Dynamical Systems and Spectral Theory

动力系统和谱理论

• 批准号: 1067988
• 负责人: David Damanik
• 金额: $ 30.3万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1067988&HistoricalAwards=false
• 关键词: Dynamical; Systems; Spectral; Theory

The objects of study are Schrödinger operators whose potentials are obtained by sampling with a continuous function along the orbits of an ergodic transformation on a compact metric space. This framework covers many examples of interest, such as almost-periodic potentials and random potentials. The spectral properties of such operators are closely linked to the dynamical behavior of an energy-indexed family of cocycles over the given ergodic transformation. The problems the proposer intends to investigate include the following: an analysis of Schrödinger cocycles over uniformly hyperbolic transformations and an analog of the Kotani Support Theorem in this context, sufficient conditions for the absence of non-uniform hyperbolicity for general cocycles over subshifts, eigenvalue statistics for the critical almost Mathieu operator, the structure of the spectrum of square Fibonacci Hamiltonian at intermediate coupling, denseness of uniform hyperbolicity for cocyles over strictly ergodic transformations beyond the continuous category, inverse spectral theory in the presence of absolutely continuous spectrum, direct and inverse spectral theory for quasi-periodic Schrödinger operators with applications to the KdV equation, and the study of measures on the unit circle associated with almost periodic Verblunsky coefficients.Quantum mechanics is a fundamental branch of physics whose foundations were established during the first half of the twentieth century. The study of quantum mechanical phenomena in disordered environments has been an area of ongoing active study since the 1950's. The mathematical study of electronic properties of disordered structures is carried out within the framework of ergodic Schrödinger operators. This project therefore has a potential impact on physics as it improves our understanding of quantum mechanical transport properties of media exhibiting certain kinds of disorder. Through the training of graduate students the project has in addition an impact on human resource development.

研究的对象是薛定谔算子，其势函数是通过沿紧度量空间上遍历变换的轨道沿着连续函数采样得到的。这个框架涵盖了许多感兴趣的例子，如准周期势和随机势。这类算子的谱性质与给定遍历变换上的能量指数余圈族的动力学行为密切相关。提议者打算调查的问题包括：一致双曲变换上的薛定谔余圈的分析和Kotani支撑定理在这方面的模拟，子移位上的一般余圈不存在非一致双曲性的充分条件，临界几乎Mathieu算子的本征值统计，中等耦合下平方Fibonacci Hamilton的谱结构，连续范畴以外严格遍历变换上的cocyles的一致双曲性的稠密性，绝对连续谱存在下的逆谱理论，拟周期薛定谔算子的正逆谱理论及其对KdV方程的应用，以及与概周期Verblunsky系数相关的单位圆上的度量的研究。量子力学是物理学的一个基本分支其基础是在二十世纪上半叶建立的。自20世纪50年代以来，无序环境中的量子力学现象的研究一直是一个持续活跃的研究领域。无序结构的电子性质的数学研究是在遍历薛定谔算子的框架内进行的。因此，这个项目对物理学有潜在的影响，因为它提高了我们对表现出某些类型的无序介质的量子力学传输特性的理解。通过培训研究生，该项目还对人力资源开发产生了影响。