A Qualitative and Quantitative Study of Self-Adjoint and Non-Self-Adjoint Sturm-Liouville Problems

Zettl技术说明。在过去五年左右的一系列论文中，这一建议的三位作者与一些合作者一起，开始系统地研究自伴Sturm-Liouville问题的谱对问题的依赖性。本项目是上述工作的自然延续。它计划扩大类self-adjointproblems覆盖（例如，通过包括所谓的“左定”问题），特别是，上述工作扩展到非自伴的情况下。后一种情况令人生畏：没有一般理论可以与自伴情况的成熟理论相比较。一种新的方法介绍了作者forthe自伴的情况下，使用工具从代数几何，产生了considerableinsight为这种情况下，预计这样做也为非自伴的情况下。Sturm-Liouville问题有一个著名的历史可以追溯到Sturm和Liouville在1836-37年的开创性论文。数学家、科学家和工程师们已经就这些问题发表了数千篇论文，但这个领域至今仍十分活跃，每年都有数十篇论文发表。这些问题在纯数学、应用数学、量子力学、科学和工程中有着广泛的应用。例如，它们可以用来研究氢原子的光谱，在大气中的涡流，中子输运等，等这个项目的主要目的是研究，定性和定量，一些主要和困难的类这样的问题知之甚少。

ZettlTechnical description. In a series of papers over the last five years or so, the three authors of this proposal, together with a number of collaborators, embarked on a systematic study of the dependence of the spectrum of self-adjoint Sturm-Liouville problems on the problem. This project is a naturalcontinuation of the above work. It is planned to enlarge the class of self-adjointproblems covered (e.g. by including the so called "left-definite" problems) and,especially, extend the above work to the non-self-adjoint case. The latter case isformidable: there is no general theory that can be compared with the well developedtheory for the self-adjoint case. A new approach introduced by the authors forthe self-adjoint case, using tools from algebraic geometry, has yielded considerableinsight for this case and is expected to do so also for the non-self-adjoint case.General description. Sturm-Liouville problems have a celebrated history dating backto the seminal papers of Sturm and Liouville in 1836-37. Thousands of papers have been written on these problems by mathematicians and by scientists, and engineers.Yet the field is still intensely active today with dozens of papers published every year. These problems have a wide range of applications in pure mathematics, applied mathematics, quantum mechanics, the sciences and engineering. For example, they can be used to study the spectrum of the hydrogen atom, eddies in the atmosphere, neutron transport, etc. etc. The main purpose of this project is to study, both qualitatively and quantitatively, some major and difficult classes of such problems about which little is known.