Positive Lyapunov Exponents for Schroedinger Cocycles

这个项目涉及动力系统理论和数学物理之间的接口的几个问题。主要的研究对象是一般基动力学上的线性上循环，特别是薛定谔上的上循环。核心问题是判断这类余循环的Lyapunov指数是否为正。预计大多数模型都会出现正的Lyapunov指数。正Lyapunov指数的谱理论结果是不存在绝对连续谱、谱局部化和动态局域化。指导研究薛定谔共周期的正Lyapunov指数流行率的一般问题如下：基础动力需要多大的随机性？共循环的光滑度属性有多大的相关性？在大耦合区域，李雅普诺夫指数总是正的吗？这些问题将借助于最近由Bonatti，Gomez-Mont和Viana推广的Furstenberg定理，一个秉承Benedicks-Carleson和Young精神的参数排除方法，以及对Kotani理论的改进和扩展来研究。正Lyapunov指数及其光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和无序的后果是至关重要的。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上。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.