Inverse scattering on non-compact manifolds with general metric

Abstract: The problems we address in this paper are the spectral theory and the inverse problems associated with Laplacians on non-compact Riemannian manifolds and more general manifolds admitting conic singularities. In particular, we study the inverse scattering problem where one observes the asymptotic behavior of the solutions of the Helmholtz equation on the manifold. These observations are analogous to Heisenberg's scattering matrix in quantum mechanics. We then show that the knowledge of the scattering matrix determines the topology and the metric of the manifold. In the paper we develop a unified approach to consider scattering problems on manifolds that can have very different type of infinities, such as regular hyperbolic ends, cusps, and cylindrical ends related to models encountered in the study of wave guides. We allow the manifold to have also conic singularities. Due to this, the studied class of manifolds include orbifolds. Such non-smooth structures arise in the study of the stability of inverse problems and of the geometrical collapse.