Spectral Geometry of Non-Compact Domains and Riemannian Manifolds

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Abstract for DMS - 0100829The principal investigator will study the spectral geometry of non-compactdomains and Riemannian manifolds in order to elucidate the geometriccontent of scattering poles. First, the PI will continue his study of thespectral geometry of hyperbolic manifolds and their perturbations. He willstudy resonances as functions on the deformation space of the underlyingdiscrete group, and define and analyze a determinant of the Laplacian.Secondly, the PI will study scattering theory for the wave equation ontwo-step nilpotent Lie groups and their quotients by discrete subgroups.New parametrices or the wave equation on the Heisenberg and Heisenberg-typegroups will be derived, and trace formula for certain quotients obtained.Riemannian submersion techniques of Gordon, Wilson, and others will be usedto obtain pairs and families of `isoscattering' manifolds which will helpdetermine the limits of geometric information which may be deduced from aknowledge of the scattering poles. Thirdly, the PI will study theisoscattering problem for exterior domains in Euclidean space.The fundamental problem of spectral geometry is to elucidate thegeometric content of the Laplace spectrum on a Riemannian manifold.For so-called scattering manifolds, the eigenvalues of the Laplaciantogether with scattering resonances constitute the spectral data for themanifold. Elucidating the geometric content of such spectral dataadvances our understanding of quantization, produces new analytic toolsfor the study of geometric objects, and provides insight into inverseproblems of a more `applied' nature where the eigenvalues and scatteringpoles are measurable quantities. The present work aims to begin withgeometrically natural examples where techniques of Lie theory, automorphicfunctions, and harmonic analysis may be used, and progress to harderproblems such as target identification by radar where such techniques are not available but the underlying mathematical problems are very similar.