Uniqueness of a Potential from Boundary Data in Locally Conformally Transversally Anisotropic Geometries

Abstract

Let $(\Omega^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is a an open set in $\Omega$ such that $g|_U$ is the Euclidean metric. Let $\Gamma= \overline{U} \cap \partial \Omega$ be connected and suppose that $U$ is the convex hull of $\Gamma$. We will study the uniqueness of an unknown potential for the Schr\"{o}dinger operator $ -\triangle_g + q $ from the associated Dirichlet to Neumann map, $\Lambda_q$. We will prove that if the potential $q$ is a priori explicitly known in $U^c$, then one can uniquely reconstruct $q$ over the convex hull of $\Gamma$ from $\Lambda_q$. We will also outline a reconstruction algorithm. More generally we will discuss the cases where $\Gamma$ is not connected or $g|_{U}$ is conformally transversally anisotropic and derive the analogous result.