Spectral theory of the Laplace operator on manifolds with generalized cusps

In this thesis we study the Laplace operator Δ acting on p-forms, defined on an n dimensional manifold with generalized cusps. Such a manifold consists of a compact piece and a noncompact one. The noncompact piece is isometric to the generalized cusp. A generalized cusp [1,∞) x N is an n dimensional noncompact manifold equipped with the warped product metric dx{2}+x{-2a}h, where N is a compact oriented manifold, h is a metric on N and a > 0 is a fixed constant. First we regard the cusp separately, where by using separation of variables we determine the spectral properties of the Laplacian and we determine explicitly the structure of the continuous part of the spectral theorem. Using this result, we meromorphically continue the resolvent of the Laplace operator to a certain Riemann surface, which we determine. By standard gluing techniques, the resolvent of the Laplace operator Δ on the manifold with cusp is meromorphically continued to the same Riemann surface. This enables us to construct the generalized eigenforms for the original manifold without boundary. That describes the continuous spectral decomposition of Δ and determines some of its important properties, like analyticity and the existence of a functional equation. We also define the stationary scattering matrix and find its analytic properties and its functional equation.