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≥ 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 −Δg+V on a domain Ω⊂ℝn 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 Gσ, σ>1. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of Ω̅. 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∞ 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.
Related
- Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds
- Wavelet resolution and Sobolev regularity of Calderón–Zygmund operators on domains
- The anisotropic Calderón problem for Riemannian metrics at high fixed frequency
- On the Uniqueness of a Nonlinear Discrete Calderón Problem
- Regularity and uniqueness to multi-phase problem with variable exponent
BibTeX
@misc{daude2026sharp,
title = {{A Sharp Regularity Threshold for Uniqueness in Riemannian Calderón-type Problems}},
author = {Thierry Daudé and Alberto Enciso and Bernard Helffer and Niky Kamran and François Nicoleau},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.21705},
eprint = {2605.21705},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.21705},
}
RIS
TY - UNPB
TI - A Sharp Regularity Threshold for Uniqueness in Riemannian Calderón-type Problems
AU - Thierry Daudé
AU - Alberto Enciso
AU - Bernard Helffer
AU - Niky Kamran
AU - François Nicoleau
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2605.21705
UR - https://arxiv.org/abs/2605.21705
AB - We prove a sharp regularity threshold for uniqueness in two anisotropic Calderón-type inverse problems in dimension n≥ 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 −Δ_g+V on a domain Ω⊂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^σ, σ>1. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of Ω̅. 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^∞ 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.
ER -