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A one-dimensional three-body resonance can be converted into a stable bound state in the continuum by tuning any one of three parameters: interaction depth, interaction range, or mass ratio.

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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.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection A clean, useful incremental map of three-body BIC pole trajectories, but the central 'touching the real axis' claim needs convergence evidence before it fully lands. the 3 major comments →

arxiv 2601.11188 v2 pith:UOSB2GA2 submitted 2026-01-16 quant-ph cond-mat.quant-gasnucl-th

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

Lucas Happ This is my paper
classification quant-ph cond-mat.quant-gasnucl-th
keywords bound states in the continuumthree-body resonancespole trajectoriescomplex scaling methodone-dimensional few-body systemsmass-imbalanced mixturesdecay widthGaussian expansion method
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 aims to show that three-body resonances in a simple one-dimensional model can be made perfectly stable by varying the right parameter. Tracing the resonance pole in the complex energy plane, the author finds that for each of the three controls—interaction depth v0, interaction range μg, and mass ratio β—the pole eventually touches the real axis, where the decay width vanishes and the resonance becomes a bound state in the continuum: a stationary state embedded in the decay continuum. The trajectories differ by control: v0 and μg produce a single such crossing, while β produces several, hinting that the stabilization mechanism is more sensitive to the system's kinematic structure than to the details of the two-body interaction. A sympathetic reader should care because the result suggests that long-lived three-body states in quasi-one-dimensional ultracold systems are not rare accidents but can be reached robustly by sweeping ordinary experimental knobs.

Core claim

The central claim is the generality of BIC formation. In a one-dimensional three-body system of two identical bosons plus a distinguishable particle interacting through Gaussian pair potentials, the author varies three dimensionless parameters one at a time and tracks the complex three-body energy E^(3). In every scan—interaction strength v0, interaction range μg, and mass ratio β—the pole trajectory approaches and touches the real axis at least once, meaning Im(E^(3)) = 0 and hence the decay width Γ = -2 Im(E^(3)) vanishes; at that parameter value the resonance is a true bound state in the continuum, embedded in the continuum of a deep dimer plus a free boson. The v0 and μg scans each show

What carries the argument

The carrying object is the three-body resonance pole E^(3) in the complex energy plane, equivalently the pair (real energy, width Γ = -2 Im E^(3)). A bound state in the continuum is defined by the pole lying on the real axis, Γ = 0. To compute the pole, the paper uses complex scaling—rotating spatial coordinates into the complex plane so that resonances become discrete eigenvalues accessible to bound-state techniques—and expands the wavefunction as a superposition of Gaussian functions with different widths. Tracing the pole as one parameter sweeps produces the trajectory; wherever the trajectory crosses the real axis, a resonance has stabilized into a BIC.

Load-bearing premise

The claim rests on reading a computed decay width below about 1e-5 as an exact zero; the paper itself notes in Sec. 4 that these minima are limited by numerical precision and sampling coarseness, so a small but nonzero width that persists under convergence would make the 'bound states' numerical near-misses rather than true bound states.

What would settle it

Compute the decay width at the claimed BIC point (around v0 ≈ -4, μg = 1, β = 5) with a systematically enlarged Gaussian basis and varied complex-rotation angle; if the minimum width converges to a positive floor instead of to zero, or if the pole trajectory stops short of the real axis at higher resolution, the BIC claim fails. A complementary observable is scattering: at an exact BIC energy, the outgoing flux into the deep-dimer-plus-free-boson channel should be exactly zero.

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

If this is right

  • At the BIC parameter values, the three-body state does not decay into the deep dimer plus a free boson, despite sitting in that continuum; its lifetime in the model is parametrically infinite.
  • Because all three parameter scans produce at least one BIC, the existence of a three-body BIC in this model is robust rather than a single fine-tuned coincidence.
  • The mass-ratio scan yields several distinct BIC locations along one trajectory, so a single family of resonances can be stabilized at multiple, separate parameter values.
  • The nearly equal real energies at which the v0 and μg trajectories hit the real axis suggest a common stabilization point tied to the two-body spectrum, not to the particular way the interaction is changed.

Where Pith is reading between the lines

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

  • The author does not derive an analytic condition for Im E^(3) = 0; a natural follow-up is to express the BIC condition in terms of two-body binding energies, which would explain why the v0 and μg trajectories touch the real axis at nearly the same energy.
  • The multiple real-axis crossings in the β scan are reminiscent of log-periodic, scale-invariant structure familiar from resonant three-body systems; if that analogy holds, the β values at successive BICs should follow a geometric progression, a simple numerical check.
  • If the kinematic mechanism is the dominant one, the same pole-touching behaviour should occur for other short-range pair potentials such as square wells or delta functions; this is a testable prediction the paper does not make.
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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 / 4 minor

Summary. The paper studies a one-dimensional three-body system of two identical bosons and a distinguishable particle interacting via Gaussian potentials. Using the complex scaling method (CSM) and the Gaussian expansion method (GEM) implemented in the author's open-source FewBodyToolkit.jl, the author tracks the three-body resonance pole in the complex energy plane as the interaction strength v0, the interaction range parameter μg, or the mass ratio β is varied. The central claim is that each of these three parameter variations produces at least one bound state in the continuum (BIC), signaled by the pole 'touching' the real axis and by decay-width minima below 10^-5. The paper also reports that the mass-ratio variation yields a more regular trajectory with multiple BIC locations, while the interaction-parameter scans each show a single BIC. The authors interpret this as evidence for the robust parametric formation of three-body BICs and suggest that the underlying mechanism is more sensitive to kinematics than to the specifics of the two-body interaction.

Significance. If the numerical claim is supported, the paper provides a useful extension of the authors' earlier mechanism for three-body resonance stabilization to a broader set of parameters in a simple, solvable model. The computation is non-circular in the sense that no fitted parameters are used; the trajectories follow directly from solving the Schrödinger equation. The reliance on an open-source toolkit is a positive feature for reproducibility. However, the central identification of BICs currently rests on near-zero computed widths without a convergence study or quantitative error estimates. Given that the paper explicitly aims to demonstrate the robustness of BIC formation across parameters, the lack of numerical-convergence evidence is the main obstacle to accepting the claim as stated.

major comments (3)
  1. [Sec. 4, Fig. 3] The central BIC claim is supported by minima in Γ = −2 Im(E(3)) that drop below 10^-5. The text itself states that these minima are limited by numerical precision and sampling coarseness, so a value at the 1e-5 level is not, by itself, evidence of an exact zero. No GEM basis parameters (number of Gaussians, width ranges, Jacobi sets) or complex-scaling angle θ are reported, and no convergence study with respect to basis size or θ is presented. In CSM, an insufficient basis can produce spurious near-zero imaginary parts when a resonance lies near a threshold or branch cut. Please provide a convergence analysis at the candidate BIC parameters showing that the imaginary part decreases systematically with basis size and θ, and state the numerical values of the closest approach for each trajectory.
  2. [Sec. 3.2, Conclusion] The generality claim 'all parameter variations led to the formation of at least one BIC' includes the mass-ratio scan, but the only evidence for the β case is the trajectory in Fig. 2, with no corresponding width scan (unlike Fig. 3 for v0 and μg) and no quantitative measure of how close the pole comes to the real axis. The statement that it 'touches' the real axis at several points needs numerical thresholds or a β-width curve. Please add a quantitative width scan for β or provide a table of minimum |Im E| values and parameter positions for each of the claimed BIC locations.
  3. [Secs. 3.1 and 4] The observation that the v0- and μg-scans produce a BIC at nearly the same real energy is used to argue robustness, but the trajectory plots show no individual data points or uncertainty estimates. To make the comparison quantitative, please include the actual computed pole positions near the closest approach (or a table of energies and widths as functions of the scanned parameters). This would also help the reader judge whether the near-degeneracy in real energy is significant or a consequence of the coarse sampling.
minor comments (4)
  1. [Abstract / Introduction] The paper describes itself as confirming the authors' earlier work [22] and extending it to 'a broader selection of system parameters.' Please clarify explicitly what new results beyond [22] are being reported, since the present paper appears to be a parametric continuation of that work.
  2. [Fig. 3(b)] The y-axis label in panel (b) appears to have a typographical error ('E3' instead of 'E(3)'). Please correct.
  3. [Sec. 2] The statement that the two-body subsystem supports two bound states would be more informative if the actual v0 and μg ranges used in the scans were given, along with the corresponding dimer binding energies. This would define the 'available parameter interval' mentioned later and improve reproducibility.
  4. [Sec. 4] The sentence about finer parameter scans likely revealing deeper minima is a useful caveat, but a quantitative estimate of the numerical precision (e.g., the typical convergence tolerance of the CSM eigenvalues) would help the reader calibrate the significance of the reported 10^-5 floor.

Circularity Check

0 steps flagged

No significant circularity: the BIC claims are direct numerical observations from solving the stated model, not reductions to fitted inputs or to the authors' prior work.

full rationale

The paper's derivation chain is self-contained in the sense required here. It specifies a one-dimensional three-body Hamiltonian (Eqs. 1-3) and solves the complex-scaled Schrödinger equation directly with GEM (Eq. 2); no parameter is fitted to the presented output. Pole trajectories are computed numerically for v0, mu_g, and beta, and the BIC identification is based on the computed width approaching the real axis (Secs. 3.1 and 4). There is no step where a fitted quantity is renamed as a prediction, and no equation is equivalent to its output by construction. The self-citations to Refs. [21], [22], and [40] are contextual: they identify the model and numerical toolkit, but the existence of the BICs is established by the new calculations reported here, not by importing a conclusion from the authors' earlier work. The statement that the results 'confirmed' [22] is a consistency remark, not a load-bearing derivation. The only substantive limitation is numerical: the minimum widths are bounded by numerical precision and sampling coarseness, so a width below 1e-5 is treated as a zero without a convergence study. This is a correctness/evidence concern that the authors themselves flag, not a circularity. Therefore the circularity score is 0.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The paper adds no fitted constants: v0, mu_g, and beta are model inputs scanned over ranges. The numerical implementation, however, silently depends on basis-set and complex-scaling parameters that are not reported, and the BIC identification assumes numerical near-zeros represent exact zeros.

free parameters (2)
  • GEM Gaussian basis parameters
    Number, widths, and centers of basis functions are chosen by hand and not reported; the accuracy of computed pole positions and widths depends on them.
  • Complex scaling angle θ
    CSM requires choosing a rotation angle; convergence with θ is not shown, though converged results should be independent of it.
axioms (4)
  • domain assumption The Gaussian two-body potential with no boson-boson interaction captures the phenomenon (V(r)=v0 exp(-mu_g (r/r0)^2)).
    Introduced in Sec. 2; all results are for this single potential shape, so generality is assumed, not shown.
  • standard math Complex scaling transforms resonances into discrete L2 eigenvalues for this system.
    Invoked in Sec. 2; standard for dilation-analytic potentials, but no verification for the truncated numerical basis.
  • domain assumption The GEM basis is sufficiently complete for converged complex-scaled eigenenergies.
    No convergence checks reported; crucial for trusting width minima at 1e-5.
  • domain assumption A pole touching the real axis in the numerical scan corresponds to a true BIC.
    The paper equates a numerical width below 1e-5 (limited by precision/sampling, Sec. 4) with 'theoretically down to zero' BIC.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of From three-body resonances to bound states in a continuum: pole trajectories." pith.science (2026). https://pith.science/paper/UOSB2GA2

@misc{pith2026260111188,
  author       = {Pith},
  title        = {Pith review of: From three-body resonances to bound states in a continuum: pole trajectories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UOSB2GA2}},
  note         = {Machine review of arXiv:2601.11188}
}
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read the original abstract

We investigate the formation of three-body bound states in the continuum by tracing pole trajectories in the complex energy plane under variation of system parameters. Using a one-dimensional model of two identical bosons and a distinguishable particle interacting via Gaussian potentials, we systematically vary the interaction strength, interaction range, and mass ratio. Our results confirm the parametric nature of few-body bound states in a continuum (BIC) and extend this characterization to a broader set of system parameters. Specifically, we find that variations of both interaction parameters and the mass ratio can lead to the formation of at least one three-body BIC. However, the exact shape of trajectories differs, and for the mass ratio variation we find a more regular pattern with multiple BIC locations. These results suggest that the mechanism of few-body BIC formation is more sensitive to the kinematic structure of the problem than to the specific details of the two-body interaction.

discussion (0)

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Reference graph

Works this paper leans on

40 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Esry, B.D., Greene, C.H., Burke, J.P.: Recombination of Three Atoms in the Ultracold Limit. Phys. Rev. Lett.83(9), 1751–1754 (1999) https://doi.org/10. 1103/PhysRevLett.83.1751

    1999

  2. [2]

    Nature440, 315 (2006)

    Kraemer, T., Mark, M., Waldburger, P., Danzl, J.G., Chin, C., Engeser, B., Lange, A.D., Pilch, K., Jaakkola, A., N¨ agerl, H.-C., Grimm, R.: Evidence for Efimov quantum states in an ultracold gas of caesium atoms. Nature440, 315 (2006)

    2006

  3. [3]

    Endo, S., Epelbaum, E., Naidon, P., Nishida, Y., Sekiguchi, K., Takahashi, Y.: Three-body forces and Efimov physics in nuclei and atoms. Eur. Phys. J. A61(1), 9 (2025) https://doi.org/10.1140/epja/s10050-024-01467-4

    doi:10.1140/epja/s10050-024-01467-4 2025

  4. [4]

    Naturwissenschaften15(1), 16–17 (1927) https://doi.org/10

    Burrau, Ø.: Berechnung des Energiewertes des Wasserstoffmolekel-Ions (H2+) im Normalzustand. Naturwissenschaften15(1), 16–17 (1927) https://doi.org/10. 1007/BF01504875

    1927

  5. [5]

    Stationary States of Nuclei

    Bethe, H.A., Bacher, R.F.: Nuclear Physics A. Stationary States of Nuclei. Rev. Mod. Phys.8(2), 82–229 (1936) https://doi.org/10.1103/RevModPhys.8.82

    doi:10.1103/revmodphys.8.82 1936

  6. [6]

    Thomas, L.H.: The Interaction Between a Neutron and a Proton and the Struc- ture of H3. Phys. Rev.47(12), 903–909 (1935) https://doi.org/10.1103/PhysRev. 47.903

    doi:10.1103/physrev 1935

  7. [7]

    Physics Letters B33(8), 563–564 (1970) https://doi.org/10.1016/ 0370-2693(70)90349-7

    Efimov, V.: Energy levels arising from resonant two-body forces in a three- body system. Physics Letters B33(8), 563–564 (1970) https://doi.org/10.1016/ 0370-2693(70)90349-7

    1970

  8. [8]

    Naidon, P., Endo, S.: Efimov physics: a review. Rep. Prog. Phys.80(5), 056001 (2017) https://doi.org/10.1088/1361-6633/aa50e8 7

    doi:10.1088/1361-6633/aa50e8 2017

  9. [9]

    Soviet Physics JETP 12, 1014 (1961)

    Faddeev, L.D.: Scattering theory for a three-particle system. Soviet Physics JETP 12, 1014 (1961)

    1961

  10. [10]

    Guo, F.-K., Hanhart, C., Meißner, U.-G., Wang, Q., Zhao, Q., Zou, B.-S.: Hadronic molecules. Rev. Mod. Phys.90(1), 015004 (2018) https://doi.org/10. 1103/RevModPhys.90.015004

    2018

  11. [11]

    Arifi, A.J., Happ, L., Ohno, S., Oka, M.: Structure of heavy mesons in the light- front quark model. Phys. Rev. D110(1), 014020 (2024) https://doi.org/10.1103/ PhysRevD.110.014020

    2024

  12. [12]

    Physics Reports347(5), 373–459 (2001) https:// doi.org/10.1016/S0370-1573(00)00107-1

    Nielsen, E., Fedorov, D.V., Jensen, A.S., Garrido, E.: The three-body problem with short-range interactions. Physics Reports347(5), 373–459 (2001) https:// doi.org/10.1016/S0370-1573(00)00107-1

    work page doi:10.1016/s0370-1573(00)00107-1 2001

  13. [13]

    Nature514(7522), 343–347 (2014) https: //doi.org/10.1038/nature13832

    Kazimierczuk, T., Fr¨ ohlich, D., Scheel, S., Stolz, H., Bayer, M.: Giant Rydberg excitons in the copper oxide Cu2O. Nature514(7522), 343–347 (2014) https: //doi.org/10.1038/nature13832

    doi:10.1038/nature13832 2014

  14. [14]

    Belov, P.A., Morawetz, F., Kr¨ uger, S.O., Scheuler, N., Rommel, P., Main, J., Giessen, H., Scheel, S.: Energy states of Rydberg excitons in finite crystals: From weak to strong confinement. Phys. Rev. B109(23), 235404 (2024) https://doi. org/10.1103/PhysRevB.109.235404

    work page doi:10.1103/physrevb.109.235404 2024

  15. [15]

    Pires, R., Ulmanis, J., H¨ afner, S., Repp, M., Arias, A., Kuhnle, E.D., Wei- dem¨ uller, M.: Observation of Efimov Resonances in a Mixture with Extreme Mass Imbalance. Phys. Rev. Lett.112(25), 250404 (2014) https://doi.org/10.1103/ PhysRevLett.112.250404

    2014

  16. [16]

    Springer, Dordrecht (1989)

    Kukulin, V.I., Krasnopol’sky, V.M., Hor´ aˇ cek, J.: Theory of Resonances. Springer, Dordrecht (1989). https://doi.org/10.1007/978-94-015-7817-2 . http: //link.springer.com/10.1007/978-94-015-7817-2

    work page doi:10.1007/978-94-015-7817-2 1989

  17. [17]

    Physics Reports302(5), 212–293 (1998) https: //doi.org/10.1016/S0370-1573(98)00002-7

    Moiseyev, N.: Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Physics Reports302(5), 212–293 (1998) https: //doi.org/10.1016/S0370-1573(98)00002-7

    doi:10.1016/s0370-1573(98)00002-7 1998

  18. [18]

    Pen’kov, F.M.: Lifetime of Efimov states of negative two-atom ions. Phys. Rev. A60(5), 3756–3763 (1999) https://doi.org/10.1103/PhysRevA.60.3756

    work page doi:10.1103/physreva.60.3756 1999

  19. [19]

    Nielsen, E., Suno, H., Esry, B.D.: Efimov resonances in atom-diatom scatter- ing. Phys. Rev. A66(1), 012705 (2002) https://doi.org/10.1103/PhysRevA.66. 012705

    work page doi:10.1103/physreva.66 2002

  20. [20]

    Pricoupenko, L.: Crossover in the Efimov spectrum. Phys. Rev. A82(4), 043633 (2010) https://doi.org/10.1103/PhysRevA.82.043633 8

    work page doi:10.1103/physreva.82.043633 2010

  21. [21]

    Few-Body Syst65(2), 38 (2024) https://doi.org/ 10.1007/s00601-024-01900-w

    Happ, L., Naidon, P., Hiyama, E.: Mass Ratio Dependence of Three-Body Reso- nance Lifetimes in 1D and 3D. Few-Body Syst65(2), 38 (2024) https://doi.org/ 10.1007/s00601-024-01900-w

    work page doi:10.1007/s00601-024-01900-w 2024

  22. [22]

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

    Happ, L., Naidon, P.: Stabilization of three-body resonances to bound states in a continuum. arXiv (2025). https://doi.org/10.48550/arXiv.2503.02037 . http: //arxiv.org/abs/2503.02037

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2503.02037 2025

  23. [23]

    Laird, E.K., Kirk, T., Parish, M.M., Levinsen, J.: Long-lived trimers in a quasi- two-dimensional Fermi system. Phys. Rev. A97(4), 042711 (2018) https://doi. org/10.1103/PhysRevA.97.042711

    work page doi:10.1103/physreva.97.042711 2018

  24. [24]

    Nature Phys1(1), 23–30 (2005) https://doi.org/10.1038/nphys138

    Bloch, I.: Ultracold quantum gases in optical lattices. Nature Phys1(1), 23–30 (2005) https://doi.org/10.1038/nphys138

    doi:10.1038/nphys138 2005

  25. [25]

    Physics Reports1042, 1–108 (2023) https: //doi.org/10.1016/j.physrep.2023.10.004

    Mistakidis, S.I., Volosniev, A.G., Barfknecht, R.E., Fogarty, T., Busch, Th., Foer- ster, A., Schmelcher, P., Zinner, N.T.: Few-body Bose gases in low dimensions—A laboratory for quantum dynamics. Physics Reports1042, 1–108 (2023) https: //doi.org/10.1016/j.physrep.2023.10.004

    work page doi:10.1016/j.physrep.2023.10.004 2023

  26. [26]

    Blume, D.: Few-body physics with ultracold atomic and molecular systems in traps. Rep. Prog. Phys.75(4), 046401 (2012) https://doi.org/10.1088/0034-4885/ 75/4/046401

    doi:10.1088/0034-4885/ 2012

  27. [27]

    Happ, L., Zimmermann, M., Betelu, S.I., Schleich, W.P., Efremov, M.A.: Uni- versality in a one-dimensional three-body system. Phys. Rev. A100(1), 012709 (2019) https://doi.org/10.1103/PhysRevA.100.012709

    work page doi:10.1103/physreva.100.012709 2019

  28. [28]

    Happ, L., Efremov, M.A.: Proof of universality in one-dimensional few-body sys- tems including anisotropic interactions. J. Phys. B: At. Mol. Opt. Phys.54(21), 21–01 (2021) https://doi.org/10.1088/1361-6455/ac3b3f

    work page doi:10.1088/1361-6455/ac3b3f 2021

  29. [29]

    Happ, L., Zimmermann, M., Efremov, M.A.: Universality of excited three-body bound states in one dimension. J. Phys. B: At. Mol. Opt. Phys.55(1), 015301 (2022) https://doi.org/10.1088/1361-6455/ac3cc8

    work page doi:10.1088/1361-6455/ac3cc8 2022

  30. [30]

    Schnurrenberger, T., Happ, L., Efremov, M.A.: Borromean states in a one- dimensional three-body system. Phys. Rev. Res.7(1), 013090 (2025) https: //doi.org/10.1103/PhysRevResearch.7.013090

    work page doi:10.1103/physrevresearch.7.013090 2025

  31. [31]

    Nat Rev Mater1(9), 1–13 (2016) https://doi.org/10.1038/ natrevmats.2016.48

    Hsu, C.W., Zhen, B., Stone, A.D., Joannopoulos, J.D., Soljaˇ ci´ c, M.: Bound states in the continuum. Nat Rev Mater1(9), 1–13 (2016) https://doi.org/10.1038/ natrevmats.2016.48

    2016

  32. [32]

    Wigner: ¨ uber merkw¨ urdige diskrete Eigenwerte

    Neumann, J., E. Wigner: ¨ uber merkw¨ urdige diskrete Eigenwerte. Phys. Z.30, 465 (1929) 9

    1929

  33. [33]

    Marinica, D.C., Borisov, A.G., Shabanov, S.V.: Bound States in the Continuum in Photonics. Phys. Rev. Lett.100(18), 183902 (2008) https://doi.org/10.1103/ PhysRevLett.100.183902

    2008

  34. [34]

    Commun Phys3(1), 91 (2020) https://doi.org/10.1038/s42005-020-0353-z

    Pankin, P.S., Wu, B.-R., Yang, J.-H., Chen, K.-P., Timofeev, I.V., Sadreev, A.F.: One-dimensional photonic bound states in the continuum. Commun Phys3(1), 91 (2020) https://doi.org/10.1038/s42005-020-0353-z

    doi:10.1038/s42005-020-0353-z 2020

  35. [35]

    Physics Reports82(2), 31–177 (1982) https://doi.org/10.1016/0370-1573(82)90014-X

    Badalyan, A.M., Kok, L.P., Polikarpov, M.I., Simonov, Yu.A.: Resonances in coupled channels in nuclear and particle physics. Physics Reports82(2), 31–177 (1982) https://doi.org/10.1016/0370-1573(82)90014-X

    doi:10.1016/0370-1573(82)90014-x 1982

  36. [36]

    In: Citro, R., Lewenstein, M., Rubio, A., Schleich, W.P., Wells, J.D., Zank, G.P

    Schleich, W.P., Tk´ aˇ cov´ a, I., Happ, L.: Insights Into Complex Functions. In: Citro, R., Lewenstein, M., Rubio, A., Schleich, W.P., Wells, J.D., Zank, G.P. (eds.) Sketches of Physics: The Celebration Collection, pp. 127–159. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-32469-7 5 . https://doi.org/10.1007/ 978-3-031-32469-7 5

    doi:10.1007/978-3-031-32469-7 2023

  37. [37]

    Progress in Particle and Nuclear Physics51(1), 223–307 (2003) https: //doi.org/10.1016/S0146-6410(03)90015-9

    Hiyama, E., Kino, Y., Kamimura, M.: Gaussian expansion method for few-body systems. Progress in Particle and Nuclear Physics51(1), 223–307 (2003) https: //doi.org/10.1016/S0146-6410(03)90015-9

    work page doi:10.1016/s0146-6410(03)90015-9 2003

  38. [38]

    Happ, L., Efremov, M.A., Nha, H., Schleich, W.P.: Sufficient condition for a quan- tum state to be genuinely quantum non-Gaussian. New J. Phys.20(2), 023046 (2018) https://doi.org/10.1088/1367-2630/aaac25

    work page doi:10.1088/1367-2630/aaac25 2018

  39. [39]

    PRX Quantum2(3), 030204 (2021) https://doi.org/10.1103/PRXQuantum.2.030204

    Walschaers, M.: Non-Gaussian Quantum States and Where to Find Them. PRX Quantum2(3), 030204 (2021) https://doi.org/10.1103/PRXQuantum.2.030204

    doi:10.1103/prxquantum.2.030204 2021

  40. [40]

    arXiv (2025)

    Happ, L.: FewBodyToolkit.jl: a Julia package for solving quantum few-body prob- lems. arXiv (2025). https://doi.org/10.48550/arXiv.2510.04447 . http://arxiv. org/abs/2510.04447 10

    Pith/arXiv arXiv doi:10.48550/arxiv.2510.04447 2025

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.