On Open Scattering Channels for Manifolds with Ends

Abstract

In the framework of time-dependent geometric scattering theory, we study the existence and completeness of the wave operators for perturbations of the Riemannian metric for the Laplacian on a complete manifold of dimension $n$. The smallness condition for the perturbation is expressed (intrinsically and coordinate free) in purely geometric terms using the harmonic radius; therefore, the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the Ricci-curvature. As an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the Riemannian metric. This stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations.