Scattering Theory on Manifolds

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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.