Time-dependent scattering theory for Schr\"odinger operators on scattering manifolds

TL;DR

This paper develops a time-dependent scattering theory for Schrödinger operators on manifolds with asymptotically conic geometry, establishing wave operator existence, completeness, and relating the scattering matrix to asymptotic eigenfunction expansions.

Contribution

It introduces a novel functional analytic approach to scattering theory on conic manifolds without microlocal analysis, using a two-space formalism with a specific reference operator.

Findings

Proves existence and completeness of wave operators.

Shows equivalence of scattering matrix to the asymptotic expansion-based matrix.

Provides a new framework for scattering on asymptotically conic manifolds.

Abstract

We construct a time-dependent scattering theory for Schr\"odinger operators on a manifold $M$ with asymptotically conic structure. We use the two-space scattering theory formalism, and a reference operator on a space of the form $R\times \partial M$, where $\partial M$ is the boundary of $M$ at infinity. We prove the existence and the completeness of the wave operators, and show that our scattering matrix is equivalent to the absolute scattering matrix, which is defined in terms of the asymptotic expansion of generalized eigenfunctions. Our method is functional analytic, and we use no microlocal analysis in this paper.