Mathematical Sciences: Research in Scattering Theory

数学科学：散射理论研究

• 批准号: 9303225
• 负责人: Ralph Phillips
• 金额: $ 4万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303225&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Scattering; Theory

This project involves three research projects in mathematical analysis. The first concerns the change in spectrum of the Laplace operator defined on automorphic functions in the upper half plane modulo a confinite subgroup of SL(2,R). The goal is to show that for the generic case, the asymptotic distribution of eigenvalues is less than the Weyl distribution calls for. The second area focuses on the circle problem in hyperbolic space which, in its simplest form, compares the number of lattice points in a plane contained in a circle with the area of the circle. Asymptotic values for the relationship have been steadily improved although comparisons with the Euclidean case are of no value. Work is proceeding on estimates of the error term in the cocompact case. Finally, work is continuing on scattering theory for symmetric spaces of noncompact type. Although this was developed in the 1970's, some of the published work contains gaps (in addition to being quite complicated). Efforts will be made to find a simpler and more coherent treatment - and to fill the gaps. Scattering theory arises in many contexts where one is able to measure the outcome of an event in which hidden obstacles are present. The theory seeks to extract information about the obstacles in terms of the initial and final data.

这个项目涉及三个数学分析方面的研究项目。第一个问题是定义在模为SL(2，R)的有限子群的上半平面上的自同构函数上的Laplace算子的谱的变化。我们的目的是证明对于一般情况，本征值的渐近分布小于Weyl分布所要求的。第二个领域集中于双曲空间中的圆问题，在其最简单的形式中，将圆中包含的平面中的格点的数目与圆的面积进行比较。虽然与欧几里德情形的比较没有价值，但该关系的渐近值一直在稳步改善。关于协紧情况下误差项的估计工作正在进行中。最后，关于非紧型对称空间的散射理论的工作还在继续。虽然这是在20世纪70年代的S发展起来的，但一些已发表的作品存在空白(除了相当复杂之外)。将努力找到一种更简单、更连贯的处理方法--并填补空白。散射理论出现在许多情况下，在这种情况下，人们能够测量存在隐藏障碍的事件的结果。该理论寻求从初始和最终数据中提取有关障碍的信息。