Anisotropic Calder\'{o}n Problem for a Non-Local Second Order Elliptic Operator

India); Susovan Pramanik (Harish-Chandra Research Institute

REVIEW 3 major objections 3 minor 1 cited by

Anisotropic Calder\'{o}n Problem for a Non-Local Second Order Elliptic Operator

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Internal data of a nonlocal order-2 operator on a shared open patch force the two manifolds to be isometric.

desk verdict A useful extension of the fractional anisotropic Calderón theorem to order 2, but the central Kannai formula has a normalization error that needs fixing. read the letter →

arxiv 2505.12255 v3 pith:CP7VFNM6 submitted 2025-05-18 math.AP

Susovan Pramanik (Harish-Chandra Research Institute , India) This is my paper · ORCID

classification math.AP MSC 35R3058J3535S0547G30

keywords anisotropicCalderónproblemnonlocalellipticoperatorfractionalpowerofbi-LaplaceheatkernelrigidityRiemannianmanifoldsisometryrecoverypseudodifferentialinternalCauchydata

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims an anisotropic Calderón theorem for the nonlocal second-order operator $L_g^{1/2} = ((-\Delta_g)^2 + m^2 I)^{1/2}$ on a smooth closed connected Riemannian manifold: if the internal Cauchy data of two such operators agree on a common open set, the manifolds are isometric. The point of the result is that a genuinely nonlocal, order-2 elliptic operator still carries complete metric information, even though the classical local anisotropic Calderón problem for the Laplacian remains open in smooth geometry. The proof works by converting the assumed data equality into equality of the associated heat kernels and then invoking a known rigidity theorem for heat kernels on Riemannian manifolds.

What carries the argument

The load-bearing chain is: (i) the fractional inverse $L_g^{-1/2}$ is written through the heat semigroup $e^{-tL_g}$ by a Gamma-integral formula, so assumed equality of data becomes vanishing of weighted $t$-integrals of the semigroup difference; (ii) integration by parts and the estimates of the bi-Laplace heat kernel (Theorem 2.1) turn those integrals into vanishing moments $\int_0^\infty \phi(s)s^k\,ds=0$; (iii) the Paley-Wiener theorem forces $\phi\equiv0$, giving equality of the bi-Laplace heat semigroups on a subdomain; (iv) Kannai's transmutation formulas formally convert this into equality of the Laplace heat kernels; and (v) a known heat-kernel rigidity theorem (Theorem 3.6) upgrades local kernel equality to a global isometry. The central object carrying the argument is the transmutation identity expressing $e^{t\Delta_g}$ (and $e^{-t\Delta_g^2}$) as an integral of $\sin(\sqrt{\tau}\,\Delta_g)/\Delta_g$ against a Gaussian weight.

What would settle it

Evaluate equation (27) on a single eigenfunction $\varphi_k$ of $\Delta_g$ with $\Delta_g\varphi_k=\lambda_k\varphi_k$; the identity as printed gives a prefactor $t^{7/6}e^{-\lambda_k^2 t}$ rather than $e^{-\lambda_k^2 t}$, so verifying the correct prefactor and weight against Kannai's statement settles whether the bridge from bi-Laplace to Laplace heat kernels is valid.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: let $(M_1,g_1)$ and $(M_2,g_2)$ be smooth closed connected Riemannian manifolds sharing an open set $(O,g)$, and suppose $L_{g_1}^{-1/2}f|_O = L_{g_2}^{-1/2}f|_O$ for every source $f\in C_0^\infty(O)$, where $L_g=(-\Delta_g)^2+m^2I$. Then a diffeomorphism $\Phi:M_1\to M_2$ exists with $\Phi^*g_2=g_1$. In other words, the full isometry class of a closed manifold is encoded in the restriction of the operator's inverse to an arbitrarily small open patch. A companion theorem (Theorem 1.4) states the same conclusion when the Cauchy data set of $A_g=L_g^{1/2}-mI$ agree.

Load-bearing premise

The proof relies on Kannai's transmutation formulas and an inverse Laplace transform step being valid on closed manifolds under the needed regularity and support conditions; the paper imports these formulas without checking those hypotheses.

Editorial extensions

If this is right

If Theorem 1.1 is correct, the assignment $(M,g)\mapsto L_g^{-1/2}$ restricted to any open patch is injective up to isometry: no two non-isometric closed manifolds can have matching internal data.
Theorem 1.4 extends the same rigidity to the operator $A_g=L_g^{1/2}-mI$ and its Cauchy data set of solutions to $A_gu=0$.
The result gives an order-2 example in the nonlocal Calderón program, whose earlier operators had orders in $(0,2)$.
Together with the heat-kernel rigidity theorem, the argument shows that local information about a nonlocal inverse problem determines global topology and geometry, not just the metric on the observed patch.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the transmutation step can be repaired with verified hypotheses, the same moment-to-heat-kernel strategy should apply to other fractional powers or functions of $\Delta_g$ that admit a sine/cosine transmutation, such as the relativistic Schrödinger operator $(-\Delta_g+m^2)^{\alpha}$.
The proof's structure suggests a quantitative cousin: replacing 'all moments vanish' with 'finitely many moments are small' might yield a stability estimate or a partial-data reconstruction statement; the paper itself does not pursue this.
Letting the mass parameter $m$ tend to zero would connect the result to the unperturbed bi-Laplacian inverse problem; the author keeps $m\neq0$ fixed and does not discuss the limit.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

What you should know: this paper proves that the Cauchy data for the nonlocal operator L_g^{1/2} = ((−Δ_g)^2 + m^2 I)^{1/2} on a closed manifold determines the isometry class. That is a genuinely new boundary case: the fractional Calderón results so far stop at order <2. The proof follows the now-standard template from Feizmohammadi–Ghosh–Krupchyk–Uhlmann [20]: show that the exterior data produce vanishing moments of the heat semigroup difference, turn that into equality of the bi-Laplace heat semigroups, then convert to equality of Laplace heat kernels, then invoke the heat kernel rigidity theorem from [20].

The good parts. The operator is new in this context and the main theorem is not in the literature. The paper is honest that the strategy is a rerun of [20] rather than a new idea; that's fine. The semigroup arguments in Section 2 are standard and mostly correct. The reliance on the heat kernel rigidity theorem from [20] is legitimate, since that's an independent established result.

The soft spots. There is a real error in the central step. Equation (27) is Kannai's transmutation formula, but as written it is false on a closed manifold. On the constant eigenfunction, the left side is e^{-tΔ^2} 1 = 1, while the right side evaluates to C t^{7/6} (the integral of sqrt(τ) e^{-τ/(4t)} gives t^{3/2}, and the prefactor contains t^{-1/3}). The correct normalization is t^{-3/2}, not t^{-1/3}. So the equality (28) does not follow from (26) as written. This is likely a typo, but it is load-bearing: everything downstream uses (28) and (34). Also, the passage from (28) to (29) applies an inverse Laplace transform without justifying the uniqueness for functions that grow like sqrt(τ) on the zero mode; that needs a short argument, but it isn't there. These are repairable, but the proof as printed is not complete.

Minor issues: bound (17) mixes L^l and L^k, the exponent in Proposition 3.2 is slightly off relative to (16)/(18), and Theorem 1.4 is stated without proof (the paper says it's similar, which is acceptable but should be detailed or flagged). These are easy fixes.

Who this is for: researchers working on nonlocal Calderón problems and anisotropic inverse problems. For that audience, the extension to order 2 is a meaningful step. The paper deserves a serious referee. I'd send it to review, asking the author to fix the Kannai normalization and justify the Laplace inversion. If those are fixed, the main theorem should hold.

Referee Report

3 major / 3 minor

Summary. This paper studies the anisotropic Calderón problem for the nonlocal second-order operator L_g^{1/2}=((−Δ_g)^2+m^2 I)^{1/2} on closed Riemannian manifolds. Theorem 1.1 claims that if two such manifolds share an open set O on which the metrics agree, and if the interior data L_{g_1}^{−1/2}f|_O = L_{g_2}^{−1/2}f|_O for all f∈C_0^∞(O), then the manifolds are isometric. The proof derives vanishing moment identities for the difference of heat semigroups of L_g, uses Paley–Wiener to conclude the semigroup difference vanishes, then invokes Kannai transmutation formulas to pass from equality of e^{−tΔ^2} to equality of e^{tΔ}, and finally applies a heat-kernel rigidity theorem from [20] to recover the diffeomorphism. A second, closely related theorem for A_g=L_g^{1/2}−mI is stated without proof.

Significance. The main result is a natural extension of the nonlocal anisotropic Calderón program to an operator of order 2, and if the proof can be repaired it would be a worthwhile contribution to the area. The paper's strategy is clear and follows the now-standard template of [20], and the final rigidity step is honestly imported from an independent source rather than reproved. The central new step, however, depends on an incorrect transmutation formula and an unsubstantiated inverse-Laplace uniqueness argument; these must be corrected before the result is reliable. The result's significance is therefore conditional on those repairs.

major comments (3)

[Section 3, Eqs. (27) and (35)] The Kannai formulas are not correct as stated on a closed manifold. For the constant function v≡1, Δ1=0 and sin(√τΔ)/Δ acts as √τ, so the right-hand side of (27) equals (1/(4√π t^{1/3}))∫_0^∞ e^{−τ/(4t)}√τ dτ = t^{7/6}, whereas e^{−tΔ^2}1=1. The same calculation applies to (35) with e^{tΔ}. Since (27) is used to derive (28) and (35) is used to obtain equality of the Laplace heat semigroups, this is a load-bearing error. The author must supply a correct transmutation identity with the proper normalization and state the hypotheses under which it holds on compact manifolds.
[Section 3, passage from (28) to (29)] The step from equality of the t-integrals in (28) to pointwise equality in σ invokes an inverse Laplace transform, but no uniqueness theorem is stated. After setting s=1/(4t), one has ∫_0^∞ e^{−sτ}F(τ)dτ=0 for all s>0 with F(τ)=[sin(√τΔ1)/Δ1−sin(√τΔ2)/Δ2]f(x). The function F is not known to lie in L^2 or to satisfy Paley–Wiener growth; on the constant component it grows like √τ. A Laplace-uniqueness theorem for polynomially growing functions or distributions, together with sufficient regularity to justify pointwise evaluation in x, must be supplied.
[Section 1, Theorem 1.4] Theorem 1.4 is a separate theorem but is followed only by 'The proof is quite similar to Theorem 1.1' and no argument is given. If this theorem is to remain in the paper, the author must either prove it or give an explicit reduction of the Cauchy data for A_g to the data used in Theorem 1.1; otherwise it should be demoted to a remark.

minor comments (3)

[Section 3, Eqs. (14), (17), and (18)] Equation (14) identifies ∂_t^l((e^{−tL_{g1}}−e^{−tL_{g2}})f) with a term involving L^k_g f; this should involve L^l_g f up to sign. Equation (17) mixes L^{2l} and L^k, and the bounds in (18) are not immediate from (17) without an additional argument. These inconsistencies appear repairable, but the displayed estimates should be corrected.
[Section 3, Proposition 3.2] Using (18) and the definition φ(s)=diff/√s gives |φ(s)|≤Cs^{(n−1)/2} for s∈(0,1), not Cs^{n/2} as printed. The conclusion φ∈L^2(0,∞) is unaffected, but the exponent should be corrected.
[Throughout] There are several typographical issues: 'psedodifferential' in Section 1; 'From (26) and (26)' before Eq. (28); a garbled displayed formula for ψ in (34); and Eq. (12) omits the harmless constant 1/Γ(1/2). These should be fixed in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper reduces the new Cauchy-data assumption to a previously established heat-kernel rigidity theorem without assuming its conclusion.

full rationale

The derivation chain goes from equality of L^{-1/2} Cauchy data to moment vanishing for the heat-semigroup difference, then via Paley-Wiener to equality of e^{-tΔ^2} heat semigroups, then via Kannai transmutation formulas to equality of Laplace heat kernels, and finally applies the rigidity theorem from [20]. None of these steps defines its conclusion in terms of its own input. The cited [20, Theorem 1.5] is an independent external result asserting that equality of Laplace heat kernels on an open set forces an isometry; its hypotheses do not include the paper's Cauchy-data assumption, and the paper genuinely derives the kernel equality rather than assuming it. The references to [20], [21], and [26] are to the same research program, with the author's advisor among the coauthors of [20], but under the stated rules this is not circular because the cited theorem does not reproduce the present theorem's conclusion from its data. No fitted parameters are introduced, no output quantity is merely a renamed input, and no uniqueness claim is imported to forbid alternatives beyond the cited rigidity theorem. The printed normalization of the Kannai formulas (27) and (35) may be a correctness or rigor issue, but it is not a circularity issue: even a mistaken external formula would not make the argument equivalent to its inputs. The paper's central claim therefore does not reduce, by construction or by self-citation, to the assumption it starts from.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central claim depends on the spectral calculus for L_g^{1/2}, external heat kernel and semigroup estimates, Kannai's transmutation formula, and a final heat kernel rigidity theorem imported from the authors' earlier work. No new physical or mathematical entities are introduced. The constant m is a fixed input parameter, not a fitted free parameter.

free parameters (1)

m = fixed nonzero real (arbitrary)
m is a fixed constant in the operator L_g = (−Δ_g)^2 + m^2 I. It is not fitted to data; it is an input parameter. The theorem holds for every fixed m ≠ 0, and the proof uses m ≠ 0 only to ensure invertibility and exponential decay of the heat semigroup.

assumptions (8)

standard math Spectral theorem and functional calculus for unbounded self-adjoint operators on L^2(M).
Used to define L_g^{1/2} and its inverse via spectral projections; standard in functional analysis.
domain assumption Heat kernel estimate for the biharmonic heat kernel K_{Δ^2_g} from He [32, Theorem 2.1].
Provides the pointwise decay (7) that underpins the boundary term vanishing in Proposition 3.1; not proved in the paper.
domain assumption Varopoulos semigroup bound [52, Theorem 1] applied to e^{-tL_g}.
Used in (17) to control the heat semigroup for large t; the application to a fourth-order semigroup is not fully justified in the text.
domain assumption Kannai's transmutation formula [34] for the heat semigroup of the bi-Laplacian.
Central to Proposition 3.5 to convert bi-Laplace heat kernel equality into Laplace heat kernel equality; the required regularity is not checked.
standard math Paley-Wiener theorem [43] for L^2(0,∞).
Used in Proposition 3.4 to conclude φ ≡ 0 from vanishing of its Fourier transform.
standard math Morera's theorem and identity theorem from complex analysis.
Used in Propositions 3.2 and 3.3 to establish holomorphy and vanishing of f.
domain assumption Unique continuation property for the heat equation on manifolds [41].
Used in Proposition 3.5 to extend equality from ω2 to O.
domain assumption Heat kernel rigidity theorem of Feizmohammadi, Ghosh, Krupchyk, Uhlmann [20, Theorem 1.5].
Final step: equality of Laplace heat kernels on an open set implies isometry; imported as a black box from overlapping authors.

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Cite this review

Pith. "Pith review of Anisotropic Calder\'{o}n Problem for a Non-Local Second Order Elliptic Operator." pith.science (2026). https://pith.science/paper/CP7VFNM6

@misc{pith2026250512255,
  author       = {Pith},
  title        = {Pith review of: Anisotropic Calder\'on Problem for a Non-Local Second Order Elliptic Operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CP7VFNM6}},
  note         = {Machine review of arXiv:2505.12255}
}

read the original abstract

This paper investigates the anisotropic Calder\'{o}n problem for a non-local elliptic operator of order 2, on closed Riemannian manifolds. We demonstrate that using the Cauchy data set, we can recover the geometry of a closed Riemannian manifold up to standard gauge.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Anisotropic Calder\'{o}n problem of a nearly Laplace-Beltrami operator of order $2+$
math.AP 2025-06 reject novelty 5.0 of 10

The paper claims an anisotropic Calderon uniqueness theorem for a logarithmic Laplacian of order 2+, but the central Paley-Wiener argument is invalid.

Reference graph

Works this paper leans on

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Pith tools

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