The Calder\'on problem for the logarithmic Schr\"odinger equation

Bastian Harrach; Tobias Weth; Yi-Hsuan Lin

REVIEW 2 major objections 6 minor 2 cited by

The Calder\'on problem for the logarithmic Schr\"odinger equation

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Boundary measurements uniquely determine the potential of a logarithmic Schrödinger equation in every dimension.

desk verdict Solid first uniqueness theorem for the logarithmic Laplacian, but the constructive part has a real gap and a sign error that need fixing. read the letter →

arxiv 2412.17775 v1 pith:QKCYOIX3 submitted 2024-12-23 math.AP

Bastian Harrach , Yi-Hsuan Lin , Tobias Weth This is my paper · ORCID

classification math.AP MSC 35R3026A3335J70

keywords CalderónproblemlogarithmicLaplacianDirichlet-to-NeumannmapuniquecontinuationRungeapproximationmonotonicitymethodnonlocalinverseproblemsboundedpotentials

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 proves a global uniqueness result for the Calderón problem associated with the logarithmic Schrödinger operator $L_{\Delta}+q$, where $L_{\Delta}$ is the logarithmic Laplacian. Specifically, if two bounded potentials $q_1,q_2$ produce the same Dirichlet-to-Neumann map (measured on small open sets outside the domain), then $q_1=q_2$ almost everywhere. This is the first such result for this near-zero-order nonlocal operator and it holds in any spatial dimension. A constructive uniqueness formula is also derived using a monotonicity method, allowing one to recover the potential from boundary measurements via a pointwise supremum over simpler potentials.

What carries the argument

The key machinery is the unique continuation property (UCP) of the logarithmic Laplacian, stated as Proposition 4.1 and imported from a companion paper: if $u\in L^1_0(\mathbb{R}^n)$ and $u=L_{\Delta}u=0$ in a nonempty open set, then $u\equiv 0$ on $\mathbb{R}^n$. This UCP is used to prove the Runge approximation property (Proposition 4.2), which asserts that solutions generated by boundary data supported in a small exterior set are dense in $L^2(\Omega)$. The Runge approximation, together with the integral identity and a localized-potentials lemma, forms the backbone of both the global uniqueness theorem and the constructive monotonicity-based recovery.

What would settle it

Find a nonzero function $u\in L^1_0(\mathbb{R}^n)$ and a nonempty open set $D$ such that $u=L_{\Delta}u=0$ in $D$ but $u\not\equiv 0$; this would directly contradict Proposition 4.1 and invalidate the Runge approximation used in the proof of Theorem 1.1.

Watch

Extended reading notes

Core claim

The central claim is that the logarithmic Laplacian, despite being a near-zero-order nonlocal operator, admits a unique continuation property and a Runge approximation property strong enough to solve the inverse problem. Under the spectral condition $\lambda_1(\Omega)+q\ge \lambda_0>0$, the paper shows that equality of the partial Dirichlet-to-Neumann maps, $\langle\Lambda_{q_1}f,g\rangle=\langle\Lambda_{q_2}f,g\rangle$ for all smooth $f,g$ supported in nonempty open sets $W_1,W_2\Subset\Omega^c$, forces $q_1=q_2$ in $\Omega$. The proof uses an integral identity that relates the difference of DN maps to the integral of $(q_1-q_2)u_1u_2$, combined with Runge approximation to localize the solutions inside $\Omega$. The paper also establishes an if-and-only-if monotonicity relation, which yields a constructive formula for recovering nonnegative potentials when $\lambda_1(\Omega)>0$.

Load-bearing premise

The proof relies on the unique continuation property for the logarithmic Laplacian: if a function and its logarithmic Laplacian both vanish in a nonempty open set, the function must vanish everywhere. If this property fails, the main uniqueness theorem collapses.

Editorial extensions

If this is right

If the main theorem is correct, then for bounded Lipschitz domains and $L^\infty$ potentials satisfying the spectral condition, the logarithmic Schrödinger operator is identified from boundary measurements in any dimension, matching the strength of known results for the fractional Laplacian.
The if-and-only-if monotonicity relation provides a constructive algorithm for recovering nonnegative potentials pointwise, using only comparisons with simpler (density-one simple function) potentials.
The paper opens the door to studying inverse problems for operators of zero or negative order, where classical elliptic regularity and CGO solutions are not available.
The UCP-based Runge approximation may be transferable to other inverse problems involving operators with logarithmic symbols, such as fractional Laplacians near $s=0$.

Reading between the lines

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

A natural extension is to ask whether the potential can be recovered from a single measurement, which the paper remarks would require a measurable UCP for the logarithmic Laplacian; this is open for rough potentials.
The monotonicity method could be adapted to detect inclusions or obstacles inside $\Omega$ from boundary measurements, analogous to known monotonicity-based obstacle detection for fractional and classical Schrödinger equations.
Because the logarithmic Laplacian approximates the fractional Laplacian as $s\to 0^+$, the uniqueness result here may inform the behavior of inverse problems for very small fractional order $s$, where positive-order theory degenerates.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Quick take: this is a real first — a Calderón-type uniqueness result for the logarithmic Laplacian, in any dimension — and the main theorem (1.1) is well proved via Runge approximation plus the known UCP. The constructive theorems (1.2 and 1.4) are another story: there's a gap in Lemma 5.1 and a sign error in the statement of Theorem 1.2, so those results need fixing before they can be trusted.

The good part: the forward problem is set up cleanly in the trace space H_T, the DN map is symmetric, and the integral identity is straightforward. Theorem 1.1 follows in a few lines from Runge approximation, which itself is a neat application of the UCP. I checked the proof of Proposition 4.2 and it's correct. This is a genuine new result, not just an exercise; the logarithmic Laplacian is a zero-order operator and doesn't fit the fractional framework directly. The reliance on the UCP from [CHW23] and the density-one lemma from [HL19] is fine; these are independent results and not circular.

Now the soft spots. Lemma 5.1 claims that for any measurable M⊂Ω one can find solutions concentrated on M and decaying on Ω\M. The proof invokes the UCP to assert that ‖~u_k‖_{L2(Ω\M)}>0 unless the function is identically zero. But the UCP (Proposition 4.1) applies only to vanishing on a nonempty open set. For a general measurable M, the complement may have empty interior, so the UCP doesn't apply. The authors themselves flag this in Remark 4.4 as an open problem. This gap affects Theorems 1.2 and 1.4, which use Lemma 5.1 with M={q1−q2≥δ}. It's not a fatal flaw for the whole paper — the main theorem doesn't use it — but it means the if-and-only-if monotonicity and the reconstruction formula are not justified as written.

There's also a sign error in Theorem 1.2. Condition (iii) is printed as ⟨(Λ_{q1}−Λ_{q2})f,f⟩≥0 and then labeled Λ_{q1}≤Λ_{q2}. That's backwards: the monotonicity inequality (5.4) gives ⟨(Λ_{q2}−Λ_{q1})f,f⟩≥0 when q2≥q1. As printed, the contradiction in (iii)⇒(i) doesn't work. This is likely a typo, but it needs correction.

Overall: the core novelty is real, the main proof is clean, and the paper deserves a serious referee. I'd recommend conditional acceptance: require the authors to either prove the measurable UCP or reformulate Lemma 5.1 and the constructive statements, and fix the sign. If they can't close the gap, the monotonicity section should be revised to state only what is actually proved.

Referee Report

2 major / 6 minor

Summary. The paper studies the Calderón problem for the logarithmic Schrödinger operator L_Δ + q, where L_Δ is the logarithmic Laplacian. After formulating a well-posed exterior boundary value problem and a Dirichlet-to-Neumann map under the spectral condition λ1(Ω)+q≥λ0>0, the authors prove a global uniqueness result (Theorem 1.1): equality of the DN maps on arbitrary nonempty open sets W1,W2⊂Ω_e forces q1=q2 in Ω. The proof uses an integral identity, the unique continuation property for L_Δ cited from [CHW23], and a Runge approximation result. The second half develops a constructive uniqueness theory: an if-and-only-if monotonicity relation between potentials and DN maps (Theorem 1.2) and a reconstruction formula for nonnegative potentials using density-one simple functions (Theorem 1.4). The paper claims all results hold in any space dimension.

Significance. If completed, the paper would provide the first global uniqueness result for the logarithmic Schrödinger operator, a genuinely near-zero-order nonlocal operator, and would extend the monotonicity-based reconstruction method to this setting. The main proof of Theorem 1.1 is concise and dimension-independent, and it rests on a clearly identified external UCP rather than on ad hoc assumptions. A particular strength is that the dependence on the UCP is made explicit in Proposition 4.1 and Remark 4.4. The constructive part is more fragile: Lemma 5.1, which is load-bearing for Theorems 1.2 and 1.4, has a proof gap and an omitted hypothesis, and Theorem 1.2 contains a sign error. These issues appear repairable within the scope of the paper, but they must be fixed before the constructive claims are justified.

major comments (2)

[§5.1, Lemma 5.1] The proof of Lemma 5.1 is not valid as written for arbitrary measurable M. It asserts that ‖~u_k‖_{L^2(Ω\M)}>0 'follows from the UCP of Proposition 4.1', but Proposition 4.1 is an open-set UCP: it applies only when the function vanishes on a nonempty open set. For a general measurable M the complement Ω\M may have empty interior, so the cited UCP does not apply; this is exactly the 'measurable UCP' that Remark 4.4 declares open. Additionally, the statement of Lemma 5.1 omits the spectral condition (1.9), although the proof invokes the Runge approximation of Proposition 4.2, which requires that condition. Since Theorem 1.2(iii)⇒(i) applies the lemma with M={x: q1(x)−q2(x)≥δ} and Theorem 1.4 relies on Theorem 1.2, the constructive uniqueness results are not justified as stated. The gap appears repairable: one can choose Runge approximations to χ_M/√|M| plus a small bump supported in Ω\M, then take a diagonal sequence, which removes the need for a measurable UCP (the case where Ω\M has measure zero can be handled by direct scaling).
[§1, Theorem 1.2 and §5.2] The sign in condition (1.12) is reversed. The monotonicity direction (ii) and the proof of (iii)⇒(i) require the inequality ⟨(Λ_{q2}−Λ_{q1})f,f⟩≥0; as printed, (1.12) is ⟨(Λ_{q1}−Λ_{q2})f,f⟩≥0. With the printed sign, condition (ii) does not imply (iii), the definition (1.13) of Λ_{q1}≤Λ_{q2} is incoherent, and the contradiction derived in (5.6) does not contradict (1.12). This is a typographical but load-bearing error: The statement and the definition must be corrected so that the monotonicity relation Λ_{q1}≤Λ_{q2} in W means ⟨(Λ_{q2}−Λ_{q1})f,f⟩≥0 for f∈C_c^∞(W).

minor comments (6)

[§2.1, eq. (2.13)] In the Cauchy-Schwarz estimate after (2.13), the second factor should be ‖v‖_{H(R^n)}, not ‖u‖_{H(R^n)}.
[§2.1, proof of Lemma 2.1] The phrase 'equivalent on H^1_0(Ω)' should read 'equivalent on H_0(Ω)'.
[§5.2, eq. (5.4)] The displayed inequality is missing a closing parenthesis: it should read ⟨(Λ_{q2}−Λ_{q1})f,f⟩, not ⟨(Λ_{q2}−Λ_{q1}f,f⟩.
[§5.1, Lemma 5.1] The statement should require M⊂Ω to have positive measure, since the target function χ_M/√(∫_M 1 dx) is otherwise undefined.
[Remark 4.4] The phrase 'positive measures' should be 'positive measure'.
[§3, proof of Lemma 3.1] The sentence beginning 'Let f ∈ H_T(Ω_e)' is missing a period before 'By definition'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniqueness argument reduces to an external UCP and standard Runge approximation, not to its own conclusion.

full rationale

The derivation chain for Theorem 1.1 is: DN-map equality, integral identity (3.14), Runge approximation (Prop. 4.2), L2-density of solutions, and then q1 = q2 a.e. Prop. 4.2 is derived from the unique continuation property in Prop. 4.1, which is cited from [CHW23, Theorem 5.1] rather than proved in this paper. That UCP is a parameter-free statement about the logarithmic Laplacian whose hypotheses (u in L1_0, u = L_Delta u = 0 in a nonempty open set) do not include the inverse problem, the DN map, or the potentials being recovered; it is therefore independent support even though one of the present authors is also an author of [CHW23]. Similarly, Theorem 1.4 uses the density-one representation lemma [HL19, Lemma 4.4]; that lemma is measure-theoretic and does not presuppose the DN-map conclusion. No parameter is fitted and no prediction is defined in terms of the quantity to be recovered, so no step reduces by construction to its own input. The genuine concerns here are correctness risks rather than circularity: Lemma 5.1 applies the open-set UCP of Prop. 4.1 to the complement Omega\M for an arbitrary measurable set M, where Omega\M may have empty interior (Remark 4.4 itself flags the missing measurable UCP), and the printed inequality (1.12) in Theorem 1.2 has the opposite sign from the monotonicity inequality (5.4) when q2 >= q1. These are substantive proof and statement issues, but they are not circularity, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It rests on several prior results, most importantly the UCP for L_Δ from [CHW23] (by co-author Weth et al.) and the density-one representation lemma from [HL19] (by co-authors Harrach and Lin). These are cited, not proved, but they are parameter-free theorems whose assumptions do not include the target conclusion.

assumptions (4)

domain assumption Unique continuation property for L_Δ: if u∈L^1_0(R^n) and u=L_Δu=0 in a nonempty open set, then u≡0 in R^n.
Cited as [CHW23, Theorem 5.1] (Proposition 4.1). This is the load-bearing premise for the Runge approximation (Proposition 4.2) and the localized-potentials lemma (Lemma 5.1).
domain assumption Spectral condition (1.9): λ1(Ω)+q(x)≥λ0>0 for a.e. x∈Ω.
Explicitly assumed in all theorems to ensure well-posedness of the forward problem (Lax-Milgram via Lemma 2.3).
domain assumption Density-one representation (5.7): q(x)=sup{φ(x): φ∈Σ_{+,0}, φ≤q} for a.e. x.
Cited as [HL19, Lemma 4.4] (Theorem 1.4 proof). A known result about density-one simple functions, not proved in this paper.
standard math Spectral theory of the logarithmic Laplacian: L_Δ has discrete eigenvalues λk(Ω) on a bounded domain, as in [CW19, Theorem 1.2].
Used to define the spectral condition and the variational characterization (2.26).

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Calder\'on problem for the logarithmic Schr\"odinger equation." pith.science (2026). https://pith.science/paper/QKCYOIX3

@misc{pith2026241217775,
  author       = {Pith},
  title        = {Pith review of: The Calder\'on problem for the logarithmic Schr\"odinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKCYOIX3}},
  note         = {Machine review of arXiv:2412.17775}
}

abstract

We study the Calder\'on problem for a logarithmic Schr\"odinger type operator of the form $L_{\Delta} +q$, where $L_{\Delta}$ denotes the logarithmic Laplacian, which arises as formal derivative $\frac{d}{ds} \big|_{s=0}(-\Delta)^s$ of the family of fractional Laplacian operators. This operator enjoys remarkable nonlocal properties, such as the unique continuation and Runge approximation. Based on these tools, we can uniquely determine bounded potentials using the Dirichlet-to-Neumann map. Additionally, we can build a constructive uniqueness result by utilizing the monotonicity method. Our results hold for any space dimension.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

Fractional anisotropic Calder\'on problem with external data
math.AP 2025-02 conditional novelty 7.0 of 10

Exterior Dirichlet-to-Neumann data for fractional Laplace-Beltrami operators determine a Euclidean-asymptotic Riemannian metric up to a diffeomorphism fixing the exterior.
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

14 extracted references · 4 canonical work pages · cited by 2 Pith papers

[4]

Fractional anisotropic Calder´ on problem on closed Rieman nian manifolds

[FGKU21] Ali Feizmohammadi, Tuhin Ghosh, Katya Krupchyk, a nd Gunther Uhlmann. Fractional anisotropic Calder´ on problem on closed Rieman nian manifolds. arXiv:2112.03480,

arXiv
[6]

Entanglement pr inciple for the fractional Laplacian with applications to inverse problems

[FL24] Ali Feizmohammadi and Yi-Hsuan Lin. Entanglement pr inciple for the fractional Laplacian with applications to inverse problems. arXiv preprint arXiv:2412.13118 ,

arXiv
[7]

The Cald er´ on problem for vari- able coeﬃcients nonlocal elliptic operators

[GLX17] Tuhin Ghosh, Yi-Hsuan Lin, and Jingni Xiao. The Cald er´ on problem for vari- able coeﬃcients nonlocal elliptic operators. Comm. Partial Diﬀerential Equations , 42(12):1923–1961,

work page 1923
[9]

The fractional anisotropic Calder\'{o}n problem for a nonlocal parabolic equation on closed Riemannian manifolds

[Lin24] Yi-Hsuan Lin. The fractional anisotropic Calder´ o n problem for a nonlocal parabolic equation on closed Riemannian manifolds. arXiv preprint arXiv:2410.17750 ,

work page Pith review arXiv
[10]

The Calder\'{o}n problem for nonlocal parabolic operators

[LLU22] Ching-Lung Lin, Yi-Hsuan Lin, and Gunther Uhlmann. The Calder´ on problem for nonlocal parabolic operators. arXiv preprint arXiv:2209.11157 ,

work page Pith review arXiv
[11]

The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonl ocal to the local

[LLU23] Ching-Lung Lin, Yi-Hsuan Lin, and Gunther Uhlmann. The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonl ocal to the local. arXiv preprint arXiv:2308.09654,

arXiv
[13]

Approximation and uniqueness results for the nonlocal diﬀuse optical tomography problem

[LZ24] Yi-Hsuan Lin and Philipp Zimmermann. Approximation and uniqueness results for the nonlocal diﬀuse optical tomography problem. arXiv preprint arXiv:2406.06226 ,

arXiv
[14]

Revisiting the anisotropic frac tional Calder´ on problem using the Caﬀarelli-Silvestre extension

[R¨ ul23] Angkana R¨ uland. Revisiting the anisotropic frac tional Calder´ on problem using the Caﬀarelli-Silvestre extension. arXiv preprint arXiv:2309.00858 ,

arXiv

Show all 14 references

[1980]

de Hoop, and Mikko Salo

[CdHS24] Giovanni Covi, Maarten V. de Hoop, and Mikko Salo. G eometrical optics for the fractional Helmholtz equation and applications to inverse problems. arXiv preprint arXiv:2412.14698,

arXiv
[2020]

The Calder´ on probl em for nonlocal operators

[GU21] Tuhin Ghosh and Gunther Uhlmann. The Calder´ on probl em for nonlocal operators. arXiv:2110.09265,

arXiv
[2021]

Unique determi nation of coeﬃcients and kernel in nonlocal porous medium equations with absorption term

[LZ23] Yi-Hsuan Lin and Philipp Zimmermann. Unique determi nation of coeﬃcients and kernel in nonlocal porous medium equations with absorption term. arXiv preprint arXiv:2305.16282,

arXiv
[2022]

Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannia n manifolds

[FKU24] Ali Feizmohammadi, Katya Krupchyk, and Gunther Uhl mann. Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannia n manifolds. arXiv preprint arXiv:2407.16866,

arXiv
[2023]

An exten sion problem for the loga- rithmic Laplacian

[CHW23] Huyuan Chen, Daniel Hauer, and Tobias W eth. An exten sion problem for the loga- rithmic Laplacian. arXiv preprint arXiv:2312.15689 ,

arXiv
[2024]

A reduction of the fractional Calder´ on problem to the local Calder´ on p roblem by means of the Caﬀarelli-Silvestre extension

[CGRU23] Giovanni Covi, Tuhin Ghosh, Angkana R¨ uland, and G unther Uhlmann. A reduction of the fractional Calder´ on problem to the local Calder´ on p roblem by means of the Caﬀarelli-Silvestre extension. arXiv preprint arXiv:2305.04227 ,

arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [4]

Fractional anisotropic Calder´ on problem on closed Rieman nian manifolds

[FGKU21] Ali Feizmohammadi, Tuhin Ghosh, Katya Krupchyk, a nd Gunther Uhlmann. Fractional anisotropic Calder´ on problem on closed Rieman nian manifolds. arXiv:2112.03480,

arXiv

[2] [6]

Entanglement pr inciple for the fractional Laplacian with applications to inverse problems

[FL24] Ali Feizmohammadi and Yi-Hsuan Lin. Entanglement pr inciple for the fractional Laplacian with applications to inverse problems. arXiv preprint arXiv:2412.13118 ,

arXiv

[3] [7]

The Cald er´ on problem for vari- able coeﬃcients nonlocal elliptic operators

[GLX17] Tuhin Ghosh, Yi-Hsuan Lin, and Jingni Xiao. The Cald er´ on problem for vari- able coeﬃcients nonlocal elliptic operators. Comm. Partial Diﬀerential Equations , 42(12):1923–1961,

work page 1923

[4] [9]

The fractional anisotropic Calder\'{o}n problem for a nonlocal parabolic equation on closed Riemannian manifolds

[Lin24] Yi-Hsuan Lin. The fractional anisotropic Calder´ o n problem for a nonlocal parabolic equation on closed Riemannian manifolds. arXiv preprint arXiv:2410.17750 ,

work page Pith review arXiv

[5] [10]

The Calder\'{o}n problem for nonlocal parabolic operators

[LLU22] Ching-Lung Lin, Yi-Hsuan Lin, and Gunther Uhlmann. The Calder´ on problem for nonlocal parabolic operators. arXiv preprint arXiv:2209.11157 ,

work page Pith review arXiv

[6] [11]

The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonl ocal to the local

[LLU23] Ching-Lung Lin, Yi-Hsuan Lin, and Gunther Uhlmann. The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonl ocal to the local. arXiv preprint arXiv:2308.09654,

arXiv

[7] [13]

Approximation and uniqueness results for the nonlocal diﬀuse optical tomography problem

[LZ24] Yi-Hsuan Lin and Philipp Zimmermann. Approximation and uniqueness results for the nonlocal diﬀuse optical tomography problem. arXiv preprint arXiv:2406.06226 ,

arXiv

[8] [14]

Revisiting the anisotropic frac tional Calder´ on problem using the Caﬀarelli-Silvestre extension

[R¨ ul23] Angkana R¨ uland. Revisiting the anisotropic frac tional Calder´ on problem using the Caﬀarelli-Silvestre extension. arXiv preprint arXiv:2309.00858 ,

arXiv