Positive Lyapunov Exponents for Schroedinger Cocycles

本项目主要研究动力系统理论与数学物理之间的一些接口问题。主要的研究对象是一般基础动力学上的线性上循环，特别强调薛定谔上循环。其核心问题是判定此类余圈的李雅普诺夫指数是否为正。预计大多数模型都会出现正的李雅普诺夫指数。正的李雅普诺夫指数的频谱理论的后果是绝对连续的频谱，频谱本地化和动力本地化的情况下。一般的问题，指导研究的流行性的积极李雅普诺夫指数forSchroedinger上循环如下：有多少随机性的基础动力学是必要的？上循环的光滑性有多相关？在大耦合区域，李雅普诺夫指数总是正的吗？这些问题将研究的帮助下，最近扩展的Furstenberg定理由于Bonatti，Gomez-Mont，和维亚纳，参数排除法的精神Benedicks-Carleson和Young，和Kotani理论的改进和扩展。积极的李雅普诺夫指数和他们的光谱和量子动力学的含义是必不可少的，以更好地理解量子力学系统中的复杂性和无序的后果。人们普遍认为，环境的随机性程度应该反映在相关量子系统的输运性质中。这种联系是研究项目的主要动机。

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.