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Spectral Theory and Integrable Systems

谱理论和可积系统

基本信息

批准号：
1700179
负责人：
Milivoje Lukic
金额：
$ 15万
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：
2017-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700179&HistoricalAwards=false
关键词：
Spectral Theory Integrable Systems

项目摘要

This project studies problems in spectral theory, the mathematical theory that describes physical notions such as energy levels of quantum systems and vibration frequencies of mechanical systems. The mathematical models considered are in regimes where there are competing influences from a disorder in the interaction and some long-range order, such as spatially slowly decaying interactions and quasi-periodic interactions. One focus is on applications of spectral theory to explain conservation laws in certain nonlinear systems and find otherwise hidden predictability in their behavior, described through the mathematical notion of integrability. The project focuses on Schrodinger operators, central to quantum mechanics, and the mathematical methods developed have the potential to illuminate other mathematical models and physical applications, such as electron conductivity in disordered materials and signal transmission using solitons. One main focus of this project is the spectral theory of almost periodic Schrodinger operators and integrability of the Korteweg-de Vries equation with almost periodic initial data. While a Lax pair representation formally rewrites the equation as an isospectral flow, rigorous characterizations of integrability, such as construction of an inverse scattering transform, are highly dependent on the phase space under consideration and require and motivate deep new results in direct and inverse spectral theory. Other topics considered in this project include estimates for the size and continuity of the solution in terms of spectral data, Schrodinger operators with slowly decaying potentials, higher-order Szego theorems, and transport properties of Schrodinger operators and quantum spin systems.

这个项目研究光谱理论中的问题，光谱理论是描述物理概念的数学理论，如量子系统的能级和机械系统的振动频率。所考虑的数学模型是在相互作用中存在无序和一些长程有序的竞争影响的区域，例如空间缓慢衰减的相互作用和准周期相互作用。其中一个重点是应用谱理论来解释某些非线性系统中的守恒定律，并通过数学上的可积性概念来描述它们的行为中隐藏的可预测性。该项目专注于量子力学的核心薛定谔算符，所开发的数学方法有可能阐明其他数学模型和物理应用，如无序材料中的电子传导性和使用孤子进行信号传输。这个项目的一个主要焦点是概周期薛定谔算子的谱理论和具有概周期初值的Korteweg-de Vries方程的可积性。虽然Lax对表示形式上将方程改写为等谱流，但对可积性的严格刻画，如构造逆散射变换，高度依赖于所考虑的相空间，并需要并激励正、逆谱理论中的深层次新结果。这个项目中考虑的其他主题包括根据谱数据对解的大小和连续性的估计、具有缓慢衰减势的薛定谔算子、高阶Szego定理以及薛定谔算子和量子自旋系统的输运性质。