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Reconstruction techniques for inverse Sturm-Liouville problems with complex coefficients

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A direct linear-system method recovers complex-valued potentials and boundary constants from inverse Sturm-Liouville data.

desk verdict A solid, useful extension of the author's NSBF inverse spectral method; the IP4 reduction has a sign error in the printed derivative formula, but the core contribution deserves serious refereeing. read the letter →

arxiv 2506.00670 v1 pith:3BSOHPP2 submitted 2025-05-31 math.CA cs.NAmath-phmath.MPmath.NAmath.SPphysics.comp-ph

Vladislav V. Kravchenko This is my paper
classification math.CAcs.NAmath-phmath.MPmath.NAmath.SPphysics.comp-ph MSC 34A5534B2465L09
keywords inverseSturm-Liouvilleproblemscomplex-valuedpotentialsNeumannseriesofBesselfunctionstwo-spectrumproblemWeylfunctionmultiplierconstantsnorminglinearalgebraicsystems
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

Each of the listed inverse Sturm-Liouville problems is reduced to computing the first coefficients of Neumann series of Bessel functions (NSBF). The paper claims that from finite spectral data — two spectra, the Weyl function, a spectrum with multiplier constants, or a spectrum with norming constants — one can solve one or two linear systems and read off the complex-valued potential as $q(x)=\frac{g_0''(x)}{g_0(x)+1}$ and the boundary constants as $h=g_0'(0)$, $H=-\psi_0'(b)$. This matters because most existing reconstruction methods require knowing the boundary conditions in advance, restrict the potential to real values, or need an extra parameter such as half the integral of the potential. The method is direct, does not iterate or require initial guesses, and comes with practical criteria for choosing the truncation order. Numerical tests on smooth and nonsmooth, real and complex potentials with noisy data are offered as evidence.

What carries the argument

The machinery is the Neumann series of Bessel functions (NSBF) representation of solutions and their derivatives, for example $\phi_h(\rho,x)=\cos\rho x+\sum_{n=0}^\infty(-1)^n g_n(x)j_{2n}(\rho x)$, with analogous series for $S$, $\psi_H$, $T$, and their derivatives. These series converge uniformly in strips of the $\rho$-plane, and their first coefficients satisfy $\phi_h(0,x)=g_0(x)+1$ and $\psi_H(0,x)=\psi_0(x)+1$, which gives the recovery formulas for $q$, $h$, and $H$. The argument is carried by identity (3.2), $\psi_H=\Delta_0\phi_h-\Delta S$, which converts the given spectral data into linear equations for the needed coefficients, and by the companion identity (3.3) used for computing additional coefficients.

What would settle it

Construct a complex-valued potential with known spectra, run the method with several truncation orders, and check whether the $N_1$ that minimizes $R(N_1)$ also minimizes the actual reconstruction error in $q$; if the correlation fails on a single smooth example such as a complex Paine-type potential with complex $h$ and $H$, the accuracy-control claim is refuted.

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Extended reading notes

Core claim

The central claim is that four standard inverse Sturm-Liouville problems with complex-valued potentials can be solved by the same two-step linear-algebra scheme. In the first step, the characteristic functions $\Delta(\rho)$ and $\Delta_0(\rho)$ are approximated from the given spectral data by solving linear systems for NSBF coefficients evaluated at the endpoint. In the second step, identity (3.2), which relates the solutions $\psi_H$, $\phi_h$, and $S$ through $\Delta_0$ and $\Delta$, is sampled at many points $\rho$ to produce another linear system for the coefficients $g_n(x)$ and $\psi_n(x)$ at each point $x$. The first coefficients then carry the whole answer: $q(x)=\frac{g_0''(x)}{g_0(x)+1}$, $h=g_0'(0)$, and $H=-\psi_0'(b)$. The paper further claims that the same construction works when the input is the Weyl function, a spectrum with multiplier constants, or a spectrum with norming constants, and that the truncation order can be selected by minimizing the residuals $R(N_1)$, $P(N_1)$, or $Q(N_1)$.

Load-bearing premise

The load-bearing premise is that the heuristic minimization of $R(N_1)$, $P(N_1)$, or $Q(N_1)$ picks a truncation order for which the truncated NSBF expansions are accurate over the whole interval; if it does not, the recovered $g_0$ and $\psi_0$ are wrong, and so is $q=g_0''/(g_0+1)$.

Editorial extensions

If this is right

  • The boundary constants $h$ and $H$ are produced by the same computation as the potential, so inverse problems no longer have to be posed with known boundary conditions.
  • Complex-valued potentials and complex boundary constants are within scope, not just real self-adjoint problems.
  • Two-spectrum data, the Weyl function, spectrum with multiplier constants, and spectrum with norming constants are all handled by the same linear-algebra machinery.
  • The residuals $R(N_1)$, $P(N_1)$, and $Q(N_1)$ give a practical, parameter-free way to choose the truncation order in place of ad hoc choices.
  • The method is direct and non-iterative, and the reported examples run in seconds even with noisy input data.

Reading between the lines

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

  • Because the recovery formulas use only the first NSBF coefficient, the same scheme should be testable on other boundary-value problems where NSBF representations exist, such as impedance-form equations or systems on graphs, without re-deriving the whole inverse theory.
  • The accuracy of $q$ will be limited where $g_0$ is differentiated; for noisy data, recovering $q$ from $\psi_0$ or from a smoothed combination of $g_0$ and $\psi_0$ may reduce endpoint errors and is a natural numerical extension.
  • The $R(N_1)$ heuristic could be validated statistically on random complex potentials; if it fails, the residual could be combined with cross-validation on a subset of the spectral data.
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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 manuscript develops a unified numerical method for several inverse Sturm-Liouville problems with complex-valued potentials: the two-spectrum problem (IP1), recovery from the Weyl function (IP2), recovery from eigenvalues and multiplier constants (IP3), and recovery from eigenvalues and norming constants (IP4). The method uses Neumann series of Bessel functions (NSBF) representations of solutions, derives linear algebraic systems for the first NSBF coefficients from identities (3.2) and (3.3), and then recovers the potential and boundary constants from the first coefficients via q(x)=g0''(x)/(g0(x)+1), h=g0'(0), H=-ψ0'(b). Numerical examples illustrate accuracy, including complex potentials and noisy data.

Significance. If the method is valid, it is a useful contribution: it treats complex-valued potentials and unknown boundary constants, unifies several inverse problems within one linear-algebra framework, and avoids iterative solvers. The derivation is transparent and mostly based on exact identities; the recovery formulas in Remark 3.3 and the reduction of IP4 to IP3 are elegant. The paper also proposes explicit residual-based criteria for choosing the truncation order, which is a practical step beyond earlier NSBF work. However, the numerical validation is partially self-referential, the truncation criterion is heuristic, and the recovery of q requires a second derivative whose numerical treatment is not specified. One displayed formula, Eq. (5.8), has a sign error that is load-bearing for IP4.

major comments (4)
  1. [§5.3, Eq. (5.8)] The derivative of the h_n term is printed with the wrong sign. Since d/dρ [h_n j_{2n}(ρb)] = h_n[(2n/ρ)j_{2n}(ρb) - b j_{2n+1}(ρb)], the last sum in Eq. (5.8) should be preceded by a minus sign; equivalently, the parentheses should read (2n/ρ)j_{2n}(ρb) - b j_{2n+1}(ρb). As printed, every n≥1 term of Δ̇(ρ_k) has the wrong sign, so β_k computed from Eq. (5.7) changes sign. Since β_k enters the linear system (5.5), a literal implementation of Eq. (5.8) would make the IP4 reconstruction fail. The agreement reported in §6.4 (difference 2.8·10^-10) therefore strongly suggests that the actual computation used a corrected formula. This is a load-bearing point for one of the four claimed inverse problems and must be corrected in the manuscript.
  2. [§4.1.3 and §4.2] The truncation-order selection is heuristic and is not tied to the actual reconstruction error. Minimizing R(N1) or P(N1) checks identity (3.2) at ρ=0 or at a finite set of points r_j, but the paper gives no argument that a small residual implies small errors in g0(x) and ψ0(x) on the whole interval. Since q is recovered by the second derivative g0''/(g0+1), any error in g0 is amplified, and the numerical differentiation procedure is not described at all. The paper also states that square systems can be ill-conditioned, yet no regularization strategy is given. These omissions do not invalidate the approach, but they are central to the claimed 'efficient accuracy control' and to reproducibility.
  3. [§6.2–§6.3 and §6.4] Part of the numerical validation is internally generated with the same NSBF machinery used by the reconstruction. In §6.2 the Weyl function is computed from NSBF-based characteristic functions; in §6.3 the multiplier constants β_k are computed from NSBF values of φ_h(ρ_k,b); and in Example 4 the eigenvalues themselves come from an NSBF-based method. These tests demonstrate internal consistency of the reduction formulas, but they are weaker than independent benchmarks. The IP1 examples computed with Matslise and the norming constants in §6.4 are the main independent checks. I recommend adding at least one fully independent benchmark for IP2 and IP3, ideally with data generated by a different numerical method.
  4. [Remark 3.3 and §4.1.1] The recovery formula q(x)=g0''(x)/(g0(x)+1) requires g0(x)+1 = φ_h(0,x) to be nonzero on (0,b), but no condition excluding zeros of this solution is stated. Similarly, Eq. (4.3) divides by Δ(μ_k), which fails if a singular value μ_k of L0 coincides with a singular value ρ_j of L; the paper assumes simple spectra but not disjointness of the two spectra. The manuscript should either state the needed nonvanishing/disjointness assumptions explicitly or explain how the method is modified in these exceptional cases.
minor comments (4)
  1. [§3, Theorem 3.1] The theorem states uniform convergence 'for every x ∈ [0,L]' but the interval is elsewhere always [0,b]; this is a typo that should be corrected.
  2. [§5.3, proof of (5.6)] The displayed derivative d/dx W[ψ_H(ρ,x), φ_h(ρ_k,x)] = (ρ^2 - ρ_k^2)ψ_H φ has the wrong sign; it should be (ρ_k^2 - ρ^2)ψ_H φ. The final identity (5.6) is nevertheless correct, but the intermediate equality as printed is not.
  3. [§6.1, Example 2] The text notes that N1=N2=13 makes system (4.2) square, while §4.1.3 warns that square systems can be ill-conditioned; a sentence explaining why this particular square case is safe would be helpful.
  4. [§7 and Data availability] The data availability statement says data are available on request, but no code is provided. Given that the method is numerical, making the MATLAB code available would substantially improve reproducibility.

Circularity Check

1 steps flagged · score 3.0 of 10

Derivation chain is non-circular: exact identities (3.2), (4.2), (4.6), (5.2) and (5.6) connect the input data to NSBF coefficients, and q is recovered from the exact relation q=g0''/(g0+1). The moderate circularity burden lies in the complex-potential numerical benchmarks, whose synthetic input data are generated with the same NSBF forward machinery that the inverse method inverts.

  1. other [Section 6.2, benchmark paragraph before Example 5; analogous in Section 6.1 Example 4 and Section 6.3]
    "The Weyl functions of the Sturm-Liouville problems considered in this subsection were computed by the formula M(ρ) = − Δ0(ρ)/Δ(ρ), where in their turn, Δ0(ρ) and Δ(ρ) were computed with the aid of the NSBF representations (3.4), (3.5), (3.11), (3.12), with the coefficients calculated following the recurrent integration procedure from [31]."

    For the complex-potential tests, the input data are produced by the same NSBF expansions that the inverse systems (4.2), (5.2), and (5.5) later solve: eigenvalues are zeros of characteristic functions built from NSBF series, Weyl values are ratios of NSBF-based Δ0 and Δ, and multiplier constants are endpoint values of NSBF-computed eigenfunctions. Thus the data lie in the finite-dimensional NSBF model class by construction; the reported agreement largely confirms that the linear systems invert their own generator, rather than independently testing whether a truncated NSBF representation faithfully captures the true potential. This is a validation circularity, not a derivation-level reduction.

full rationale

The derivation chain itself is not circular. Theorem 3.1 is cited from the previously published [31]; although the author overlaps, the theorem is a parameter-free convergence statement that does not include the inverse result. The systems (4.1)-(4.6), (5.2), and (5.5) are obtained by substituting exact NSBF representations into elementary identities (3.2), (3.3), and (5.1), and q is recovered from the exact formula q = g0''/(g0+1) (Remark 3.3), not from a fitted parameter. No uniqueness result is imported from the authors' own work. The main circularity-adjacent issue is that several numerical validations for complex-valued potentials generate their test data with the same NSBF forward machinery (Example 4 uses [30], Section 6.2 uses NSBF characteristic functions, Section 6.3 uses NSBF-computed multiplier constants); this weakens the empirical independence of those examples, but it does not make the mathematical reduction of the inverse problem equivalent to its inputs. I also note, as a correctness issue rather than a circularity, that Eq. (5.8) appears to have the wrong sign for the derivative of the h_n j_{2n}(ρb) terms; as written the IP4 reduction is defective, and the 2.8e-10 agreement in Section 6.4 suggests the actual code used the corrected derivative. This does not affect the circularity assessment.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. Its free parameters are numerical truncation orders and collocation sets, chosen by hand or heuristic. The main axiomatic input is the NSBF representation theory from the author's earlier work, plus the simplicity assumption on spectra. Conditioning of the resulting linear systems is not analyzed.

free parameters (5)
  • N1 (truncation order) = 7, 13, 31, 22, 33, 49 in the examples
    Truncation index for systems (4.1), (4.2), (4.4), (4.5). Chosen by minimizing the heuristic functionals R(N1) or P(N1) (or Q(N1) for IP2). No theorem guarantees that the minimum gives the best reconstruction.
  • N2 = equal to N1 in all examples
    Truncation order for system (4.2). Set equal to N1 for simplicity.
  • N3 = equal to N1 in all numerical tests
    Truncation order for system (4.6). The paper says 'The parameter N3 here and in the other numerical tests was chosen equal to N1'.
  • Collocation point set {r_j} for system (4.6) = J = 1501 points, logarithmically spaced on [0.01, 1000] (and analogous sets for IP2)
    The points r_j and their count J define the linear system (4.6). The paper uses the same distribution in most tests, but does not derive a general criterion for J or the range.
  • Strip height a in NSBF error estimates = a = 0 in the numerical tests (points chosen real)
    The error estimates in Theorem 3.1 depend on a strip |Im ρ| ≤ a. The text notes a should not be too large, but no quantitative guidance is given.
assumptions (5)
  • domain assumption NSBF representations for solutions and their derivatives (Theorem 3.1 of [31])
    The entire algorithm is built on these series expansions and their convergence in strips of the complex plane. They are cited from the author's prior work, not re-proved in this paper.
  • domain assumption The spectra of problems L and L0 are simple
    Stated in Sections 1 and 2. The method does not address multiple eigenvalues.
  • domain assumption Uniqueness and stability of the two-spectrum problem for complex potentials (results of [8], [10], [11])
    The inverse problem is considered well-posed in theory on the basis of these cited results; the numerical method itself does not enforce stability.
  • ad hoc to paper The linear systems (4.1)-(4.6), (5.2), (5.5) are nonsingular for the chosen truncation and collocation points
    The paper notes that square systems can be ill-conditioned and proposes a heuristic to pick N1, but gives no conditioning analysis for the systems actually solved.
  • domain assumption The input spectral data are exact for the reconstructions (noise is only introduced in the dedicated tests)
    The method is derived assuming exact data; the noise tests are empirical.

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Cite this review

Pith. "Pith review of Reconstruction techniques for inverse Sturm-Liouville problems with complex coefficients." pith.science (2026). https://pith.science/paper/3BSOHPP2

@misc{pith2026250600670,
  author       = {Pith},
  title        = {Pith review of: Reconstruction techniques for inverse Sturm-Liouville problems with complex coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BSOHPP2}},
  note         = {Machine review of arXiv:2506.00670}
}
read the original abstract

A variety of inverse Sturm-Liouville problems is considered, including the two-spectrum inverse problem, the problem of recovering the potential from the Weyl function, as well as the recovery from the spectral function. In all cases the potential in the Sturm-Liouville equation is assumed to be complex valued. A unified approach for the approximate solution of the inverse Sturm-Liouville problems is developed, based on Neumann series of Bessel functions (NSBF) representations for solutions and their derivatives. Unlike most existing approaches, it allows one to recover not only the complex-valued potential but also the boundary conditions of the Sturm-Liouville problem. Efficient accuracy control is implemented. The numerical method is direct. It involves only solving linear systems of algebraic equations for the coefficients of the NSBF representations, while eventually the knowledge only of the first NSBF coefficients leads to the recovery of the Sturm-Liouville problem. Numerical efficiency is illustrated by several test examples.

Figures

Figures reproduced from arXiv: 2506.00670 by the authors.

Figure 1
Figure 1. P(N1) and R(N1) computed for Example 1, with 10 eigenpairs given. Both indicate the choice of N1 = 7. 6.1 Solution of two-spectrum inverse problem (IP1) Example 1. Consider equation (2.1) with q(x) = x 2 , 0 < x < 1, (6.1) and the boundary conditions (2.2) with h = 10 and H = π. For the first test we recover this Sturm￾Liouville problem from ten eigenpairs:  λk, λ0 k 9 k=0. The “exact” eigenvalues were computed by … view at source ↗
Figure 2
Figure 2. Potential from Example 1, recovered from 10 eigenp [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. First Paine potential (Example 2) recovered from 1 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Same as Fig. 3, but recovered from 15 noisy eigenpai [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Complex valued otential (6.3) recovered from 15 no [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Complex valued potential from Example 3, recovere [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Complex valued potential from Example 3, recovere [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Mathieu potential from Example 4, recovered from t [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Mathieu potential from Example 4, recovered from t [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Mathieu potential from Example 4, recovered from [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Fig.11 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 11
Figure 11. Figure 11: Mathieu potential from Example 4, recovered from [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Weyl function of Sturm-Liouville problem from Ex [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Potential from Example 5 recovered from M(ρ) given at 2000 points logarithmically spaced along the segment [0.01, 1000]. The maximum absolute error was 1.57 attained at the origin (the relative error was approximately 0.0157). The constants h and H were recovered with…
Figure 14
Figure 14. Figure 14: Potential from Example 6 recovered from the Weyl fu [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: Complex valued potential from Example 3 recovere [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: Complex valued Mathieu potential from Example 4 r [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]

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