Scattering theory for Riemannian Laplacians

TL;DR

This paper develops a scattering theory for the Laplace-Beltrami operator on non-compact Riemannian manifolds, establishing conditions for wave operator existence and spectral analysis without requiring specific asymptotic metric behavior.

Contribution

It introduces a novel scattering framework for Riemannian Laplacians that does not depend on asymptotic metric conditions, expanding spectral theory understanding.

Findings

Conditions for wave operator existence and completeness are established.

Spectral analysis includes identification of continuous spectrum.

Proves absence of singular continuous spectrum under given conditions.

Abstract

In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace-Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.