Localisation pour des opérateurs de Schrödinger aléatoires dans $L^2\left({ℝ}^d\right)$ : un modèle semi-classique

Klopp, Frédéric

Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 265-316.

Dans $L^2({ℝ}^d)$, nous démontrons un résultat de localisation exponentielle pour un opérateur de Schrödinger semi-classique à potentiel périodique perturbé par de petites perturbations aléatoires indépendantes identiquement distribuées placées au fond de chaque puits. Pour ce faire, on montre que notre opérateur, restreint à un intervalle d’énergie convenable, est unitairement équivalent à une matrice aléatoire infinie dont on contrôle bien les coefficients. Puis, pour ce type de matrices, on prouve un résultat de type localisation d’Anderson. On applique aussi ce résultat pour prouver la localisation à grande énergie ou grand désordre, pour des modèles d’Anderson discrets à longue portée.

In $L^2({ℝ}^d)$, we prove exponential localization for a semi-classical periodic Schrödinger operator perturbated by small independant identically distributed random perturbations put in each well of the periodic potential. To do this, we first show that our operator, restricted to some suitably chosen energy interval, is unitarily equivalent to an infinite random matrix with coefficients we can control. Then, for this type of random matrices, we prove an Anderson localization theorem. We also apply this result to prove localization at large energy or large disorder, for long range discrete Anderson models.

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