Uniqueness of a Potential from Local Boundary Measurements

Abstract: Let $(\Omega^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is an open set in $\Omega$ such that $g|_U$ is the Euclidean metric. Let $\Gamma= \overline{U} \cap \partial \Omega$ be non-empty, connected, strictly convex and 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 local Dirichlet to Neumann map, $C_q^{\Gamma,\Gamma}$. Indeed, we will prove that if the potential $q$ is a priori explicitly known in $U^c$, then one can uniquely reconstruct $q$ from the knowledge of $C^{\Gamma,\Gamma}_q$.