  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.
