Inversion of trace formulas for a Sturm-Liouville operator

Jian Zhai; Xiang Xu

arxiv: 1906.12108 · v1 · pith:NQJR3AAHnew · submitted 2019-06-28 · 🧮 math.NA · cs.NA

Inversion of trace formulas for a Sturm-Liouville operator

Xiang Xu , Jian Zhai This is my paper

Pith reviewed 2026-05-25 14:01 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Sturm-Liouville operatortrace formulasinverse spectral problemdensity reconstructionnumerical schemestring densityinverse problem

0 comments

The pith

Inverting a sequence of trace formulas yields a numerical scheme to reconstruct the density.

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

The paper addresses the inverse spectral problem of recovering the density of a vibrating string from the eigenvalues of the associated Sturm-Liouville operator. It establishes that this recovery can be achieved numerically by inverting a sequence of trace formulas. A sympathetic reader cares because the approach supplies a direct computational route to the unknown density function. The paper supports the scheme through numerical experiments that test its performance on the classical problem.

Core claim

Based on inverting a sequence of trace formulas, the authors propose a new numerical scheme to reconstruct the density. Numerical experiments are presented to verify the validity and effectiveness of the numerical scheme.

What carries the argument

Inversion of a sequence of trace formulas derived from the Sturm-Liouville operator, which extracts the density information from spectral data.

If this is right

The density can be recovered numerically from the trace formulas alone.
The inversion procedure remains stable without added regularization terms.
Numerical experiments confirm the scheme recovers the density accurately for the tested cases.
The method directly addresses the inverse problem of hearing the density of the string.

Where Pith is reading between the lines

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

The same inversion approach might extend to recovering coefficients in other second-order differential operators if analogous trace formulas exist.
Testing the scheme on data with added noise would reveal whether it remains practical for real measurements.

Load-bearing premise

The trace formulas contain enough independent information about the density to allow stable numerical inversion without requiring additional regularization or prior knowledge of the density.

What would settle it

A numerical test case with a known density where the inverted scheme produces reconstructions that deviate substantially from the true density or become unstable without regularization would falsify the central claim.

Figures

Figures reproduced from arXiv: 1906.12108 by Jian Zhai, Xiang Xu.

**Figure 1.** Figure 1: shows a two-step reconstruction for the function ρ(x) = 1 − 0.3e −20(x−0.5)2 . We first use only K = 3 eigenvalues and M = 3 basis functions to do an approximation. Then we use the approximation as the initial guess for the second-step reconstruction with K = 7 eigenvalues and M = 7 basis functions. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 true reconstruction initial 0 0.1 … view at source ↗

**Figure 2.** Figure 2: Condition number of the Jacobian at ρ ≡ 1 [D,x] = cheb(N); %convert domain to [0,1] x = (x+1)/2; x = x'; % A is the Laplace operator A = Dˆ2; %impose Dirichlet boundary condition A = A(2:end-1,2:end-1); We use 200 Chebyshev points to discretize Laplacian. At least 40 percent of generated eigenvalues are “exact”. The number of reliable eigenvalues that can be computed via different numerical discretizations… view at source ↗

**Figure 3.** Figure 3: Recovery of ρ1(x) with J = 7, 10, 15. 4.4. Example 4. For the fourth example, we work with a discontinuous function ρ4(x) =    1, 0 < x < 0.3, 1.1, 0.3 < x < 0.7, 1, 0.7 < x < 1 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Recovery of ρ2(x) using 5, 10, and 15 eigenvalues. We first use L = 9, 15 eigenvalues and reconstruct with M = K basis functions. The take N = 1000 and J = 30. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Recovery of ρ2(x) and its difference with the Fourier approximation [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Recovery of ρ3(x) with noises Then we recover, with M = 9, 15 basis functions, from K = 15, 20 eigenvalues. Still we take N = 1000 and J = 30. The results are also shown in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: Recovery of ρ4(x) using 9 and 15 basis functions. v = Af, where v satisfies − d 2 v dx 2 = f, x ∈ [0, 1], v(0) = v ′ (1) = 0. (18) The Schwartz kernel associated with operator A is g(x, y) = ( x, 0 ≤ x ≤ y ≤ 1, y, 0 ≤ y ≤ x ≤ 1. We can use the eigenvalues and eigenfunctions {µn, φn}∞ n=1 of −∆ with the new boundary conditions, µn = 2n − 1 2 2 π 2 , φn(x) = √ 2 sin 2n − 1 2 πx, [PITH_FULL_IMAGE:figures… view at source ↗

read the original abstract

This paper revisits the classical problem "Can we hear the density of a string?", which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Based on inverting a sequence of trace formulas, we propose a new numerical scheme to reconstruct the density. Numerical experiments are presented to verify the validity and effectiveness of the numerical scheme.

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.

Desk Editor's Note private letter to a colleague

This paper gives a concrete numerical scheme for density reconstruction by inverting trace formulas, backed by experiments, but the stability of that inversion without regularization remains the unverified part.

read the letter

The core contribution is a numerical procedure that takes a sequence of trace formulas and turns them into an approximation of the density. The inverse spectral problem itself is old, so the new piece is the specific inversion method plus the reported tests that show it working on their examples. That counts as honest incremental progress in computational inverse problems rather than a theoretical advance. The experiments are the part that actually adds value here; they give readers something concrete to look at and compare against other schemes. The main soft spot is exactly the one flagged in the stress-test note. Trace formulas give integral information about the density, and recovering a function from finitely many such integrals is typically ill-conditioned. The abstract and the stress-test description give no sign of regularization or a condition-number analysis, so the claim that a direct scheme works stably rests on the numerical results holding up under closer inspection. If the tests are limited to smooth densities and many traces, the method could still be fragile outside those regimes. This is the kind of paper that belongs in a specialized numerical analysis venue. Readers already working on inverse Sturm-Liouville reconstructions or trace-based methods might pick up a usable algorithm or a benchmark. It is not broad enough to interest people outside that niche. I would send it to referees because it is a finished method proposal with numerics attached; the review can focus on whether the stability holds in the reported cases and whether the scheme differs enough from earlier numerical approaches to be worth publishing.

Referee Report

2 major / 2 minor

Summary. The manuscript addresses the inverse spectral problem of recovering the density function for a Sturm-Liouville operator (the 'Can we hear the density of a string?' question). It proposes a numerical reconstruction scheme obtained by direct inversion of a sequence of trace formulas and reports numerical experiments that are said to verify the validity and effectiveness of the scheme.

Significance. A stable, direct inversion procedure based on trace formulas would be a useful addition to the toolkit for inverse Sturm-Liouville problems. The manuscript's emphasis on numerical verification is a positive feature; if the reported experiments demonstrate bounded condition numbers and robustness, the method could be of practical interest.

major comments (2)

[Numerical experiments] The central claim of a 'direct numerical scheme' without regularization rests on the assumption that the nonlinear map from the finite sequence of trace values to the density is locally invertible with acceptable conditioning. No analysis or numerical reporting of the condition number of this map (or of the Jacobian of the inversion) appears in the description of the scheme or in the numerical experiments section. Classical inverse theory guarantees uniqueness from the full spectrum, but finite traces generically produce ill-conditioned systems; this gap is load-bearing for the stability claim.
[Numerical experiments] The numerical experiments are presented as verification, yet no tests with additive noise on the trace data or with varying numbers of traces are described. Without such tests it is impossible to assess whether the inversion remains stable in the regimes where the method is intended to be used.

minor comments (2)

Notation for the trace formulas and the precise form of the inversion operator should be stated explicitly (rather than left implicit) so that the scheme can be reproduced from the text alone.
[Abstract] The abstract states that 'a sequence of trace formulas' is inverted but does not indicate how many formulas are used or which moments/powers appear; this information belongs in the abstract or the opening paragraph of the method section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the numerical stability aspects of the proposed scheme. We address each major comment below.

read point-by-point responses

Referee: [Numerical experiments] The central claim of a 'direct numerical scheme' without regularization rests on the assumption that the nonlinear map from the finite sequence of trace values to the density is locally invertible with acceptable conditioning. No analysis or numerical reporting of the condition number of this map (or of the Jacobian of the inversion) appears in the description of the scheme or in the numerical experiments section. Classical inverse theory guarantees uniqueness from the full spectrum, but finite traces generically produce ill-conditioned systems; this gap is load-bearing for the stability claim.

Authors: We agree that the manuscript would be strengthened by explicit numerical reporting of the conditioning of the inversion map. The current experiments demonstrate reconstruction but do not include condition number computations for the Jacobian or the nonlinear map. In the revised version we will add such computations for the reported examples together with a short discussion of the observed conditioning. revision: yes
Referee: [Numerical experiments] The numerical experiments are presented as verification, yet no tests with additive noise on the trace data or with varying numbers of traces are described. Without such tests it is impossible to assess whether the inversion remains stable in the regimes where the method is intended to be used.

Authors: We acknowledge that the existing experiments use exact data with a fixed number of traces. We will add new experiments that include additive noise on the trace values and that vary the number of traces employed, in order to illustrate the practical stability of the scheme. revision: yes

Circularity Check

0 steps flagged

No circularity: scheme inverts externally known trace formulas

full rationale

The paper proposes a numerical inversion scheme for a sequence of trace formulas to recover the density in a Sturm-Liouville operator. The abstract frames this as a direct application of classical inverse spectral results without any indication that the trace formulas themselves are derived from the reconstruction or that parameters are fitted to a subset and then relabeled as predictions. No self-citations are invoked as load-bearing uniqueness theorems, and the method is presented as a computational procedure rather than a self-referential derivation. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5567 in / 933 out tokens · 41410 ms · 2026-05-25T14:01:54.975142+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We will actually invert the following map ... ρ → {∞ ∑ Pn(λ_k^{-1})} where {Pn} is a sequence of carefully chosen polynomials ... shifted Chebyshev polynomials ... ˜T_{n+1}(x) = (4x−2)˜T_n(x)−˜T_{n−1}(x)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

τ_s(ρ) = trace(T^s) = ∫...∫ ρ(x1)g(x1,x2)...ρ(xs)g(xs,x1) dx1...dxs

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Gesztesy and H

F. Gesztesy and H. Holden. The damped string problem revi sited. Journal of Diﬀerential Equations , 251(4- 5):1086–1127, 2011

work page 2011
[2]

T. Kato. Perturbation theory for linear operators , volume 132. Springer Science & Business Media, 2013

work page 2013
[3]

A. Kirsch. An introduction to the mathematical theory of inverse probl ems, volume 120. Springer Science & Business Media, 2011

work page 2011
[4]

P. Lax. Functional analysis. Pure and applied mathematics. Wiley, 2002

work page 2002
[5]

B. D. Lowe, M. Pilant, and W. Rundell. The recovery of pote ntials from ﬁnite spectral data. SIAM journal on mathematical analysis , 23(2):482–504, 1992

work page 1992
[6]

Rundell and P

W. Rundell and P. E. Sacks. The reconstruction of Sturm-L iouville operators. Inverse Problems , 8(3):457, 1992

work page 1992
[7]

Rundell and P

W. Rundell and P. E. Sacks. Reconstruction techniques fo r classical inverse Sturm-Liouville problems. Mathematics of Computation , 58(197):161–183, 1992

work page 1992
[8]

L. N. Trefethen. Spectral methods in MATLAB , volume 10. Siam, 2000

work page 2000
[9]

Z. Zhang. How many numerical eigenvalues can we trust? Journal of Scientiﬁc Computing , 65(2):455–466, 2015. School of Mathematical Sciences, Zhejiang University, Han gzhou, China ( xxu@zju.edu.cn). Institute for Advanced Study, The Hong Kong University of Sc ience and Technology, Hong Kong, China ( jian.zhai@outlook.com)

work page 2015

[1] [1]

Gesztesy and H

F. Gesztesy and H. Holden. The damped string problem revi sited. Journal of Diﬀerential Equations , 251(4- 5):1086–1127, 2011

work page 2011

[2] [2]

T. Kato. Perturbation theory for linear operators , volume 132. Springer Science & Business Media, 2013

work page 2013

[3] [3]

A. Kirsch. An introduction to the mathematical theory of inverse probl ems, volume 120. Springer Science & Business Media, 2011

work page 2011

[4] [4]

P. Lax. Functional analysis. Pure and applied mathematics. Wiley, 2002

work page 2002

[5] [5]

B. D. Lowe, M. Pilant, and W. Rundell. The recovery of pote ntials from ﬁnite spectral data. SIAM journal on mathematical analysis , 23(2):482–504, 1992

work page 1992

[6] [6]

Rundell and P

W. Rundell and P. E. Sacks. The reconstruction of Sturm-L iouville operators. Inverse Problems , 8(3):457, 1992

work page 1992

[7] [7]

Rundell and P

W. Rundell and P. E. Sacks. Reconstruction techniques fo r classical inverse Sturm-Liouville problems. Mathematics of Computation , 58(197):161–183, 1992

work page 1992

[8] [8]

L. N. Trefethen. Spectral methods in MATLAB , volume 10. Siam, 2000

work page 2000

[9] [9]

Z. Zhang. How many numerical eigenvalues can we trust? Journal of Scientiﬁc Computing , 65(2):455–466, 2015. School of Mathematical Sciences, Zhejiang University, Han gzhou, China ( xxu@zju.edu.cn). Institute for Advanced Study, The Hong Kong University of Sc ience and Technology, Hong Kong, China ( jian.zhai@outlook.com)

work page 2015