Dynamical Systems and Spectral Theory

研究的对象是薛定谔算子，它的位势是在紧致度量空间上沿着遍历变换的轨道用连续函数采样得到的。这个框架涵盖了许多感兴趣的例子，例如概周期势和随机势。这种算子的谱性质与给定遍历变换上的能量指标族余循环族的动力学行为密切相关。作者打算研究的问题包括：一致双曲变换上的薛定谔余循环的分析和Kotani支撑定理的类比，子移位上一般上循环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，中间耦合的平方斐波那契哈密顿量的谱结构，连续范畴外严格遍历变换上余周期的一致双曲性的稠密性，绝对连续谱存在的逆谱理论，准周期薛定谔算子的正逆谱理论及其在KdV方程中的应用对与几乎周期的Verblunsky系数有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，其基础建立于二十世纪上半叶。无序环境中的量子力学现象的研究自20世纪50年代S以来一直是一个活跃的领域。无序结构的电子性质的数学研究是在遍历薛定谔算符的框架内进行的。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无序的介质的量子力学输运性质的理解。通过培训研究生，该项目还对人力资源开发产生了影响。

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.