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

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.