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H\"older stability of an inverse spectral problem for the magnetic Schr\"odinger operator on a simple manifold

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Pith's one-line read Magnetic Schrödinger potentials recover Hölder stably from boundary spectral data on simple manifolds.

desk verdict First Hölder stability for the solenoidal magnetic potential from boundary spectral data; structurally sound, with a real proof gap in Lemma 5.5(b) that should be fixable. read the letter →

arxiv 2507.13619 v1 pith:S75IN3B2 submitted 2025-07-18 math.AP

Boya Liu , Hadrian Quan , Teemu Saksala , Lili Yan This is my paper · ORCID
classification math.AP MSC 35R3058J5035P10
keywords inversespectralproblemmagneticSchrödingeroperatorHölderstabilityboundarydataDirichlet-to-NeumannmapgeodesicraytransformsimplemanifoldGel'fand
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

The paper proves that on a simple Riemannian manifold, the electric potential and the solenoidal (gauge-invariant) part of a magnetic potential can be recovered stably from boundary spectral data—the Dirichlet eigenvalues and the Neumann traces of the eigenfunctions. The stability is Hölder-type, meaning the $L^2$ error of the recovered potentials is controlled by a power of the $\ell^1$-weighted difference of the spectral data. This is the first stability result for the magnetic potential from spectral data alone, without assuming it is known near the boundary or globally. A key intermediate step shows the boundary spectral data stably determine the hyperbolic Dirichlet-to-Neumann map, and the proof reduces the main recovery to the stability of the geodesic ray transform.

What carries the argument

The proof has two main components, presented in reverse order. First, it constructs geometric optics (GO) solutions to the hyperbolic equation $\partial_t^2 u - \Delta_{g,A}u + qu = 0$ of the form $u(t,x)=e^{i(\psi(x)-t)/h}\alpha(t,x)\beta_A(t,x)+r(t,x)$, where the phase $\psi$ solves an eikonal equation and the amplitudes $\alpha,\beta_A$ solve transport equations; this reduces stability of lower-order coefficients to the stable inversion of the geodesic ray transform of one-forms and functions. Second, it proves that the boundary spectral data stably determine the hyperbolic Dirichlet-to-Neumann map via an elliptic Dirichlet-to-Neumann map $\Pi_{A,q}(z)$ defined in lower-regularity Sobolev spaces, leading to a bound on $\|\Lambda^\sharp_{A_1,q_1}-\Lambda^\sharp_{A_2,q_2}\|$ by a power of $\delta$ through a Taylor expansion and optimization over $z<0$.

What would settle it

A concrete way to test the result would be numerical or analytic computation of the stability exponent for a specific simple manifold (e.g., Euclidean ball) where the geodesic ray transform is well understood: if the observed error between recovered and true potentials decays slower than any power of the spectral data difference, it would contradict the Hölder claim. Alternatively, one can check whether the constant in Lemma 5.5(b) indeed remains uniform for $z\in(-\infty,-1]$ for all $q$ with $\|q\|_{H^1}\le N$; if it grows like $|z|^\beta$, the exponent $\theta$ in Theorem 1.1 would change.

Watch

Extended reading notes

Core claim

The central result (Theorem 1.1) asserts that if $(M,g)$ is a simple Riemannian manifold of dimension $n \ge 2$, and if two magnetic potentials $A_1,A_2$ and electric potentials $q_1,q_2$ agree on the boundary and lie in bounded admissible classes, then there exist constants $C>0$ and $\theta,\sigma_2\in(0,1)$ (depending only on the manifold, $n$, a regularity parameter $m$, and $N$) such that $\|A_1^s - A_2^s\|_{L^2} + \|q_1 - q_2\|_{L^2} \le C(\delta+\delta^\theta)^{\sigma_2}$, where $\delta$ is the weighted $\ell^1$ distance between the eigenvalue sequences and Neumann trace sequences. This directly establishes a quantitative (Hölder-stable) version of the Gel'fand inverse spectral problem for the magnetic Schrödinger operator, up to the unavoidable gauge invariance that only the solenoidal part of $A$ is recoverable.

Load-bearing premise

The proof of Lemma 5.5(b) uses an unstated condition $z < -2\|q\|_{L^\infty}$ to get a uniform bound on the elliptic solution in $L^2$, but the lemma is later applied for all $z \le -1$ without verifying that this condition holds uniformly over the admissible class of $q$.

Editorial extensions

If this is right

  • If correct, the result gives the first quantitative stability estimate for recovering a magnetic potential's solenoidal part from spectral data, filling a gap in the literature where only uniqueness was known.
  • It provides a unified proof that reduces the stable recovery of both electric and magnetic potentials to the stable invertibility of geodesic ray transforms, suggesting the method may extend to other geometries where such transforms are stably invertible.
  • The intermediate step, connecting boundary spectral data to the hyperbolic Dirichlet-to-Neumann map with Hölder stability, applies to all smooth compact manifolds, not just simple ones, and could be reused in other inverse spectral problems.
  • The lower-regularity setup for the elliptic and hyperbolic Dirichlet-to-Neumann maps may allow stability results under weaker a priori regularity assumptions on the potentials.

Reading between the lines

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

  • The Hölder exponents $\theta$ and $\sigma_2$ come from optimizing powers of $|z|$ in the elliptic-to-hyperbolic reduction, so they are likely not optimal; a sharper analysis could yield explicit exponents in terms of $n$ and $s$.
  • The stability estimate for $A^s$ in the $L^2$ norm, combined with the gauge invariance, suggests that any practical reconstruction algorithm would recover the magnetic field $dA$ (or the solenoidal part) rather than the full one-form, and the Hölder modulus quantifies the resolution limit of such algorithms.
  • The method's reliance on geodesic ray transform stability means the result should extend to non-simple manifolds (e.g., with trapped sets) as soon as analogous stability estimates for the ray transform become available, as the authors hint.
  • A concrete testable extension would be to check whether the exponent $\sigma_2$ can be improved to the same value as in the electric-only case (e.g., $1/12$ vs. $1/16$) by optimizing the choices of amplitudes in Section 4.
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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 claims a Hölder stability estimate for the magnetic Schrödinger operator on a simple Riemannian manifold: from boundary spectral data (eigenvalues and Neumann traces of eigenfunctions) one can recover the solenoidal part of the magnetic potential and the electric potential with a modulus of continuity of the form (δ+δ^θ)^σ2. The proof proceeds in two stages. First, using geometric optics solutions for the hyperbolic wave-type equation and stability of the geodesic ray transform of functions and one-forms, the authors prove in Theorem 1.2 a Hölder estimate from the hyperbolic Dirichlet-to-Neumann map. Second, in Section 5 the authors show that the boundary spectral data control the hyperbolic Dirichlet-to-Neumann map through a family of elliptic Dirichlet-to-Neumann maps, using a Taylor expansion in the spectral parameter and weighted ℓ1 estimates on the spectral differences.

Significance. If the proof is correct, the paper would be the first Hölder stability result for the magnetic potential from boundary spectral data, and the technical reduction of spectral data to hyperbolic data through elliptic Dirichlet-to-Neumann maps in negative-order Sobolev spaces appears to be a genuine contribution. The paper is carefully structured, and the main architecture is coherent: Lemma 5.2 connects spectral data to elliptic DN maps, Lemma 5.4 connects elliptic DN maps to the hyperbolic DN map, and Lemmas 4.1–4.4 reduce the hyperbolic inverse problem to the stable inversion of the geodesic ray transform via the known stability results of Stefanov–Uhlmann. No machine-checked proofs or code are supplied; the argument is analytic and relies on external theorems as stated. The central claimed novelty is significant for the inverse spectral problem literature, but the proof as written contains a load-bearing regularity gap that must be addressed before the stated theorems are established.

major comments (3)
  1. [Section 5.2, Lemma 5.5(b) and Proposition 5.6] Lemma 5.5(b) is stated for every z<0, but its proof introduces the unstated condition z<−2‖q‖_{L∞} before the estimate (5.26). This condition is used to control the solution w2 of (5.24) and is not assumed anywhere in the statement of Theorem 1.1. The lemma feeds directly into Proposition 5.6, where the estimates (5.29), (5.30), and (5.35) are applied for arbitrary z<0, and later into (5.43), where the minimum over x=|z|≥1 is taken. Since q is only assumed to lie in Q(N), which controls H^1(M) and not L∞(M) when n≥2, there is no uniform bound on ‖q‖_{L∞} within the admissible class. Smooth functions on the unit disk with prescribed H^1 norm but arbitrarily large L∞ norm show that the condition z<−2‖q‖_{L∞} cannot be verified uniformly for z near 0 or at z=−1. Thus the bound (5.22), the exponent s/2+1/4 in Proposition 5.6, and the Hölder exponent θ in Theorem 1.1 are not justified by the stated hypotheses. This is a load-bearing gap, not a local presentation issue.
  2. [Section 1.1, Proposition 3.2, and Theorem 1.1] There is a systematic mismatch between the regularity assumed in the main theorem and the regularity required by the proofs. Theorem 1.1 assumes q1,q2∈Q(N) with Q(N)={q: ‖q‖_{H^1(M)}≤N}, while Proposition 3.2 requires q∈L∞(Q) (and A∈W^{1,∞}(Q)) for the existence and estimates of geometric optics solutions. For n≥2, H^1(M) is not contained in L∞(M), so the GO solutions used in Section 4 and in Proposition 5.8 are not available under the hypotheses of Theorem 1.1. The same issue affects Lemma 5.5, whose proof uses ‖q‖_{L∞} in (5.26), and Proposition 5.6, whose constants are claimed to depend only on j, N, and M. A repair would require either adding a uniform L∞ or higher-order Sobolev bound to the admissible class Q(N), or proving an approximation/density argument that maintains all constants uniformly; neither is present in the manuscript.
  3. [Section 5.2, Taylor expansion leading to (5.43)] Even if one attempted to fix Lemma 5.5(b) by choosing |z| large, the Taylor formula for P^{(j)}(0) in Proposition 5.7 integrates P^{(n+1)}(τ) over τ∈(z,0), and the bound (5.42) is used uniformly for τ in that interval. The hidden condition z<−2‖q‖_{L∞} would then need to hold for all τ in (z,0), including arbitrarily small negative τ, which cannot be ensured under the stated admissibility assumptions. Consequently, the estimate (5.36) and the subsequent minimization leading to (5.44) do not follow from the arguments as written.
minor comments (3)
  1. [Section 5.2, equation (5.21)] In equation (5.21) and the sentence before it, the second operator appears as Λ^♯_{A1,q2}; this should presumably be Λ^♯_{A2,q2}, since the difference of the two hyperbolic DN maps is being written.
  2. [Section 4.1, Lemma 4.2] In the statement of Lemma 4.2, the norms on the right-hand side of (4.23) are written as H^2(S^+_yM_1), but the estimate is integrated over y∈∂M1 in the proof. The norms should be H^2(∂+SM1) or the statement should clarify the integrated norm.
  3. [Section 3, Proposition 3.2] The assumption A∈W^{1,∞}(Q) and q∈L∞(Q) should be stated on M rather than Q, since the coefficients are time-independent; this is a notational/clarity issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof reduces boundary spectral data to stable geodesic ray-transform inversion via external results, with no target quantity fitted into the hypotheses.

full rationale

The paper's derivation chain is not circular. Theorem 1.2 is proved first from the hyperbolic Dirichlet-to-Neumann map by constructing geometric optics solutions and reducing the recovery to stable inversion of geodesic ray transforms, where the ray-transform stability estimates are taken from the external works of Stefanov-Uhlmann [64] and Sharafutdinov [61]. Theorem 1.1 is then obtained in Section 5 by a separate, independent reduction from boundary spectral data to the hyperbolic Dirichlet-to-Neumann map via the elliptic Dirichlet-to-Neumann family; this follows the external framework of [1, 18, 23] and extends it to the magnetic case. The boundary spectral data enter only as the input distance δ in Lemma 5.2 and Proposition 5.7, and the inequality (5.44) is an upper bound on the hyperbolic map in terms of δ, not a definitional identity: (5.21) is a concrete series formula for the map difference, and Proposition 5.7 estimates the resulting terms by δ after applying Weyl asymptotics and trace bounds. The final Hölder exponents come from minimizing x^{s/2+1/4}+δ x^{n+1}, an elementary computation independent of the unknowns being recovered. No parameter is fitted to a subset of the data and then renamed a prediction; the geometric optics amplitudes are chosen from a priori phase and transport equations, not from the potentials being recovered. The only self-citations, [49] and [50], appear in the literature survey as pointers to accounts of uniqueness for time-dependent hyperbolic equations and play no role in any proof, so they are not load-bearing. The reader-flagged issue in Lemma 5.5(b), namely the hidden condition z < -2||q||_{L∞}, concerns whether a constant is uniform over the admissible class; that is a correctness or robustness gap in the proof, not a circularity, because the lemma does not assume the conclusion of Theorem 1.1 or define any target quantity in terms of the spectral data. Thus there is no circular step to report.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data; all constants are existential. The axioms are the geometric simplicity, the black-box ray transform stability, the boundary agreement of coefficients, and standard PDE/spectral theory.

assumptions (6)
  • domain assumption The manifold (M,g) is simple, i.e., strictly convex boundary and unique distance-minimizing geodesics between any two points.
    Assumed in Theorems 1.1 and 1.2; guarantees a simple extension, GO solutions in geodesic polar coordinates, and stable inversion of ray transforms.
  • domain assumption Stable inversion estimates for the normal operators of geodesic ray transforms hold on simple manifolds, quoted from Stefanov-Uhlmann [64, Theorems 3 and 4].
    Used as the external benchmark to convert ray-transform estimates into L2 stability for A^s and q.
  • domain assumption A1 = A2 and q1 = q2 on the boundary of M.
    Stated in Theorem 1.1; needed to extend the differences by zero to a larger simple manifold M1 and to use global ray transforms.
  • domain assumption T > diam(M1) for a simple extension M1 of M.
    Needed for the support of the cutoff phi in the GO amplitude alpha(t,r,theta) = rho^{-1/4} phi(t-r) Psi; see Section 3.
  • standard math Standard elliptic regularity, trace theorems, Weyl asymptotics, and resolvent expansions from [26, 39, 61] are valid.
    Invoked throughout Sections 3-5 without proof.
  • domain assumption The shifted operator trick (Remark 5.1) allows assuming q non-negative and the operator positive definite without loss of generality.
    Used in Lemma 5.2 and Proposition 5.7 to justify spectral expansions; the shift does not change the difference of spectral data.

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Pith. "Pith review of H\"older stability of an inverse spectral problem for the magnetic Schr\"odinger operator on a simple manifold." pith.science (2026). https://pith.science/paper/S75IN3B2

@misc{pith2026250713619,
  author       = {Pith},
  title        = {Pith review of: H\"older stability of an inverse spectral problem for the magnetic Schr\"odinger operator on a simple manifold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S75IN3B2}},
  note         = {Machine review of arXiv:2507.13619}
}
read the original abstract

We show that on a simple Riemannian manifold, the electric potential and the solenoidal part of the magnetic potential appearing in the magnetic Schr\"odinger operator can be recovered H\"older stably from the boundary spectral data. This data contains the eigenvalues and the Neumann traces of the corresponding sequence of Dirichlet eigenfunctions of the operator. Our proof contains two parts, which we present in the reverse order. (1) We show that the boundary spectral data can be stably obtained from the Dirichlet-to-Neumann map associated with the respective initial boundary value problem for a hyperbolic equation, whose leading order terms are a priori known. (2) We construct geometric optics solutions to the hyperbolic equation, which reduce the stable recovery of the lower order terms to the stable inversion of the geodesic ray transform of one-forms and functions.

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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. Full citation record

  1. Inverse spectral problems with sparse data and applications to passive imaging on manifolds

    math.AP 2025-07 accept novelty 7.0 of 10

    Two theorems show that a potential on a closed Riemannian manifold is uniquely recovered from sparse eigenvalue data and eigenfunction restrictions to an open set, with applications to single-measurement passive imaging.

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