Research on Hyperbolic Equations and Scattering Theory

双曲方程与散射理论研究

• 批准号: 9970229
• 负责人: Antonio Sa Barreto
• 金额: $ 7.3万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-15 至 2002-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970229&HistoricalAwards=false
• 关键词: Research; Hyperbolic; Equations; Scattering; Theory

The proposed research of Antonio Sa Barreto is divided in three main parts.In the first topic, the investigator intends to study theexistence of resonances for second order self-adjoint perturbations of theEuclidean Laplacian. Secondly he intends to continue his research onScattering theory for asymptotically hyperbolicmanifolds.Particular examples of such manifolds are the hyperbolic space and itsquotientsby certain discrete groups of linear fractional transformations. The methodsused here can also be applied to study scattering for perturbations of theDe Sitter-Schwarzschild model of black holes. The principal investigatorwill study the distributions of resonances andthe connection between the resonances and the trace of the wave group.In the third topic, the investigator proposes to study what type ofinformation can be obtained from the scattering matrix in severaldifferent situations.He proposes to consider the question of existenceof transparent metric perturbations of the Euclidean metric, i.e those thatare not determined by the scatteringmatrix at a fixed energy. He also intends to consider the question ofdetermination of a non-compactly supported metric perturbation, that havesufficient decay at infinity, from theknowledge of the scattering matrix at all energies.The same questions can be posed in non-Euclidean settings, as for example,for metric and potentials defined on asymptotically hyperbolic manifolds.As metioned above, this is in some sense the case ofthe De Sitter-Schwarzschild model of black holes. One would like to know ifa time independent perturbation of this model by a metric and potential can berecovered from the scattering matrix at all energies.The general idea of the first project in this proposal is to study, indifferent situations, the effects that certainperturbations have on a medium. For example how certaindisturbances will affect the propagation of light or sound. Resonancescan be seen as a damping effect the perturbation has on the propagation ofwaves. The third project concerns the reciprocal of this question, i.eknowing the effects that an unknown perturbation has on a medium, determinethe the perturbation. In the second project the investigator will studyanalogous questions for the case where the medium is ``asymptoticallyhyperbolic''. A closely related and important example is theDe Sitter-Schwarzschild model of black holes.

Antonio Sa Barreto提出的研究分为三个主要部分。在第一个主题中，研究者打算研究欧几里得拉普拉斯二阶自伴扰动的共振的存在性。 其次，他打算继续研究渐近双曲流形的散射理论。这种流形的具体例子是双曲空间及其由某些离散线性分数变换组得到的商。这里使用的方法也可以应用于研究黑洞德西特-史瓦西模型扰动的散射。 首席研究员将研究共振的分布以及共振与波群轨迹之间的联系。在第三个主题中，研究者建议研究在几种不同情况下可以从散射矩阵中获得什么类型的信息。他建议考虑欧几里得度量的透明度量扰动的存在问题，即那些在某个时刻不是由散射矩阵确定的扰动。 固定能量。他还打算根据所有能量的散射矩阵的知识，考虑确定非紧支持的度量扰动的问题，该扰动在无穷远处具有足够的衰减。在非欧几里得设置中可以提出相同的问题，例如，对于渐近双曲流形上定义的度量和势。如上所述，从某种意义上来说，这就是 De 的情况。 黑洞的西特-史瓦西模型。人们想知道是否可以从所有能量的散射矩阵中恢复度量和势对该模型的时间无关扰动。本提案中第一个项目的总体思路是研究在不同情况下某些扰动对介质的影响。例如，某些干扰将如何影响光或声音的传播。 共振可以被视为扰动对波传播的阻尼效应。第三个项目涉及这个问题的倒数，即了解未知扰动对介质的影响，确定扰动。 在第二个项目中，研究者将研究介质为“渐近双曲”情况的类似问题。 一个密切相关且重要的例子是黑洞的德西特-史瓦西模型。