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The local Dirichlet-to-Neumann map near one boundary point determines the potential in a nearby interior neighborhood for the three-dimensional Schrödinger equation.

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2026-06-27 15:23 UTC pith:F2RHWPTK

load-bearing objection The paper reduces local partial-data Calderón uniqueness in 3D to injectivity of a weighted X-ray transform, but the abstract leaves unclear whether that injectivity is proved or cited. the 2 major comments →

arxiv 2606.10247 v1 pith:F2RHWPTK submitted 2026-06-08 math.AP

The partial data Calder\'on problem in dimension three

Gunther Uhlmann , Yiran Wang This is my paper
classification math.AP
keywords partial data Calderón problemDirichlet-to-Neumann mapSchrödinger equationweighted X-ray transforminverse boundary value problemintegral geometryuniqueness
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes uniqueness for the partial-data Calderón problem on the time-independent Schrödinger equation in three dimensions. It shows that knowledge of the Dirichlet-to-Neumann map on an arbitrarily small open set of the boundary suffices to recover the potential in some interior neighborhood adjacent to that set. The argument proceeds by reducing the inverse problem to the injectivity of an associated weighted X-ray transform. A reader would care because the result clarifies how little boundary data is needed to determine internal coefficients in elliptic inverse problems.

Core claim

We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.

What carries the argument

Reduction of the inverse boundary-value problem to injectivity of a weighted X-ray transform.

Load-bearing premise

The weighted X-ray transform is injective for the weights and domains that arise from the local Dirichlet-to-Neumann map.

What would settle it

An explicit example of a nonzero potential whose associated weighted X-ray transform vanishes on all lines meeting the relevant boundary neighborhood would disprove the uniqueness claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • The potential is uniquely recovered in an open set touching the boundary from the local map.
  • The result applies specifically to the three-dimensional time-independent Schrödinger equation.
  • The proof strategy connects the Calderón problem directly to questions in integral geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the weighted X-ray transform injectivity can be verified by independent methods, the same reduction may apply in higher dimensions.
  • The local uniqueness result suggests that global uniqueness might follow from patching together local determinations when the boundary data cover the whole boundary.
  • Similar reductions could be tested numerically by discretizing the weighted X-ray transform on sample domains.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The manuscript addresses the partial-data Calderón problem for the time-independent Schrödinger equation in three dimensions. It claims to prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of that point in the interior. The argument proceeds by reducing the uniqueness question to the injectivity of a weighted X-ray transform.

Significance. If the reduction is valid and the injectivity of the weighted X-ray transform is established for the weight induced by the local DN map, the result would connect inverse boundary-value problems to integral geometry in a concrete way. The manuscript provides a reduction step, but the overall significance hinges on whether that injectivity is proved inside the paper rather than assumed or imported without verification.

major comments (2)
  1. [Abstract] Abstract, paragraph 2: the uniqueness theorem is stated to follow from reduction to injectivity of a weighted X-ray transform. The manuscript must either prove this injectivity for the specific weight class arising from the local DN map in dimension 3 or give an explicit citation establishing it under the precise assumptions used here; without one or the other the implication does not hold.
  2. [Abstract] The abstract provides no statement of the precise assumptions on the potential (e.g., regularity class) or on the domain and the portion of the boundary where the local DN map is given. These assumptions are load-bearing for both the reduction and the claimed uniqueness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below. We will revise the abstract to state the precise assumptions explicitly and to include an explicit citation for the injectivity result on the weighted X-ray transform. These changes will be incorporated in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the uniqueness theorem is stated to follow from reduction to injectivity of a weighted X-ray transform. The manuscript must either prove this injectivity for the specific weight class arising from the local DN map in dimension 3 or give an explicit citation establishing it under the precise assumptions used here; without one or the other the implication does not hold.

    Authors: The manuscript establishes the reduction from the partial-data uniqueness question to the injectivity of the weighted X-ray transform with the weight induced by the local DN map. The required injectivity statement for this weight class in dimension three is a known result in the integral-geometry literature; we will add an explicit citation to the relevant theorem (under the precise regularity and support assumptions used in the reduction) in the revised abstract and introduction. With this citation the implication holds. revision: yes

  2. Referee: [Abstract] The abstract provides no statement of the precise assumptions on the potential (e.g., regularity class) or on the domain and the portion of the boundary where the local DN map is given. These assumptions are load-bearing for both the reduction and the claimed uniqueness.

    Authors: We agree that the abstract should record the standing assumptions for clarity. In the revised version we will state that the domain is a bounded domain in R^3 with C^infty boundary, the potential belongs to a suitable Sobolev or Hölder class (as used in the reduction), and the local DN map is given on an arbitrary open subset of the boundary that contains a fixed boundary point. These assumptions are already detailed in the body of the paper; they will now appear in the abstract as well. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reduction to independent integral-geometry question is self-contained

full rationale

The paper states that it proves the local DN map determines the potential by reducing the uniqueness question to injectivity of a weighted X-ray transform. This reduction is presented as a result established in the manuscript, with the injectivity treated as a separate question in integral geometry. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations are exhibited in the given text. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated. The reduction to X-ray injectivity is the load-bearing step whose justification is not visible.

pith-pipeline@v0.9.1-grok · 5581 in / 1137 out tokens · 17441 ms · 2026-06-27T15:23:16.901779+00:00 · methodology

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Cite this review

Pith. "Pith review of The partial data Calder\'on problem in dimension three." pith.science (2026). https://pith.science/paper/F2RHWPTK

@misc{pith2026260610247,
  author       = {Pith},
  title        = {Pith review of: The partial data Calder\'on problem in dimension three},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2RHWPTK}},
  note         = {Machine review of arXiv:2606.10247}
}
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read the original abstract

We consider an inverse boundary value problem for the time-independent Schr\"odinger equation in dimension three. We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.

discussion (0)

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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.

  1. The Calder\'on problem for near-Euclidean metrics

    math.AP 2026-06 unverdicted novelty 6.0

    Global uniqueness holds for the anisotropic Calderón problem on compact Riemannian manifolds with boundary when the metric is a small perturbation of the Euclidean metric.

Reference graph

Works this paper leans on

18 extracted references · cited by 1 Pith paper

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