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 $-Δ_g+V$ on a domain $Ω\subset\mathbb{R}<sup>n$ 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 $G<sup>σ$, $σ&gt;1$. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of $\overlineΩ$. 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 $C<sup>\infty$ 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.

• The paper proves uniqueness modulo natural gauge transformations for analytic coefficients, while every non-analytic Gevrey class Gσ with σ>1 admits arbitrarily close, non-isometric coefficients with identical Dirichlet-to-Neumann maps.
• The paper constructs counterexamples using localized Gevrey perturbations, conformal metric changes, frequency tuning, and a compactly supported prescribed-Jacobian method in dimensions n≥3.
• The paper shows analyticity is the exact rigidity threshold for fixed-potential Riemannian Schrödinger and fixed nonzero-frequency anisotropic Calderón problems, while constant potentials, dimension two, and resonant frequencies remain open or excluded cases.

Overview and main results

This paper, by Daudé, Enciso, Helffer, Kamran, and Nicoleau (2605.21705), establishes a sharp regularity threshold for global uniqueness in two anisotropic Calderón-type inverse boundary value problems in dimension $n \ge 3$. The first problem is a Riemannian Schrödinger inverse problem in which the scalar potential $V$ is prescribed (nonconstant) and the metric $g$ is unknown; the second is the anisotropic Calderón problem for a conductivity $\gamma$ at a fixed nonzero frequency $\lambda_0 \neq 0$. In both settings the paper proves that real-analytic coefficients are uniquely determined by the Dirichlet-to-Neumann (DN) map modulo the natural gauge group, while nonuniqueness holds densely in every non-analytic Gevrey class $G^\sigma$, $\sigma > 1$, and hence in $C^\infty$.

The central claim is stronger than mere gauge nonuniqueness: the counterexamples are not isometric — there is no diffeomorphism $\Psi:\overline\Omega\to\overline\Omega$, whether or not it fixes the boundary or preserves $V$, with $V$0. This upgrades the finite-regularity ($V$1) counterexamples of the authors' earlier work to infinite regularity, and isolates analyticity as the exact dividing line.

The fixed-potential Riemannian Schrödinger problem

The first setting considers $V$2 on a bounded smooth domain $V$3, with $V$4 fixed and nonconstant, and the DN map defined weakly via the Dirichlet energy plus potential term. Because $V$5 is fixed a priori, the natural gauge is not the full boundary-fixing diffeomorphism group but the subgroup $V$6 of diffeomorphisms satisfying $V$7. The authors note this problem has not previously been isolated as a separate global Calderón-type problem; it sits between the classical anisotropic Calderón problem (unknown geometry, no zeroth-order term) and the fixed-geometry Schrödinger inverse problem (unknown potential).

The nonuniqueness theorem states that if $V$8 is analytic (or $V$9), $g$0 is nonconstant, and $g$1, then every $g$2-neighborhood of $g$3, for any $g$4, contains infinitely many pairs of uniformly elliptic metrics $g$5 with identical DN maps $g$6 but which are not related by the pushforward of any diffeomorphism of $g$7.

The mechanism is elementary and does not require a prescribed-Jacobian theorem. Since $g$8 is nonconstant, there is an interior box where $g$9, so $\gamma$0 is an analytic coordinate chart. One takes a compactly supported nonnegative Gevrey function $\gamma$1 supported there, sets $\gamma$2, and conformally perturbs $\gamma$3. Under the Liouville transformation $\gamma$4, the effective potential becomes $\gamma$5. A key lemma shows that no nontrivial localized conformal factor preserves the effective potential (by boundary unique continuation), so a compensating diffeomorphism is genuinely needed. Because $\gamma$6 serves as a local coordinate, the compatibility condition $\gamma$7 is solved explicitly by modifying only the $\gamma$8-coordinate. Non-isometry is detected by total volume: $\gamma$9 since pushforward preserves volume, while $\lambda_0 \neq 0$0 strictly because $\lambda_0 \neq 0$1 with strict inequality on a set of positive measure. An explicit example with linear potential $\lambda_0 \neq 0$2 is worked out.

The construction uses the hypothesis that $\lambda_0 \neq 0$3 is nonconstant only locally; no Morse or level-set conditions are imposed. When $\lambda_0 \neq 0$4 is constant, the mechanism fails: unique continuation forces any localized conformal factor preserving the effective potential to be trivial, so the constant-potential case reduces to the classical anisotropic Calderón problem, which remains open in general.

The fixed nonzero-frequency anisotropic Calderón problem

For $\lambda_0 \neq 0$5 and $\lambda_0 \neq 0$6, the DN map at frequency $\lambda_0 \neq 0$7 is invariant under volume-preserving boundary-fixing diffeomorphisms $\lambda_0 \neq 0$8, since pulling back the equation produces the factor $\lambda_0 \neq 0$9 multiplying $G^\sigma$0. The main theorem asserts dense nonuniqueness in every $G^\sigma$1, $G^\sigma$2: pairs of conductivities $G^\sigma$3 arbitrarily close to a given analytic (or smooth) background, with equal DN maps at $G^\sigma$4, yet not related by any diffeomorphism.

The construction combines three ingredients:

• A constrained Gevrey test function: a function $G^\sigma$5 with $G^\sigma$6, vanishing zeroth moment and weighted moment against $G^\sigma$7, and prescribed Dirichlet energy $G^\sigma$8. The two moment conditions respectively normalize the induced frequency and kill the leading term of the determinant invariant.
• Conformal factor and frequency tuning: with $G^\sigma$9 and $\sigma > 1$0, the frequency $\sigma > 1$1 defined through the compatibility equation satisfies $\sigma > 1$2; a scaling argument then rescales both conductivities to recover exactly the prescribed frequency $\sigma > 1$3.
• A compactly supported prescribed-Jacobian lemma in Gevrey spaces: given small $\sigma > 1$4 with zero integral, one constructs $\sigma > 1$5 equal to the identity near $\sigma > 1$6 with $\sigma > 1$7. This is proved by an explicit fiberwise right inverse for divergence on a box (with quantitative Gevrey norm estimates and a recursive dimension-one constant), followed by a Moser flow argument using a Gevrey flow theorem with radius loss. Notably, the authors avoid a global Gevrey version of Dacorogna–Moser/Rivière–Ye theory; locality suffices because all perturbations are supported in an interior box.

Non-isometry follows from the determinant invariant $\sigma > 1$8, which equals the Riemannian volume of the associated metric $\sigma > 1$9 and is invariant under pushforward. Equality of the invariant for the constructed pair would force $C^\infty$0; expanding in $C^\infty$1 and using the moment conditions leaves a nonzero leading coefficient proportional to $C^\infty$2, provided $C^\infty$3 and $C^\infty$4 — both excluded by the choice of $C^\infty$5. This yields a contradiction for all sufficiently small $C^\infty$6.

Analytic rigidity

Both problems admit positive uniqueness results at the analytic endpoint, obtained by a minor adaptation of the Lassas–Uhlmann reconstruction theorem. For the fixed-potential problem, equality of DN maps for analytic metrics implies $C^\infty$7 with $C^\infty$8 analytic, boundary-fixing, and $C^\infty$9. For the fixed-frequency problem, the conductivity equation rewrites as a zero-energy Schrödinger equation on the metric $\Psi:\overline\Omega\to\overline\Omega$0 with potential $\Psi:\overline\Omega\to\overline\Omega$1; reconstructing the pair $\Psi:\overline\Omega\to\overline\Omega$2 and comparing volume densities then forces $\Psi:\overline\Omega\to\overline\Omega$3, i.e., $\Psi:\overline\Omega\to\overline\Omega$4. The nonzero frequency is essential here: at $\Psi:\overline\Omega\to\overline\Omega$5 the potential vanishes and the determinant cannot be pinned down.

Together with the nonuniqueness theorems, this establishes that analyticity is the exact threshold for uniqueness modulo the natural gauge in both problems. The sharpness hinges on the Denjoy–Carleman nonquasianalyticity of $\Psi:\overline\Omega\to\overline\Omega$6 for $\Psi:\overline\Omega\to\overline\Omega$7: compactly supported Gevrey functions exist precisely when $\Psi:\overline\Omega\to\overline\Omega$8, and this localization property is the only essential use of the strict inequality.

Limitations and open questions

Several restrictions qualify the results. Both constructions require $\Psi:\overline\Omega\to\overline\Omega$9; the conformal exponent $V$0 becomes singular in dimension two, and the two-dimensional case is untouched. The fixed-potential result requires $V$1 to be nonconstant; for constant $V$2 the gauge enlarges to the full boundary-fixing diffeomorphism group and the present mechanism collapses, leaving the classical anisotropic Calderón problem in $V$3 unresolved. The spectral assumptions $V$4 and $V$5 exclude resonant frequencies, and the counterexamples are perturbative — they lie arbitrarily close to a given background rather than being globally arbitrary. Whether the full anisotropic Calderón problem admits similar Gevrey counterexamples, or whether uniqueness can be restored under additional geometric hypotheses on the background metric, remains open.

Conclusion

The paper identifies a precise regularity dichotomy for two anisotropic Calderón-type problems with zeroth-order terms: uniqueness modulo the natural gauge holds in the analytic category via Lassas–Uhlmann reconstruction, while every non-analytic Gevrey class carries dense families of non-isometric counterexamples with identical DN maps. The two constructions exploit different scalar mechanisms — the potential itself as a local coordinate versus a localized prescribed-Jacobian lemma — but share the structural feature that a zeroth-order term breaks the full anisotropic gauge, after which analyticity alone remains rigid.