{"id":"95ca2436-04dc-435b-ba72-7455f0b7a744","arxiv_id":"2507.13619","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On simple Riemannian manifolds, the electric potential and the solenoidal magnetic potential of the magnetic Schrödinger operator are recovered Hölder stably from eigenvalues and Neumann traces of Dirichlet eigenfunctions.","lead":"This paper proves that on simple Riemannian manifolds, the electric potential and the solenoidal part of the magnetic potential in a magnetic Schrödinger operator can be recovered with Hölder stability from boundary spectral data. It is the first stability result for an unknown magnetic potential in this boundary inverse spectral problem, opening the way to quantitative reconstruction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniform L^∞ control of q on the admissible class is missing; Lemma 5.5(b) invokes the hidden condition z < -2‖q‖∞, and Q(N) only bounds q in H^1.","rationale":"The reader correctly identified the unstated z < -2‖q‖∞ condition in Lemma 5.5(b), and this is a genuine defect in the written proof. However, the more structural problem is that Q(N) is only an H^1 bound on q, while the proof repeatedly needs a uniform L∞ control on q: Proposition 3.2 assumes q ∈ L∞, and Lemma 5.5(b) uses ‖q‖∞. On compact manifolds of dimension n ≥ 2, H^1 does not embed into L∞, so the constants in Propositions 5.6 and 5.7 are not shown to depend only on N. The main claim may still be true after either correcting Lemma 5.5(b) or strengthening the admissible class, so the appropriate outcome remains conditional acceptance pending a rigorous repair.","tokens_in":46545,"tokens_out":24442,"duration_ms":310969,"concrete_test":"Take M the unit disk and q_ε smooth truncations of log log(1/|x|) with ‖q_ε‖_{H^1} ≤ 1 but ‖q_ε‖_{L∞} → ∞. Solve (5.1) with A = 0, z = -1, and a fixed boundary value h, and compute the best constant in (5.22). If the constant grows with ε, Lemma 5.5(b) cannot be uniform over Q(N), so the constant in Theorem 1.1 cannot depend only on N; if the constant stays bounded, the z-condition in the proof is a fixable defect and the remaining issue is only to state the corrected proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weak point is Lemma 5.5(b), which Section 5 uses to control the elliptic Dirichlet-to-Neumann map at every z < 0. The lemma claims (5.22) for all z < 0, but its proof requires the unstated condition z < -2‖q‖_{L∞}. Proposition 5.6 applies the lemma at arbitrary z < 0, and (5.43) then minimizes over x = |z| ≥ 1; if q is large, z = -1 need not satisfy the hidden condition. If the constant in (5.22) depends on z, the exponents s/2 + 1/4 in Proposition 5.6 and the Hölder exponent θ in Theorem 1.1 are not justified. This is compounded by the admissible class: Q(N) only bounds q in H^1(M), and on a compact manifold of dimension n ≥ 2, H^1 does not control L∞. There are smooth q on the unit disk with ‖q‖_{H^1} ≤ N but ‖q‖_{L∞} arbitrarily large. Hence even a z-independent repair of Lemma 5.5(b) would require a q-bound not supplied by the hypotheses of Theorem 1.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":46796,"tokens_out":6316,"duration_ms":75879,"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":[{"comment":"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.","section":"Section 5.2, Lemma 5.5(b) and Proposition 5.6"},{"comment":"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.","section":"Section 1.1, Proposition 3.2, and Theorem 1.1"},{"comment":"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.","section":"Section 5.2, Taylor expansion leading to (5.43)"}],"minor_comments":[{"comment":"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.","section":"Section 5.2, equation (5.21)"},{"comment":"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.","section":"Section 4.1, Lemma 4.2"},{"comment":"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.","section":"Section 3, Proposition 3.2"}],"recommendation":"major_revision","confidential_remarks":"The gap concerning Lemma 5.5(b) is exactly the kind of hidden-assumption issue that should be checked carefully in revision: the condition z<−2‖q‖_{L∞} is used in the proof but never appears in the statement, and the admissibility class Q(N) does not control L∞ norms in dimension n≥2. The self-citations [49,50] appear only in the literature review and do not drive the argument, so I see no circularity concern. The paper’s main idea and structure are sound and novel; with an appropriately strengthened admissibility assumption or a uniform approximation argument, the result can likely be repaired within the scope of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: this is a genuinely new result. Theorem 1.1 is the first Hölder stability estimate for the solenoidal part of an unknown magnetic potential from boundary spectral data, and the paper handles the gauge issue correctly by recovering only A^s. The architecture is coherent: boundary spectral data to a negative-index elliptic DN map, then to the hyperbolic DN map, then GO solutions, then ray-transform stability. The negative-index Sobolev setting in Section 5 is a real technical contribution, and the changed exponents in Proposition 5.6 are handled with care. The self-citations are confined to the literature review and are not load-bearing. The ray-transform stability from Stefanov–Uhlmann is used as a black box, which is appropriate.\n\nThe soft spot is exactly where the stress-test points. Lemma 5.5(b) is stated for every z < 0, but the proof invokes the unstated condition z < -2||q||_L∞. The admissible class Q(N) only controls q in H^1, which on compact manifolds of dimension n ≥ 3 does not give a uniform L∞ bound. Proposition 5.6 then applies Lemma 5.5(b) at arbitrary z < 0, including z = -1, so as written the proof does not justify the claimed uniformity. I suspect the estimate in Lemma 5.5(b) is actually true without the hidden condition—the solution should decay as q grows—so this is a repairable gap rather than a counterexample. But it is a gap in a load-bearing step, and the exponents in Theorem 1.1 depend on it.\n\nA smaller issue: Proposition 3.2 is stated for A in W^{1,∞} and q in L∞, while the admissible classes only give H^{⌈n/2⌉+1} and H^1. Sobolev embedding probably makes the GO estimates go through, but the paper does not say exactly where this is justified.\n\nThis paper is for researchers in inverse spectral theory and the BC-method. It deserves a serious referee: the main claim is novel, the strategy is sound, and the identified gaps are local and likely fixable. I would send it to peer review, with the referee asked to demand a corrected proof of Lemma 5.5(b) and a short remark on how the GO construction runs under the admissible regularity.","headline":"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.","tokens_in":47325,"tokens_out":8407,"would_cite":true,"duration_ms":104013,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","58J50","35P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Magnetic Schrödinger potentials recover Hölder stably from boundary spectral data on simple manifolds.","keywords":["inverse spectral problem","magnetic Schrödinger operator","Hölder stability","boundary spectral data","Dirichlet-to-Neumann map","geodesic ray transform","simple manifold","Gel'fand problem"],"falsifier":"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.","tokens_in":46343,"feed_emoji":"📐","tokens_out":2019,"duration_ms":24136,"temperature":0.7,"pith_summary":"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.","feed_headline":"Magnetic potentials recover Hölder-stably from spectral data","feed_subtitle":"New theorem on simple manifolds gives first quantitative stability for the magnetic part, reducing the proof to geodesic ray transforms.","key_machinery":"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 $z<0$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the stability estimates for the geodesic ray transform of one-forms and functions on simple manifolds, which are the final step in recovering $A^s$ and $q$ from the hyperbolic DtN map.","marker":"[64]"},{"why":"Establishes Hölder stability for the electric potential from the hyperbolic DtN map on simple manifolds using GO solutions, a template this paper extends to include magnetic potentials.","marker":"[12]"},{"why":"Gives the original Hölder-type stability for the Schr\\\"odinger potential from boundary spectral data in the Euclidean setting, and its methods are adapted for Lemma 5.4's reduction.","marker":"[1]"},{"why":"Provides a stability estimate for the electric potential in the electromagnetic wave equation, serving as a baseline for the magnetic case and a source of the elliptic DtN map approach.","marker":"[18]"},{"why":"Supplies the Helmholtz decomposition of the magnetic potential into solenoidal and gradient parts, which is essential for defining gauge-invariant recovery and for the ray-transform arguments.","marker":"[61]"},{"why":"Gives the well-posedness and energy estimates for the hyperbolic initial boundary value problem, used throughout for the existence of solutions and boundary trace estimates.","marker":"[39]"},{"why":"Provides the Green's identity for the magnetic Laplacian and the gauge-invariance of the DtN map, used in deriving the integral identity in Section 4.","marker":"[8]"}],"fun_headline_variants":["Hölder-stable recovery from boundary spectral data","Magnetic and electric potentials recovered stably from spectra","Stable inversion of magnetic Schrödinger spectral data","Spectral data yield Hölder stability for potentials on simple manifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $q$.","fun_headline_variants_meta":{"raw":{"variants":["Hölder-stable recovery from boundary spectral data","Magnetic and electric potentials recovered stably from spectra","Stable inversion of magnetic Schrödinger spectral data","Spectral data yield Hölder stability for potentials on simple manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000495,"raw_usage":{"total_tokens":2417,"prompt_tokens":922,"completion_tokens":1495,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":1430}},"tokens_in":538,"tokens_out":1495,"duration_ms":10748,"temperature":1.0,"reasoning_tokens":1430,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:20:53.246022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Stefanov and G","cited_arxiv_id":null,"evidence_quote":"Provides the stability estimates for the geodesic ray transform of one-forms and functions on simple manifolds, which are the final step in recovering $A^s$ and $q$ from the hyperbolic DtN map."},{"cited_title":"Bellassoued and D","cited_arxiv_id":null,"evidence_quote":"Establishes Hölder stability for the electric potential from the hyperbolic DtN map on simple manifolds using GO solutions, a template this paper extends to include magnetic potentials."},{"cited_title":"Alessandrini and J","cited_arxiv_id":null,"evidence_quote":"Gives the original Hölder-type stability for the Schr\\\"odinger potential from boundary spectral data in the Euclidean setting, and its methods are adapted for Lemma 5.4's reduction."},{"cited_title":"Ben Joud","cited_arxiv_id":null,"evidence_quote":"Provides a stability estimate for the electric potential in the electromagnetic wave equation, serving as a baseline for the magnetic case and a source of the elliptic DtN map approach."},{"cited_title":"Sharafutdinov","cited_arxiv_id":null,"evidence_quote":"Supplies the Helmholtz decomposition of the magnetic potential into solenoidal and gradient parts, which is essential for defining gauge-invariant recovery and for the ray-transform arguments."},{"cited_title":"Katchalov, Y","cited_arxiv_id":null,"evidence_quote":"Gives the well-posedness and energy estimates for the hyperbolic initial boundary value problem, used throughout for the existence of solutions and boundary trace estimates."},{"cited_title":"Bellassoued and I","cited_arxiv_id":null,"evidence_quote":"Provides the Green's identity for the magnetic Laplacian and the gauge-invariance of the DtN map, used in deriving the integral identity in Section 4."}],"review_version":1}