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Broken isospectrality in inverted black hole potentials and the emergence of Efimov scaling

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Umer Shahzad, Sebastian H. Völkel

General Relativity and Quantum Cosmology · Submitted 19 Oct 2026 · 43 references

Pith's verdict: Solid numerical demonstration that the inverted Regge-Wheeler and Zerilli potentials are not isospectral, with careful error analysis and honest limitations.

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arXiv ID2610.27102 (version 1)
Pith ID5LYGCZY4, Pith's permanent record of this paper and its review
CitationSebastian H. V\"olkel, Umer Shahzad. "Broken isospectrality in inverted black hole potentials and the emergence of Efimov scaling." arXiv:2610.27102 (2026). https://pith.science/paper/5LYGCZY4
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The paper

The paper claims

Inverted black-hole potentials break a long-held spectral symmetry and reveal geometric level spacing.

The reading

Previous work established that the Regge-Wheeler and Zerilli potentials for Schwarzschild black hole perturbations have identical quasinormal-mode spectra (isospectrality). This paper shows that when those potentials are inverted into wells—the setting of the bound-state approach to quasinormal modes—the isospectrality breaks: the bound-state spectra of the inverted Regge-Wheeler and Zerilli potentials are distinct. The first direct computation of the inverted Zerilli spectrum is presented, with relative energy differences of order 10⁻⁴ to 10⁻⁵ that are numerically robust. The paper also demonstrates that inversion destroys the first-order Darboux transformation linking the original barriers. Despite the spectral split, both spectra exhibit geometric scaling at large quantum numbers, a pattern known from Efimov three-body physics.

What carries the argument

The central comparison is between the bound-state spectra of the inverted Regge-Wheeler potential (Eq. 4) and the inverted Zerilli potential (Eq. 5). The numerical methods are a shooting method (adaptive Runge-Kutta) and a pseudospectral collocation method, cross-validated against each other and against previously obtained analytical asymptotics for the Regge-Wheeler case. The interpolation V_s(𝐫, ε) (Eq. 8) with ε∈[0,1] allows systematic study of the spectral shift. The large-n analysis uses the asymptotic geometric scaling relation E_n ∼ E_* exp[-2π n/(s_∞)] with s_∞ = √{ℓ(ℓ+1)-1/4}.

Extended reading notes

Core claim

The paper establishes that the bound-state spectra of the inverted Regge-Wheeler and Zerilli potentials are not identical, contrary to what one might expect from the isospectrality of the original potential barriers. The relative energy difference is approximately 6.5×10⁻⁵ for ℓ=2,n=0 and grows to about 4.7×10⁻⁴ at large n, decreasing with increasing ℓ. Potential inversion also eliminates the first-order Darboux relation that provided a simple proof of isospectrality for the barriers. A linear interpolation between the two potentials shows that the largest energy shifts occur in the interior rather than at the endpoints, requiring second-order perturbation theory to explain. At large n, both spectra approach a geometric progression controlled by the leading attractive 1/x² tail, a signature of Efimov physics, but with a potential-dependent offset that keeps the spectra distinct even asymptotically.

Load-bearing premise

The numerical shoot and pseudospectral methods resolve relative energy differences as small as ~10⁻⁷ with sufficient accuracy to exclude systematic bias, and the absence of an analytic closed form for the inverted Zerilli spectrum means the central claim relies on these numerical data.

Editorial extensions

If this is right

  • The broken isospectrality implies that the bound-state mapping from potential barriers to wells does not preserve the isospectral property; different barrier representations may yield different bound-state spectra.
  • Any first-order Darboux transformation between the original barriers becomes invalid after inversion; higher-order transformations, if they exist, would be needed to restore isospectrality in the wells.
  • The common Efimov scaling suggests that black-hole bound states and few-body Efimov effects share a structural feature—an attractive inverse-square tail—but the spectral offset encodes the potential's short-range details.
  • Numerical methods that compute quasinormal modes from bound states (e.g., via analytic continuation) must account for the spectral splitting; the choice of potential (Regge-Wheeler vs. Zerilli) may affect the accuracy of the continued spectra.

Reading between the lines

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

  • The existence of distinct bound-state spectra for isospectral barriers raises the question of whether the analytic continuation from bound states to quasinormal modes is unique: two different bound-state spectra might map to the same quasinormal mode spectrum under the complex coordinate transformation, or they might not. This has implications for the reliability of recent high-precision methods b
  • The Efimov analogy invites a deeper search: if the spectral offset δ encodes short-range physics, perhaps black-hole parameters (mass, spin, charge) play a role analogous to the scattering length in cold atoms. A systematic study of δ across alternative theories of gravity could test whether the offset is a sensitive probe of deviations from general relativity.
  • The interpolation analysis shows that linear perturbation theory fails even near the eikonal limit; this suggests that semiclassical approximations (WKB) for the inverted potentials may require second-order corrections to capture the spectral splitting, which could affect the accuracy of QNM estimates from the bound-state method.
Original abstract

Are the inverted Regge-Wheeler and Zerilli potentials arising in Mashhoon's bound-state approach to black-hole quasinormal modes isospectral? Extending previous numerical and analytical studies of the inverted Regge-Wheeler potential, we present the first direct computation of the bound-state spectrum of the inverted Zerilli potential. Although Chandrasekhar proved that the original potentials have identical quasinormal-mode spectra, we show that their inverted counterparts have distinct bound-state spectra. We also demonstrate that potential inversion destroys their first-order Darboux relation, used by Glampedakis et al. to provide a simpler proof of isospectrality. A continuous interpolation between the potentials reveals substantially larger energy shifts in the interior than at the endpoints, indicating cancellations that suppress the endpoint spectral splitting. Explaining this behavior requires second-order perturbative contributions, even towards the eikonal limit. Despite these differences, both spectra exhibit asymptotic geometric scaling characteristic of Efimov physics, first discovered in three-boson systems. This scaling also applies to other inverted potentials with the same leading tail, including the scalar-field case studied here. Beyond their theoretical implications, our results raise questions for recently developed methods that compute quasinormal modes with high precision from numerical bound states.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-21: 1 request, 6,901 input tokens.

What the abstract itself states, as probabilities: claims first 0.86 · states limitation 0.35 · human participants 0.01 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Are the inverted Regge-Wheeler and Zerilli potentials arising in Mashhoon's bound-state approach to black-hole quasinormal modes isospectral? Extending previous numerical and analytical studies of the inverted Regge-Wheeler potential, we present the first direct computation of the bound-state spectrum of the inverted Zerilli potential. Although Chandrasekhar proved that the original potentials have identical quasinormal-mode spectra, we show that their inverted counterparts have distinct bound-state spectra. We also demonstrate that potential inversion destroys their first-order Darboux relation, used by Glampedakis et al. to provide a simpler proof of isospectrality. A continuous interpolation between the potentials reveals substantially larger energy shifts in the interior than at the endpoints, indicating cancellations that suppress the endpoint spectral splitting. Explaining this behavior requires second-order perturbative contributions, even towards the eikonal limit. Despite these differences, both spectra exhibit asymptotic geometric scaling characteristic of Efimov physics, first discovered in three-boson systems. This scaling also applies to other inverted potentials with the same leading tail, including the scalar-field case studied here. Beyond their theoretical implications, our results raise questions for recently developed methods that compute quasinormal modes with high precision from numerical bound states.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Are the inverted Regge-Wheeler and Zerilli potentials arising in Mashhoon's bound-state approach to black-hole quasinormal modes isospectral? background 0.85 0.05 0.03 unclear 0.98
s1 Extending previous numerical and analytical studies of the inverted Regge-Wheeler potential, we present the first direct computation of the bound-state spectrum of the inverted Zerilli potential. contribution 0.99 0.95 0.06 asserted 0.99
s2 Although Chandrasekhar proved that the original potentials have identical quasinormal-mode spectra, we show that their inverted counterparts have distinct bound-state spectra. result 1.00 0.92 0.78 asserted 1.00
s3 We also demonstrate that potential inversion destroys their first-order Darboux relation, used by Glampedakis et al. to provide a simpler proof of isospectrality. result 0.92 0.83 0.18 asserted 1.00
s4 A continuous interpolation between the potentials reveals substantially larger energy shifts in the interior than at the endpoints, indicating cancellations that suppress the endpoint spectral splitting. result 0.99 0.92 0.10 hedged possibility 0.56
s5 Explaining this behavior requires second-order perturbative contributions, even towards the eikonal limit. interpretation 0.86 0.77 0.06 asserted 0.99
s6 Despite these differences, both spectra exhibit asymptotic geometric scaling characteristic of Efimov physics, first discovered in three-boson systems. result 0.94 0.84 0.29 asserted 1.00
s7 This scaling also applies to other inverted potentials with the same leading tail, including the scalar-field case studied here. result 0.94 0.75 0.10 asserted 1.00
s8 Beyond their theoretical implications, our results raise questions for recently developed methods that compute quasinormal modes with high precision from numerical bound states. interpretation 0.96 0.35 0.07 hedged possibility 0.67

What kind of paper the classifier reports this is: theoretical result 0.81 · evidence from simulation or computation 0.97.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1. Relative difference between bound states of inverted
Figure 2. FIG. 2. Relative energy shift
Figure 3. shows s s ℓn, s 8 ℓ , and δ s ℓn for all three poten￾tials and different ℓ. In each case, s s ℓn rapidly approaches s 8 ℓ , with slower convergence as ℓ increases, consistent with the more rapid delocalization of eigenfunctions at smaller ℓ. The phases also rapidly approach constant values. The Regge-Wheeler and Zerilli phases nearly co￾incide and are clearly distinct from the scalar phase, with smaller differences between the potentials as ℓ increases. The ratio of the Zerilli and Regge-Wheeler bound-state energies therefore approaches an n-independent limit, limnÑ8 EZ ℓ
Figure 4. FIG. 4. Adjacent-multiplier relative energy differences,
Figure 6. FIG. 6. Spectral-grid self-convergence of the inverted Regge
Figure 8. FIG. 8. Relative deviations of the shooting-method eigenval
Figure 10. shows the Regge-Wheeler and Zerilli eigen￾functions for selected ℓ and n. At fixed n, the eigenfunc￾tions become increasingly similar as ℓ increases, while increasing n leads to rapid delocalization. The deeper wells at larger ℓ also yield more strongly bound states at fixed n. The small relative displacement between the potentials is reflected in their eigenfunctions. Although both potentials share the same large-r asymptotic form, their differences cannot be eliminated by a simple shift. −0.15 −0.10 −0.05 0.00 V(x), En ℓ = 2 −0.3 −0.2 −0.1 0.0 V(x), En ℓ = 3 −20 −10 0 10 2

Showing 7 of the paper's 10 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.27102 by the authors.

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