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

Abstract

Let $(Ω^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is a an open set in $Ω$ such that $g|_U$ is the Euclidean metric. Let $Γ= \overline{U} \cap \partial Ω$ 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 $ -\triangle_g + q $ from the associated Dirichlet to Neumann map, $Λ_q$. 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 $Γ$ from $Λ_q$. We will also outline a reconstruction algorithm. More generally we will discuss the cases where $Γ$ is not connected or $g|_{U}$ is conformally transversally anisotropic and derive the analogous result.