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A note on the scattering theory of Kato-Ricci manifolds

Batu Güneysu, Maxime Marot

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.

A note on the scattering theory of Kato-Ricci manifolds

Abstract

In this note we prove a new 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.

Paper Structure

This paper contains 2 theorems, 31 equations.

Key Result

Theorem 1

Assume that $g,h$ are complete and quasi-isometric Riemannian metrics on $M$ such that $(M,g)$ and $(M,h)$ are Kato-Ricci with Then the wave operators $W_\pm(-\Delta_h, -\Delta_g;I_{g,h})$ exist and are complete.

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2