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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.

本研究课题主要研究无序环境中量子力学现象的理论。该研究旨在扩展遍历薛定谔算子框架内无序结构电子性质的数学分析。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，在物理学方面具有潜在的影响。该项目研究遍历薛定谔算子的谱特性。这些薛定谔算子的势函数是通过在紧度量空间上沿遍历变换的轨道沿着连续函数采样得到的。这个框架涵盖了许多感兴趣的例子，如准周期势和随机势。该项目的主要目标是调查以下方面：一维拟周期薛定谔算子的正谱和逆谱理论及其在Korteweg-de弗里斯方程中的应用，高维拟周期薛定谔算子绝对连续谱的存在，能隙长度衰减与势的光滑性之间的关系，与动态定义的Cantor集上的平衡测度相关的Jacobi参数的几乎周期性，一般算子重整化方程的范围及其应用，超越双曲动力学范围的一维自相似势，周期薛定谔算子扰动的束缚态和本质谱之间的联系，以及概周期雅可比矩阵的正谱和逆谱理论之间的接口。