Abstract
We study the spectrum of random Schrödinger operators acting onL 2(R d) of the following type\(H = - \Delta + W + \sum _{x \in \mathbb{Z}^d } t_x V_x \). The\((t_x )_{x \in \mathbb{Z}^d } \) are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.
Résumé
Dans ce travail, nous étudions le spectre d'opérateurs de Schrödinger aléatoires agissant surL 2(R d) du type suivant\(H = - \Delta + W + \sum _{x \in \mathbb{Z}^d } t_x V_x \). Les\((t_x )_{x \in \mathbb{Z}^d } \) sont des variables aléatoires i.i.d. Sous de faibles hypothèses surV, nous démontrons que le bord inférieur du spectre deH n'est composé que de spectre purement ponctuel et, que les fonctions propres associées sont exponentiellement décroissantes. Pour ce faire nous donnons une nouvelle preuve de l'estimée de Wegner valable sans hypothèses de signe surV.
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Communicated by T. Spencer
U.R.A. 760 C.N.R.S.
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Klopp, F. Localization for some continuous random Schrödinger operators. Commun.Math. Phys. 167, 553–569 (1995). https://doi.org/10.1007/BF02101535
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DOI: https://doi.org/10.1007/BF02101535