Spectral Theory of Ergodic Operators

本研究项目研究无序环境中量子力学现象的理论。这项研究的目的是在遍历薛定谔算符的框架内扩展无序结构电子性质的数学分析。该项目通过提高对呈现某种无序的介质的量子力学输运性质的理解，在物理学上具有潜在的影响。该项目研究遍历薛定谔算子的谱性质。这些薛定谔算子的位势是通过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。该项目的主要目的是研究下列内容：一维拟周期薛定谔算子的正逆谱理论及其在Korteweg-de Vries方程中的应用，高维拟周期薛定谔算子绝对连续谱的存在，间隙长度的衰减与势的光滑性之间的关系，与动态定义的康托集上的平衡测量有关的雅可比参数的几乎周期性，一般算符重整化方程及其应用的范围，双曲动力学范围之外的一维自相似势，周期薛定谔算子扰动的束束态与本质谱之间的联系，以及概周期雅可比矩阵的正逆谱理论之间的界面。

This research project studies the theory of quantum mechanical phenomena in disordered environments. The research aims to extend mathematical analysis of the electronic properties of disordered structures within the framework of ergodic Schrödinger operators. The project has potential impact in physics through improvement of the understanding of quantum mechanical transport properties of media exhibiting certain kinds of disorder. The project investigates spectral properties of ergodic Schrödinger operators. The potentials of these Schrödinger operators 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 primary objectives of the project are to investigate the following: direct and inverse spectral theory for quasi-periodic Schrödinger operators in one dimension with applications to the Korteweg-de Vries equation, the presence of absolutely continuous spectrum for quasi-periodic Schrödinger operators in higher dimensions, the relationship between decay of gap lengths and smoothness of the potential, the almost periodicity of the Jacobi parameters associated with balanced measures on dynamically defined Cantor sets, the scope of general operator renormalization equations and their applications, one-dimensional self-similar potentials beyond the reach of hyperbolic dynamics, connections between bound states and essential spectrum for perturbations of periodic Schrödinger operators, and the interface between direct and inverse spectral theory for almost periodic Jacobi matrices.