Scattering Theory on Manifolds

流形上的散射理论

• 批准号: 0901334
• 负责人: Antonio Sa Barreto
• 金额: $ 30.16万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901334&HistoricalAwards=false
• 关键词: Scattering; Theory; Manifolds

In this project the principal investigator will study scattering and inverse scattering on asymptotically Euclidean and asymptotically hyperbolic manifolds. Examples of asymptotically hyperbolic manifolds include quotients of hyperbolic space by certain groups of motion. The de Sitter-Schwarzschild model of the exterior of a black hole can be viewed as an asymptotically hyperbolic manifold with two ends, while the Schwarzschild model of the exterior of a black hole can be viewed as a manifold with two ends that is asymptotically hyperbolic on one end and asymptotically Euclidean on the other. The principal investigator will study the asymptotic behavior of solutions of the wave equation on nontrapping asymptotically hyperbolic manifolds and the Schwarzschild model of a black hole. In both cases the long-time behavior of waves is associated with the distribution of resonances, or scattering poles, and the project will address this question as well. The principal investigator will also study inverse scattering, including the question of determining an asymptotically Euclidean manifold from the scattering matrices at all energies. This requires the proof of a support theorem that generalizes to this setting the well-known support theorem for the Radon transform in Euclidean space.One central problem in the study of partial differential equations is to understand how geometric properties of a medium influence the way waves propagate on it. The principal investigator will study the long time-behavior of solutions to the wave equation on spaces that are motivated by examples from physics. The "inverse problem" consists of using measurements obtained from waves that propagate on a certain medium to determine some of the medium's geometric properties. For example, by knocking on the surface of an object and listening to how it responds to this signal, one can obtain information about the object's interior; by measuring the time a wave travels between two points on the surface of the object one would like to obtain information about the variable speed of propagation of the waves in the interior of the object. The principal investigator will study problems of this nature in the spaces indicated above. The techniques that need to be developed might also have applications in other areas, such as tomography.

在这个项目中，主要研究者将研究渐近欧氏流形和渐近双曲流形上的散射和逆散射。渐近双曲流形的例子包括某些运动组的双曲空间的商。黑洞外部的de Sitter-Schwarzschild模型可以看作是一个两端渐近双曲的流形，而黑洞外部的Schwarzschild模型可以看作是一个两端渐近双曲、另一端渐近欧几里得的流形。主要研究者将研究非陷阱渐近双曲流形上的波动方程的解的渐近行为和黑洞的Schwarzschild模型。在这两种情况下，波的长期行为都与共振或散射极点的分布有关，该项目也将解决这个问题。首席研究员还将研究逆散射，包括从所有能量下的散射矩阵确定渐近欧几里德流形的问题。这需要证明一个支持定理，它推广了欧氏空间中著名的Radon变换支持定理。研究偏微分方程组的一个中心问题是了解介质的几何性质如何影响波在其上传播的方式。主要研究人员将研究空间上波动方程的解的长时间行为，这些行为是由物理学中的例子引起的。“逆问题”包括使用从在特定介质上传播的波获得的测量结果来确定该介质的某些几何性质。例如，通过敲击物体的表面并聆听它对该信号的响应，可以获得关于物体内部的信息；通过测量波在物体表面的两个点之间传播的时间，人们希望获得关于波在物体内部传播的可变速度的信息。首席调查员将在上述空间研究这种性质的问题。需要开发的技术也可能在其他领域应用，如断层成像。