Mathematical Physics
Dynamical bounds for Sturmian Schrödinger operators
Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1191-1196.

The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a Sturmian potential with respect to the golden number has been investigated intensively in recent years. Damanik and Tcheremchantsev developed a method and found a non-trivial dynamical upper bound for transport exponents for this model. This method can be generalized to obtain results for almost all irrational numbers. As a counter example, we exhibit a pathological irrational number with no possible better bound. Moreover, we establish a global lower bound for the lower box dimension of the spectrum that could be used to obtain a dynamical lower bound for irrational numbers with bounded density.

Le modèle de Fibonacci, c'est-à-dire un opérateur de Schrödinger associé à un potentiel Sturmien dépendant du nombre d'Or, a fait l'objet de nombreuses études ces dernières années. Cette note a pour objet de généraliser les résultats obtenus par Damanik et Tcheremchantsev sur les exposants de transport du modèle de Fibonacci au même opérateur associé à d'autres nombres irrationnels. Avec leur méthode, nous donnons une borne dynamique supérieure pour presque tout nombre irrationnel. Nous donnons un contre exemple pour lequel aucune nouvelle borne n'est possible. Enfin nous donnons une borne inférieure pour la dimension de boîte du spectre de l'opérateur. Cela nous donnera pour les nombres irrationnels à densité bornée une borne dynamique inférieure pour les exposants de transports.

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DOI: 10.1016/j.crma.2008.09.019
Marin, Laurent 1

1 UMR 6628-MAPMO, Université d'Orléans, B.P. 6759, 45067 Orléans cedex, France
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Marin, Laurent. Dynamical bounds for Sturmian Schrödinger operators. Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1191-1196. doi : 10.1016/j.crma.2008.09.019. http://www.numdam.org/articles/10.1016/j.crma.2008.09.019/

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