Scattering theory of the Hodge–Laplacian under a conformal perturbation

Abstract

Let $g$ and $\tilde{g}$ be Riemannian metrics on a noncompact manifold $M$, which are conformally equivalent. We show that under a very mild first order control on the conformal factor, the wave operators corresponding to the Hodge–Laplacians ${\Delta }_g$ and ${\Delta }_{\tilde{g}}$ acting on differential forms exist and are complete. We apply this result to Riemannian manifolds with bounded geometry and more specically, to warped product Riemannian manifolds with bounded geometry. Finally, we combine our results with some explicit calculations by Antoci to determine the absolutely continuous spectrum of the Hodge–Laplacian on $j$-forms for a large class of warped product metrics.