Research on Hyperbolic Equations and Scattering Theory

安东尼奥·萨·巴雷托提出的研究分为三个主要部分。在第一个课题中，研究者打算研究欧几里得拉普拉斯算子的二阶自伴随微扰共振的存在性。其次，他打算继续研究渐近双曲流形的散射理论。这种流形的特殊例子是双曲空间及其由某些线性分数变换的离散群所构成的流形。这里使用的方法也可以应用于研究黑洞de Sitter-Schwarzschild模型的扰动散射。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，研究者提出研究在几种不同的情况下散射矩阵可以获得什么类型的信息。他提出考虑欧几里得度规的透明度规摄动的存在性问题，即那些在固定能量下不由散射矩阵决定的摄动。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处有充分衰减的非紧支持度规微扰的问题。同样的问题也可以在非欧几里得环境中提出，例如，对于在渐近双曲流形上定义的度规和势。如上所述，在某种意义上，这就是黑洞的德西特-史瓦西模型的情况。我们想知道这个模型在所有能量下，是否可以从散射矩阵中得到与时间无关的度规和势的扰动。本提案第一个项目的总体思路是，在不同情况下，研究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一种阻尼效应。第三个项目涉及这个问题的倒数，即知道未知扰动对介质的影响，确定扰动。在第二个项目中，研究者将研究介质是“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德西特-史瓦西模型。

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.