Bornes dynamiques pour des opérateurs de Schrödinger quasi-périodiques

Laurent Marin 1
1 Laboratoire mathématiques - analyse, probabilités, modélisation
MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
Abstract : In this thesis, we study the dynamics of discrete, one-dimensional, sturmian Schrödinger operators. The main result is a dynamical bound from above for transport exponents that valuate speed of the wavepacket spreading. This bound is true for almost every sturmian potential and is sub-ballistic for a coupling constant big enough. This bound is valid with respect to a diophantine condition on the irrational number that define the potential. This condition is true for almost every irrational numbers. We show an example of irrational number with ballistic motion at any coupling constant. We study the fractal dimension of the spectrum of these operators which can bound from below, under more restrictive assumptions, transport exponents.We get a new bound from below for the box dimension of the spectrum. Assumptions needed to use this bound on dynamical purpose are the initial condition to be cyclic and the potential associated to a bounded means irrational number. In the last part of the thesis, we show that the spectrum of the operator associated to the so-called silver mean ß = [2, 2, . . . ] has a hyperbolic structure. The spectrum can be express as the non wandering set of a dynamical system. Using Markov partition method, we conjugate its dynamics to a symbolic one. The dynamical system behave like a Smale horseshoe. We derive from hyperbolicity spectral information, especially on fractal dimension. For example, we get that Hausdorff and box dimensions coincide for this operator.
Document type :
Theses
Mathematics [math]. Université d'Orléans, 2009. French. <NNT : 2009ORLE2042>


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Laurent Marin. Bornes dynamiques pour des opérateurs de Schrödinger quasi-périodiques. Mathematics [math]. Université d'Orléans, 2009. French. <NNT : 2009ORLE2042>. <tel-00482512>

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