{"id":"98408dfb-7a13-476e-af24-b2e5e5badf3b","arxiv_id":"2507.15560","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Approximate eigen-data on an open subset determine a closed Riemannian manifold up to Lipschitz distance epsilon^(1/12) and a Lipschitz potential up to epsilon^(1/(80n)), giving double-logarithmic stability.","lead":"An inverse spectral problem: roughly knowing the eigenvalues and eigenfunctions on a small patch of a curved space determines the whole space and the potential, with a double-logarithmic stability rate. This is the first such general-geometry stability result for the inverse interior spectral problem with a nonzero potential.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quantitative unique continuation estimate of Theorem 2.2, imported from the unpublished preprint [55] and extended to nonzero potentials by a one-line assertion, is the linchpin of both metric and potential reconstruction; without verifying it at H^1 regularity for Lipschitz q, the main…","rationale":"The central claim is a uniform double-logarithmic stability estimate, and Theorem 2.2 is the only conduit from approximate spectral data on U to quantitative control inside M. I checked the parts of the manuscript that are developed in detail—the Toponogov-based distance coordinates (§3), the slicing algebra (§4.2), and the graph-Laplacian discretization (§5)—and found no internal contradiction: the inclusion-exclusion via joint tails makes Lemma 4.3 plausible, and the error bookkeeping with ρ_2 = ε^{1/(64n)} yields the stated ε^{1/(80n)} exponent. The weak point is the imported UCP estimate and the one-sentence extension to q ∈ C^{0,1}. The reader identified the same dependency; this stress-test agrees and can add only that the H^1/L^2 regularity of the estimate is exactly the regularity used in (4.9)–(4.12), so any hidden regularity requirement propagates directly into Proposition 4.1 and the final rate. Thus the manuscript should not be accepted as fully established until [55] is available or the estimate is proved in the appendix; the stated verdict CONDITIONAL remains appropriate.","tokens_in":30540,"tokens_out":29836,"duration_ms":332227,"concrete_test":"Obtain or independently reproduce the proof of [55, Thm 1.3] for the operator ∂_t^2 − Δ_g + q with q ∈ C^{0,1}; check that the estimate holds for u ∈ H^1(M × [−T,T]) with f ∈ L^2 and that the constants depend only on ∥q∥_{C^0}, not on ∥q∥_{C^{0,1}} or on higher norms. In particular, verify the reduction 'adding a zero-th order potential does not affect the proof' against [17, Thm 1.2]; if the proof needs f ∈ H^1 or u ∈ H^2, recompute the δ-dependence in (4.10)–(4.12) and Proposition 4.1 to see whether Theorem 2's ω(δ) = (log|log δ|)^{−C_4} remains valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof rests on Theorem 2.2 (Section 2), a quantitative unique continuation estimate for ∂_t^2 − Δ_g + q. The theorem is stated with u ∈ H^1, f ∈ L^2, q ∈ C^{0,1}, and constants depending only on ∥q∥_{C^0}, but it is quoted from the unpublished preprint [55], whose stated theorem is for q = 0. The paper justifies the q-term by referring to [17, Thm 1.2] in one sentence without reproducing the argument. This estimate is used in Proposition 4.1 and Lemma 4.2 to convert δ-approximate spectral data into approximate L^2 norms of χ_{M_α} φ1 on every domain of influence; these feed Lemma 4.5 (slicing), Proposition 4.8 (metric reconstruction), and Section 5 (graph-Laplacian reconstruction of q). If [55] actually requires u ∈ H^2 or f ∈ H^1, or if the constants degrade with ∥q∥_{C^{0,1}} instead of ∥q∥_{C^0}, the δ-dependence in (4.10)–(4.12) and Proposition 4.1 changes, and the double-logarithmic rate in Theorem 2 loses its stated exponent. No proof or counterexample is supplied in this manuscript that would rule this out.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative stability theorem for Gel'fand's inverse interior spectral problem for Schrödinger operators on closed Riemannian manifolds with bounded geometry. Given a small approximation of the eigenvalues and of the restrictions of orthonormal eigenfunctions to a fixed open subset U, the author reconstructs a manifold close to the true one in Lipschitz distance and a discrete potential close to q on an epsilon-net, with explicit rates of order epsilon^{1/12} for the metric and epsilon^{1/(80n)} for the potential. The main theorem is then used to derive a double-logarithmic stability estimate, a stability rate that is consistent with known lower bounds in related inverse wave problems. The method combines quantitative unique continuation for the wave operator, a Boundary-Control-style slicing argument, distance coordinates, and a new graph-Laplacian discretization to evaluate the potential without knowing the manifold structure in advance.","tokens_in":30740,"tokens_out":6370,"duration_ms":77779,"significance":"If the main results are correct, this is the first quantitative stability estimate for Gel'fand's interior spectral problem in general geometry with a nonzero potential, and it confirms that double-logarithmic stability is the natural rate in this generality. The paper's constructive aspects are a genuine strength: the reconstruction procedure is algorithmic in structure, the constants are uniform over a geometric class, and the a posteriori formula q = lambda_1 + (Delta phi_1)/phi_1 is a true identity rather than a fitted relation. The use of disjoint slices to enable potential recovery and the graph-Laplacian approximation of the elliptic operator are also substantive contributions. The main caveat is that a central analytical input, the quantitative unique continuation estimate, is imported from an unpublished preprint and extended to nonzero potentials only by a one-sentence assertion.","major_comments":[{"comment":"The quantitative unique continuation estimate is stated as a theorem, but its proof is not supplied: it is quoted from the unpublished preprint [55], whose stated result is for q=0, and the extension to Lipschitz q is justified only by the sentence 'adding a zero-th order potential term does not affect the proof except that the constants would additionally depend on ||q||_{C^0}, see [17, Thm. 1.2].' This estimate is load-bearing: it is used in Proposition 4.1 and Lemma 4.2 to convert approximate spectral data into approximate L^2 norms of chi_{M_alpha} phi_1, and it determines the double-logarithmic rate in Theorem 2. The manuscript must either prove the q-dependent version at the stated regularity (u in H^1, f in L^2, q in C^{0,1}, constants depending only on ||q||_{C^0}) or cite a published theorem with exactly these hypotheses. If [55] actually requires u in H^2 or f in H^1, or if the constants degrade with ||q||_{C^{0,1}}, then the delta-dependence in (4.10)-(4.12) and the final rate in Theorem 2 would change.","section":"Section 2, Theorem 2.2"},{"comment":"Theorem 2.2 assumes that U is an open subset with smooth boundary, but in the applications U is only assumed to be an open set containing a ball, and the sets U_k in (4.1) are arbitrary disjoint open subsets, not necessarily with smooth boundary. Since Theorem 2.2 is invoked with these U and U_k, the manuscript should specify how to reduce to domains with smooth boundary (for example by choosing geodesic balls with smooth boundary) or prove the estimate for Lipschitz domains. Without such a reduction, the unique continuation estimate is not directly applicable as stated.","section":"Section 2 and Section 4, smoothness of U"},{"comment":"Proposition 4.1 is a central step, but its proof is not given: the text says that 'using Theorem 2.2 and following the proof in [23, Sec. 4] or [18]' gives the result, and only the definitions of U^a are written out. The adaptation is not entirely formal because here the potential q is nonzero, the wave operator includes q, and the approximate data enter through (4.10)-(4.12). The manuscript should provide at least a concise proof of how Theorem 2.2 converts the smallness of the approximate wave on U_k into the L^2 closeness of u^a to chi_{M_alpha} u, including the dependence of J and delta on sigma and epsilon. This is required to make the later error accounting in Lemma 4.2 and Section 5.3 checkable.","section":"Section 4.1, Proposition 4.1"},{"comment":"The error propagation after the choice sigma = 2^{-L} epsilon^{4L} is only sketched. In particular, the statement that bphi(x_i)^2 is 'determined up to error of order (D/epsilon)^L epsilon^{4L} <= C epsilon^{3L}' needs a formal accounting, because the number of sets V_j is of order (D/epsilon)^L and the constants in the earlier lemmas depend on L, C3, C7, and c1; moreover, delta = delta(sigma, epsilon) is reused at each stage without an explicit composition. The conclusion is plausible, but the current text leaves the delta(epsilon) dependence and the iteration of the estimates implicit. A short error-propagation lemma would remove the need for the reader to reconstruct the argument.","section":"Section 5.3, equation (5.14)"}],"minor_comments":[{"comment":"The notation for the geometric classes is inconsistent: M(n,D,K,v0) is defined in (2.4) with the bound |Sec| <= K2, while the notation uses K rather than K2, and M(n,D,K,K1,v0) then introduces K1. This makes it difficult to track which curvature constants appear in Lemma 2.1, Proposition 3.5, and Proposition 4.8. Please align the notation in one place.","section":"Section 2, notation (2.4)-(2.5)"},{"comment":"The title contains the typo 'Gel'f and's' for 'Gel'fand's'; the abstract and title should be corrected.","section":"Title and Abstract"},{"comment":"The sentence 'q0 is not a cut point of x0 because one can always extend [x0q0] to a longer minimizing geodesic' is correct but terse; since the nearest-point geometry is used here as a uniformisation argument, it would help to add one sentence explaining why the extension can be chosen to remain minimizing for a uniform length in the limit.","section":"Lemma 3.3"},{"comment":"The map Psi in (6.6) is defined via a Voronoi decomposition and is not continuous; the paper correctly calls it an isometry in the metric sense. It would be clearer to state explicitly that Psi is a measurable, not necessarily continuous, epsilon-isometry, since some readers may expect a homeomorphism from the word 'isometry'.","section":"Proof of Theorem 2"}],"recommendation":"major_revision","confidential_remarks":"The main barrier to acceptance is the unresolved dependency on the unpublished preprint [55] for the quantitative unique continuation estimate. The remainder of the argument appears coherent, and the paper contains a substantial new idea in the graph-Laplacian reconstruction of q, but I would not recommend acceptance until the q-dependent UCP theorem is either proved in the manuscript or replaced by a published reference with the required regularity and constant dependence. The notation inconsistency in the geometric classes should also be fixed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me cut to the chase: this is a genuine advance. It gives the first quantitative stability estimate for Gel'fand's interior spectral problem with a nonzero potential in general geometry, and the double-logarithmic rate matches the known ill-posedness. The core argument is coherent and the error accounting throughout is consistent: the metric reconstruction runs at epsilon^{1/12}, the potential at epsilon^{1/(80n)}, and the two logarithms come from the unique continuation estimate.\n\nThe genuinely new pieces are worth naming. The distance-coordinate bi-Lipschitz property is proved via Toponogov comparison and angle control, needing only one covariant derivative of curvature rather than the higher derivatives used in [18,23]. That avoids a third logarithm. The slicing procedure is modified so slices are disjoint, which is necessary for the potential step. And the graph-Laplacian reconstruction of q is a real new idea; the formula q = lambda_1 + (Delta_X phi_hat)/phi_hat is a posteriori and not circular. The paper also openly flags what it does not solve, whether the curvature conditions can be relaxed to Ricci bounds, which I read as honest.\n\nNow the soft spots, in proportion. The load-bearing import is Theorem 2.2, the quantitative unique continuation estimate for the wave equation with a potential, taken from [55], an unpublished preprint (arXiv:2404.16448). The stated result there is for q=0; the paper extends to Lipschitz q with one sentence citing [17]. If [55] actually needs u in H^2 or has constants degrading with ||q||_{C^{0,1}} instead of ||q||_{C^0}, the exponents would shift. I see no evidence of that, but the paper does not supply the proof either. That is a verification gap, not a demonstrated flaw. A referee should check this point carefully. The rest of the toolchain, involving [23], [41], and [28], is published and standard in this area.\n\nOverall, this deserves a serious referee. I would send it out, and I would specifically ask the referee to verify the q-extension in Theorem 2.2. If that holds, the paper is a strong contribution. If not, the main theorem would need reworking. I would cite it in my own work once the q-extension is confirmed.","headline":"Strong first double-logarithmic stability for nonzero potentials in Gel'fand's interior spectral problem; the main risk is an imported unique continuation estimate from an unpublished source.","tokens_in":31373,"tokens_out":3170,"would_cite":true,"duration_ms":34063,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","58J50","53B21"],"pacs":[],"model":"deepseek-v4-flash","headline":"Approximate spectral data on an open subset determine the whole closed manifold and its Schrödinger potential, at a double-logarithmic rate.","keywords":["Gel'fand's inverse problem","inverse interior spectral problem","Schrödinger operator","quantitative stability","quantitative unique continuation","graph Laplacian","Gromov-Hausdorff distance","Lipschitz distance"],"falsifier":"Take two flat tori with Lipschitz potentials supported outside a common geodesic ball $U$ that is isometric in both manifolds, and compute the first $J$ eigenvalues and eigenfunction traces for $J\\sim\\delta^{-1}$; if the potential difference or the Lipschitz distance of the manifolds can remain larger than $C_1(\\log|\\log\\delta|)^{-C_4}$ while the spectral data are $\\delta$-close, the main estimate is false.","tokens_in":30219,"feed_emoji":"🧮","tokens_out":11614,"duration_ms":112417,"temperature":0.7,"pith_summary":"This paper establishes a quantitative stability estimate for Gel'fand's inverse interior spectral problem: the eigenvalues of $-\\Delta_g + q$ on a closed Riemannian manifold, together with the restrictions of the eigenfunctions to a fixed open set $U$, determine the metric $g$ and the potential $q$, and the determination is stable against measurement error. If the spectral data are known only up to error $\\delta$, the paper constructs a manifold close to the true one in Lipschitz and Gromov-Hausdorff distance, together with a discrete function that approximates $q$ on a net, with explicit error rates. Two manifolds whose data are $\\delta$-close are then forced to be diffeomorphic, nearly isometric, and to have potentials close under that isometry. The modulus is double-logarithmic in $\\delta$, and this is the first such estimate in general geometry for a nonzero potential; the proof runs through quantitative unique continuation and a graph discretization of the Laplace-Beltrami operator.","feed_headline":"Approximate spectra on a patch recover manifold and potential","feed_subtitle":"Eigenvalue and eigenfunction traces on an open set pin down the metric and Schrödinger potential at double-log accuracy.","key_machinery":"Two devices carry the argument. The first is the distance coordinate $\\Phi_L(x) = (d(x,z_1),\\ldots,d(x,z_L))$ formed from a maximal $r_L$-separated set in the known ball $B(p,r_0/2)$; Proposition 3.5 proves it is bi-Lipschitz on $M\\setminus U$ using Toponogov comparison and a uniform cut-locus-free ball, which avoids a third logarithm. The second is the graph Laplacian on the reconstruction points $X=\\{x_i\\}$: $(\\Delta_X\\hat\\phi)(x_i) = \\frac{2(n+2)}{\\mathrm{vol}^a(B(x_i,\\rho))\\rho^2}\\sum_{j:\\hat d_{ij}<\\rho}(\\hat\\phi(x_j))^{-1}\\int_{V_j}\\phi^2 - \\frac{2(n+2)}{\\rho^2}\\hat\\phi(x_i)$, where the $V_j$ are disjoint slices of the form $\\{x: d(x,U_k)\\in[\\beta_k\\varepsilon-\\varepsilon,\\beta_k\\varepsilon)\\}$ and $\\hat d_{ij}$, $\\mathrm{vol}^a$ are computed from the approximate spectral data. Proposition 5.11 shows this operator approximates $\\Delta_g\\phi_1$ pointwise with error $O(\\varepsilon^{1/(80n)})$; the key identity is the second-order Jacobian estimate $|J_x(v)-1|\\leq C|v|^2$ in geodesic normal coordinates together with uniform $C^{2,\\alpha}$ bounds on the first eigenfunction. The potential is then recovered from $\\hat q_i = \\lambda_1^a + (\\Delta_X\\hat\\phi)(x_i)/\\hat\\phi(x_i)$.","core_discovery":"The paper's central claim is Theorem 1: for any $(M,g)$ in the bounded-geometry class $\\mathcal{M}(n,D,K,K_2,v_0)$ with a Lipschitz potential $q$, any $\\delta$-approximation of the spectral data on an open set $U$ determines a Riemannian manifold $(\\hat M,\\hat g)$ diffeomorphic to $M$ with $d_L((M,g),(\\hat M,\\hat g)) \\leq C_1\\varepsilon^{1/12}$, and numbers $\\hat q_i$ attached to a $C_3\\varepsilon$-net $\\{x_i\\}$ in $M$ such that $|\\hat q_i - q(x_i)| \\leq C_2\\varepsilon^{1/(80n)}$; the constants depend only on the geometric bounds and $\\|q\\|_{C^{0,1}}$. Theorem 2 restates this as stability of the inverse problem: if the spectral data of $(M_1,g_1,q_1)$ and $(M_2,g_2,q_2)$ on an isometric open set are $\\delta$-close, then the manifolds are diffeomorphic, $d_L(M_1,M_2) \\leq C_1(\\log|\\log\\delta|)^{-C_4}$, and there is a $\\omega(\\delta)$-isometry $\\Psi$ with $\\|q_1 - q_2\\circ\\Psi\\|_{L^\\infty} \\leq C_2\\omega(\\delta)$, where $\\omega(\\delta) = C_1(\\log|\\log\\delta|)^{-C_4}$. The double logarithm enters through quantitative unique continuation for the wave operator; the reconstruction itself is explicit and discrete, producing a finite metric space and a discrete potential rather than relying on the manifold structure.","pith_inferences":["The graph-Laplacian reconstruction is likely implementable numerically: on a flat torus or sphere one can discretize the known patch $U$, form slices from approximate distances, and evaluate the explicit formula for $\\hat q_i$, checking the predicted exponents $1/12$ and $1/(80n)$ against direct eigensolves.","The same slicing-plus-graph-Laplacian pipeline may transfer to other inverse problems with a source-to-solution map on closed manifolds, such as fractional Calderón-type problems, where geometric optics is unavailable.","The dependence on the second covariant derivative of curvature looks like an artifact of the Taylor-expansion proof; if that step can be replaced by second-order calculus available under mere Ricci bounds, the author's open question about the Ricci class would likely have a positive answer.","Because the constants enter through $\\exp(h^{-C})$ factors, the number of eigenfunctions needed for a target accuracy is enormous; this may be intrinsic, and information-theoretic bounds could show that the double-logarithmic rate is not an artifact."],"forward_implications":["A finite block of about $\\delta^{-1}$ eigenpairs already determines a discrete model of the manifold with error $C_1\\varepsilon^{1/12}$, so the inverse problem is algorithmically accessible without knowing the manifold in advance.","Two manifolds with $\\delta$-close spectral data on an isometric open subset must be diffeomorphic, with Lipschitz distance at most $C_1(\\log|\\log\\delta|)^{-C_4}$, and the same near-isometry transfers $q_1$ to $q_2$ within $C_2\\omega(\\delta)$ in $L^\\infty$.","The constants are uniform over the geometric class $\\mathcal{M}(n,D,K,K_2,v_0)$, so the stability estimate does not degrade as the manifold and potential vary within the class.","Recovering the potential does not require complex geometric optics or analytic continuation; only the first eigenfunction's values and Laplacian, approximated by a graph Laplacian, are needed.","In this general class a Hölder modulus is not expected, since a related wave inverse problem is exponentially unstable, making the double-logarithmic rate the natural target."],"supporting_citations":[{"why":"Supplies the quantitative unique continuation estimate for the wave operator (Theorem 2.2) that converts small spectral data into control of eigenfunctions outside U.","marker":"[55]"},{"why":"Provides the quantitative geometric boundary-control and slicing framework that the metric reconstruction follows.","marker":"[18]"},{"why":"Provides the quantitative stability algorithm for Gel'fand's inverse boundary problem whose slicing and Fourier-coefficient arguments are adapted here.","marker":"[23]"},{"why":"Gives the uniform lower bound on the first eigenfunction used to make slice detection via L2 norms robust.","marker":"[41]"},{"why":"Turns an approximate Hausdorff approximation of interior distance functions into a reconstructed Riemannian manifold in Proposition 4.8.","marker":"[28]"},{"why":"Supplies distance-difference, angle, and cut-locus estimates used to prove the bi-Lipschitz distance coordinate without high derivative bounds.","marker":"[44]"},{"why":"Provides the graph discretization of the Laplace-Beltrami operator that motivates and justifies the weighted graph Laplacian in Definition 5.10.","marker":"[20]"}],"fun_headline_variants":["Patch spectra fix manifold and potential up to double-log","Double-log stability: inverse Schrödinger spectra on a subset","Reconstructing manifolds from a few eigenfunctions on a patch","Local spectral data recover metric and potential quantitatively","Quantitative stability for inverse interior spectral problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument leans on an imported quantitative unique-continuation estimate for the wave equation, which says that small data on $U$ force small data in a neighbourhood with a specific $h$-dependence; if that estimate is not valid at the stated metric regularity, or its constants cannot be tracked, none of the stability bounds follow.","fun_headline_variants_meta":{"raw":{"variants":["Patch spectra fix manifold and potential up to double-log","Double-log stability: inverse Schrödinger spectra on a subset","Reconstructing manifolds from a few eigenfunctions on a patch","Local spectral data recover metric and potential quantitatively","Quantitative stability for inverse interior spectral problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1687,"prompt_tokens":1066,"completion_tokens":621,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":682,"completion_tokens_details":{"reasoning_tokens":545}},"tokens_in":682,"tokens_out":621,"duration_ms":7494,"temperature":1.0,"reasoning_tokens":545,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:29:56.414778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two flat tori with Lipschitz potentials supported outside a common geodesic ball $U$ that is isometric in both manifolds, and compute the first $J$ eigenvalues and eigenfunction traces for $J\\sim\\delta^{-1}$; if the potential difference or the Lipschitz distance of the manifolds can remain larger than $C_1(\\log|\\log\\delta|)^{-C_4}$ while the spectral data are $\\delta$-close, the main estimate is false.","supporting_citations":[{"cited_title":"Inverse Spectral Problems for Collapsing Manifolds II: Quantitative Stability of Reconstruction for Orbifolds","cited_arxiv_id":"2404.16448","evidence_quote":"Supplies the quantitative unique continuation estimate for the wave operator (Theorem 2.2) that converts small spectral data into control of eigenfunctions outside U."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the quantitative geometric boundary-control and slicing framework that the metric reconstruction follows."},{"cited_title":"Burago, S","cited_arxiv_id":null,"evidence_quote":"Provides the quantitative stability algorithm for Gel'fand's inverse boundary problem whose slicing and Fourier-coefficient arguments are adapted here."},{"cited_title":"Honda, Elliptic PDEs on compact Ricci limit spaces and applications, Mem","cited_arxiv_id":null,"evidence_quote":"Gives the uniform lower bound on the first eigenfunction used to make slice detection via L2 norms robust."},{"cited_title":"Ivanov, Distance difference representations of Riemannian manifolds, Geom","cited_arxiv_id":null,"evidence_quote":"Supplies distance-difference, angle, and cut-locus estimates used to prove the bi-Lipschitz distance coordinate without high derivative bounds."},{"cited_title":"Burago, S","cited_arxiv_id":null,"evidence_quote":"Provides the graph discretization of the Laplace-Beltrami operator that motivates and justifies the weighted graph Laplacian in Definition 5.10."}],"review_version":1}