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

点击翻译按钮获取中文摘要

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.