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

DMS-9971031摘要本研究将在几个方向上扩展Weyl-Titchmarsh-Levinson方法。 重点的第一个领域将是构造和分析的m-函数，以了解光谱行为，特别是系统的内部奇点和准周期系统。此外，分析涉及跨光谱带的m-函数的解析延拓将被追求，以理解共振现象。下一个目标将是利用m函数来理解受各种光谱行为影响的散射行为，特别是共存的绝对连续，奇异连续和点连续光谱，以及它们的扰动。最后的重点将涉及使用的m-函数，以获得各种估计，如格林函数估计，这是预期将被用来开发一个m-函数为基础的潜在的恢复计划从散射数据。由于微分方程是许多自然系统建模和后续研究的基础，因此微分方程的研究是科学分析和理解各种物理现象的基础。 在这个项目中解决的问题是在斯特姆-刘维尔方程，这在各种设置，如量子力学，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，和生物种群建模出现的光谱和散射理论表达;所获得的结果特别是预计将显着影响这些领域。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。本科生参与这项研究进一步确保了持续的知识基础，这是先进技术发展的重要基础，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.