On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calder\'on problem for the Conformal Laplacian

TL;DR

This paper establishes that on asymptotically hyperbolic manifolds with constant scalar curvature, the scattering matrix at a specific energy uniquely determines the boundary metric's jet, linking inverse scattering and the conformal Laplacian.

Contribution

It introduces a new inverse scattering result connecting the scattering matrix to boundary metric determination on AH manifolds with constant scalar curvature.

Findings

Scattering matrix at energy (n+1)/2 determines boundary metric jet.

Utilizes relation between eigenvalue problem and conformal Laplacian.

Extends inverse scattering theory to asymptotically hyperbolic manifolds.

Abstract

In this short note, we use the relation obtained by Guillarmou--Guillop\'e and Chang--Gonz\'alez between the generalized eigenvalue problem for asymptotically hyperbolic (AH) manifolds and the Conformal Laplacian, to obtain a new inverse scattering result: on an AH manifold of dimension $n+1$ with constant scalar curvature $-n(n+1)$, we show that the scattering matrix at energy $\frac{n+1}{2}$ determines the jet of the metric on the boundary, up to a diffeomorphism and conformal factor.