Research on Hyperbolic Equations and Scattering Theory

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德拉普拉斯方程的二阶自伴扰动的共振的存在性。其次，他打算继续他对渐近双曲流形的散布理论的研究。这种流形的具体例子是双曲空间及其由某些离散的线性分式变换群构成的商群。本文所用方法也可用于研究黑洞的de Sitter-Schwarzschild模型的微扰散射。主要研究人员将研究共振的分布以及共振与波群轨迹之间的关系。在第三个主题中，研究人员建议研究在几种不同情况下从散射矩阵中可以获得什么类型的信息。他建议考虑欧几里德度规的透明度规扰动的存在问题，即在固定能量下不由散射矩阵决定的那些信息。他还打算考虑从所有能量的散射矩阵的知识中确定在无穷远有足够衰减的非紧支撑度规微扰的问题。同样的问题可以在非欧几里德环境中提出，例如，对于定义在渐近双曲流形上的度规和势。如上所述，在某种意义上这是黑洞的de Sitter-Schwarzschild模型的情况。我们想知道这个模型的时间无关的微扰是否可以从所有能量的散射矩阵中被度规和势覆盖。本方案第一个项目的总体思想是在不同的情况下研究确定的微扰对介质的影响。例如，确定性扰动将如何影响光或声音的传播。共振可以看作是微扰对波传播的一种衰减效应。第三个项目涉及这个问题的倒数，即知道未知微扰对介质的影响，确定微扰。在第二个项目中，研究人员将研究介质为“渐近双曲”的情况下的类似问题。一个密切相关且重要的例子是黑洞的Thede Sitter-Schwarzschild模型。

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.