Uniqueness of a Potential from Local Boundary Measurements

TL;DR

This paper proves that an unknown potential in a Schrödinger operator on a Riemannian manifold can be uniquely reconstructed from local boundary measurements, given the potential is known outside a specific region.

Contribution

It establishes a uniqueness result for the inverse boundary value problem for Schrödinger operators on Riemannian manifolds with local boundary data.

Findings

Unique reconstruction of potential from local boundary measurements.

Potential known outside a region allows for full recovery within the region.

Results applicable to manifolds with convex boundary segments.

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 $C^{\Gamma,\Gamma}_q$.