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 \emph{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 a bounded geometry and more specifically, to warped product Riemannian manifolds with a 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.

A new section with several examples has been added, including the calculation of absolutely continuous k-form spectra on a large class of warped product metrics

References in corpus (1)

• On Open Scattering Channels for Manifolds with Ends