Research On Wave Equations and Scattering Theory

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PI: Antonio Sa Barreto, Purdue UniversityDMS-0140657Abstract of the project: Research on Wave Equations and Scattering Theoryon Manifolds In the first part of this project, the investigator will seek an improvement of the currently known lower bounds for the counting function of resonance for arbitrary second order self-adjoint perturbations of the Laplacian in Euclidean space. The second part consists of studyingscattering theory in asymptotically hyperbolic manifolds. Establishing upper, and possibly lower, bounds for the counting function of resonance in these spaces, and also giving a dynamical definition of the scattering matrix in terms of the radiation fields. The latter part is also expectedto be possible in the asymptotically Euclidean case. The third part consists of studying problems of recovering information about an asymptotically hyperbolic, or asymptotically Euclidean, manifold, and its Riemannian structure, from the scattering matrix. It is expectedthat the dynamical definition of the scattering matrix will also play an important role in this part.This project concerns the investigation of how waves propagate in a medium, and reciprocally, what type of information about a medium can be extracted from a certain knowledge on the propagation of waves in it. For example, how a certain perturbation of a known medium, like anobstacle in space, affects the propagation of waves, and, vice-versa, determine properties of the obstacle from information obtained from waves reflected by it. It also concerns the study of the wave equation on spaces that are geometrically different from the usual Euclideanspace, e.g., spaces that resemble the hyperbolic space at infinity.