  In this paper we consider scattering theory on manifolds with special
cusp-like metric singularities of warped product type g=dx^2 + x^(-2a)h, where
a>0. These metrics form a natural subset in the class of metrics with warped
product singularities and they can be thought of as interpolating between
hyperbolic and cylindrical metrics. We prove that the resolvent of the Laplace
operator acting on p-forms on such a manifold extends to a meromorphic function
defined on the logarithmic cover of the complex plane with values in the
bounded operators between weighted L^2-spaces. This allows for a construction
of generalized eigenforms for the Laplace operator as well as for a meromorphic
continuation of the scattering matrix. We give a precise description of the
asymptotic expansion of generalized eigenforms on the cusp and find that the
scattering matrix satisfies a functional equation.
