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Anisotropic Calder\'{o}n problem of a nearly Laplace-Beltrami operator of order $2+$

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Observation data for a log-Laplacian recover the whole manifold up to isometry.

desk verdict Genuine new result and a clean setup, but the proof's decisive step from vanishing moments to φ=0 is invalid, so Theorem 1.1 is not established as written. read the letter →

arxiv 2506.12535 v2 pith:J5I5Q5NT submitted 2025-06-14 math.AP

Susovan Pramanik (Harish-Chandra Research Institute , India) This is my paper · ORCID
classification math.AP MSC 35R3058J3535S0558J40
keywords anisotropicCalderónproblemlogarithmicLaplaciannearLaplace–BeltramioperatorCauchydatasetheatkernelpseudodifferentialclosedRiemannianmanifoldinverseproblems
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 a positive answer to the anisotropic Calderón problem for an operator that sits at order 2+, just past the Laplace–Beltrami operator: the logarithmic Laplacian $L_g = (-\Delta_g + mI)\ln(-\Delta_g + mI)$. The claim is that on any closed connected Riemannian manifold of dimension greater than two, knowing the Cauchy data of $L_g$ on a small open set, together with the metric on that set, pins down the entire manifold up to diffeomorphism. A sympathetic reader would care because this moves nonlocal inverse problems from fixed fractional orders below 2 to operators arbitrarily close to the classical Laplacian, where the anisotropic Calderón problem remains open in smooth geometry. The proof works by converting the Cauchy data into equality of heat semigroups on the observation set and then invoking a known result that heat-kernel equality implies isometry.

What carries the argument

The central object is the logarithmic Laplacian operator $L_g = (-\Delta_g + mI)\ln(-\Delta_g + mI)$, whose spectral action multiplies the $k$-th Fourier mode by $(\lambda_k+m)\ln(\lambda_k+m)$; its principal symbol is $(|\xi|^2_g + m)\ln(|\xi|^2_g + m)$, giving an operator of order $2+\varepsilon$ for every $\varepsilon>0$. The load-bearing mechanism is a chain of reductions: equality of Cauchy data implies vanishing integral identities involving the heat semigroups of the two manifolds; after integration by parts, Hardy's inequality, and a change of variables, the Paley–Wiener theorem is invoked to turn those vanishing moment integrals into vanishing Taylor coefficients of a holomorphic function, forcing the heat semigroup difference to vanish on $O$; from there the equality of heat kernels follows and a known theorem converts heat-kernel equality into an isometry.

What would settle it

One concrete check is to look for a nonzero $L^2(0,\infty)$ function $\psi$ with $\int_0^\infty \tau^j \psi(\tau)\, d\tau = 0$ for every $j\ge 0$ (moment-indeterminate functions of this kind are known to exist) and to determine whether $\psi$ can have the specific form of the rescaled heat-semigroup difference between equations (3.17) and (3.18); if such a difference exists, the proof's central vanishing inference fails. Short of that, checking whether the Paley–Wiener transform of the relevant $\psi$ is analytic in a full neighborhood of $z=0$ would settle the disputed step.

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Extended reading notes

Core claim

The paper's central claim, Theorem 1.1, states that if two closed connected Riemannian manifolds $(M_1, g_1)$ and $(M_2, g_2)$ of dimension greater than two agree on a non-empty open set $O$, and if their Cauchy data sets for the operator $L_g = (-\Delta_g + mI)\ln(-\Delta_g + mI)$ agree on $O$ for a fixed $m>1$, then there is a diffeomorphism $\Phi: M_1 \to M_2$ with $\Phi^* g_2 = g_1$. The operator is defined spectrally through the eigenvalues $(\lambda_k+m)\ln(\lambda_k+m)$, and its symbol $(|\xi|^2_g + m)\ln(|\xi|^2_g + m)$ places it in the pseudodifferential class of order $2+\varepsilon$ for every $\varepsilon>0$, which is what makes it a nearly Laplace–Beltrami operator of order $2+$. The theorem is established by deriving from the Cauchy data a family of integral identities, using Hardy's inequality and the Paley–Wiener theorem to conclude that the heat semigroups coincide on $O$, and then invoking the heat-kernel uniqueness theorem quoted as Theorem 3.1 of the paper's reference [18].

Load-bearing premise

The proof depends on the step where vanishing of all the integrals $\int_0^\infty \varphi(t)/t^{1+k}\, dt$ for $k=0,1,2,\dots$ is taken to force $\varphi$ to be identically zero; that inference requires a Paley–Wiener transform to be analytic at a boundary point, and the cited Paley–Wiener theorem does not provide that analyticity.

Editorial extensions

If this is right

  • If the theorem is correct, the Cauchy data of the logarithmic Laplacian on any open set determine the isometry class of a closed Riemannian manifold of dimension greater than two.
  • The result extends the nonlocal anisotropic Calderón program from fractional Laplace–Beltrami operators of order $2\alpha<2$ to an operator of order $2+$, bringing it closer to the unresolved smooth local problem.
  • Because the proof reaches equality of heat kernels, all heat-kernel invariants of the manifold become observable from the Cauchy data, not just the conformal class.
  • The same reduction, if valid, could be applied to other functions of the shifted Laplacian whose symbols grow like order $2+$.
  • The method suggests that knowledge of the Cauchy data for one such nearly local operator may be enough to recover the full metric, without boundary measurements on the whole manifold.

Reading between the lines

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

  • A natural companion question, not addressed in the paper, is whether the same scheme works for the operator obtained by replacing $\ln(-\Delta_g + mI)$ with any slowly varying complete Bernstein function of the Laplacian, since only the order-$2+$ symbol growth appears to be used.
  • If the moment-vanishing step at the boundary can be made rigorous, the proof actually yields a reconstruction route: recover the heat semigroup on $O$ from the Cauchy data, then apply boundary-control or heat-kernel methods to recover the metric.
  • The example of the logarithmic Laplacian may serve as a testbed for whether local anisotropic Calderón uniqueness can be obtained as a limit of order-$2+$ nonlocal problems as the logarithmic correction is scaled away.
  • A direct way to stress-test the paper's central inference is to study the flat-torus or round-sphere case, where heat kernels are explicit, and check whether two different metrics can produce heat-semigroup differences whose moment integrals all vanish.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper investigates the anisotropic Calderón problem for the logarithmic Laplacian L_g = (-Δ_g + mI) log(-Δ_g + mI) on closed Riemannian manifolds of dimension greater than two. The main theorem (Theorem 1.1) claims that equality of the Cauchy data sets on a nonempty open set where the metrics agree forces the two manifolds to be isometric. The proof strategy is to show that the equality of Cauchy data implies certain integral identities for the difference of heat semigroups, to conclude the heat semigroups agree on the open set, then to deduce equality of the heat kernels, and finally to invoke the rigidity theorem [18, Theorem 1.5]. A central analytic step in this chain is not justified.

Significance. If correct, the result would be a valuable contribution to the program of extending nonlocal Calderón-type inverse problems to operators of order 2+, thereby connecting the fractional and local anisotropic cases. The paper contains a careful spectral and semigroup setup for the logarithmic Laplacian, and the overall strategy is natural and worth pursuing. However, the decisive step in the proof is invalid, and the main theorem is not established by the arguments presented.

major comments (3)
  1. [Section 3, between (3.17) and (3.18)] The Paley-Wiener theorem guarantees that the Fourier transform \hat{\psi}(\xi+i\eta) = \int_0^\infty e^{i\tau(\xi+i\eta)}\psi(\tau)\,d\tau is holomorphic only in the upper half-plane H_+. The Taylor expansion of this function at an interior point \zeta_0 \in H_+ has coefficients of the form \int_0^\infty (i\tau)^j e^{i\tau\zeta_0}\psi(\tau)\,d\tau, not the power moments \int_0^\infty \tau^j \psi(\tau)\,d\tau. Writing the Taylor series with coefficients \int_0^\infty \tau^j \psi(\tau)\,d\tau would require analyticity of \hat{\psi} at \zeta=0, which Paley-Wiener does not provide. Consequently, the assertion that the identity (3.8) forces \hat{\psi} \equiv 0 is not justified.
  2. [Section 3, implication from (3.8) to (3.16)] The implication 'if \int_0^\infty \tau^j \psi(\tau)\,d\tau = 0 for every j \geq 0, then \psi \equiv 0' is false in general. For example, \psi_0(\tau)=e^{-\tau^{1/4}}\sin(\tau^{1/4}) belongs to L^2(0,\infty) and satisfies \int_0^\infty \tau^j \psi_0(\tau)\,d\tau = 4\Gamma(4j+4)2^{-2j-2}\sin((j+1)\pi)=0 for every j\geq 0, yet \psi_0 is not identically zero. Therefore the vanishing of the moment integrals (3.8) cannot, by itself, yield (3.16). The equality of heat semigroups (3.18) and the heat kernels (3.19) do not follow from the given proof.
  3. [Section 3, derivation of heat kernel equality (3.19)-(3.24)] The deduction of the heat kernel equality and the final appeal to [18, Theorem 1.5] rests entirely on (3.18), which in turn depends on the unjustified (3.16). Since the step from (3.8) to (3.16) is invalid, the proof does not supply any alternative route to the heat kernel equality. Thus Theorem 1.1 is unsupported by the present argument.
minor comments (3)
  1. [Abstract] The abstract contains typographical errors ('Logarithemic', 'close' for 'closed') and should be proofread.
  2. [Section 3, equation (3.13)] In the Taylor formula (3.13), the denominator '(m-1)!' appears to be a misprint for '(l-1)!'.
  3. [Section 3, regularity discussion] The notation H(M) is overloaded: it is defined as D(L_g) in (1.4) but later used to denote Sobolev spaces H^l(M_i). Using distinct notation would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof reduces Cauchy data to heat-kernel equality via operator identities and cites an external rigidity theorem; the identified flaw is an invalid analytic step, not a circular reduction.

full rationale

The derivation chain is not circular. Starting from the assumed equality of Cauchy data (3.2), the paper defines the operators Lg and Ag spectrally and via the heat semigroup, derives the integral identities (3.5), (3.7), and (3.8), and then attempts to conclude the heat semigroup equality (3.18). The only fragile step is between (3.17) and (3.18), where the Paley-Wiener transform of ψ is expanded in a Taylor series at the boundary point z = 0 with power moments as coefficients. Paley-Wiener only guarantees holomorphy in the upper half-plane, not in a neighborhood of 0, and the expansion used is not justified. This is an analytic validity/correctness gap in the proof, not circularity: no parameter is fitted, no quantity is defined in terms of the target quantity, and the conclusion is not an input by construction. The cited rigidity result [18, Theorem 1.5] is external work by Feizmohammadi, Ghosh, Krupchyk, and Uhlmann, not the present author's own theorem, and the author's earlier work [43] appears only as motivation and is not load-bearing. There is no self-referential reduction, no fitted input renamed as prediction, and no ansatz smuggled in via citation. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted; the proof is analytic. The load-bearing external input is the heat-kernel rigidity theorem. The invalid Paley-Wiener step is not an axiom but a flawed inference, flagged in red_flags.

assumptions (5)
  • standard math Spectral decomposition and functional calculus for A_g = -Delta_g + mI define L_g and its logarithm.
    Used throughout Sections 1 and 2; standard for self-adjoint elliptic operators on compact manifolds.
  • standard math Gaussian upper bound for the heat kernel, Theorem 2.1 from Grigor'yan.
    Used to justify convergence of the semigroup integrals in Section 2.1.
  • standard math Hardy's inequality on (0, infinity).
    Used to infer psi = phi(1/t) is in L2 from the finiteness of integral |phi'|^2 in Section 3.
  • standard math Paley-Wiener theorem for L2(0, infinity).
    Used to define the holomorphic transform of psi; the paper's application goes beyond the theorem's scope.
  • domain assumption Heat kernel rigidity: equality of heat kernels on an open set implies isometry, from [18, Thm 1.5].
    External theorem from the fractional Calderon literature, quoted without proof; it is the final bridge to the isometry conclusion.

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Pith. "Pith review of Anisotropic Calder\'{o}n problem of a nearly Laplace-Beltrami operator of order $2+$." pith.science (2026). https://pith.science/paper/J5I5Q5NT

@misc{pith2026250612535,
  author       = {Pith},
  title        = {Pith review of: Anisotropic Calder\'on problem of a nearly Laplace-Beltrami operator of order $2+$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5I5Q5NT}},
  note         = {Machine review of arXiv:2506.12535}
}
read the original abstract

This paper investigates the anisotropic Calder\'{o}n problem for Logarithemic Laplacian, on closed Riemannian manifolds, which could be considered as near Laplace operator. We demonstrate that the Cauchy data set recovers the geometry of a closed Riemannian manifold up to standard gauge.

Discussion (0). Continue with ORCID to comment.

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