The scattering matrix for the p-form Laplacian on asymptotically conic manifolds

Abstract

We give an explicit description of the scattering matrix for the Hodge-Laplacian on co-closed p-forms on asymptotically conic manifolds of dimension $n\ge 3$. We develop generalized eigenfunctions and a functional calculus to describe the result, a departure from the Fourier Integral Operators used in the scalar case. Starting from the Hodge decomposition at infinity, we construct generalized eigenforms which are co-closed and establish a spectral representation for the p-form Hodge Laplacian. The special case of dimension 3 for $p=1$ characterises the electric field for Maxwell's equations.