Forward and inverse scattering on manifolds with asymptotically cylindrical ends

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 2(dy)+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 2(dy)+h(x,dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energies, we show that these two manifolds are isometric.