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 Schrödinger operator $-{△}_g+q$ 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 $C_q^{\Gamma ,\Gamma }$.

14 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1802.02645