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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.

本课题研究无序环境中的量子力学现象理论。该研究旨在在遍历Schrödinger算子的框架内扩展无序结构的电子性质的数学分析。该项目通过提高对表现出某些无序的介质的量子力学输运性质的理解，对物理学具有潜在的影响。该项目研究遍历Schrödinger算子的频谱特性。这些Schrödinger算子的势是通过紧化度量空间上沿遍历变换轨道的连续函数进行采样得到的。这个框架涵盖了许多有趣的例子，比如几乎周期势和随机势。该项目的主要目标是调查以下方面：一维拟周期Schrödinger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Schrödinger算子的绝对连续谱的存在，间隙长度衰减与势平滑的关系，动态定义Cantor集上平衡测度相关的Jacobi参数的概周期性，一般算子重整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，周期Schrödinger算子摄动的束缚态与本质谱之间的联系，以及几乎周期Jacobi矩阵的正逆谱理论之间的接口。