Inverse scattering with disjoint source and observation sets on asymptotically hyperbolic manifolds

Abstract

In this note, the scattering operator of an asymptotically hyperbolic manifold is only allowed to act on functions supported on the source set where is an open subset of the boundary, and the resulting functions are then restricted to the observation set where Γ is open. When Γ is the complement of the closure of we call the corresponding operator the off-diagonal scattering operator with respect to We prove that for a non-empty open proper subset such that the intersection of its closure and its complement is not empty, the off-diagonal scattering operator with respect to determines the manifold modulo isometries which are equal to the identity at the boundary. We also prove there is no analog of the L2 boundary controllability from an open subset of the boundary for radiation fields, which presents a possible obstacle to extending the inverse result for arbitrary disjoint subsets and