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Stability of Gel'fand's inverse interior spectral problem for Schr\"odinger operators

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Pith's one-line read Approximate spectral data on an open subset determine the whole closed manifold and its Schrödinger potential, at a double-logarithmic rate.

desk verdict 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potentials in Gel'fand's interior spectral problem; the main risk is an imported unique continuation estimate from an unpublished source. read the letter →

arxiv 2507.15560 v1 pith:7FILVUL5 submitted 2025-07-21 math.AP math.DG

Jinpeng Lu This is my paper · ORCID
classification math.APmath.DG MSC 35R3058J5053B21
keywords Gel'fand'sinverseprobleminteriorspectralSchrödingeroperatorquantitativestabilityuniquecontinuationgraphLaplacianGromov-HausdorffdistanceLipschitz
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 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.

What carries the argument

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 $\hat q_i = \lambda_1^a + (\Delta_X\hat\phi)(x_i)/\hat\phi(x_i)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 4 minor

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.

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 (4)
  1. [Section 2, Theorem 2.2] 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.
  2. [Section 2 and Section 4, smoothness of U] 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.
  3. [Section 4.1, Proposition 4.1] 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.
  4. [Section 5.3, equation (5.14)] 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.
minor comments (4)
  1. [Section 2, notation (2.4)-(2.5)] 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.
  2. [Title and Abstract] The title contains the typo 'Gel'f and's' for 'Gel'fand's'; the abstract and title should be corrected.
  3. [Lemma 3.3] 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.
  4. [Proof of Theorem 2] 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'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reconstruction is a genuine data-to-model map, and the two imported same-author theorems are external black boxes with independent support.

full rationale

The derivation chain does not assume its conclusion. The metric reconstruction proceeds from approximate spectral data through quantitative unique continuation (Theorem 2.2), distance-function approximation (Lemma 4.5), and an external manifold-reconstruction theorem (Proposition 4.8 via [28, Thm. 1.2]); none of these steps is the target inverse stability result. The potential reconstruction uses the a posteriori identity q = lambda_1 + (Delta_X phi_hat)/phi_hat, in which every quantity is computed from the spectral data and the reconstructed metric, so it is a genuine reconstruction rather than a tautology. Theorem 2.2 is attributed to the author's own preprint [55], but the paper explicitly notes that the q = 0 case was proved independently in [16,17,56] and cites [17, Thm. 1.2] for the zero-order potential perturbation, so the load-bearing estimate is not justified only by self-citation. Proposition 4.8 is imported from [28], which is a separate result on reconstructing manifolds from distance observations, not a reformulation of the theorem being proved here. No parameter is fitted to the spectral data and then renamed a prediction: delta is chosen explicitly in terms of epsilon and a priori bounds, and the constants depend only on the bounded-geometry class. The admitted limitations, such as the extra curvature regularity in (1.2) and the open Question about Ricci bounds, are correctness/open-problem concerns, not circularity. Overall, the paper is self-contained against external benchmarks as far as circularity is concerned.

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

The ledger is clean: no new particles, forces, or fitted parameters. The proof imports several substantial analytic and geometric theorems, including an unpublished preprint [55] by the same author circle, and the paper's own Lemma 2.1 is the main bootstrapping device.

assumptions (6)
  • domain assumption Quantitative unique continuation for the wave operator with Lipschitz potential (Theorem 2.2, from [55, Thm 1.3])
    Black-box input for Proposition 4.1; all reconstruction of interior distance functions from the U-data passes through this estimate.
  • domain assumption A priori geometric class M(n,D,K,K2,v0) with diam, |Sec|, vol, and first and second covariant derivative bounds on curvature (1.1)-(1.2)
    Defines the compact class; gives injectivity radius lower bound, C^{1,alpha}/C^{2,alpha} compactness, and uniform C^2 geodesic normal coordinate estimates needed in Proposition 5.11.
  • domain assumption Potential bounds and spectral shift (2.1): C0^{-1} <= q <= C0 with ||q||_{C^{0,1}} <= C0
    Makes lambda_1 comparable to C0 and permits positivity of the first eigenfunction; the paper shows this is without loss of generality by adding a constant.
  • standard math Uniform positive lower bound for the L^2-normalized first eigenfunction (Lemma 2.1)
    Proved in the paper via compactness and [41, Thm 1.5]; it fixes the constants c1 and c* that determine which candidate slices enter R*_out.
  • domain assumption Manifold reconstruction and interpolation theorem [28, Thm 1.2]
    Converts the Hausdorff approximation R* of interior distance functions into a diffeomorphic manifold with Lipschitz distance C1 epsilon^(1/12); used in Proposition 4.8.
  • standard math Toponogov comparison and angle continuity results ([3, Thm 8.41], [44, Cor 3.2])
    Basis of the bi-Lipschitz distance coordinate in Proposition 3.5, which avoids an extra logarithm and is essential for the slice partition in Section 5.

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Cite this review

Pith. "Pith review of Stability of Gel'fand's inverse interior spectral problem for Schr\"odinger operators." pith.science (2026). https://pith.science/paper/7FILVUL5

@misc{pith2026250715560,
  author       = {Pith},
  title        = {Pith review of: Stability of Gel'fand's inverse interior spectral problem for Schr\"odinger operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FILVUL5}},
  note         = {Machine review of arXiv:2507.15560}
}
abstract

We study Gel'fand's inverse interior spectral problem of determining a closed Riemannian manifold $(M,g)$ and a potential function $q$ from the knowledge of the eigenvalues $\lambda_j$ of the Schr\"odinger operator $-\Delta_g + q$ and the restriction of the eigenfunctions $\phi_j|_U$ on a given open subset $U\subset M$, where $\Delta_g$ is the Laplace-Beltrami operator on $(M,g)$. We prove that an approximation of finitely many spectral data on $U$ determines a finite metric space that is close to $(M,g)$ in the Gromov-Hausdorff topology, and further determines a discrete function that approximates the potential $q$ with uniform estimates. This leads to a quantitative stability estimate for the inverse interior spectral problem for Schr\"odinger operators in the general case.

Figures

Figures reproduced from arXiv: 2507.15560 by the authors.

Figure 1
Figure 1. Setting of Lemma 3.4. The point q is a nearest point in the closure of B = B(p, r0/2) from x. The blue ball B(q, ρ0) does not intersect with the cut locus of x for a constant ρ0 determined in Lemma 3.3. The red ball has radius ρ0/4 centered at some point q ′ . If Γ is any rL-net in the red ball with rL determined in Lemma 3.1, then there exists a point z1 ∈ Γ such that |∠z1xy − π/2| ≥ θ0. To proceed further, we need… view at source ↗
Figure 2
Figure 2. An illustration of sets Vi and corresponding points xi (for L = 2). The middle region enclosed by red lines is Vi of the form M∗ τ for τ = (β1, β2, 0, · · · , 0) ∈ A. The multi-index β satisfies the criterion (4.17) and hence Mε β⟨l⟩ ̸= ∅ for all l. The corresponding point xi to Vi is chosen as any one point of Mε β⟨l⟩ in the blue region enlarged by scale ε 2 . It could happen that Vi may be empty for some i, or xi … view at source ↗

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Cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.