Regge poles of analogous rotating black holes in binary Bose-Einstein condensates: The gapped excitations

Da-Shin Lee; Tien Hsieh; Wei-Can Syu

REVIEW 3 major objections 6 minor 52 references

Regge poles of analogous rotating black holes in binary Bose-Einstein condensates: The gapped excitations

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A localized mass-shell perturbation splits each Regge pole of a draining bathtub vortex into inner and outer branches, linking black-hole spectral instability to a tabletop Bose-Einstein condensate scattering experiment.

desk verdict A credible within-subfield extension of Regge-pole instability to a binary BEC analog, with a genuinely new co-rotating vs counter-rotating stability asymmetry, but the numerics need convergence documentation before I would bet on the bifurcations. read the letter →

arxiv 2507.15403 v2 pith:3TUOBQPI submitted 2025-07-21 gr-qc cond-mat.quant-gas

Wei-Can Syu , Tien Hsieh , Da-Shin Lee This is my paper · ORCID

classification gr-qccond-mat.quant-gas

keywords drainingbathtubvortexReggepolesquasinormalmodesbinaryBose-Einsteincondensategappedexcitationsspectralinstabilitycomplexangularmomentumanalogblackholes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a two-component Bose-Einstein condensate shaped as a draining bathtub vortex can be used as a tabletop model of a rotating black hole surrounded by a shell of matter. It claims that a localized energy-gap perturbation, written as $\epsilon\,\mathrm{sech}^2[\alpha(r-r_0)]$ and generated by a tunable Rabi coupling, splits the Regge pole spectrum--the complex angular momenta at which resonant scattering occurs--into inner and outer branches, and that this bifurcation is a genuine spectral destabilization according to the criterion $|\Delta\lambda/\lambda|\gg\epsilon$. If correct, the instability of black-hole quasinormal spectra under environmental perturbations becomes observable in a cold-atom experiment through the scattering interference pattern, with counter-rotating modes destabilizing more readily than co-rotating ones.

What carries the argument

The central object is the Regge pole: a complex angular momentum $m=\mathrm{Re}\,m+i\,\mathrm{Im}\,m$ at which the incoming-wave coefficient at infinity vanishes for fixed real frequency, so the scattering amplitude has a pole after converting the partial-wave sum into a contour integral. The machinery that carries the argument is a modified continued-fraction method built on a radial-series expansion about a matching point $b>r_0+1/\alpha$ outside the bump, with initial coefficients obtained by integrating the radial equation from the horizon to $b$ and a four-term recurrence reduced by Gaussian elimination to the continued-fraction condition. The paper judges stability by comparing the relative pole displacement $|\Delta\lambda_{\pm,n}^{(\epsilon)}/\lambda_{\pm,n}^{(0)}|$ with the perturbation amplitude $\epsilon$. A separate semiclassical calculation perturbs null geodesics by a small constant mass term and produces the deflection angle and interference-wavelength shift used to interpret the scattering imprints.

What would settle it

Repeat the modified continued-fraction computation with explicit truncation order and convergence tolerance over the same parameter grid ($\omega=0.1,1.0$; $r_0=10$ to $100$; $\epsilon=10^{-3}$ to $10^{-1}$; $\alpha=3/2$). If the bifurcation positions or overtaking-jump locations move with truncation order, or if the pole-residue sum fails to reproduce the partial-wave scattering amplitude, the claimed destabilization is a numerical artifact. In the laboratory, a low-frequency planar-wave scattering experiment on a binary-BEC vortex with a localized Rabi-coupling bump should show the large-angle deviation from the unperturbed interference pattern; its absence over the full angular range would falsify the paper's central prediction.

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Extended reading notes

Core claim

In the complex angular momentum plane, the unperturbed Regge poles split into an inner branch, with smaller $|\mathrm{Re}\,m|$, and an outer branch, with larger $|\mathrm{Re}\,m|$, as soon as the bump perturbation is switched on. The paper takes this bifurcation as a sign of destabilization: a pole is unstable when $|\Delta\lambda_{\pm,n}^{(\epsilon)}/\lambda_{\pm,n}^{(0)}|\gg\epsilon$. As the perturbation amplitude grows from $10^{-3}$ to $10^{-1}$, the onset of bifurcation moves from higher overtones toward the fundamental mode; in the rotating background it reaches the fundamental for counter-rotating modes while the co-rotating modes remain more resistant. Shifting the bump position $r_0$ drives large migrations and overtaking jumps, with relative pole displacements as large as $70\epsilon$ for the second overtone at $\omega=1.0$ and $50\epsilon$ for the first overtone at $\omega=0.1$. These events leave imprints on the scattering amplitude: overtone migration shows up at small scattering angles and fundamental-pole migration at large scattering angles, which the paper connects to the scattering interference pattern in the condensate.

Load-bearing premise

The whole bifurcation picture rests on the unverified assumption that the numerical procedure finds the true resonance positions for every displayed bump position, width, strength, and frequency; the paper reports no convergence checks.

Editorial extensions

If this is right

For $\ell=0$ the perturbed spectrum stays symmetric, $m_{+n}^{(\epsilon)}=-m_{-n}^{(\epsilon)}$; for $\ell=1$ the rotation breaks that symmetry and the counter-rotating branch destabilizes first.
The destabilization criterion separates stable from unstable poles: ratios far above $\epsilon$ mark bifurcated poles, and as $\epsilon$ rises the bifurcation onset travels from high overtones down to the fundamental mode.
In the rotating background, counter-rotating fundamental poles show overtaking jumps near $r_0=30,60,95$ at low frequency, while co-rotating fundamental poles trace closed trajectories with much smaller relative displacements.
Complex angular momentum decomposition attributes the extra scattering oscillations to the outer-branch poles and the fundamental pole; the inner-branch contribution is subdominant in the rotating case.
For $\omega=0.1$, the perturbed scattering amplitude deviates from the unperturbed one at essentially all scattering angles, with the largest effects at small angles from overtone destabilization and at large angles from fundamental destabilization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the bifurcation picture holds, the onset curve in the $(\epsilon,r_0)$ plane becomes a sharp experimental prediction: a cold-atom experiment could map where the counter-rotating fundamental pole first jumps and compare that curve with the continued-fraction calculation.
The semiclassical wavelength shift, $\lambda\approx\pi/(2\omega r_E)-(\epsilon/E^2)(\hat a_-/\bar a_-+\hat a_+/\bar a_+)\pi/(8\omega^2 r_E^2)$, is an independent observable; phase-sensitive imaging of the interference pattern could probe the mass-shell effect even where individual pole bifurcations are hard to resolve.
Because the destabilization criterion is scale-free, the same environmental-bump model applied to astrophysical rotating black holes would predict analogous overtaking jumps in the retrograde channel; the BEC vortex is the only controlled system in which that prediction could be tested.
A natural extension would vary both $\epsilon$ and $\omega$ at fixed $\alpha=3/2$ to map the frequency boundary where the fundamental Regge pole first destabilizes, checking whether the co-rotating/counter-rotating asymmetry persists across the whole frequency range.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid, within-subfield extension of the dirty-black-hole Regge pole program to a binary BEC analog, and the co-rotating versus counter-rotating stability asymmetry is a genuinely new observation. The main claim—that a localized mass-shell bump bifurcates the Regge poles and that migration or overtaking leaves imprints in the scattering amplitude—is credible but not yet independently checkable from the text.

What is new: Refs. [30,31] studied scattering by a draining bathtub vortex without a mass-shell perturbation; Refs. [32,33] studied dirty black holes but not the BEC analog or gapped excitations; Ref. [22] treated gapped excitations but not Regge poles. This paper puts those pieces together. The unperturbed Regge pole spectrum and the low-frequency scattering profiles are benchmarked against Ref. [31], which is the right external check. The semiclassical section is a clean extension of Ref. [31] to a homogeneous mass term, and the wavelength shift in Eq. (43) is derived in an appendix rather than asserted.

Soft spots, in order of seriousness. First, the numerical backbone—the modified continued fraction method in Appendix A—does not report truncation order, Gaussian-elimination depth, the chosen matching point b, or accuracy estimates. The minimal-solution condition b/2 < r0 + 1/alpha < b is passed off with a citation to Ref. [52]. That matters because the destabilization criterion uses epsilon down to 1e-3, so the claimed 50-70 epsilon effects require root accuracy around 1e-4, and no such accuracy is demonstrated. This is exactly the kind of gap that can hide a numerical artifact; I see no evidence of one, but I also cannot rule it out. Second, the semiclassical section replaces the spatially dependent mass shell with a homogeneous m^2 = epsilon and says it qualitatively agrees with the full numerics without showing a direct comparison. Minor, since the semiclassical part is explicitly heuristic. Third, the experimental observability claim has no noise budget. Minor for a theory paper, but it softens the claim that the interference pattern can be observed.

If I were refereeing, I would ask for: (i) convergence data in m, omega, and r0 for representative cases; (ii) explicit truncation order and b-independence checks; (iii) one direct comparison between the semiclassical cross section and the full numerical result. These are addressable. The central bifurcation observation looks plausible and internally consistent across the figures, and the co-rotating versus counter-rotating asymmetry is a real result that deserves scrutiny. Yes, send it to review; a serious referee can push for the missing numerical detail, but this is not a desk-reject.

Referee Report

3 major / 6 minor

Summary. The manuscript studies Regge poles in the draining bathtub vortex analog of a rotating black hole, modeled in a two-component Bose-Einstein condensate with a spatially dependent Rabi-induced mass term mu_s^2(r)=epsilon sech^2[alpha(r-r0)]. The authors first compute a semiclassical scattering amplitude for a homogeneous mass perturbation, then numerically obtain the Regge-pole spectrum using a modified continued fraction method. They report a bifurcation of the spectrum into inner and outer branches, interpret this as spectral destabilization under the criterion |Delta lambda/lambda| >> epsilon, and find that corotating modes are more stable than counterrotating ones. They also track pole migration with changing r0, identify overtaking jumps, and show corresponding imprints on the differential scattering amplitude at small and large scattering angles.

Significance. If the numerical results are robust, the paper provides a concrete tabletop analog system for black-hole spectral instability, with falsifiable predictions: the epsilon-scaling criterion, the angular-region dependence of the scattering imprint, and the co/counterrotating asymmetry. The use of the complex angular momentum method to connect pole branches to scattering observables is a strength, as is the benchmarking of the unperturbed Regge poles against Ref. [31]. However, the central bifurcation and overtaking-jump claims rest on a numerical method whose convergence and accuracy are not documented, and the semiclassical comparison is asserted rather than demonstrated. These gaps prevent the claims from being fully supported at present.

major comments (3)

[Appendix A, Figs. 4-7] The central numerical claim rests entirely on the modified continued fraction method, but no convergence tests are reported. The paper does not state the truncation order of the continued fraction (A14), the number of Gaussian-elimination steps, the chosen matching point b (only constrained by b>r0+1/alpha), or the accuracy of the horizon-to-b integration that sets a0 and a1 in (A7)-(A8). The reference to Ref. [52] for the minimal-solution condition is not a demonstration for the present potential and parameter range, which includes r0 up to 100, omega=0.1 and 1.0, and epsilon down to 1e-3. Since bifurcations and overtaking jumps in Figs. 4-7 could in principle be numerical artifacts of an inadequately converged or improperly matched series solution, please provide convergence tests with respect to truncation order, matching point b, and integration tolerance, and state the achieved absolute accuracy of the Regge-pole locations.
[Section III, after Eq. (43)] The text states that the semiclassical results with a small spatially independent mass perturbation 'qualitatively agree with the full numerical studies,' but no comparison is shown. Fig. 3 is a semiclassical, homogeneous-mass calculation, while Figs. 8 and 11 are full numerical calculations with the spatially dependent bump (9) that vanishes at infinity. Because the semiclassical interpretation is used to explain the scattering-angle dependence of the perturbation effect, this agreement should be quantified by overlaying the semiclassical and numerical cross sections for the same parameters, or the claim should be explicitly qualified as an analogy rather than an agreement.
[Section IV B, Figs. 5-7] The identification of 'overtaking jumps' is inferred from sequences of roots as r0 is increased by Delta r0=1, but the paper does not describe the root-tracking procedure or how it distinguishes a genuine pole crossing or branch switch from a discontinuity caused by the numerical solver. This distinction is load-bearing for the destabilization claim. Please specify the tracking algorithm, and demonstrate that the jumps persist under increased truncation and varied matching points, for example by verifying continuity of the residues or by cross-checking a few trajectories with an independent method.

minor comments (6)

[Eq. (46)] The definition lambda_(epsilon)_(±,n)=m_(epsilon)_(±,n) l is problematic for the nonrotating case l=0, where it would give lambda=0 and an undefined ratio. This is presumably a typo, and the definition should be clarified.
[Section IV A] The phrase 'modified continuous fraction method' should be 'modified continued fraction method.'
[Appendix A] The reference to 'series expansion (S34)' should refer to Eq. (A1).
[Appendix A, Eqs. (A3)-(A6)] The notation for the azimuthal index and the circulation is inconsistent: l and m appear in the recurrence coefficients in a way that is not clearly connected to the l and m used in Eq. (8). Please make the notation uniform.
[Fig. 3] The axis labels in the right panel appear garbled ('log10|dphi/dsigma|||dphi/dsigma|epsilon-...'); please fix the LaTeX.
[Section V] For the unperturbed Regge poles and scattering amplitudes, the paper says they are consistent with Ref. [31] but gives no quantitative comparison. A brief statement of the achieved accuracy would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: RP bifurcations are computed outputs, not fitted inputs; self-citations are prior independent model/benchmark sources.

full rationale

The derivation chain is not circular. The central result — bifurcation, migration, and overtaking jumps of Regge poles under a sech^2 mass-shell perturbation — is obtained by numerically solving Eq. (7) with the modified continued fraction method (A2)-(A14). The quantities |Delta(lambda)/lambda| ~ 50-70 epsilon are outputs of the root-finding procedure, not parameters fitted to those values; epsilon, alpha, r0, and omega are inputs fixed in advance. The destabilization criterion |Delta(lambda)/lambda| >> epsilon is an external benchmark imported from Ref. [39], and the interpretation of bifurcation as destabilization is taken from Refs. [32,33]; neither is derived from the paper's own fitted values. The unperturbed Regge pole spectrum and the low-frequency scattering amplitudes are checked against the independent external results of Ref. [31]. The BEC model is cited from the authors' own Ref. [22], but that is a prior refereed publication with independent experimental grounding in Refs. [23-26], and it is not the target result of this paper; no uniqueness theorem or ansatz is smuggled in through self-citation. The only flagged weakness is numerical validation: Appendix A states 'Regarding to the issue of convergence of series expansion (S34), we refer the study of Ref. [52] where mentioned that a_k is a minimal solution to the recurrence relation when b/2 < r0 + 1/alpha < b', but no truncation order, matching-point convergence, or accuracy estimates are shown. That is a potential numerical-artifact risk, not a circular reduction, and therefore does not raise the circularity score. Accordingly, the paper receives 0: no significant circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model has five hand-set input parameters (epsilon, alpha, r0, omega, ell); none are fitted to external data, and no new physical entity is introduced. The main axioms are the BEC-to-massive-Klein-Gordon reduction, the constant-density Thomas-Fermi limit, the weak-coupling perturbation assumption, and the imported destabilization criterion from the dirty-black-hole literature.

free parameters (5)

epsilon (mass-shell amplitude) = 10^-2 (also 10^-3 and 10^-1 in Fig. 4)
Sets the effective mass term mu_s^2(r)=epsilon sech^2[alpha(r-r0)] and the homogeneous mass perturbation m^2=epsilon in Sec. III. Chosen by hand; the destabilization threshold and bifurcation onset depend on it.
alpha (inverse bump width) = 3/2
Fixed throughout; controls the spatial width of the mass shell and appears in the matching condition b > r0 + 1/alpha.
r0 (bump position) = 10 to 100 (scanned)
Migration and overtaking jumps are studied as r0 increases; the onset of bifurcation shifts with r0 and affects which overtone destabilizes first.
omega (frequency) = 1.0 and 0.1
High- and low-frequency regimes; the Regge pole spectra and scattering amplitudes are computed at these two values.
ell (vortex winding number or circulation) = 0 and 1
Sets the nonrotating versus rotating background; the metric cross term depends on C=ell/(m_a c), and the co-rotating/counter-rotating stability comparison uses ell=1.

assumptions (5)

domain assumption The binary BEC equations (1) reduce to a single massive Klein-Gordon equation (2) under equal densities and couplings rho1=rho2=rho, theta1=theta2=theta, g11=g22=g, g12<g, with effective mass m_eff^2=2 m_a c Omega/rho.
Invoked in Sec. II; if the miscibility or equal-density conditions fail, the single-field gapped equation and the acoustic metric (5) do not follow. The authors point to Ref. [22] for details.
domain assumption Thomas-Fermi approximation and constant density rho~rho_infty far from the vortex core, r >> R_TF, with c^2~(g-g12)rho/m_a.
Used to write the draining bathtub metric (5). Near the core the density profile differs and the analogue geometry is not valid.
ad hoc to paper Small Rabi coupling: Omega is small so epsilon=2d Omega0=10^-2 is a perturbation, and in Sec. III the energy regime E~1/r_E >> sqrt(epsilon) holds so E^2~omega^2.
The paper fixes epsilon=10^-2 and alpha=3/2 and treats the mass shell as a perturbation; the semiclassical expansion to order epsilon relies on this smallness.
domain assumption The Regge pole destabilization criterion |Delta lambda/lambda| >> epsilon from Refs. [32,39] transfers from Schwarzschild and dirty black holes to the draining bathtub analog system.
Used in Sec. IV to classify poles as destabilized. The authors do not derive this criterion for their system; they import it from the black hole spectral instability literature.
standard math Minimal-solution convergence of the continued fraction (A14) for b/2 < r0+1/alpha < b, per Ref. [52].
Appendix A relies on this to ensure the series expansion around r=b converges to the desired wavefunction solution.

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Cite this review

Pith. "Pith review of Regge poles of analogous rotating black holes in binary Bose-Einstein condensates: The gapped excitations." pith.science (2026). https://pith.science/paper/3TUOBQPI

@misc{pith2026250715403,
  author       = {Pith},
  title        = {Pith review of: Regge poles of analogous rotating black holes in binary Bose-Einstein condensates: The gapped excitations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3TUOBQPI}},
  note         = {Machine review of arXiv:2507.15403}
}

read the original abstract

In this paper, we study the spectrum of the Regge poles (RPs), which are the counterparts of quasinormal modes, in a draining bathtub vortex within a two-component Bose-Einstein condensate (BEC) system. We study the gapped excitations of the condensate with the spatially dependent energy gap term using a spatially tunable Rabi coupling, which will be treated as a perturbation. This model serves as an analogue of a rotating black hole surrounded by an environmental mass shell. We first compute the semiclassical scattering amplitude with the spatially independent mass effect due to the orbital interference. In the case of the mass-shell, bifurcation of the spectrum is observed, resulting in the destabilization of the RPs. We also study the migration of RPs by shifting the bump position. Our results show that the RPs of the co-rotating modes exhibit greater stability than those of the counter-rotating modes. Large migration and overtaking jumps of the overtone (fundamental RP) leave an imprint on the scattering amplitude at small (large) scattering angles. This can be observed in the scattering interference pattern in experiments.

Figures

Figures reproduced from arXiv: 2507.15403 by the authors.

**Figure 3.** shows the deflection cross section in (42) with/without perturbations for nonrotating/rotating backgrounds. Based on (41), for the corotating (counterrotating ) orbit, the deflection angle of 0 < ϕ < π (π < ϕ < 2π) is with the impact parameter b ≫ b+c ( b ≫ b−c) where the inverse of dΘ+ db ( dΘ− db ) is large that dominates the total cross section at 0 < ϕ < π ( π < ϕ < 2π). The interference term induces the oscill… view at source ↗

**Figure 4.** FIG. 4. The RP spectra are shown under various perturba [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (8 more)

**Figure 5.** Figure 5: FIG. 5. Migration behavior for RPs illustrated by the case of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: FIG. 7. The plots show the relative deviation [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 6.** Figure 6: FIG. 6. Migration behavior for RPs illustrated by the case [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 8.** Figure 8: FIG. 8. The logarithm of the squared scattering amplitude [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. The deviation of the scattering amplitude due to the [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. Distinct pole branches contribute the scattering amplitude in their respective angular regions in both nonrotating [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. The logarithm of the squared scattering amplitude [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 12.** Figure 12: FIG. 12. The deviation of the scattering amplitude due to [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

Discussion (0). Continue with ORCID to comment.

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