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.

研究的对象是通过在紧凑的度量空间上沿着沿着梯形离子的轨道进行采样的。与给定的ergodic转换上的无抗指数家族的动态性联系在一起。对临时统计的一般共生的非均匀双曲线，几乎是MATHIEU操作员的统计数据，正方形的纤维纤维纤维菌在中间的耦合中的纤维纤维纤维性均具有统一的耦合。具有应用于KDV方程的系统运算符，以及与几乎周期性的Verblunsky系数相关的T圆的测量研究的研究，在疾病环境中的量子机械现象的研究一直在进行中，正在进行中的OFA正在进行的研究由于。在ErgodicSchrödinger运营商的框架内，对物理的框架进行了数学研究。对人力资源发展的影响。