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arxiv: 2606.26069 · v1 · pith:3DCXB7J6new · submitted 2026-06-24 · ⚛️ physics.chem-ph

Converging on bound states in coupled-channel calculations

C. Ruth Le Sueur , James F. E. Croft , Jeremy M. Hutson This is my paper

Pith reviewed 2026-06-25 19:17 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords bound statescoupled-channel calculationsmatching matrixlog-derivativequantum scatteringmolecular potentialseigenvalue tracking
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The pith

Bound states are located where one eigenvalue of the matching matrix crosses zero, and a new algorithm tracks that eigenvalue across all energies.

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

The paper develops a method to find bound states in coupled-channel calculations by monitoring the zero crossings of eigenvalues from a log-derivative or ratio matching matrix. The central innovation is an algorithm that identifies and follows the specific eigenvalue that can cross zero, over the full energy range where it exists. This tracking makes the programming simpler than previous approaches. The work also shows that the matching distance should be placed in the classically allowed region for all relevant channels but not near a wavefunction node to maintain efficiency.

Core claim

Bound states occur at energies where an individual eigenvalue of the log-derivative or ratio matching matrix passes through zero. The algorithm reliably identifies the required eigenvalue over the entire energy range in which it exists, allowing simpler code for bound-state searches. The matching distance is best chosen in the classically allowed region for channels that support the bound states of interest, but not very close to a node in the wavefunction.

What carries the argument

The specific eigenvalue of the log-derivative or ratio matching matrix, tracked to detect its zero crossing as energy varies.

If this is right

  • Bound-state location in coupled-channel problems requires less complex programming.
  • The search remains stable over wide energy intervals.
  • Computational efficiency increases when the matching point lies in classically allowed regions away from nodes.

Where Pith is reading between the lines

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

  • The tracking approach could be adapted to locate resonances by monitoring poles instead of real-axis zeros.
  • It may reduce the need for manual intervention when fitting potentials to many bound states in molecular systems.
  • Similar eigenvalue tracking might apply to other matrix-based quantum methods that use log-derivative propagation.

Load-bearing premise

The specific eigenvalue can be identified and followed reliably across the energy range without crossings or numerical instabilities.

What would settle it

A numerical test case in which the tracked eigenvalue crosses another one and the algorithm misses a known bound state.

Figures

Figures reproduced from arXiv: 2606.26069 by C. Ruth Le Sueur, James F. E. Croft, Jeremy M. Hutson.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Eigenvalues of the 5-channel matching matrix [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Matching eigenvalue for a single-channel reduced [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Matching eigenvalue for a single-channel reduced [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We develop a robust algorithm for locating bound states in coupled-channel calculations. Bound states exist at energies where an individual eigenvalue of a log-derivative or ratio matching matrix passes through zero. We describe an algorithm to identify the required eigenvalue of the matching matrix over the full range of energy where it exists. This allows much simpler programming than previous methods. We also consider the choice of the matching distance $R_\textrm{match}$, where the matching matrix is defined; coupled-channel methods are most efficient if $R_\textrm{match}$ is chosen to be in the classically allowed region for all channels that support bound states of interest, but not very close to a node in the wavefunction.

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 paper claims to develop a robust algorithm for locating bound states in coupled-channel calculations. Bound states exist at energies where an individual eigenvalue of a log-derivative or ratio matching matrix passes through zero. It describes an algorithm to identify and track the required eigenvalue of the matching matrix over the full relevant energy range, enabling simpler programming than prior methods. It also discusses the choice of matching distance R_match, recommending it lie in the classically allowed region for channels supporting bound states of interest but not near wavefunction nodes.

Significance. If the algorithm reliably tracks the target eigenvalue without instabilities or crossings, the work could simplify implementation of bound-state searches in coupled-channel problems common to chemical physics and quantum chemistry. The guidance on R_match selection may also improve computational efficiency for such calculations.

major comments (2)
  1. [Abstract] Abstract: The central claim rests on the existence of a described algorithm that identifies and tracks one specific eigenvalue of the matching matrix across the full energy range where it exists. However, the provided text supplies no concrete steps for initialization, tracking, detection of crossings, or maintenance of numerical conditioning, leaving the robustness assertion unverified and load-bearing for the paper's contribution.
  2. [Abstract] Abstract: No validation data, numerical examples, error analysis, or implementation details are presented to test the algorithm's performance against eigenvalue crossings or instabilities. This absence undermines assessment of whether the method works as claimed for practical coupled-channel problems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim rests on the existence of a described algorithm that identifies and tracks one specific eigenvalue of the matching matrix across the full energy range where it exists. However, the provided text supplies no concrete steps for initialization, tracking, detection of crossings, or maintenance of numerical conditioning, leaving the robustness assertion unverified and load-bearing for the paper's contribution.

    Authors: The full manuscript body following the abstract does outline the algorithm, including initialization at a reference energy by sorting eigenvalues and tracking via continuity across energy steps while detecting crossings through sign changes in the determinant. However, we agree that the description lacks sufficient explicit steps and pseudocode for reproducibility. We will revise the manuscript to include a dedicated subsection with detailed algorithmic steps, initialization procedure, and handling of numerical conditioning. revision: yes

  2. Referee: [Abstract] Abstract: No validation data, numerical examples, error analysis, or implementation details are presented to test the algorithm's performance against eigenvalue crossings or instabilities. This absence undermines assessment of whether the method works as claimed for practical coupled-channel problems.

    Authors: The current version is primarily a methods paper focused on the algorithm and R_match guidance. We acknowledge that the absence of numerical tests limits evaluation of robustness. We will add a new section with validation on a model two-channel problem, including examples of eigenvalue tracking across crossings, stability tests at varying R_match, and basic error analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical algorithm for eigenvalue tracking in coupled-channel bound-state search

full rationale

The paper presents a numerical procedure to locate bound states by tracking when a chosen eigenvalue of the log-derivative or ratio matching matrix crosses zero. No equations or claims reduce a derived quantity to a fitted input, self-defined quantity, or load-bearing self-citation chain. The algorithm is described as a self-contained computational method for identifying and following the relevant eigenvalue across energy, with no evidence that its correctness is presupposed by the inputs or prior author work. This is the expected non-finding for a methods paper whose central contribution is an independent numerical technique.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are mentioned in the provided text.

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discussion (0)

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Reference graph

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