On Open Scattering Channels for Manifolds with Ends

TL;DR

This paper investigates the existence, completeness, and stability of scattering channels for Laplacians on manifolds with ends under small geometric perturbations, using intrinsic geometric conditions.

Contribution

It introduces a geometric smallness condition for metric perturbations ensuring wave operator completeness and scattering matrix stability on manifolds with ends.

Findings

Wave operators are shown to exist and be complete under geometric smallness conditions.

The scattering matrix remains stable under small metric perturbations.

Scattering channels preserve their interaction properties despite perturbations.

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.