Lyapunov exponents of ergodic Schrödinger operators

Abstract : In this thesis we are interested in the Lyapunov exponent of ergodic Schrödinger cocycles. These cocycles occur in the analysis of solutions to the Schrödinger equation where the potential is defined with ergodic dynamics. We study two distinct aspects related to the the Lyapunov exponent for different kinds of dynamics. First we focus on a large deviation theorem for quasi-periodic cocycles and then for potentials defined by the doubling map. We prove that estimates of Bourgain & Goldstein type are granted if an upper estimate involved in the theorem is true on a strip of the complex plane. Then we establish a new technique to prove this upper bound in the quasi-periodic setting, based on subharmonic arguments suggested by Avila, Jitomirskaya & Sadel. We adapt afterwards the method to the doubling map and prove a large deviation theorem for the inverse branches of this dynamics. In the second part, we establish an asymptotic development similar to the results of Figotin & Pastur and Sadel & Schulz-Baldes for the Lyapunov exponent of Schrödinger cocycles at small coupling when the potential is a mixture of quasi-periodic and random. The analysis distinguishes the cases when the energy is either diophantine or resonant with respect to the frequency of the quasi-periodic part of the potential.
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Florian Metzger. Lyapunov exponents of ergodic Schrödinger operators. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066127⟩. ⟨tel-01632410⟩

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