Positive Lyapunov Exponents for Schroedinger Cocycles

薛定谔循环的正李亚普诺夫指数

• 批准号: 0653720
• 负责人: David Damanik
• 金额: --
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-15 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653720&HistoricalAwards=false
• 关键词: Positive; Lyapunov; Exponents; Schroedinger; Cocycles

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.

摘要该项目处理动态系统理论与数学物理学之间的接口上的几个问题。研究的主要对象是一般基础动力学上的线性共生，特别强调了施罗辛格共生。核心问题是确定此类共生的Lyapunov指数是否为正。预计大多数模型的阳性Lyapunov指数会发生。阳性Lyapunov指数的光谱理论后果是没有绝对连续的光谱，光谱定位和动态定位。一般性问题指导了关于阳性Lyapunov指数的流行率的研究，如下：需要多少基础动力学的随机性？合过程的平滑性特性有多相关？ Lyapunov指数在大耦合方面总是积极的吗？这些问题将在由于Bonatti，Gomez-Mont和Viana引起的弗斯滕堡定理的最新扩展的帮助下进行调查，戈麦斯 - 蒙特和Viana是本尼迪克斯 - 卡勒森和Young精神的参数排除方法，以及对kotani理论的改进和扩展。对阳性和综合性的综合性和量子的综合性是更好的综合性，是对综合性的综合性，是A a频谱和综合性的综合性，即在A的综合范围内，是A an A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A S A A A S A A A A S A A S S A A组在上讲而进行。机械系统。人们普遍认为，环境的随机性应反映在相关量子系统的运输特性中。这种联系是研究项目的主要动机。