Inverse scattering on Riemannian manifolds and Lorentizian manifolds

We talk about two recent results on inverse scattering problems on Riemannian manifolds (a joint work with M. Lassas) and Lorentzian manifolds (with S. Alxakis, M. Lassas and T. Tyni). 

In both cases, the boundary control method plays an important role geometrically, and the issue is centered around the passage from the S-matrix to the boundary value problem on a bounded domain.

In the first topic, we consider a non-compact Riemannian manifold $\mathcal M = \mathcal K \cup \mathcal M_1 \cup \cdots \cup \mathcal M_N$, where $\mathcal K$ is relatively compact, $\mathcal M_i$ is diffeomorphic to $(0,\infty)\times M_i$ and $M_i$ is $(n-1)$-dimensional. Assuming that the Riemannian metric is asymptotically equal to a warped product metric on each end $\mathcal M_i$, we show that the whole manifold is determined by the knowledge of one diagonal component of the S-matrix for all energies. The key-fact is the Rellich type theorem showing the critical decay rate of scattering solutions at infinity. 

The second topic is concerned with the non-linear wave equation 

$\square_{g}u + a(x)u^{\kappa} = 0$ on non-compact Lorentzian manifold. The idea consists in Penrose's compactification which embeds the Minkowski space into a locally perturbed Lorentzian manifold. We show that the knowledge of scattering functional determines the conformal type of Lorentzian metric and the non-linear term.