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arxiv: 2510.27293 · v3 · submitted 2025-10-31 · ⚛️ nucl-th

Double pole S-matrix singularity in the continuum of ⁷Be

David Cardona Ochoa , Marek P{\l}oszajczak , Nicolas Michel , Simin Wang This is my paper

Pith reviewed 2026-05-18 03:26 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords exceptional pointS-matrix singularityGamow shell modelberyllium-7nuclear resonancescontinuumresonance doublet
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The pith

The 5/2- resonances in 7Be form a double-pole singularity in the S-matrix at an exceptional point.

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

This paper identifies a double pole singularity, known as an exceptional point, in the S-matrix for the 5/2- doublet of resonances in beryllium-7. It uses the Gamow shell model to show this by the merging of the two resonance wave functions and spectral functions. The singularity also appears in the behavior of spectroscopic factors and electromagnetic transitions. A reader would care because this reveals a precise mechanism by which nuclear resonances interact and coalesce in the presence of the scattering continuum.

Core claim

The double pole singularity of the S-matrix, the so-called exceptional point, associated with the 5/2^- doublet of resonances in the spectrum of 7Be has been identified in the framework of the Gamow shell model. The exceptional point singularity is demonstrated by the coalescence of wave functions and spectral functions of the two resonances, as well as by the singular behavior of spectroscopic factors and electromagnetic transitions.

What carries the argument

The exceptional point, defined as the double pole S-matrix singularity where two resonances coalesce, demonstrated through wave function coalescence and singular spectroscopic factors.

Load-bearing premise

The chosen Gamow shell model basis and effective interaction correctly reproduce the continuum coupling and resonance coalescence without creating artificial singularities.

What would settle it

Varying a parameter in the model and observing whether the two 5/2- resonance poles merge into a single double pole with coinciding wave functions at that point.

Figures

Figures reproduced from arXiv: 2510.27293 by David Cardona Ochoa, Marek P{\l}oszajczak, Nicolas Michel, Simin Wang.

Figure 1
Figure 1. Figure 1: The experimental spectrum of 7Be is compared with the GSM-CC spectrum. The widths (in keV) of resonances are given in brackets. On the right-hand side, one shows the spectrum for a Hamiltonian H(λEP ) for which the two 5/2 − states coalesce [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The energy eigenvalue trajectories (left panel) and the partial widths of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The phase rigidity of 5/2 − resonances is plotted as a function of their separation energy. The phase rigidity becomes equal zero at the EP [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Spectral function of both 5/2 − resonances for different values of the phase rigidity. two states. At the EP, the spectral functions perfectly coincide, indicating that the two resonances are identical and have lost their individuality. The more striking effects of the self-orthogonality can be seen in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the real part (upper panels) and imaginary part [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the real part (left panel) and imaginary part (right panel) of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

The double pole singularity of the $S$-matrix, the so-called exceptional point, associated with the $5/2^-$ doublet of resonances in the spectrum of $^{7}$Be has been identified in the framework of the Gamow shell model. The exceptional point singularity is demonstrated by the coalescence of wave functions and spectral functions of the two resonances, as well as by the singular behavior of spectroscopic factors and electromagnetic transitions.

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 / 2 minor

Summary. The manuscript claims that a double-pole S-matrix singularity (exceptional point) associated with the 5/2^- resonance doublet in 7Be has been identified within the Gamow shell model. The singularity is demonstrated through coalescence of the two resonances' wave functions and spectral functions, together with singular behavior in spectroscopic factors and electromagnetic transition matrix elements.

Significance. If the central claim is confirmed, the work would supply a concrete nuclear-physics realization of an exceptional point in the continuum, illustrating how resonance coalescence arises from explicit continuum coupling. The Gamow shell model is a suitable framework for this investigation, and the qualitative signatures listed are consistent with the expected phenomenology of non-Hermitian degeneracies.

major comments (2)
  1. [Results and Discussion] The central claim requires that coalescence observed inside a finite Berggren-basis diagonalization corresponds to a true double-pole singularity of the physical S-matrix. No explicit reconstruction of the S-matrix on the real axis or comparison against an independent continuum method (e.g., R-matrix or hyperspherical harmonics) is reported; without such a cross-check the coalescence could be an artifact of basis truncation or the chosen effective interaction.
  2. [Numerical results] No numerical value is given for the parameter (or set of parameters) at which the exceptional point occurs, nor are error estimates or sensitivity tests with respect to contour discretization and basis size provided. These data are necessary to establish that the reported coalescence is robust rather than an accidental feature of the particular numerical setup.
minor comments (2)
  1. The abstract would be clearer if it stated the specific Hamiltonian parameter that is varied to reach the exceptional point.
  2. [Method] Notation for the Berggren basis states and the contour deformation should be defined explicitly in the methods section to avoid ambiguity for readers unfamiliar with the Gamow shell model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below and indicate the revisions we plan to make to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Results and Discussion] The central claim requires that coalescence observed inside a finite Berggren-basis diagonalization corresponds to a true double-pole singularity of the physical S-matrix. No explicit reconstruction of the S-matrix on the real axis or comparison against an independent continuum method (e.g., R-matrix or hyperspherical harmonics) is reported; without such a cross-check the coalescence could be an artifact of basis truncation or the chosen effective interaction.

    Authors: We appreciate this comment, which highlights an important aspect of validating the physical relevance of our findings. The Gamow shell model is formulated to directly capture the S-matrix poles through the solution of the Schrödinger equation with outgoing boundary conditions in the complex energy plane using the Berggren basis. The observed coalescence of poles, wave functions, and singular behavior in observables is thus a manifestation of the exceptional point in this non-Hermitian framework. While an explicit reconstruction of the S-matrix on the real axis or a direct comparison with R-matrix calculations would indeed provide further cross-validation, such analyses are beyond the scope of the current work and would require significant additional computational effort. We note that the GSM has been extensively validated against other methods in previous studies for similar systems. In the revised manuscript, we will add a discussion clarifying why the observed coalescence in the complex plane corresponds to the physical S-matrix singularity and include additional convergence tests to address concerns about basis truncation. revision: partial

  2. Referee: [Numerical results] No numerical value is given for the parameter (or set of parameters) at which the exceptional point occurs, nor are error estimates or sensitivity tests with respect to contour discretization and basis size provided. These data are necessary to establish that the reported coalescence is robust rather than an accidental feature of the particular numerical setup.

    Authors: We agree with the referee that specifying the precise parameter value at which the exceptional point is encountered, together with quantitative error estimates and sensitivity tests, is important for demonstrating the robustness of the result. In our calculations, the exceptional point is located by varying the strength of the residual interaction or a scaling parameter in the Hamiltonian. We will include in the revised manuscript the specific numerical value of this parameter at which the two 5/2^- resonances coalesce. Additionally, we will provide tables showing the dependence on the number of basis states and the discretization of the complex contour, along with estimated uncertainties in the pole positions. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical observation of resonance coalescence in GSM is independent of the exceptional-point claim

full rationale

The paper solves the non-Hermitian GSM Hamiltonian in a Berggren basis and reports numerical coalescence of the two 5/2- poles together with singular behavior in wave functions, spectral functions, spectroscopic factors and EM transitions. This is a direct computational result obtained by diagonalization and parameter variation; it does not reduce by definition to the input Hamiltonian or to any self-citation. No fitted parameter is relabeled as a prediction, no uniqueness theorem is invoked from prior self-work, and the continuum approximation is an explicit modeling choice rather than a hidden tautology. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on the domain assumption that the Gamow shell model correctly captures continuum effects for this resonance doublet; no explicit free parameters or invented entities are named.

axioms (1)
  • domain assumption The Gamow shell model with its chosen effective interaction and basis accurately represents the continuum coupling and resonance doublet in 7Be.
    This assumption is required to interpret the observed coalescence as a physical exceptional point rather than a model artifact.

pith-pipeline@v0.9.0 · 5600 in / 1218 out tokens · 34671 ms · 2026-05-18T03:26:42.515152+00:00 · methodology

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Reference graph

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