Unique Reconstruction of a Potential from the Dirichlet to Neumann Map in Locally CTA Geometries

Let (M, g) be a compact smooth Riemannian manifold with smooth boundary and suppose that U is an open set in M such that g|U is Euclidean. Let Γ = U ∩ ∂M be connected and suppose that U is the convex hull of Γ. We will study the uniqueness of an unknown potential for the Schrödinger operator −4g + q from the associated Dirichlet to Neumann map, Λq. Indeed, we will prove that if the potential q is a priori explicitly known in U c then one can uniquely reconstruct q from Λq. We will also give a reconstruction algorithm for the potential. More generally we will also discuss the cases where Γ is not connected or g|U is conformally transversally anisotropic and derive the analogous result.