Scattering theory of the Hodge-Laplacian under a conformal perturbation

TL;DR

This paper establishes the existence and completeness of wave operators for the Hodge-Laplacian under conformal metric perturbations, and analyzes the spectrum on warped product manifolds with bounded geometry.

Contribution

It proves wave operator existence and completeness for conformal perturbations of the Hodge-Laplacian and determines the spectrum on certain warped product manifolds.

Findings

Wave operators exist and are complete under mild conformal control.

Results apply to manifolds with bounded geometry and warped products.

Explicit spectrum characterization for a class of warped product metrics.

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.