A note on the scattering theory of Kato-Ricci manifolds

TL;DR

This paper introduces a new $L^1$ criterion for wave operators' existence and completeness on Kato-Ricci manifolds, based on heat semigroup estimates related to Ricci curvature.

Contribution

It establishes a novel $L^1$ criterion for wave operators on Kato-Ricci manifolds, connecting heat kernel estimates with geometric curvature conditions.

Findings

New $L^1$ criterion for wave operators

Heat semigroup estimates depend on Ricci curvature in Kato class

Results apply to noncompact Riemannian manifolds

Abstract

In this note we prove a new $L^1$ criterion for the existence and completeness of the wave operators corresponding to the Laplace-Beltrami operators corresponding to two Riemannian metrics on a fixed noncompact manifold. Our result relies on recent estimates on the heat semigroup and its derivative, that are valid if the negative part of the Ricci curvature is in the Kato class - so called Kato-Ricci manifolds.