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arxiv: 2606.09405 · v1 · pith:XTG3HU2Wnew · submitted 2026-06-08 · 🧮 math.SP

Uniform stability of recovering the Sturm-Liouville operator on a star-graph

Maria Kuznetsova This is my paper

Pith reviewed 2026-06-27 14:03 UTC · model grok-4.3

classification 🧮 math.SP
keywords Sturm-Liouville operatorstar-graphinverse spectral problemuniform stabilityWeyl vectorLipschitz estimatestransmutation operator
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The pith

Recovering Sturm-Liouville potentials on a star-graph from the Weyl vector is uniformly stable, with Lipschitz constants depending only on a bound for the potentials' norms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves uniform stability for the inverse problem of recovering the Sturm-Liouville operator on a star-graph from the Weyl vector. It extends uniqueness and constructive recovery results previously known for trees and supplies Lipschitz estimates whose constant depends solely on an a priori bound on the L2-norms of the potentials. A reader would care because such stability justifies that the inverse problem is well-posed and supports the development of reliable numerical methods. The work also derives auxiliary uniform stability for the direct problem and for the partial derivatives of the transmutation operator kernel.

Core claim

The map from the vector of potentials on the edges of the star-graph to the associated Weyl vector is uniformly stable in suitable function-space norms. This yields explicit Lipschitz estimates controlled only by the number bounding the potentials' norms. The result extends the tree-graph case and includes uniform stability of the direct spectral map together with stability of the relevant derivatives of the transmutation kernel.

What carries the argument

The Weyl vector on the star-graph, which encodes the spectral data and serves as the input for the constructive recovery procedure based on transmutation operators.

If this is right

  • Inverse problems reducible to the Weyl-vector problem inherit the same uniform Lipschitz stability.
  • The constructive recovery procedure remains stable under small perturbations of the Weyl vector when potentials stay within a fixed norm bound.
  • Existence of stable numerical algorithms for the inverse problem on star-graphs is justified by the Lipschitz control.
  • Stability constants are uniform across all potentials whose norms are bounded by the same number.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the reduction technique extends to graphs with cycles, the same Lipschitz stability may apply to a wider class of quantum graphs.
  • The uniform stability of transmutation-kernel derivatives supplies error bounds that could be used in approximation schemes for numerical reconstruction.
  • The result indicates that moving from an interval or tree to a star-graph does not introduce additional instability beyond what is already controlled by the potential norms.

Load-bearing premise

The reduction of other spectral-data problems to the Weyl-vector problem on the star-graph inherits the constructive method from the tree case without introducing new instabilities.

What would settle it

A sequence of potentials with uniformly bounded L2 norms whose Weyl vectors converge in the target norm but whose potentials diverge by an amount exceeding any fixed multiple of the Weyl-vector difference.

Figures

Figures reproduced from arXiv: 2606.09405 by Maria Kuznetsova.

Figure 1
Figure 1. Figure 1: The contours γ and γm By (2.6) and (2.9), the poles of the function Mk(λ) lie inside γ. Applying the method of spectral mappings analogously to [33, §1.6.1], for each fixed x ∈ (0, π), we obtain the main equation of Inverse problem 2: S˜ k(x, λ) = Sk(x, λ) − 1 2πi Z γ D˜ k(x, λ, µ)Mˆ k(µ)Sk(x, µ) dµ. (4.2) Equations of such type are the key step for proving the uniqueness of solution of inverse spectral pr… view at source ↗
read the original abstract

In the paper, we study the problem of recovering the Sturm-Liouville operator on a star-graph from the Weyl vector. It generalizes the problem of recovering the classical Sturm-Liouville operator on an interval from the Weyl function, and the problems of recovering from other spectral data can be reduced to this problem. The uniqueness and the constructive method for solving the problem under study were previously obtained by V.A. Yurko in the case of a tree (Inverse Problems, 2005). Here, we prove its uniform stability, which includes Lipschitz estimates with a constant depending only on the number bounding the norms of the potentials. Stability results are necessary for justifying the well-posedness of the problem statement, and they are important for developing numerical methods. As auxiliary results, we obtain the uniform stability of the direct problem, as well as the uniform stability of the partial derivatives of the transmutation operator kernel related to the classical Sturm-Liouville operator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proves uniform stability for the inverse Sturm-Liouville problem on a star-graph, recovering the potentials from the Weyl vector. It establishes Lipschitz estimates whose constant depends only on the a priori bound M on the L²-norms of the potentials. The work generalizes Yurko’s uniqueness and constructive method from trees to star-graphs and asserts that other spectral-data inverse problems reduce to the Weyl-vector problem while inheriting the same uniform stability. Auxiliary results include uniform stability of the direct problem and of the partial derivatives of the transmutation-operator kernel for the classical interval case.

Significance. If the claimed M-only Lipschitz constants are established, the result supplies a concrete justification of well-posedness for a class of inverse problems on graphs and supports the design of stable numerical reconstruction algorithms. The auxiliary stability statements for the direct problem and transmutation kernels are independently useful. The significance is reduced by the absence of explicit verification that the reduction maps from other spectral data preserve the M-only dependence.

major comments (2)
  1. Abstract: the claim that 'the problems of recovering from other spectral data can be reduced to this problem' and inherit the uniform (M-only) stability is asserted without any estimate showing that the Lipschitz constant of the composite map remains independent of the number of edges, the vertex degree, or the norms of the auxiliary spectral data. This dependence is load-bearing for the central assertion of uniform stability.
  2. The reduction step from spectra-plus-norming-constants (or Dirichlet-to-Neumann data) to the Weyl vector is stated to inherit the constructive method from the tree case, yet no modulus-of-continuity estimate controlling the reduction is supplied; without it the M-only Lipschitz bound for the original data cannot be guaranteed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable feedback on our manuscript. The comments highlight an important aspect regarding the inheritance of uniform stability through reductions from other spectral data. We address these points below and will make the necessary revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: Abstract: the claim that 'the problems of recovering from other spectral data can be reduced to this problem' and inherit the uniform (M-only) stability is asserted without any estimate showing that the Lipschitz constant of the composite map remains independent of the number of edges, the vertex degree, or the norms of the auxiliary spectral data. This dependence is load-bearing for the central assertion of uniform stability.

    Authors: We agree with the referee that the abstract makes a strong claim about the reduction inheriting the M-only Lipschitz stability without providing explicit supporting estimates. In the paper, we generalize Yurko's results from trees to star-graphs for the Weyl vector problem and state that other inverse problems reduce to it. To address this, we will revise the manuscript by adding a new subsection that derives the modulus of continuity for the reduction maps (from spectra and norming constants to the Weyl vector), showing that the Lipschitz constants depend only on the a priori bound M and not on the number of edges or other variable parameters. This will be done by carefully tracking the dependencies in the constructive algorithm adapted from the tree case. revision: yes

  2. Referee: The reduction step from spectra-plus-norming-constants (or Dirichlet-to-Neumann data) to the Weyl vector is stated to inherit the constructive method from the tree case, yet no modulus-of-continuity estimate controlling the reduction is supplied; without it the M-only Lipschitz bound for the original data cannot be guaranteed.

    Authors: The referee correctly notes the absence of an explicit modulus-of-continuity estimate for the reduction. While the constructive method is inherited and the uniqueness holds, the stability constant's independence from graph parameters requires verification. We will include in the revision a proof that the reduction is Lipschitz with constant depending only on M, by bounding the differences in the spectral data and showing how they translate to the Weyl vector components uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity; stability estimates are an independent new contribution

full rationale

The paper cites Yurko (2005) solely for uniqueness and the constructive recovery method on trees, then supplies a separate proof of uniform (Lipschitz) stability whose constant depends only on the a-priori bound M. No equation in the abstract or described derivation reduces the stability claim to a fitted parameter, a self-definition, or a self-citation chain; the reduction of other spectral data to the Weyl-vector problem is asserted as a generalization but does not alter the logical status of the new estimates. Auxiliary stability results for the direct problem and transmutation kernels are likewise presented as fresh. The derivation chain is therefore self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions from inverse spectral theory for graphs and the prior uniqueness theorem; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence and analytic properties of the Weyl vector for the Sturm-Liouville operator on a star-graph
    Invoked as the data for the inverse problem; assumed from the direct problem theory referenced in the abstract.
  • domain assumption The transmutation operator kernel and its partial derivatives admit uniform stability estimates under bounded potentials
    Stated as an auxiliary result obtained in the paper; relies on classical properties for the interval case.

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    for allx∈[0, T]and a.e.y∈[a(x), b(x)],there existsf ′(x, y)such that f ′ ∈L(S),S:= (x, y):x∈[0, T], y∈[a(x), b(x)] ,

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    on[0, T]and b′(x)f(x, b(x)), a ′(x)f(x, a(x))∈L(0, T)

    the functionsf(x, a(x))andf(x, b(x))are defined a.e. on[0, T]and b′(x)f(x, b(x)), a ′(x)f(x, a(x))∈L(0, T). Then,F∈AC[0, T]and F ′(x) =b ′(x)f(x, b(x))−a ′(x)f(x, a(x)) + Z b(x) a(x) f ′(x, y)dy.(6.10) To prove the proposition, one has to integrate the right-hand side of (6.10) from0tox0 and apply the Fubini–Tonelli theorem. Using the Newton–Leibniz formu...