Weyl-Titchmarsh-Levinson Spectral Analysis of Sturm-Liouville Expressions

Sturm-Liouville 表达式的 Weyl-Titchmarsh-Levinson 谱分析

• 批准号: 9971031
• 负责人: Dominic Clemence
• 金额: $ 5.1万
• 依托单位: North Carolina Agricultural & Technical State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2002-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971031&HistoricalAwards=false
• 关键词: Weyl; Titchmarsh; Levinson; Spectral; Analysis

DMS-9971031ABSTRACTThis research will extend the Weyl-Titchmarsh-Levinson methodology in several directions. The first area of focus will be the construction and analysis of the m-function to understand spectral behavior, particularly of systems with interior singularities and quasi-periodic systems. Further, analysis involving analytic continuation of the m-function across spectral bands will be pursued to understand resonance phenomena. The next goal will be the exploitation of the m-function to understand scattering behavior subject to various spectral behavior, with particular regard for coexisting absolutely continuous, singular continuous, and point continuous spectrum, as well as their perturbations. The final focus will involve using the m-function to obtain various estimates, such as Greens' function estimates, which are expected to be employed to develop an m-function based potential recovery scheme from scattering data. Since they are the basis for the modeling and subsequent study of many natural systems, the study of differential equations is the foundation for the scientific analysis and understanding of a variety of physical phenomena. The problems addressed in this project are couched in the spectral and scattering theory of Sturm-Liouville equations, which arise in a variety of settings such as quantum mechanics, fluid wave mechanics, crystal structures, acoustics, quantum chaos, optic fiber design, photodissociation, shell structure deformation, and biological population modeling; the results obtained are in particular expected to significantly impact these areas. The theory facilitates probing deeply into the mathematical foundations upon which these physical systems are based, thereby increasing our scientific understanding of the various systems. The involvement of undergraduate students in this research further ensures a continued knowledge base, which is the essential foundation for developments in advanced technology, and hence the advancement of civilization.

摘要本研究将在几个方向上扩展Weyl-Titchmarsh-Levinson方法。第一个重点领域将是m函数的构建和分析，以理解光谱行为，特别是具有内部奇点和准周期系统的系统。此外，涉及跨谱带m函数解析延拓的分析将被用于理解共振现象。下一个目标将是利用m函数来理解各种光谱行为下的散射行为，特别是共存的绝对连续、奇异连续和点连续光谱，以及它们的摄动。最后的重点将涉及使用m函数来获得各种估计，例如格林函数估计，预计将用于开发基于m函数的散射数据潜在恢复方案。由于微分方程是许多自然系统建模和后续研究的基础，因此对微分方程的研究是科学分析和理解各种物理现象的基础。本项目解决的问题是在Sturm-Liouville方程的光谱和散射理论中表达的，这些理论在量子力学、流体波动力学、晶体结构、声学、量子混沌、光纤设计、光解离、壳结构变形和生物种群建模等各种环境中出现；预计所取得的结果将对这些领域产生重大影响。该理论有助于深入探索这些物理系统所基于的数学基础，从而增加我们对各种系统的科学理解。本科生参与这项研究进一步确保了一个持续的知识基础，这是先进技术发展的必要基础，从而促进了文明的进步。