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

DMS-9971031ABSTRACT本研究将在多个方向上扩展Weyl-Titchmarsh-Levinson方法学。第一个重点领域将是m函数的构造和分析，以了解谱行为，特别是具有内部奇点的系统和准周期系统的谱行为。此外，还将进行涉及跨频带的m函数的解析延拓的分析，以了解共振现象。下一个目标将是利用m函数来理解受各种谱行为影响的散射行为，特别是关于共存的绝对连续谱、奇异连续谱和点连续谱及其扰动。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，这些估计有望被用来从散布数据中开发基于m-函数的潜在恢复方案。由于它们是许多自然系统建模和后续研究的基础，所以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。这个项目中涉及的问题是Sturm-Liouville方程的光谱和散射理论，这些方程在各种背景下出现，如量子力学、流体波力学、晶体结构、声学、量子混沌、光纤设计、光解、壳结构变形和生物种群模拟；所获得的结果尤其有望对这些领域产生重大影响。该理论有助于深入探讨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必要基础，因此也是文明进步的基础。

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.