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Spectral Theory of Ergodic Operators

遍历算子的谱论

基本信息

批准号：
1700131
负责人：
David Damanik
金额：
$ 20.1万
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：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700131&HistoricalAwards=false
关键词：
Spectral Theory Ergodic Operators

项目摘要

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.

该研究项目研究无序环境中的量子力学现象理论。该研究旨在在遍历薛定谔算子的框架内扩展对无序结构电子特性的数学分析。该项目通过提高对表现出某种无序的介质的量子力学传输特性的理解，对物理学产生潜在影响。该项目研究遍历薛定谔算子的光谱特性。这些薛定谔算子的势是通过沿着紧度量空间上的遍历变换的轨道使用连续函数采样来获得的。该框架涵盖了许多有趣的例子，例如近周期势和随机势。该项目的主要目标是研究以下内容：一维准周期薛定谔算子的正谱理论和逆谱理论及其在 Korteweg-de Vries 方程中的应用、更高维度准周期薛定谔算子的绝对连续谱的存在、间隙长度衰减与势平滑度之间的关系、相关雅可比参数的几乎周期性动态定义的康托集上的平衡测度、一般算子重正化方程的范围及其应用、超出双曲动力学范围的一维自相似势、束缚态与周期性薛定谔算子扰动的基本谱之间的联系，以及几乎周期性雅可比矩阵的正谱理论和逆谱理论之间的接口。