Forward and inverse scattering on manifolds with asymptotically cylindrical ends

TL;DR

This paper investigates an inverse scattering problem on non-compact manifolds with cylindrical ends, proving that identical scattering matrices imply the manifolds are isometric, thus establishing a uniqueness result in geometric inverse problems.

Contribution

It demonstrates that the scattering matrix on a cylindrical end uniquely determines the manifold's geometry among a class of asymptotically cylindrical manifolds.

Findings

Identical scattering matrices imply manifolds are isometric.

The inverse problem has a unique solution under the given conditions.

The result extends inverse scattering theory to manifolds with cylindrical ends.

Abstract

We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties : On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form $(dy)^2 + h(x,dx)$, $h(x,dx)$ being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to $(dy)^2 + h(x,dx)$. Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energy, we show that these two manifolds are isometric.