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REVIEW 3 major objections 3 minor 2 cited by

FewBodyToolkit.jl claims to be a general, open-source Julia solver for quantum two- and three-body bound states and resonances.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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FewBodyToolkit.jl is a documented, open-source Julia solver for two- and three-body quantum bound and resonant states that matches reference results for Coulomb, low-dimensional universal, nuclear resonance, and positronium-ion benchmarks.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A well-executed software paper for a useful Julia few-body toolkit; the main caveat is that parameter sensitivity can silently produce wrong states, and that deserves a clearer look. the 3 major comments →

arxiv 2510.04447 v2 pith:6IUDKRSF submitted 2025-10-06 quant-ph physics.comp-ph

FewBodyToolkit.jl: a Julia package for solving quantum few-body problems

Lucas Happ This is my paper
classification quant-ph physics.comp-ph
keywords few-body physicsGaussian expansion methodJulia packagebound statesresonancescomplex scalingthree-body problemSchrödinger equation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper presents FewBodyToolkit.jl, a Julia package that solves the Schrödinger equation for two- and three-body systems using the Gaussian expansion method, with support for one, two, and three spatial dimensions, arbitrary pair interactions, and both bound and resonant states. The author's central claim is that this single package makes general few-body calculations accessible: a simple API, documentation, and automatic handling of angular momentum coupling and identical-particle symmetries. Validated benchmarks support the claim: Coulomb eigenvalues reproduced to roughly 10^-5 for states up to n=40, resonance positions and widths for a phenomenological nuclear potential within about 0.5 percent, universal one-dimensional three-body energy ratios to four digits, and the positronium negative ion binding energy within 0.073 percent. If these results hold, researchers gain a documented open-source reference implementation for few-body bound and resonant calculations without writing their own solver.

Core claim

The paper introduces FewBodyToolkit.jl as an open-source Julia implementation of the Gaussian expansion method for general two- and three-body quantum systems. The wavefunction is expanded in Gaussian basis functions whose widths form a geometric progression, and an optional complex range adds oscillatory structure to reach highly excited states and resonances. The Schrödinger equation becomes a generalized eigenvalue problem, H c = E S c, with mostly analytic matrix elements; resonances are obtained by complex scaling. The package is organized into three solver modules—a two-body solver, a one-dimensional three-body solver, and a three-dimensional three-body solver—and is validated against

What carries the argument

Gaussian expansion method: expand each relative-motion wavefunction as a superposition of Gaussians with widths in a geometric progression, using products of two such Gaussians in Jacobi coordinates for three-body Faddeev components. This converts the Schrödinger equation into a dense generalized eigenvalue problem with analytic overlap, kinetic, and (for several potentials) interaction matrix elements; complex-ranged Gaussians and complex scaling extend the same machinery to resonances.

Load-bearing premise

The solver's accuracy rests on the user supplying converging numerical parameters (the number and widths of the Gaussian basis functions), and the paper itself shows that unoptimized defaults can miss bound states and generate spurious eigenvalues.

What would settle it

Recompute the two-body Coulomb example with the paper's non-optimized parameters (nmax=10, r1=0.1, rnmax=30): the paper predicts exactly four bound states with energies near -0.499876, -0.124543, -0.054437, -0.028644 and two spurious positive eigenvalues near +0.007342 and +0.603176; any materially different pattern would contradict the reported behavior.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Researchers can compute bound and resonant states for two- and three-body systems with arbitrary central pair potentials, including potentials given only as numerical tables.
  • Complex-ranged Gaussian bases make highly excited states and resonances tractable with modest basis sizes, as shown by the Coulomb spectrum through n=40.
  • Automatic Faddeev decomposition and identical-particle symmetrization in the three-body modules remove a major manual step in setting up three-body calculations.
  • The package offers a reproducible reference for teaching, benchmarking, and testing new few-body methods.
  • If adopted, it could become a common baseline that different few-body approaches compare against, similar to standard packages in other numerical disciplines.

Where Pith is reading between the lines

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

  • Editorial inference: the accessibility claim depends on users selecting good Gaussian width parameters; the paper's own Table 2 shows that defaults can miss bound states and produce spurious positive-energy eigenvalues, so automatic basis optimization would materially strengthen the package.
  • Editorial inference: because the Gaussian expansion is variational for bound states, the same solver output could be extended to give rigorous upper-bound certificates and convergence estimates for each eigenvalue.
  • Editorial inference: the interpolation scheme used for three-body interaction integrals could be adapted to nonlocal or momentum-dependent potentials, a natural extension the paper lists as future work.
  • Editorial inference: the one-dimensional three-body module's reproduction of universal energy ratios suggests the package could serve as a cheap testbed for universal few-body predictions across mass ratios and interaction ranges.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper introduces FewBodyToolkit.jl, an open-source Julia package implementing the Gaussian expansion method for two- and three-body quantum few-body problems. It describes three solver modules (GEM2B, GEM3B1D, ISGL), covering bound states and resonances via complex scaling, several spatial dimensions, symmetries, and general pairwise interaction shapes. The package is validated against exact Coulomb energies (up to n = 40, differences of order 10^-6 or smaller), resonance positions and widths for a nuclear-similar potential (agreement to ~0.5% or better), universal 1D three-body energy ratios (four digits), and Ps− bound-state properties (binding energy within 0.073%, radii within 0.45%). The paper claims that the package is an accessible, documented general few-body solver suitable for research and teaching.

Significance. The package addresses a genuine gap—there is no widely available, documented general few-body solver with this feature combination. The manuscript's positive evidence is strong and appropriately external: the Coulomb comparison uses the independent Antique.jl code, resonances are checked against Lazauskas [46], and Ps− against Frolov [47]. The inclusion of runnable examples, documentation, and the honest reporting of non-converged default parameters in Tables 2 and 3 are commendable. If the caveats discussed below are addressed, the paper would be a useful resource for benchmark studies, method development, and teaching. The central method itself is standard and the derivation is sound.

major comments (3)
  1. [§5.2, Tables 2–3] The accessibility claim is undercut by the behavior shown in Tables 2 and 3. With the non-optimized parameters in the worked example (nmax=10, r1=0.1, rnmax=30), the Coulomb solver returns only 4 of the first 6 bound states and two spurious positive-energy eigenvalues (+0.0073, +0.603) without any warning or diagnostic. The 2-body optimizer GEM_Optim_2B restores state 6 but degrades the ground state from -0.499876 to -0.489956 (~2% error). Section 4.2 lists input validation but no spectral-quality/convergence check, and no optimizer or automatic diagnostic is described for GEM3B1D or ISGL. The paper should either provide automated guidance/diagnostics for choosing Gaussian ranges and detecting spurious states in all modules, or explicitly frame the package as requiring expert parameter tuning and narrow the 'accessible' claim accordingly.
  2. [§3.2, §7, and abstract] The abstract's 'arbitrary pair-interactions' is stronger than what is implemented. Section 3.2 restricts interactions to those satisfying f(r→0)∼r^{−α} with α<d+l+l′, and Section 7 concedes that broad strongly repulsive interactions remain challenging. None of the benchmarks probes such a boundary case (all use Coulomb, exponential-screened, Gaussian, or contact potentials). Please replace 'arbitrary' with a precise scope statement, or add a benchmark/example that demonstrates the advertised flexibility at the stated boundary.
  3. [Table 1 and §5] Table 1 lists 'Resonances via CSM: Yes' for all three modules, and the abstract promises calculation of resonant states. Yet Section 5 demonstrates complex scaling only for GEM2B (Table 5); no three-body resonance calculation is shown for GEM3B1D or ISGL. Because three-body resonances are a core motivation of the package, please add at least one three-body resonance benchmark/example, or explicitly state in Table 1 and the text that three-body CSM is implemented but not yet validated.
minor comments (3)
  1. [§5.3, Table 6] The 'Reference' column in Table 6 is from Ref. [32], a previous publication by the same author. The agreement is valuable but is not a fully independent validation; please state this explicitly or add an independent reference.
  2. [General (codebase publication)] The manuscript does not specify a version number or archived DOI for the released package. For reproducibility, please cite a tagged release (e.g., Zenodo) in addition to the GitHub repository.
  3. [§5.2 code snippet] The variables simax and redind are defined in the example script but not used; removing them would make the pedagogical code clearer.

Circularity Check

0 steps flagged

No significant circularity: all central validations are against external benchmarks; the one self-benchmark (Ref. [32]) is independent substantive support.

full rationale

The paper is a software/methods paper, not a derivation of new physical predictions. Its load-bearing validations are external: the Coulomb spectrum is compared with exact values from Antique.jl (Tables 2-4), the complex-scaling resonances with results of Lazauskas [46] (Table 5), and the Ps- binding energy and radii with results of Frolov [47] (Table 7). These references do not depend on the package's inputs or fitted parameters. The only benchmark against work co-authored by the present author is Table 6, which uses universal three-body energy ratios from Ref. [32]; those ratios are published universal quantities, not fitted numbers, and the package reproduces them from independently constructed Gaussian and contact interactions. No equation in the paper defines a predicted quantity in terms of the fitted parameters, and the optimization example in Tables 2-3 is explicitly presented as a trade-off rather than as a prediction. The 'arbitrary pair-interactions' claim is explicitly scoped by the boundedness condition in Sec. 3.2 and by admitted challenges in Sec. 7, which is a limitation, not a circular step. The Gaussian expansion method itself is the standard method of Ref. [19] and is not derived from the package's outputs. Accordingly, no circular step is present.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper adds no new physics entities, forces, or fitted constants; its ledger is methodological. The central fragility is that GEM's accuracy depends entirely on user-chosen basis parameters — demonstrated by the paper's own Table 2, where default parameters miss two bound Coulomb states and yield spurious positive energies. The two-body scale in the universality example is a chosen input, not a fitted prediction. The 'arbitrary interaction' feature is subject to the alpha < d + l + l' boundedness condition stated in Sec 3.2.

free parameters (3)
  • Two-body binding energy scale (target_e2 = -1e-3) = -1e-3
    Sec 5.3: the Gaussian/contact potential strength is scaled (v0GEMOptim) so the two-body energy is -10^-3, matching the universal regime of Ref [32]. This is a problem-defining input for the benchmark, not a hidden fit of the predicted three-body ratios; the three-body energies then come out as genuine solver output.
  • Optimized Gaussian ranges (r1, rnmax) = r1=0.871259, rnmax=45.664907
    Sec 5.2: GEM_Optim_2B adjusts the two-body basis ranges to reproduce the 6th Coulomb state, trading accuracy on the ground state (-0.489956 vs exact -0.5). Demonstrates the package's optimization feature; a methodological choice, disclosed in the text.
  • Complex-scaling angle theta_csm = 40.0 degrees
    Sec 5.2 resonance example: user-chosen rotation angle of the complex-scaling contour; resonance extraction via the is_in_triangle helper is delegated to a script whose selection criterion is not shown in the paper.
axioms (4)
  • domain assumption Center-of-mass motion separates from internal dynamics (no external forces, or harmonic confinement only)
    Sec 2.1, Eqs. (2)-(3): the package solves the internal Hamiltonian only; if external forces beyond harmonic are present the method is approximate.
  • domain assumption Gaussian basis expansion converges for the target states with tractable basis sizes
    Sec 2.2 / Sec 7: GEM is well-established (Ref [19]) but the paper itself notes challenges for broad strongly repulsive interactions; Tables 2-3 show that convergence depends critically on user-provided basis parameters.
  • domain assumption Complex scaling (CSM) exposes resonance poles without distorting bound-state results
    Sec 5.2 uses theta_csm=40 degrees per Ref [44] theory; the method is standard, but the complex-ranged Gaussian implementation (omega_cr=1.5) is an extension described in prose, without a formal derivation in this paper.
  • domain assumption Integrability/boundedness condition f(r→0) ~ r^-alpha with alpha < d + l + l'
    Sec 3.2: the 'arbitrary potential' claim is restricted to potentials satisfying this short-distance condition; potentials with harder singularities are out of scope.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of FewBodyToolkit.jl: a Julia package for solving quantum few-body problems." pith.science (2026). https://pith.science/paper/6IUDKRSF

@misc{pith2026251004447,
  author       = {Pith},
  title        = {Pith review of: FewBodyToolkit.jl: a Julia package for solving quantum few-body problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6IUDKRSF}},
  note         = {Machine review of arXiv:2510.04447}
}
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read the original abstract

Few-body physics explores quantum systems of a small number of particles, bridging the gap between single-particle and many-body regimes. To provide an accessible tool for such studies, we present FewBodyToolkit.jl, a Julia package for quantum few-body simulations. The package supports general two- and three-body systems in various spatial dimensions with arbitrary pair-interactions, and allows to calculate bound and resonant states. The implementation is based on the well-established Gaussian expansion method and we illustrate the package's capabilities through benchmarks and research examples. The package comes with documentation and examples, making it useful for research, teaching, benchmarking, and method development.

Figures

Figures reproduced from arXiv: 2510.04447 by Lucas Happ.

Figure 2
Figure 2. Figure 2: Densities of s-wave eigenstates of the Coulomb potential. Numerical solu￾tions obtained via FewBodyToolkit.jl are shown as solid lines, exact values provided by Antique.jl as markers. The resulting radial densities r 2 |ψ(r)| 2 are displayed in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scaling of runtime (left) and memory usage (right) with basis size 3 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Stabilization of three-body resonances to bound states in a continuum

    cond-mat.quant-gas 2025-03 unverdicted novelty 5.0

    A two-channel model demonstrates stabilization of three-body resonances into bound states in the continuum via parameter tuning, shown in 1D mass-imbalanced and 3D Efimov systems with magnetic field control.

  2. From three-body resonances to bound states in a continuum: pole trajectories

    quant-ph 2026-01 conditional novelty 4.0

    In a 1D three-body model, each of three parameter scans—interaction strength, range, and mass ratio—drives a resonance pole onto the real axis, forming at least one bound state in the continuum.

Reference graph

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    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.