Research on Hyperbolic Equations and Scattering Theory

Antonio Sa Barreto的研究主要分为三个部分：第一部分，研究了欧氏拉普拉斯算子的二阶自伴扰动的共振存在性. 其次，他打算继续他的研究对散射理论的渐近hyperbolicmanifolds.特别的例子，这种流形是双曲空间和它的quotientsby某些离散群的线性分式变换.本文所用的方法也可用于研究De Sitter-Schwarzschild黑洞模型扰动的散射。 主要研究者将研究共振的分布以及共振和波群迹之间的联系。在第三个主题中，研究者提出研究在几种不同的情况下，从散射矩阵中可以获得什么类型的信息。他提出考虑欧几里德度量的透明度量扰动的存在问题，即那些在固定能量下不由散射矩阵确定的。他还打算考虑的问题ofdetermination的非紧支持度规扰动，有足够的衰减在无穷远，从知识的散射矩阵在所有的能源。同样的问题可以提出在非欧设置，例如，为度量和势定义在渐近双曲流形。如上所述，这是在某种意义上的情况下的德西特-史瓦西模型的黑洞。人们想知道这个模型是否能从散射矩阵中得到度规和势的与时间无关的扰动。这个建议的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. 共振可以看作是扰动对波传播的阻尼效应。第三个方案是关于这个问题的倒数，即已知未知扰动对介质的影响，确定扰动的大小。 在第二个项目中，研究者将研究介质为“渐近双曲”的情况下的类似问题。 一个密切相关的重要例子是德西特-史瓦西黑洞模型。

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.