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.

DMS-9971031Abstractthis研究将将Weyl-Titchmarsh-Levinson方法扩展到多个方向。 重点的第一个领域是对M功能的构建和分析，以了解光谱行为，尤其是室内奇异性和准周期系统的系统。此外，将进行涉及跨光谱频段的M功能的分析延续的分析，以了解共振现象。下一个目标将是对M功能的开发来理解以各种光谱行为为生的散射行为，特别考虑共存绝对连续的，奇异的连续和点连续光谱以及它们的扰动。最终焦点将涉及使用M功能来获得各种估计值，例如绿色功能估计值，这些估计值有望从散射数据中开发出基于M功能的潜在恢复方案。由于它们是对许多自然系统进行建模和随后研究的基础，因此对微分方程的研究是对各种物理现象的科学分析和理解的基础。 该项目中解决的问题是在Sturm-liouville方程式的光谱和散射理论中，这些方程式在各种环境中出现，例如量子力学，流体波力力学，晶体结构，声学，声学，量子混乱，光纤纤维设计，光纤设计，壳结构的造成壳结构变形和生物学种群模型；特别是预期获得的结果会显着影响这些领域。该理论促进了这些物理系统所基于的数学基础，从而促进了我们对各种系统的科学理解。本科生参与这项研究的参与进一步确保了持续的知识基础，这是先进技术发展的基础，因此是文明的发展。