Research On Wave Equations and Scattering Theory

Pi：Antonio sa Barreto，普渡大学DMS-0140657项目摘要：波动方程和散射理论流形的研究在这个项目的第一部分，研究者将寻求对欧氏空间中任意二阶自伴扰动的共振计数函数的下界的改进。第二部分研究渐近双曲流形中的散射理论。建立了这些空间中共振计数函数的上界和可能的下界，并根据辐射场给出了散射矩阵的动力学定义。在渐近欧几里得情形下，后半部分也是可能的。第三部分研究了从散射矩阵中恢复关于渐近双曲流形或渐近欧氏流形及其黎曼结构的信息的问题。预计散射矩阵的动力学定义也将在这一部分发挥重要作用。这个项目研究波如何在介质中传播，以及从关于波在介质中传播的特定知识中可以提取关于介质的哪种类型的信息。例如，已知介质的某种扰动，如空间中的障碍物，如何影响波的传播，反之亦然，根据从障碍物反射的波获得的信息来确定障碍物的性质。它还涉及到在几何上与通常的欧几里德空间不同的空间上的波动方程的研究，例如类似于无穷大处的双曲空间的空间。

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.