Scattering theory for Schr\"{o}dinger equations on manifolds with asymptotically polynomially growing ends

TL;DR

This paper develops a scattering theory for Schrödinger equations on manifolds with polynomially growing ends, establishing spectral properties and wave operator completeness using Mourre and Kato theories.

Contribution

It introduces a novel framework combining Mourre and Kato theories for scattering on manifolds with polynomially growing ends, including two-space scattering analysis.

Findings

Spectral properties of Schrödinger operators are characterized.

Existence and asymptotic completeness of wave operators are proven.

Two-space scattering with a simple reference system is analyzed.

Abstract

We study a time-dependent scattering theory for Schr\"{o}dinger operators on a manifold with asymptotically polynomially growing ends. We use the Mourre theory to show the spectral properties of self-adjoint second-order elliptic operators. We prove the existence and the asymptotic completeness of wave operators using the smooth perturbation theory of Kato. We also consider a two-space scattering with a simple reference system.