Dynamical Localization II with an Application to the Almost Mathieu Operator |
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Authors: | Germinet François |
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Affiliation: | (1) UFR de Mathématiques and LPTMC, Université Paris VII, Denis Diderot, 75251 Paris Cedex, 05, France;(2) Present address: UFR de Mathématiques and URA, USTL, 59655 V. d'Ascq, France |
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Abstract: | Several recent works have established dynamical localization for Schrödinger operators, starting from control on the localization length of their eigenfunctions, in terms of their centers of localization. We provide an alternative way to obtain dynamical localization, without resorting to such a strong condition on the exponential decay of the eigenfunctions. Furthermore, we illustrate our purpose with the almost Mathieu operator, H, , =–+ cos(2(+x)), 15 and with good Diophantine properties. More precisely, for almost all , for all q>0, and for all functions 2() of compact support, we show thatThe proof applies equally well to discrete and continuous random Hamiltonians. In all cases, it uses as input a repulsion principle of singular boxes, supplied in the random case by the multi-scale analysis. |
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Keywords: | dynamical localization random Schrö dinger operator almost Mathieu model multiscale analysis uniform exponential localization |
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