An inverse obstacle problem for the magnetic Schr\"odinger equation

Hiroshi Takase; Mourad Choulli

REVIEW 3 major objections 4 minor 9 references

An inverse obstacle problem for the magnetic Schr\"odinger equation

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Two boundary measurements determine magnetic Schrödinger obstacle data

desk verdict A solid, genuinely new stability result for the magnetic Schrödinger inverse obstacle problem; the core Carleman estimate has one asserted-but-unshown step (the a=0 reduction) that should be expanded before acceptance. read the letter →

arxiv 2411.15792 v1 pith:SUUWR5DX submitted 2024-11-24 math.AP

Mourad Choulli , Hiroshi Takase This is my paper · ORCID

classification math.AP MSC 35R3035Q4158J35

keywords inverseobstacleproblemmagneticSchrödingerequationLipschitzstabilitylogarithmicCarlemaninequalityboundarymeasurementsestimatewithdegenerateweightsCauchydata

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 establishes quantitative stability for an inverse obstacle problem for the magnetic Schrödinger equation: the unknown function $f$ prescribed on the boundary $\Gamma$ of an obstacle is determined, with explicit error bounds, from two measurements on a surrounding surface $\partial\Omega$, namely the solution $u(f)$ and its normal derivative $\partial_{\nu_g}u(f)$. The main theorem is a Lipschitz-type estimate locally in time, $$\|f\|_{$H^{1}$(\Gamma\times(\varepsilon,T-\varepsilon))}\le C $e^{{c/\varepsilon}}$\left(\|u(f)\|_{$H^{1}$(\Sigma_0)}+\|\partial_{\nu_g}u(f)\|_{$L^{2}$(\Sigma_0)}\right),$$ valid for every admissible $f$ and every $\varepsilon\in(0,T/2)$. A global-in-time $L^2$ estimate follows by interpolation and gives a logarithmic modulus of continuity. This matters because it turns uniqueness into stability: small changes in the far-field measurements force small changes in the obstacle boundary data, with the control deteriorating exponentially as the observation window approaches the initial and final times. The authors state this is the first stability result for this inverse obstacle problem for the magnetic Schrödinger equation.

What carries the argument

The carrying object is a Carleman estimate for the dynamical magnetic Schrödinger operator $P=i\partial_t+L$ on the space-time cylinder $D\times(0,T)$, with the degenerate weight $\phi(x,t)=(e^{\gamma(\varphi(x)+2m)}-e^{4\gamma m})\ell(t)$ and $\ell(t)=[t(T-t)]^{-1}$; here $\varphi$ is a $C^4$ function whose positive level set has no critical points and satisfies a strong convexity condition, so the weight blows up at the initial and final times. The estimate controls weighted first-order norms of $u$ in the interior and weighted normal-derivative data on the inner boundary $\Sigma$ in terms of $\|Pu\|^2$ plus boundary terms on the outer boundary $\Sigma_0$ and tangential/time-derivative terms on $\Sigma$. In the proof the conjugated operator is split into self-adjoint and skew-adjoint parts and the cross term is expanded; the large parameters $\gamma$ and $s$ absorb lower-order terms, and the magnetic potential $a$ enters as a lower-order perturbation. The admissible-set bounds on $f$ are then used to absorb the time and tangential derivatives of $u$ on $\Gamma$ into the diagonal Carleman weight on the left-hand side. The same machinery, together with a Hardy-type inequality, yields the global logarithmic version.

What would settle it

Perform the missing absorption step for a nonzero magnetic potential: take a constant $a$ in Euclidean space with a simple annulus geometry and track every term involving $a$ through the splitting of the conjugated operator. If any magnetic term survives at order comparable to the leading Carleman weight, for instance a term of order $s^2\gamma^2\xi^2|z|^2$ or a boundary term not controlled by $\sigma(|\partial_{\nu}z|^2+\sigma^2|z|^2)$, then the stated inequality with constants independent of $a$ fails. A direct numerical evaluation of Proposition 2.1 with constant $a$ and a smooth exact solution would be a concrete test.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.1 and the Carleman mechanism behind it. For the admissible class $F$ of boundary functions $f$ satisfying a lower bound on their $L^2$ norm and an upper bound on time and tangential derivatives, the two boundary measurements $(u(f)|_{\Sigma_0},\partial_{\nu_g}u(f)|_{\Sigma_0})$ dominate $f$ on the inner boundary $\Gamma$ over any open time interval strictly inside $(0,T)$. The inequality has the explicit form displayed above; it is local in time because intervals near $t=0$ and $t=T$ are discarded. Two corollaries are drawn directly: if $f(x,t)=a(x)b(t)$ with $b$ known, then $a$ is Lipschitz stable on $\Gamma$ from the same measurements; and by a Hardy-type inequality the full time interval is recovered with a logarithmic modulus of continuity. The proof route is a Carleman estimate for $i\partial_t+L$, followed by an absorbing argument on the inner boundary that uses the admissible-class bounds to discard lower-order terms.

Load-bearing premise

The central estimate is proved only for the magnetic potential $a=0$; the paper asserts that the large parameters $\gamma$ and $s$ make the magnetic terms harmless, but it does not display the absorption calculation, and the final constants depend on $a$ and must remain uniform for the theorem to hold.

Editorial extensions

If this is right

For every $\varepsilon\in(0,T/2)$, the Cauchy data on the outer boundary determine $f$ on $\Gamma\times(\varepsilon,T-\varepsilon)$, with the error growing at most like $e^{c/\varepsilon}$ as $\varepsilon\to 0$.
When $f$ separates as $a(x)b(t)$ with $b$ known, the spatial factor $a$ is recovered Lipschitz-continuously on the whole obstacle boundary $\Gamma$.
Globally in time, $f$ is determined from the same two measurements with a logarithmic modulus of continuity, not merely a uniqueness statement.
The two-measurement inverse obstacle strategy, previously applied to elliptic, parabolic, and hyperbolic equations, now covers the magnetic Schrödinger equation as well.

Reading between the lines

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

Because the magnetic terms are treated as lower-order perturbations, the same proof route should extend to time-dependent magnetic potentials or to lower-order complex potentials, provided the same absorption can be made uniform.
The explicit $e^{c/\varepsilon}$ factor signals severe ill-posedness near $t=0$ and $t=T$; a natural next step is to check whether this rate is optimal by constructing explicit solutions, for example with a flat metric and a constant magnetic field, whose boundary traces saturate the estimate.
The Carleman inequality itself may be reusable as a black box for other inverse problems for the magnetic Schrödinger equation, such as recovering a magnetic or electric potential from boundary observations, since the weight is independent of $a$.

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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

Short version: this is a genuine first — stability for an inverse obstacle problem for the magnetic Schrödinger equation — and most of the machinery is assembled with care. But the key Carleman estimate contains one real gap: the magnetic potential is dismissed by fiat (“due to the large parameters γ and s...”), and the computation that follows is for a = 0. The reduction is probably fixable, but it is asserted, not demonstrated, and Theorem 1.1 sits on it.

What is actually new: the boundary-to-boundary stability estimates for P = i∂_t + L_a, with exponential-in-1/ε Lipschitz-type control locally in time and a logarithmic modulus globally via the interpolation argument. The Carleman inequality in Proposition 2.1, featuring time-derivative and tangential-derivative terms on the inner boundary, is new for the dynamic Schrödinger operator, and its derivation from the nine I_k terms is mostly clean algebra. The step from the Carleman estimate to Theorem 1.1 is the right argument: negative weight on the obstacle boundary, admissible-class control of ∂_t f and ∇_τ f by (β/α)‖f‖, and the ℓ(t) ≥ Tε/2 cut. The semigroup well-posedness appendix is standard and fine.

Where I push back: the a = 0 reduction is the one spot I want written out. The magnetic perturbation contributes bulk terms of size s‖a‖|∇z|² and s³‖a‖|∇φ||∇z||z|, plus lower-order terms, and — more delicately — boundary integrals on Σ with ∂_ν z weighted by a. Large γ and s plausibly absorb all of this, since σ = sγξ can dominate ‖a‖ uniformly, but the positivity structure on the inner boundary is the delicate part of the estimate, and that is exactly where the paper says “enough” instead of showing it. Because the constants in Theorem 1.1 carry the W^{3,∞} norm of a, the absorption has to be uniform. I would not call this fatal; I would call it a necessary expansion.

Minor: Corollary 1.3 imports a Hardy-type inequality from the authors’ own prior works [5] and [2]. Those are independently published results, not the target theorem, so the self-citation is not a problem; just note the global result leans on it.

Audience: people doing Carleman-based stability for Schrödinger equations or inverse obstacle problems. It deserves a serious referee, with the request to expand the magnetic absorption. I agree with the conditional verdict, leaning positive.

Referee Report

3 major / 4 minor

Summary. The paper establishes stability inequalities for an inverse obstacle problem for the magnetic Schrödinger equation on an exterior-type domain. The unknown quantity is the Dirichlet trace f on the obstacle boundary Γ×(0,T), and the data are the trace and normal derivative of the solution on an outer boundary Σ0. The main result, Theorem 1.1, gives a Lipschitz stability estimate on the time interval (ε,T−ε) with constant Ce^{c/ε}, and Corollary 1.3 converts this into a global logarithmic stability estimate via an external Hardy-type inequality. The proof relies on a Carleman estimate for the magnetic Schrödinger operator with a degenerate weight, Proposition 2.1, whose proof occupies Section 3, and on a well-posedness result for the forward IBVP in Appendix A.

Significance. If the proof is completed, this appears to be the first quantitative stability result for an inverse obstacle problem for the magnetic Schrödinger equation, and the Carleman estimate with boundary terms involving only time and tangential derivatives on the inner boundary is potentially reusable. The proof is largely self-contained, with the main external ingredient being a Hardy-type inequality from the authors' prior work [2,5], which is independently published. The paper also gives a clean semigroup argument for well-posedness. However, the central Carleman estimate is currently not fully justified for the magnetic operator, so the main theorem is not yet supported as written.

major comments (3)

[Section 3, proof of Proposition 2.1] The Carleman estimate is proved only for a=0. The sentence 'Due to the large parameters γ and s, it is enough to establish the expected inequality when a = 0' is asserted but not demonstrated. For the magnetic operator P = i∂t + Δ_g + 2ia·∇ + V_a, conjugation with e^{sϕ} produces additional terms of the form 2ia·∇z − 2s(a·∇ϕ)z + V_a z, where V_a contains first derivatives of a and the quadratic term in a. In the bilinear estimates these give bulk terms of size s|a||∇z|², s³|a||∇ϕ||∇z||z|, and s|V_a||z|², and, more delicately, boundary integrals on Σ involving ∂ν z with coefficients depending on a. Absorbing these terms requires σ = sγξ to dominate |a| uniformly and requires the signed boundary term on Σ to control the new normal-derivative contributions. Because the constants in Theorem 1.1 depend on a through ζ, this absorption must hold uniformly in the admissible class; otherwise the main stability estimate is unsupported. This gap is localized and fixable, but the perturbation calculation must be written out before the proof can be accepted.
[Section 2, definition of the weight functions] The parameter m appearing in ϕ(x,t) = (e^{γ(φ(x)+2m)} − e^{4γm})ℓ(t) and ξ(x,t) = e^{γ(φ(x)+2m)}ℓ(t) is never defined. In the proof of Theorem 1.1, the estimate e^{2sϕ}ω³ ≤ C e^{−csℓ} on Σ0 is essential for bounding the boundary-data terms by the data norms. This estimate is true only if m is chosen so that φ < 2m on ∂Ω, making ϕ negative on Σ0. Without an explicit definition or condition on m, the stated upper bound does not follow and the final stability inequality is not justified. Please add the required condition on m and include m in the parameter set ζ if needed.
[Section 3, integration by parts in time] In the derivation of I3, the identity I3 = (1/2)∫_Q sϕ′′|z|² is obtained by integrating by parts in t and dropping the boundary terms at t=0 and t=T. Since ϕ ∼ ℓ(t) and ℓ is singular at the endpoints, these boundary terms vanish only if z decays sufficiently fast as t→0,T, for instance because e^{sϕ}→0 at the endpoints. This requires the same condition on m as in the previous comment and should be stated explicitly. For the function class u ∈ L²((0,T);H²(D)) ∩ H¹((0,T);H¹(D)) with no vanishing condition at t=0,T, the integration by parts is not automatically justified.

minor comments (4)

[Introduction] There is a typo in 'non-homogenuous' on page 2; it should be 'non-homogeneous'.
[Corollary 1.3] The Hardy-type inequality from [5, Corollary 3.1] is cited rather than stated. Since the corollary is load-bearing for the global logarithmic stability, it would improve the paper to state the precise inequality and the hypotheses needed for its application to f.
[Section 3, final substitution] When passing from the z-variable to u = e^{−sϕ}z at the end of the proof of Proposition 2.1, the cross terms generated by z′ = e^{sϕ}(u′ + sϕ′u) and ∂ν z = e^{sϕ}(∂ν u + s∂νϕ u) are not displayed. These are standard to absorb using the large parameters, but a short justification would make the proof fully detailed.
[Appendix A] The uniqueness claim for the IBVP with f=0 and u0=0 is asserted after the semigroup construction; it would be clearer to state explicitly that it follows from the contraction semigroup property, although this is standard.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stability estimate is derived from an explicit Carleman inequality; self-citations are auxiliary and not load-bearing.

full rationale

The derivation chain is self-contained with respect to its stated inputs. Theorem 1.1 is proved directly from the Carleman estimate in Proposition 2.1, and that estimate is derived by explicit integration by parts for the principal operator, with the magnetic potential handled only by the asserted large-parameter reduction in Section 3. The self-citations [2,5] are used only for an auxiliary Hardy-type inequality in Corollary 1.3; that inequality is an external, published result and does not smuggle in the target stability estimate. No parameter is fitted to the measurements and then renamed a prediction: the constants C and c depend only on the problem data zeta, not on f or on the boundary measurements. The only notable weakness is the sentence 'Due to the large parameters gamma and s, it is enough to establish the expected inequality when a = 0,' which is an unverified perturbation argument for the magnetic potential. That is a correctness or completeness gap, not circularity: the a = 0 estimate is not equivalent by definition to the magnetic estimate, and the paper does not define the magnetic Carleman estimate in terms of its own conclusion. Consequently, the central claim is not circular.

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

No free parameters are fitted to data; the theorem's constants (C, c, γ*, s*) are generic constants from Carleman estimates, not fitted values. Assumptions are the regularity of the metric and magnetic potential, the existence of the weight function φ, and the admissibility conditions on f. No invented entities are introduced.

assumptions (5)

domain assumption The metric g = (g_kℓ) ∈ W^{3,∞} is uniformly elliptic, and the magnetic potential a ∈ W^{3,∞}; these regularities underpin the Carleman estimate and IBVP well-posedness.
Invoked in Section 1 and Appendix A to define the operator L and obtain a contraction semigroup.
domain assumption There exists φ ∈ C^4(R^n) satisfying (1.1): |∇_g φ|_g > 0 and ∇_g^2 φ positive definite on {φ>0}, with {φ<0} bounded.
This weight function is required for the Carleman estimate; its existence is cited to [3, Example 1.1].
ad hoc to paper For large parameters γ and s the magnetic potential terms in the Carleman estimate can be absorbed, so it suffices to prove Proposition 2.1 for a=0.
Asserted at the start of Section 3 without a written perturbation argument.
standard math Hardy-type inequality from [5, Corollary 3.1] (also [2, Corollary 1.16]) bounds the L2 norms of f near t=0,T by ε^r ||f||_{H^1((0,T);L2(Γ))}.
Used in the proof of Corollary 1.3 to obtain global logarithmic stability.
standard math Semigroup theory results: Lumer-Phillips theorem and [9, Proposition 3.7.2] on m-dissipative operators.
Used in Appendix A to establish well-posedness of the IBVP.

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Cite this review

Pith. "Pith review of An inverse obstacle problem for the magnetic Schr\"odinger equation." pith.science (2026). https://pith.science/paper/SUUWR5DX

@misc{pith2026241115792,
  author       = {Pith},
  title        = {Pith review of: An inverse obstacle problem for the magnetic Schr\"odinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SUUWR5DX}},
  note         = {Machine review of arXiv:2411.15792}
}

read the original abstract

We establish stability inequalities of an inverse obstacle problem for the magnetic Schr\"odinger equation. We mainly study the problem of reconstructing an unknown function defined on the obstacle boundary from two measurements performed on the boundary of a domain surrounding the obstacle. We show for the inverse problem a Lipschitzian stability locally in time and a logarithmic stability globally in time.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

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Choulli and M

M. Choulli and M. Yamamoto. Global stability result for p arabolic cauchy problems. Journal of Inverse and Ill-Posed Problems , 29:895–915, 12 2021

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M. Tucsnak and G. W eiss. Observation and Control for Operator Semigroups . Birkh¨ auser Verlag, 2009. Universit´e de Lorraine, 34 cours L ´eopold, 54052 Nancy cedex, France Email address : mourad.choulli@univ-lorraine.fr Institute of Mathematics for Industry, Kyushu Universit...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

Choulli and M

M. Choulli and M. Yamamoto. Global stability result for p arabolic cauchy problems. Journal of Inverse and Ill-Posed Problems , 29:895–915, 12 2021

work page 2021

[2] [2]

M. Choulli. An Introduction to the Uniqueness of Continuation of Second Order Partial Dif- ferential Equations . to appear

work page

[3] [1]

Baudouin and J.-P

L. Baudouin and J.-P. Puel. Uniqueness and stability in a n inverse problem for the schr¨ odinger equation. Inverse Problems , 18:1537–1554, 2002

work page 2002

[4] [3]

An inverse hyperbolic obstacle problem

M. Choulli and H. Takase. An inverse hyperbolic obstacle problem. arXiv:2407.05662, 2024

work page Pith review arXiv 2024

[5] [4]

Choulli and H

M. Choulli and H. Takase. Lipschitz stability for an elli ptic inverse problem with two mea- surements. arXiv:2404.13901, 2024

arXiv 2024

[6] [6]

Huang, Y

X. Huang, Y. Kian, E. Soccorsi, and M. Yamamoto. Carleman estimate for the schr¨ odinger equation and application to magnetic inverse problems. Journal of Mathematical Analysis and Applications, 474:116–142, 2019

work page 2019

[7] [7]

Mercado and R

A. Mercado and R. Morales. Exact controllability for a sc hr¨ odinger equation with dynamic boundary conditions. SIAM Journal on Control and Optimization , 61:3501–3525, 2023

work page 2023

[8] [8]

Mercado, A

A. Mercado, A. Osses, and L. Rosier. Inverse problems for the schr¨ odinger equation via carle- man inequalities with degenerate weights. Inverse Problems , 24(1), 015017:18pp, 2008

work page 2008