A Sharp Regularity Threshold for Uniqueness in Riemannian Calderón-type Problems

Abstract

We prove a sharp regularity threshold for uniqueness in two anisotropic Calderón-type inverse problems in dimension $n\ge 3$. The main setting is the Riemannian Schrödinger problem with fixed scalar potential: for a prescribed nonconstant analytic function $V$, we study whether the Dirichlet-to-Neumann map of $-\hat{I}{”}_g+V$ on a domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ determines the unknown metric $g$. The natural gauge is the group of boundary-fixing diffeomorphisms preserving $V$. We show that, while analytic metrics are uniquely determined modulo this gauge by a minor adaptation of the Lassas--Uhlmann reconstruction theorem, uniqueness fails densely in every non-analytic Gevrey class $G^{\ddot{I}}ƒ$, $\ddot{I}ƒ>1$. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of $\overline{\hat{I}}\text{c◯}$. We also prove the analogous sharp threshold for the anisotropic Calderón problem at fixed nonzero frequency, thereby upgrading the previously known finite-regularity counterexamples to Gevrey and $C^{\infty }$ regularity. The two constructions use different scalar mechanisms: for fixed potentials, the nonconstant potential itself provides a local coordinate, while at nonzero frequency one uses a compactly supported prescribed-Jacobian lemma in Gevrey spaces. Thus analyticity is the exact threshold for uniqueness in both problems.