In this book we are interested in manifolds with cusp like singularities that are in between the cases of cylindrical end and of hyperbolic cusp. More precisely, we study the Laplace operator acting on p-forms, defined on an n-dimensional manifold with generalized cusp. Such a manifold consists of a compact piece and a noncompact one. The noncompact piece is isometric to the generalized cusp. A generalized cusp is an n-dimensional noncompact manifold equipped with a parameter dependent warped product metric. When the positive parameter goes to zero, the cusp becomes a cylinder, and when it goes to infinity, it could be thought of as approaching the n-dimensional hyperbolic cusp. In such a manifold we construct the generalized eigenforms of the Laplacian. Thus, we give a description of the continuous spectral decomposition of the Laplace operator and we determine 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.

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