97-430 F. Germinet, S. De Bi\`evre

Dynamical Localization for Discrete and Continuous Random Schr\"odinger Operators (64K, Latex) 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 tight-binding 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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