Spectral Theory of Ergodic Operators

量子力学是物理学的一个基本分支，其基础是在20世纪上半叶建立起来的。自20世纪50年代以来，无序环境中量子力学现象的研究一直是一个活跃的研究领域。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无序的介质的量子力学输运特性的理解。通过培养研究生，该项目还对人力资源开发产生了影响。该项目将研究拟周期薛定谔算子的正反谱理论及其在KdV方程中的应用，中间耦合时平方斐波那契哈密顿量的谱结构和态密度测量的类型，量子系统中的输运和决定相关输运指数的机制，一维弱耦合伯努利-安德森模型态密度测量的绝对连续性，周期薛定谔算子的束缚态和本质谱之间的联系，以及几乎周期Jacobi矩阵的正逆谱理论之间的接口。谱理论和动力系统的方法将用于实现这些目标。

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 Schrodinger 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. The project will study direct and inverse spectral theory for quasi-periodic Schrodinger operators with applications to the KdV equation, the structure of the spectrum and the type of the density of states measure of the square Fibonacci Hamiltonian at intermediate coupling, transport in quantum systems and mechanisms that determine the associated transport exponents, the absolute of continuity of the density of states measure of the weakly coupled Bernoulli-Anderson model in one dimension, connections between bound states and essential spectrum for perturbations of periodic Schrodinger operators, and the interface between direct and inverse spectral theory for almost periodic Jacobi matrices. Methods from spectral theory and dynamical systems will be used in pursuing these goals.