Research On Wave Equations and Scattering Theory

波动方程与散射理论研究

• 批准号: 0140657
• 负责人: Antonio Sa Barreto
• 金额: $ 13.35万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140657&HistoricalAwards=false
• 关键词: Research; Wave; Equations; Scattering; Theory

PI: Antonio Sa Barreto, Purdue UniversityDMS-0140657Abstract of the project: Research on Wave Equations and Scattering Theoryon Manifolds In the first part of this project, the investigator will seek an improvement of the currently known lower bounds for the counting function of resonance for arbitrary second order self-adjoint perturbations of the Laplacian in Euclidean space. The second part consists of studyingscattering theory in asymptotically hyperbolic manifolds. Establishing upper, and possibly lower, bounds for the counting function of resonance in these spaces, and also giving a dynamical definition of the scattering matrix in terms of the radiation fields. The latter part is also expectedto be possible in the asymptotically Euclidean case. The third part consists of studying problems of recovering information about an asymptotically hyperbolic, or asymptotically Euclidean, manifold, and its Riemannian structure, from the scattering matrix. It is expectedthat the dynamical definition of the scattering matrix will also play an important role in this part.This project concerns the investigation of how waves propagate in a medium, and reciprocally, what type of information about a medium can be extracted from a certain knowledge on the propagation of waves in it. For example, how a certain perturbation of a known medium, like anobstacle in space, affects the propagation of waves, and, vice-versa, determine properties of the obstacle from information obtained from waves reflected by it. It also concerns the study of the wave equation on spaces that are geometrically different from the usual Euclideanspace, e.g., spaces that resemble the hyperbolic space at infinity.

主要研究者：Antonio Sa Barreto，Purdue UniversityDMS-0140657项目摘要：流形上波动方程和散射理论的研究在该项目的第一部分，研究者将寻求对目前已知的共振计数函数的下界的改进，用于任意二阶自伴扰动的拉普拉斯在欧几里得空间中。 第二部分研究了渐近双曲流形上的散射理论。 建立上，并可能较低，在这些空间中的共振计数功能的界限，并给出了一个动态定义的散射矩阵的辐射场。后一部分也有望在渐近欧几里德的情况下是可能的。第三部分是研究从散射矩阵中恢复关于渐近双曲或渐近欧几里得流形及其黎曼结构的信息的问题。散射矩阵的动力学定义也将在这一部分中发挥重要作用，本项目涉及研究波在介质中的传播，以及从关于波在介质中传播的某些知识中可以提取关于介质的什么类型的信息，例如，已知介质的某种扰动，如空间中的障碍物，影响波的传播，反之亦然，从反射波获得的信息中确定障碍物的属性。它还涉及在几何上不同于通常的欧几里得空间的空间上的波动方程的研究，例如，类似于无穷远双曲空间的空间。