Spectral theory of the p-form Laplacian on manifolds with generalized cusps

Spectral theory and scattering theory on non-compact manifolds with ends of various shapes have had a huge influence on mathematics and physics. One important and extensively studied family consists of manifolds with hyperbolic cusps. These manifolds first appeared in mathematics in the context of number theory as quotients of the upper half plane by arithmetic lattices. It was discovered that the spectral theory on such constant negative curvature surfaces is equivalent to the theory of automorphic functions and that their scattering theory may be used to meromorphically continue Eisenstein series [13, 9]. Later, methods of scattering theory were applied to the more general case of manifolds with hyperbolic cusps that are of negative curvature outside a compact set [9, 10, 12]. The continuous spectrum of the Laplace operator on such manifolds is known to be [1/4,∞) and its generalized eigenfunctions are given by the meromorphically continued generalized Eisenstein functions. The multiplicity of the continuous spectrum is constant and equals the number of cusps. Another important family of examples are manifolds with cylindrical ends. The spectral theory and scattering theory of Dirac type operators on such manifolds plays an important role in the Atiyah Patodi Singer index theorem [2] and scattering theory can be successfully applied to describe the spectral subspaces explicitly [11]. In addition, manifolds with cylindrical ends serve as models for waveguides and their scattering has thus also been studied recently [8]. The continuous spectrum of the Laplace operator on functions for a manifold with cylindrical end is [0,∞); its multiplicity at λ > 0 is the number of eigenvalues of the Laplace operator on the boundary that are smaller than λ. In this paper we are interested in manifolds with cusp-like singularities that can be thought of as interpolating between these two cases. Let (N, h) be a closed