Mathematical Sciences: Research in Scattering Theory

本项目涉及三个数学分析方面的研究项目。第一部分讨论了在上半平面自同构函数上定义的拉普拉斯算子对SL（2，R）的有限子群模的谱变化。目标是证明对于一般情况，特征值的渐近分布小于Weyl分布所要求的。第二个领域的重点是双曲空间中的圆问题，其最简单的形式是比较一个圆中包含的平面上的点阵点的数量与圆的面积。虽然与欧几里得情况的比较没有价值，但这种关系的渐近值已经稳步提高。在紧致情况下，误差项的估计工作正在进行中。最后，继续研究非紧型对称空间的散射理论。虽然这是在20世纪70年代发展起来的，但一些已发表的工作包含空白（除了相当复杂之外）。将努力寻找一种更简单和更连贯的治疗方法，并填补空白。散射理论出现在许多情况下，当人们能够测量存在隐藏障碍的事件的结果时。该理论试图从初始和最终数据中提取有关障碍的信息。

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.