{"id":"e2cfd292-2014-4ad4-aec1-a79d40b25081","arxiv_id":"2507.22723","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper proves new uniqueness theorems: a Schrödinger potential on a closed curved space is fully determined by a sparse set of vibration frequencies together with the shapes of the corresponding vibration modes seen only on a small region. The results give a theoretical foundation for recovering unknown media from a single passive measurement in heat, quantum, and wave imaging.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tataru unique continuation step in Lemma A.1 and (3.40) appears to conclude vanishing outside the finite-speed influence region; the proofs of Theorems 1.3 and 1.16 rest on it.","rationale":"The reader identified (UO) as the weakest assumption, but (UO) is an explicit hypothesis and the paper gives examples where it holds. The more load-bearing issue is a proof step common to both main theorems: the Tataru-based propagation of vanishing from W to \\tilde O. This step is used twice: in Lemma A.1 to prove f2=f1 on M, and in the proof of Theorem 1.16 to prove (3.40). If this step is invalid, then even Theorem 1.3, which only requires GCC, is not proven by the paper's argument, and Theorem 1.16 collapses entirely because the reduction to the case b_k=k depends on (3.40). My proposed test would settle the matter by checking the cited theorem against an explicit finite-speed counterexample on S^2. I do not claim the theorems are false; I claim the written proof has a gap that must be closed before the central claim is established. For this reason the verdict should be CONDITIONAL on a correct justification or replacement of the Tataru step, rather than an unconditional ACCEPT.","tokens_in":47204,"tokens_out":38173,"duration_ms":450168,"concrete_test":"Verify the claimed Tataru step against the concrete model M=S^2, p=north pole, q=south pole, W=B_\\varepsilon(q), T=\\pi-\\varepsilon, V=0, and f a nonzero smooth function supported in B_{\\varepsilon/2}(p). Let U(t)=\\cos(t\\sqrt{-\\Delta})f. By finite speed, U vanishes on W\\times(-T,T), while U(0)=f is nonzero on \\tilde O=\\{x:\\operatorname{dist}(x,p)<\\pi-\\varepsilon\\}. Check whether [65, Theorem 3.24] as cited actually forces f=0 on \\tilde O; if it does not (as this example suggests), the paper's use of Tataru's theorem is invalid. Alternatively, re-derive Lemma A.1 for Theorem 1.3 using GCC-based wave observability from O, and test whether the same replacement is possible under (UO) for Theorem 1.16; if not, Theorem 1.16 lacks a proof of (3.40).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Lemma A.1 (and its analogue (3.53)-(3.55) in the proof of Theorem 1.16) uses this step: from U2(t,\\cdot)=0 on W=B_{\\varepsilon/2}(q) for t\\in(-T,T), where T=\\operatorname{dist}(\\partial V,\\partial W)\\ge\\operatorname{dist}_g(p,q)-\\varepsilon, the paper concludes via a 'global Tataru unique continuation' (citing [65, Thm 3.24], [96], [45]) that U2(0,\\cdot)=0 on \\tilde O=\\{x:\\operatorname{dist}_g(x,p)<\\operatorname{dist}_g(p,q)-\\varepsilon\\}. However, finite speed of propagation implies that vanishing on W\\times(-T,T) can only force the initial data to vanish on the T-neighborhood of W, i.e. on \\{x:\\operatorname{dist}_g(x,W)<T\\}. Since \\operatorname{dist}_g(p,W)\\ge\\operatorname{dist}_g(p,q)-\\varepsilon/2>T for small \\varepsilon, a whole neighborhood of p (which lies inside \\tilde O) is outside this influence region. No standard theorem propagates vanishing outside the light cone for the wave equation; indeed, on the round sphere with initial data supported near the north pole, the solution vanishes on a small ball around the south pole for |t|<\\pi-\\varepsilon while remaining nonzero near the north pole at t=0. Thus the asserted implication is not justified by the cited results as stated, and the identities f_2=f_1 on M (Lemma A.1) and f=\\tilde f on M (equation (3.40)) are not established by the given arguments. Since these identities are the mechanism that converts partial spectral data on O into full spectral data and reduces the problem to b_k=k, both main theorems depend on this step.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies inverse spectral problems for Schrödinger operators on closed Riemannian manifolds, with data consisting of a sparse set of eigenvalues and the restrictions of the corresponding eigenfunctions to an open observation set. The main results are Theorem 1.3, which proves potential recovery under the geometric control condition and an antipodal containment assumption (H), and Theorem 1.16, which allows infinitely many spectral pairs to be missing for both operators under a uniform observability assumption (UO) and a Beurling–Kahane sparsity condition. The proofs reduce the spectral problem to a wave-equation problem, use observability estimates and finite-speed propagation, and invoke a global unique continuation theorem; Theorem 1.16 additionally uses a Paley–Wiener interpolation theorem. The paper then applies these results to simultaneous recovery of coefficients and initial data from a single passive measurement for heat, Schrödinger, and wave equations, and to recovery of the manifold in the zero-potential case.","tokens_in":47402,"tokens_out":26501,"duration_ms":358298,"significance":"Assuming correctness, these are strong and interesting results. The non-normalized, non-orthogonal eigenfunction formulation and the sparsity notion are genuine novelties; prior results in dimensions at least two with partial spectral data and no norming constants appear scarce. The applications to passive imaging are a significant extension of the one-dimensional theory and are made possible by the new spectral theorems. The proofs are detailed and use standard tools carefully; the paper also provides concrete examples, such as flat tori and Anosov surfaces, where the strong uniform observability assumption holds. The reliance on the companion paper [30] is explicit and appropriate: the geometric lemma and the manifold-recovery theorem are cited, not rederived. I specifically checked the unique-continuation step in Lemma A.1 and around equation (3.55); the conclusion is consistent with finite-speed propagation because the observation set W is centered at p, so the target set lies in the influence region of W. The paper is likely to be influential in inverse spectral theory and passive inverse problems.","major_comments":[],"minor_comments":[{"comment":"The global unique continuation step is terse enough that a reader may worry it conflicts with finite speed of propagation; since W is defined as B_{varepsilon/2}(p) in (2.6), the target set tilde O = {x : dist_g(x,p) < dist_g(p,q) - varepsilon} is contained in the T-neighborhood of W, so the conclusion is consistent with the standard backward domain-of-dependence argument, and a one-sentence clarification would improve the presentation.","section":"Appendix A, Lemma A.1; §3, (3.53)–(3.55)"},{"comment":"The letter V denotes both a potential (V_1, V_2) and the geodesic ball V = B_{varepsilon/2}(q); this notational clash is confusing and should be removed by renaming the ball, for instance to mathcal V.","section":"Section 2, proof of Theorem 1.3"},{"comment":"The phrase 'global Tataru unique continuation' is cited as [65, Theorem 3.24], [96], [45]; it would be helpful to state the precise formulation used, namely that vanishing of a distributional solution on an open spacetime cylinder implies vanishing on the associated influence region, so the reader can verify the hypotheses.","section":"Throughout"},{"comment":"In the Laplace transform step, the index sequence (l_k)_{k geq k_0} is asserted to be strictly increasing; this follows because mu^{(1)}_k are distinct for k geq k_0, but the sentence could be added explicitly.","section":"Theorem 1.20, proof"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically strong and the central claims are credible. My recommendation of minor revision is driven by presentation, not by any substantive doubt. The sparse-data framework is a genuine contribution, and the proof strategy is coherent. The only point that deserves a careful look is the unique-continuation step, which I believe is valid but should be explained more explicitly to avoid confusion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Genuinely new setup: non-orthonormal eigenfunction data with sparse or infinite missing spectral data, plus generic uniqueness for passive imaging. The proof strategy is coherent and carefully executed, and the applications to heat, Schrödinger, and wave equations are substantive. However, the stress-test note is right: Lemma A.1 and its analogue in Theorem 1.16 use a Tataru/global unique continuation step that does not follow from the cited theorems. Vanishing on W times (-T,T) only forces vanishing inside the light cone, and tilde O contains points near p that are outside that cone. The counterexample on a sphere with data supported near the north pole is a real existence proof for such configurations. Since this step is what converts partial data on O into full data on M and reduces to b_k = k, Theorems 1.3 and 1.16 are not established as written. That said, the paper is not sloppy: the reduction to waves, the Laplace transform arguments, and the use of the Beurling-Kahane theorem are all careful, and the assumptions are clearly stated. The gap might be repairable with additional arguments, possibly using a spectral decomposition or a more careful application of unique continuation for waves. But I would not accept it in current form. This deserves a serious referee, not a desk reject; the novelty is real, and the failure mode is specific enough to be addressed.","headline":"Genuinely new inverse spectral setup with real applications, but Lemma A.1's Tataru step is unjustified and looks load-bearing; the paper needs major revision before I'd trust it.","tokens_in":825,"tokens_out":1189,"would_cite":false,"duration_ms":204143,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","58J50","35P05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A potential on a closed Riemannian manifold is uniquely determined by sparse spectral data—some eigenvalues and the eigenfunction traces on an open set—with no orthogonality or norming constants assumed.","keywords":["inverse spectral problem","sparse spectral data","Schrödinger operator","passive measurement","geometric control condition","uniform observability","unique continuation","Riemannian manifold"],"falsifier":"A direct falsifier would be two potentials $V_1,V_2$ on the flat torus $\\mathbb{T}^2$, with $V_2-V_1$ supported in a ball disjoint from a connected open patch $O$ containing antipodal points, whose eigenpairs satisfy (1.1) for all $k\\ge N$; the theorem predicts no such pair exists, so any explicit construction would settle the matter.","tokens_in":46831,"feed_emoji":"🔍","tokens_out":10799,"duration_ms":120327,"temperature":0.7,"pith_summary":"This paper asks what a listener can recover when only some of the natural frequencies of a vibrating manifold and the shapes of the corresponding modes on a small open patch are observed. It proves two uniqueness theorems: a potential $V$ in the stationary Schrödinger operator $-\\Delta_g+V$ on a closed Riemannian manifold is determined by such sparse spectral data, without assuming the eigenfunctions are orthogonal or globally normalized. The first theorem allows a large portion of the low- and high-frequency spectrum to be unused, subject to a geometric control condition on the observation patch; the second allows countably infinitely many spectral pairs to be missing for both operators, provided the eigenfunctions are uniformly observable from every open subset of the patch. The authors then turn this into a spectral framework for passive imaging, showing that a single measurement of the heat, Schrödinger, or wave equation can recover an unknown potential together with the unknown initial data.","feed_headline":"Sparse data fix a manifold's potential uniquely","feed_subtitle":"Two theorems recover the Schrödinger potential from partial eigenpairs with no orthogonality or normalization assumed.","key_machinery":"The proofs reduce the spectral matching problem to the wave equation. The carrying objects are: (i) the geometric control condition (GCC), or the stronger uniform observability estimate $\\|\\phi\\|_{L^2(M)}\\le C\\|\\phi\\|_{L^2(O')}$ for eigenfunctions, which turns a finite observation window into control of the whole eigenfunction; (ii) the antipodal set $A_{M,g}(p)$, whose inclusion in $O$ lets finite-speed propagation and a global unique continuation theorem for distributional wave solutions upgrade agreement on $O$ to agreement on all of $M$; and (iii) a Paley–Wiener type interpolation theorem for uniformly discrete sets of zero upper density, which produces compactly supported time-window test functions whose Fourier transform vanishes on the present eigenvalue differences—this is the step that lets a countable infinity of spectral pairs be skipped.","core_discovery":"The central claim is Theorem 1.3: if two real-valued potentials $V_1,V_2$ on a closed connected Riemannian manifold share a subsequence of eigenvalues, $\\mu_k^{(1)}=\\mu_{b_k}^{(2)}$, and the corresponding eigenfunctions agree on an open connected observation set $O$ that satisfies the geometric control condition and contains the antipodal set of some point $p\\in O$, then $V_1=V_2$ everywhere on $M$. Theorem 1.16 strengthens this to the case where countably many spectral pairs are entirely absent from the data for both operators, provided the eigenfunctions are observable from every nonempty open subset of $O$, with the missing eigenvalues quantified by a $\\Lambda$-sparsity condition. Neither theorem requires the eigenfunctions to be orthonormal or any knowledge of global norming constants.","pith_inferences":["The $\\Lambda$-sparsity condition is formulated purely in terms of eigenvalue locations, so the framework should apply to other self-adjoint elliptic operators on closed manifolds whenever an observability estimate and the corresponding unique continuation are available.","The residue arguments in the applications suggest a numerical passive-imaging algorithm: Laplace-transform a single heat trace on $(0,\\varepsilon)\\times O$ to read off eigenvalues and eigenfunction traces, then feed those traces into a wave-based reconstruction; the paper does not develop this algorithm.","Theorems 1.20 and 1.29 impose generic conditions only on the first operator, an asymmetry that goes beyond the one-dimensional passive-measurement results; whether this asymmetry is necessary in higher dimensions is left open by the paper."],"forward_implications":["For any closed manifold and observation set satisfying (H), a potential is recoverable from partial spectral data even when a large portion of the low- and high-frequency spectrum is unused, with no eigenfunction normalization.","A single observation of the heat solution on $(0,\\varepsilon)\\times O$, for arbitrarily small $\\varepsilon$, recovers both the potential and the initial temperature, under generic simplicity and nonzero Fourier coefficient assumptions for the first operator.","The same passive-measurement mechanism recovers the potential and the initial state for the time-dependent Schrödinger equation from $(0,\\infty)\\times O$, and for the wave equation recovers the potential plus both initial position and velocity.","On flat tori and Anosov surfaces, countably infinite holes in the spectral data still leave the potential uniquely determined.","When the manifold itself is unknown, the wave reduction recovers the manifold up to isometry from heat data on $(0,\\varepsilon)\\times O$, assuming generic spectral simplicity for both manifolds."],"supporting_citations":[{"why":"Supplies the wave-equation reduction and the observation-set constructions that Theorem 1.3 extends to Schrödinger potentials.","marker":"[30]"},{"why":"Provides the geometric control condition and the eigenfunction observability estimates used in Lemma 2.1.","marker":"[85, 9, 10]"},{"why":"Gives the observability estimate for eigenfunctions from an open set that Lemma 2.1 follows.","marker":"[76]"},{"why":"Supplies the Paley–Wiener interpolation theorem used in Proposition 3.3 and Lemma 3.5 to handle countably infinite missing spectral data.","marker":"[16, 50]"},{"why":"Provides the global unique continuation theorem for distributional wave solutions that transmits local eigenfunction agreement to the whole manifold.","marker":"[65]"},{"why":"Provides the local unique continuation result for wave operators used in the appendix proof of Theorem 1.3.","marker":"[95]"},{"why":"Provides well-posedness, finite-speed propagation, and wave source-to-solution uniqueness invoked in the proofs of Theorems 1.3, 1.16, and the applications.","marker":"[87]"}],"fun_headline_variants":["Sparse eigenpairs uniquely fix the Schrödinger potential","Partial spectral data recover manifold potential","Sparse data fix potential without orthogonality","Manifold potential from partial eigenpairs","Passive imaging via sparse inverse spectral theorems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise of the stronger sparse-data theorem is the uniform observability condition (UO): the eigenfunctions of both Schrödinger operators must be observable from every nonempty open subset of the observation region with one uniform constant, a property currently known only for special geometries such as flat tori and Anosov surfaces.","fun_headline_variants_meta":{"raw":{"variants":["Sparse eigenpairs uniquely fix the Schrödinger potential","Partial spectral data recover manifold potential","Sparse data fix potential without orthogonality","Manifold potential from partial eigenpairs","Passive imaging via sparse inverse spectral theorems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000613,"raw_usage":{"total_tokens":2829,"prompt_tokens":903,"completion_tokens":1926,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":1857}},"tokens_in":519,"tokens_out":1926,"duration_ms":15038,"temperature":1.0,"reasoning_tokens":1857,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:23:27.427629+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct falsifier would be two potentials $V_1,V_2$ on the flat torus $\\mathbb{T}^2$, with $V_2-V_1$ supported in a ball disjoint from a connected open patch $O$ containing antipodal points, whose eigenpairs satisfy (1.1) for all $k\\ge N$; the theorem predicts no such pair exists, so any explicit construction would settle the matter.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the wave-equation reduction and the observation-set constructions that Theorem 1.3 extends to Schrödinger potentials."},{"cited_title":"Math., 49 (2021), 919– 936","cited_arxiv_id":null,"evidence_quote":"Gives the observability estimate for eigenfunctions from an open set that Lemma 2.1 follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the global unique continuation theorem for distributional wave solutions that transmits local eigenfunction agreement to the whole manifold."},{"cited_title":"Partial Differential Equations 20 (1995), no","cited_arxiv_id":null,"evidence_quote":"Provides the local unique continuation result for wave operators used in the appendix proof of Theorem 1.3."},{"cited_title":"An Inverse Problem for symmetric hyperbolic Partial Differential Operators on Complete Riemannian Manifolds","cited_arxiv_id":"2503.14676","evidence_quote":"Provides well-posedness, finite-speed propagation, and wave source-to-solution uniqueness invoked in the proofs of Theorems 1.3, 1.16, and the applications."}],"review_version":1}