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

泽特技术描述。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问题谱对该问题的依赖性。该项目是上述工作的自然延续。计划扩大所涵盖的自伴问题的类别(例如，通过包括所谓的“左定”问题)，特别是将上述工作扩展到非自伴情况。后一种情况是可怕的：对于自伴情况，没有可以与发展良好的理论相比较的一般理论。作者利用代数几何中的工具，针对自伴情况介绍了一种新的方法，对这种情况有了很好的认识，并有望对非自伴情况也这样做。斯图姆-刘维尔问题有一段著名的历史，可以追溯到1836-37年斯图姆和刘维尔的开创性论文。数学家、科学家和工程师已经就这些问题撰写了数千篇论文。然而，该领域至今仍十分活跃，每年都会发表数十篇论文。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、大气中的涡旋、中子输运等。这个项目的主要目的是定性和定量地研究一些对这些问题知之甚少的主要和困难类别。

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.