Alexander Strohmaier (Leeds)

Title: Scattering theory for differential forms and its relation to cohomology
Abstract: I will consider spectral theory of the Laplace operator on a manifold that is Euclidean outside a compact set. An example of such a setting is obstacle scattering where several compact pieces are removed from $R^d$. The spectrum of the operator on functions is absolutely continuous. In the case of general $p$-forms eigenvalues at zero may exist, the eigenspace consisting of L^2-harmonic forms. The dimension of this space is computable by cohomological methods. I will present some new results concerning the detailed expansions of generalised eigenfunctions, the scattering matrix, and the resolvent near zero. These expansions contain the L^2-harmonic forms so there is no clear separation between the continuous and the discrete spectrum. This can be used to obtain more detailed information about the L^2-cohomology as well as the spectrum. If I have time I will explain an application of this to physics. (joint work with Alden Waters)

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