi$ and $P_I(H_\omega)$ is the spectral
projector of $H(\omega)$ corresponding to the interval $I$. The
result is obtained through the control of the decay of the
eigenfunctions of $H(\omega)$ and covers, in the discrete case,
the Anderson tightbinding model with Bernouilli potential
(dimension $\nu=1$) or singular potential ($\nu>1$), and in the
continuous case Anderson as well as random Landau Hamiltonians.
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