Positive Lyapunov Exponents for Schroedinger Cocycles

本课题研究动力系统理论与数学物理之间的接口问题。研究的主要对象是一般基动力学上的线性环，特别强调薛定谔环。核心问题是确定这些环的Lyapunov指数是否为正。预计在大多数模型中会出现正的李雅普诺夫指数。正Lyapunov指数的谱理论结果是绝对连续谱的缺失、谱局域化和动态局域化。指导研究薛定谔环的正Lyapunov指数普遍性的一般问题如下：需要多少基本动力学的随机性？循环的平滑性有多重要？李雅普诺夫指数在大耦合状态下总是正的吗？这些问题将在Bonatti， Gomez-Mont和Viana最近对Furstenberg定理的扩展，Benedicks-Carleson和Young精神的参数排除方法，以及Kotani理论的改进和扩展的帮助下进行研究。正李雅普诺夫指数及其谱和量子动力学含义对于更好地理解量子力学系统中复杂性和无序性的后果至关重要。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.