Inverse problems for a nonlinear dynamical Schr\"odinger operator with magnetic potential

Boya Liu; Mandeep Kumar; Manmohan Vashisth

Inverse problems for a nonlinear dynamical Schr\"odinger operator with magnetic potential

T0 review · 0 major / 3 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read The Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials for a nonlinear dynamical Schrödinger operator under analyticity assumptions.

desk verdict The paper shows uniqueness of time-dependent magnetic and electric potentials from the DN map for a nonlinear dynamical Schrödinger operator under analyticity, plus forward well-posedness at optimal Sobolev regularity, for both full and partial data. read the letter →

arxiv 2606.18212 v1 pith:HH65CVVF submitted 2026-06-16 math.AP

Mandeep Kumar , Boya Liu , Manmohan Vashisth This is my paper · ORCID

classification math.AP

keywords inverseproblemsDirichlet-to-NeumannmapnonlinearSchrödingerequationmagneticpotentialanalyticcontinuationpartialdatauniqueness

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 shows that the Dirichlet-to-Neumann map associated with a nonlinear Schrödinger equation containing time-dependent magnetic and electric potentials can recover those potentials uniquely. The uniqueness result applies both when full boundary data is available and when only partial data is measured on small boundary subsets, assuming the potentials are known near the boundary. Analyticity of the potentials is used to extend local information globally via analytic continuation. The work also establishes well-posedness of the forward problem, achieving optimal Sobolev regularity for the solutions.

What carries the argument

The Dirichlet-to-Neumann map for the nonlinear dynamical Schrödinger operator with magnetic and electric potentials, which maps boundary inputs to boundary outputs to recover the interior potentials.

What would settle it

Finding two distinct sets of analytic potentials that produce identical Dirichlet-to-Neumann maps for the same nonlinear equation would falsify the uniqueness result.

Watch

Extended reading notes

Core claim

Under suitable analyticity assumptions on the potentials, the Dirichlet-to-Neumann map determines the time-dependent magnetic and electric potentials uniquely from both full and partial boundary measurements for the nonlinear dynamical Schrödinger operator.

Load-bearing premise

The potentials satisfy suitable analyticity assumptions that enable uniqueness via analytic continuation.

Editorial extensions

If this is right

The time-dependent magnetic and electric potentials are uniquely recovered from full Dirichlet-to-Neumann data.
Uniqueness extends to partial data when potentials are known near the boundary and Neumann measurements are taken on arbitrarily small open boundary subsets.
The forward problem admits solutions with optimal Sobolev regularity.
The results apply to both complete and incomplete boundary observations.

Reading between the lines

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

Relaxing analyticity might require Carleman estimates or other microlocal techniques to retain uniqueness.
The partial-data setup could apply to limited-access settings such as medical or geophysical imaging.
The magnetic potential recovery suggests extensions to vector-potential problems in quantum mechanics.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that analyticity on the coefficients lets the Dirichlet-to-Neumann map recover the time-dependent magnetic and electric potentials uniquely, and this holds for full data as well as partial data measured on arbitrarily small boundary sets when the potentials are already known near the boundary. They also prove the forward nonlinear problem is well-posed with optimal Sobolev regularity.

What stands out as new is the combination of the nonlinear dynamical setting with a magnetic potential and explicit time dependence, plus the partial-data result under those analyticity assumptions. The partial-data version with small measurement regions is a concrete improvement over earlier full-data statements in this line of work.

The analyticity assumption is the clearest limitation. It is a standard tool for continuation arguments in inverse problems, but it restricts the result to a narrow class of coefficients and leaves open whether the same uniqueness holds without it. The forward well-posedness claim is load-bearing for the inverse part; if the regularity estimates check out, that part is useful on its own. Nothing in the abstract or claim description points to circular reasoning or mismatched assumptions.

This is written for people already working on inverse problems for nonlinear Schrödinger operators or magnetic potentials. A reader focused on partial-data uniqueness or time-dependent coefficients will find the statements directly relevant. A broader PDE audience will probably see it as too specialized.

The paper deserves a serious referee. The claims are specific, the technical pillars are named clearly, and the partial-data extension is worth checking in detail.

Referee Report

0 major / 3 minor

Summary. The manuscript studies inverse problems for a nonlinear dynamical Schrödinger operator with magnetic and electric potentials. Under suitable analyticity assumptions on the potentials, it claims to prove that the Dirichlet-to-Neumann map uniquely determines the time-dependent magnetic and electric potentials, both from full boundary data and from partial data (with potentials known near the boundary and Neumann measurements on arbitrarily small boundary subsets). It also claims to establish well-posedness of the forward nonlinear problem, obtaining optimal Sobolev regularity for the solutions.

Significance. If the uniqueness theorems hold, the work would extend existing results on inverse problems for Schrödinger operators to the nonlinear dynamical setting with magnetic potentials and would strengthen partial-data results via analytic continuation. The forward well-posedness statement with optimal regularity would provide a useful technical foundation for such inverse problems. The analyticity assumption is a standard device in this literature and is used explicitly for continuation arguments.

minor comments (3)

The abstract states that the forward problem admits solutions with 'optimal Sobolev regularity,' but the precise Sobolev indices (e.g., H^s for which s) are not indicated; this should be stated explicitly in the introduction or the well-posedness theorem statement.
Notation for the magnetic potential A(t,x) and electric potential q(t,x) should be introduced once and used consistently; the current abstract alternates between 'magnetic and electric potentials' and 'time-dependent magnetic and electric potentials' without a clear first definition.
The partial-data result assumes the potentials are known near the boundary; this assumption should be stated as a numbered hypothesis in the main theorem rather than only in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so there are no points requiring point-by-point rebuttal. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in uniqueness proof

full rationale

The paper establishes uniqueness of time-dependent potentials from the DN map under analyticity assumptions, plus forward well-posedness at optimal Sobolev regularity. Analytic continuation is a standard external device; the forward regularity claim is independent technical content. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation chain remains self-contained against external benchmarks and does not collapse to its inputs by construction.

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

The uniqueness statements rest on the analyticity assumption for the potentials and on the well-posedness of the forward problem; both are stated without derivation in the abstract.

assumptions (1)

domain assumption The magnetic and electric potentials are analytic in space and time.
Analyticity is invoked to recover the potentials from boundary data via analytic continuation.

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Cite this review

Pith. "Pith review of Inverse problems for a nonlinear dynamical Schr\"odinger operator with magnetic potential." pith.science (2026). https://pith.science/paper/HH65CVVF

@misc{pith2026260618212,
  author       = {Pith},
  title        = {Pith review of: Inverse problems for a nonlinear dynamical Schr\"odinger operator with magnetic potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HH65CVVF}},
  note         = {Machine review of arXiv:2606.18212}
}

read the original abstract

We study two inverse problems for a nonlinear dynamical Schr\"odinger operator with magnetic and electric potentials. Under suitable analyticity assumptions, we show that the Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials. We establish the uniqueness of these potentials from both full data and partial data. In particular, for the partial data problem, the desired uniqueness is established by assuming that the potentials are known near the boundary, and the Neumann data is measured on arbitrarily small open subsets of the boundary. In addition, we establish the well-posedness of the forward problem, where we obtain the optimal Sobolev regularity for solutions.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

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