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 Schrödinger operator $-{△}_g+q$ 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.

27 pages