Inverse scattering on conformally compact manifolds

TL;DR

This paper investigates inverse scattering on conformally compact manifolds, demonstrating that the scattering matrix at fixed energies uniquely determines the boundary behavior of the metric and potential.

Contribution

It establishes a novel result that the scattering matrix at fixed energies uniquely recovers the boundary Taylor series of the metric and potential on conformally compact manifolds.

Findings

Scattering matrix at fixed energies determines boundary metric behavior.

Potential and metric Taylor series at the boundary are uniquely recoverable.

Results apply to manifolds with variable sectional curvature.

Abstract

We study inverse scattering for $\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(\lambda_1,$ $\lambda_2)$ in a suitable subset of $\mc$, determines $\alf,$ and the Taylor series of both the potential and the metric at the boundary.