Positive Lyapunov Exponents for Schroedinger Cocycles

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ABSTRACT This project deals with several problems on the interface between dynamical systems theory and mathematical physics. The main object of study are linear cocycles over general base dynamics, with special emphasis on Schroedinger cocycles. The core problem is to decide whether the Lyapunov exponents of such cocycles are positive. It is expected that positive Lyapunov exponents occur for the majority of models. Spectral theoretical consequences of positive Lyapunov exponents are absence of absolutely continuous spectrum, spectral localization, and dynamical localization. General questions guiding the research on the prevalence of positive Lyapunov exponents forSchroedinger cocycles are the following: How much randomness of the base dynamics is needed? How relevant are smoothness properties of the cocycle? Are Lyapunov exponents always positive in the large-coupling regime? These questions will be investigated with the help of a recent extension of Furstenberg's Theorem due to Bonatti, Gomez-Mont, and Viana, a parameter exclusion method in the spirit of Benedicks-Carleson and Young, and refinements and extensions of Kotani theory.Positive Lyapunov exponents and their spectral and quantum dynamicalimplications are essential to a better understanding of the consequences of complexity and disorder in quantum mechanical systems. It is widely expected that the degree of randomness of the environment should be reflected in the transport properties of the associated quantum system. This connection is the main motivation for the research project.