pith. sign in

arxiv: 2512.02110 · v2 · pith:M6C22XS2new · submitted 2025-12-01 · 🌀 gr-qc · astro-ph.HE· hep-th

Exceptional Points and Resonance in Black Hole Ringdown

Rodrigo Panosso Macedo , Takuya Katagiri , Kei-ichiro Kubota , Hayato Motohashi This is my paper

Pith reviewed 2026-05-21 17:33 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords exceptional pointsquasinormal modesblack hole ringdownresonanceavoided crossingsgravitational waveshyperboloidal framework
0
0 comments X

The pith

Exceptional points make the average of resonant modes the relevant frequency for black hole ringdown signals.

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

The paper proposes an exceptional-point framework to characterize resonances in black hole ringdown that standard quasinormal-mode analysis misses. In a phenomenological model of environmental black holes treated with hyperboloidal coordinates, quasinormal-mode frequencies and eigenfunctions nearly coalesce at exceptional points. This produces stronger time-domain contributions from the modes and clear departures from simple exponential decay. The frequency at the exceptional point itself, defined as the average of the two resonant frequencies, emerges as the quantity that best describes the observable signal in this regime.

Core claim

We propose an exceptional-point (EP) framework for black-hole ringdown beyond the standard quasinormal-mode (QNM) paradigm. It provides a first-principles characterization of the resonance associated with avoided crossings near EPs, an effect that conventional QNM analysis cannot fully capture. Employing a phenomenological environmental black-hole model with the hyperboloidal framework, we identify near-coalescence of both QNM eigenvalues and eigenfunctions, and directly demonstrate that the resonance produces enhanced mode contributions in the time domain, resulting in characteristic departures from exponentially damped oscillations. Our formulation further reveals that the EP frequency, a

What carries the argument

The exceptional point in the quasinormal-mode spectrum of a phenomenological environmental black-hole model, where eigenvalues and eigenfunctions coalesce and the hyperboloidal framework tracks the resulting resonance.

If this is right

  • Resonance near exceptional points produces enhanced mode contributions in the time-domain ringdown waveform.
  • Ringdown signals exhibit characteristic departures from purely exponentially damped oscillations.
  • The exceptional-point frequency, defined as the average of the resonant modes, is the physically relevant observable for signal extraction.
  • This framework supplies a foundation for modeling resonances that standard quasinormal-mode analysis cannot capture.

Where Pith is reading between the lines

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

  • The approach could be applied to extract environmental parameters from observed gravitational-wave ringdown signals.
  • It suggests that avoided crossings and exceptional points may appear in a wider range of perturbed black-hole systems beyond the specific model used here.
  • Extending the hyperboloidal treatment to full numerical-relativity waveforms would test whether exceptional-point resonances survive in more complete spacetimes.

Load-bearing premise

The phenomenological environmental black-hole model with the hyperboloidal framework accurately captures the physics that produces exceptional points and the associated resonance in realistic black-hole ringdown.

What would settle it

Numerical evolution of a realistic black-hole spacetime or analysis of actual gravitational-wave ringdown data that shows neither enhanced mode amplitudes nor departures from exponential decay near the predicted exceptional-point frequency would falsify the claim.

Figures

Figures reproduced from arXiv: 2512.02110 by Hayato Motohashi, Kei-ichiro Kubota, Rodrigo Panosso Macedo, Takuya Katagiri.

Figure 1
Figure 1. Figure 1: Panels (a) and (b): Migration of the QNMs for parameter range a/rh ∈ [0, 15] with ϵ = ϵ∗(≃ 0.00204 r −2 h ). The fundamental mode n = 0 (purple) and the overtones n = 1–3 (green, blue, yellow) are displayed, with black markers denoting the initial Schwarzschild values. One observes overtaken transitions between overtones and the avoided crossing between the resonant n = 0 and n = 1 (originally n = 3) modes… view at source ↗
Figure 3
Figure 3. Figure 3: Extracted frequencies obtained using the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Wave signal at future null infinity I +. Panel (a): Decomposition of the reconstructed resonant waveform that evolves from the constant initial data (Ψ, ∂τΨ)|τ=0 = (1, 0). The fundamental mode and first overtone are significantly ex￾cited compared to the others. The inset highlights the funda￾mental mode and first overtone, showing they are nearly out of phase. Panel (b): Comparison among the full signal (… view at source ↗
read the original abstract

We propose an exceptional-point (EP) framework for black-hole ringdown beyond the standard quasinormal-mode (QNM) paradigm. It provides a first-principles characterization of the resonance associated with avoided crossings near EPs, an effect that conventional QNM analysis cannot fully capture. Employing a phenomenological environmental black-hole model with the hyperboloidal framework, we identify near-coalescence of both QNM eigenvalues and eigenfunctions, and directly demonstrate that the resonance produces enhanced mode contributions in the time domain, resulting in characteristic departures from exponentially damped oscillations. Our formulation further reveals that the EP frequency, given by the average of the resonant modes, emerges as the physically relevant observable in the near-EP regime, and offers a robust foundation for modeling and extracting resonant ringdown signals.

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 / 1 minor

Summary. The paper proposes an exceptional-point (EP) framework for black-hole ringdown beyond the standard quasinormal-mode paradigm. Using a phenomenological environmental black-hole model within the hyperboloidal framework, it identifies near-coalescence of QNM eigenvalues and eigenfunctions near EPs, directly demonstrates enhanced time-domain mode contributions from resonance leading to departures from pure exponential damping, and concludes that the EP frequency (average of the resonant modes) emerges as the physically relevant observable for modeling and extracting resonant ringdown signals.

Significance. If validated, the work offers a new first-principles approach within the chosen model for capturing resonance effects near avoided crossings that standard QNM analysis misses, with the EP frequency identified as a robust observable. The parameter-free character of the derivation inside the model and the explicit time-domain demonstration of enhanced contributions are strengths that could aid signal extraction in perturbed black-hole spacetimes, though the phenomenological setup limits immediate applicability to realistic astrophysical scenarios.

major comments (2)
  1. Abstract: the central claim that the framework 'provides a first-principles characterization' and 'offers a robust foundation' for realistic ringdown rests on the unverified assertion that the phenomenological environmental black-hole model with the hyperboloidal framework accurately reproduces near-EP behavior; no direct comparison to full numerical-relativity simulations is reported to confirm this.
  2. Abstract: the demonstration of 'enhanced mode contributions in the time domain' and 'characteristic departures from exponentially damped oscillations' is performed inside the model, but the load-bearing step linking these to physical black-hole ringdown requires independent verification against numerical relativity, which is absent.
minor comments (1)
  1. Clarify the precise definition of the hyperboloidal framework and any boundary conditions used in the model to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and recommendation for major revision. We address each point below, agreeing to revise the abstract to better reflect the phenomenological nature of the model and the absence of direct numerical-relativity comparisons.

read point-by-point responses

  1. Referee: Abstract: the central claim that the framework 'provides a first-principles characterization' and 'offers a robust foundation' for realistic ringdown rests on the unverified assertion that the phenomenological environmental black-hole model with the hyperboloidal framework accurately reproduces near-EP behavior; no direct comparison to full numerical-relativity simulations is reported to confirm this.

    Authors: We agree that the manuscript does not report direct comparisons to full numerical-relativity simulations. The first-principles characterization refers to the EP analysis of eigenvalues, eigenfunctions, and resonance within the chosen phenomenological environmental black-hole model using the hyperboloidal framework. The model is designed to capture effects leading to avoided crossings and EPs. To address the concern, we will revise the abstract to qualify the language, emphasizing that the results and foundation apply to this model setup rather than claiming direct applicability to realistic ringdown without further validation. revision: yes

  2. Referee: Abstract: the demonstration of 'enhanced mode contributions in the time domain' and 'characteristic departures from exponentially damped oscillations' is performed inside the model, but the load-bearing step linking these to physical black-hole ringdown requires independent verification against numerical relativity, which is absent.

    Authors: We acknowledge that the explicit demonstrations of enhanced time-domain contributions and departures from pure exponential damping are performed inside the phenomenological model. The link to physical black-hole ringdown is based on the model's ability to reproduce resonance effects near EPs that standard QNM analysis misses. We will revise the abstract to specify the model context for these demonstrations and add a brief note in the conclusions acknowledging that independent verification against numerical relativity would be required to confirm broader applicability to perturbed black-hole spacetimes. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper applies the hyperboloidal framework to a phenomenological environmental black-hole model and derives the EP frequency as the average of resonant modes from the eigenvalue coalescence analysis. This emerges directly from the model's equations without reducing to a fitted input renamed as prediction or depending on load-bearing self-citations. The framework is presented as first-principles within the chosen setup, with no quoted steps showing self-definitional reduction or ansatz smuggling. The central claim of resonance and enhanced time-domain contributions follows from the near-EP eigenvalue behavior rather than presupposing the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a phenomenological environmental black-hole model whose detailed assumptions are not enumerated in the abstract; no explicit free parameters, axioms, or invented entities are listed, but the model itself functions as an ad-hoc domain assumption.

axioms (1)
  • domain assumption The phenomenological environmental black-hole model with hyperboloidal framework captures the essential non-Hermitian physics near exceptional points.
    Invoked to identify near-coalescence of eigenvalues and eigenfunctions and to demonstrate resonance effects.

pith-pipeline@v0.9.0 · 5674 in / 1364 out tokens · 31183 ms · 2026-05-21T17:33:48.017875+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 9 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Finite-Window Centered Organization of Neighboring Poles

    gr-qc 2026-04 unverdicted novelty 7.0

    Near-degenerate poles on finite windows organize into a centered carrier-plus-jet structure with κ and η² controlling the two-scale error hierarchy, verified via toy numerics and Kerr quasinormal modes.

  2. Axial Oscillations of Viscous Neutron Stars

    gr-qc 2026-04 unverdicted novelty 7.0

    Viscous neutron stars have new families of axial oscillation modes without perfect-fluid counterparts, featuring mode avoidance and long-lived modes.

  3. Detectability of avoided crossings in black hole ringdowns

    gr-qc 2026-05 unverdicted novelty 6.0

    Bayesian analysis finds individual QNM frequencies near avoided crossings hard to resolve even under optimistic conditions, though collective AC waveform signatures may remain detectable if those modes dominate and sl...

  4. Quasinormal modes and continuum response of de Sitter black holes via complex scaling method

    hep-th 2026-05 unverdicted novelty 6.0

    Complex scaling turns the outgoing boundary problem for de Sitter black hole perturbations into a spectral problem, enabling unified computation of quasinormal modes and continuum response for scalar, electromagnetic,...

  5. Pole Structure of Kerr Green's Function

    gr-qc 2026-05 unverdicted novelty 6.0

    Homogeneous solutions and connection coefficients in the radial Teukolsky equation for Kerr black holes exhibit simple poles at Matsubara frequencies that cancel in the Green's function, along with canceling zero-freq...

  6. Bilinear products and the orthogonality of quasinormal modes on hyperboloidal foliations

    gr-qc 2026-04 unverdicted novelty 6.0

    Bilinear products for black hole quasinormal modes on hyperboloidal foliations are divergent due to CPT transformations but can be regularized to define orthogonal modes and excitation coefficients.

  7. Quasinormal modes and continuum response of de Sitter black holes via complex scaling method

    hep-th 2026-05 unverdicted novelty 5.0

    Complex scaling unifies quasinormal modes and continuum response for black-hole perturbations in four-dimensional Schwarzschild-de Sitter spacetimes.

  8. Finite-Window Centered Organization of Neighboring Poles

    gr-qc 2026-04 unverdicted novelty 5.0

    A centered first-jet basis for neighboring quasinormal modes in finite time windows replaces the ill-conditioned sum of two resolved damped exponentials with a carrier plus t exp(-i omega_c t) term when the dimensionl...

  9. Complex scaling approach to quasinormal modes of Schwarzschild and Reissner--Nordstr\"om black holes

    hep-th 2026-04 unverdicted novelty 5.0

    Complex scaling converts outgoing boundary conditions into eigenvalue problems to compute quasinormal frequencies for Schwarzschild and Reissner-Nordström black holes, including the extremal limit.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · cited by 7 Pith papers · 12 internal anchors

  1. [1]

    Zur Quantentheorie des Atomkernes,

    G. Gamow, “Zur Quantentheorie des Atomkernes,” Z. Phys.51, 204 (1928)

    work page 1928

  2. [2]

    J. R. Taylor,Scattering Theory: The Quantum Theory of Nonrelativistic Collisions(John Wiley & Sons, Inc., New York, 1972)

    work page 1972

  3. [3]

    V. I. Kukulin, V. M. Krasnopol’sky, and J. Horáček, Theory of Resonances: Principles and Applications, Rei- del Texts in the Mathematical Sciences (Springer Dor- drecht, 1989)

    work page 1989

  4. [4]

    H.BreuerandF.Petruccione,The Theory of Open Quan- tum Systems(Oxford University Press, 2002)

    work page 2002

  5. [5]

    N.Moiseyev,Non-Hermitian Quantum Mechanics(Cam- bridge University Press, 2011)

    work page 2011

  6. [6]

    Non- Hermitian physics and PT symmetry,

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non- Hermitian physics and PT symmetry,” Nature Physics 14, 11 (2018)

    work page 2018

  7. [7]

    Exceptional topology of non-Hermitian systems,

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, “Exceptional topology of non-Hermitian systems,” Rev. Mod. Phys.93, 015005 (2021), arXiv:1912.10048 [cond- mat.mes-hall]

    work page arXiv 2021

  8. [8]

    Non-Hermitian physics,

    Y. Ashida, Z. Gong, and M. Ueda, “Non-Hermitian physics,” Adv. Phys.69, 249 (2021), arXiv:2006.01837 [cond-mat.mes-hall]

    work page arXiv 2021

  9. [9]

    Kato,Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1995)

    T. Kato,Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1995)

    work page 1995

  10. [10]

    Parity- time symmetry and exceptional points in photonics,

    Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity- time symmetry and exceptional points in photonics,” Na- ture Materials18, 783 (2019)

    work page 2019

  11. [11]

    Review of exceptional point-based sensors,

    J. Wiersig, “Review of exceptional point-based sensors,” Photon. Res.8, 1457 (2020)

    work page 2020

  12. [12]

    Non-hermitian and topological photonics: optics at an exceptional point,

    M. Parto, Y. G. N. Liu, B. Bahari, M. Khajavikhan, and D. N. Christodoulides, “Non-hermitian and topological photonics: optics at an exceptional point,” Nanophoton- ics10, 403 (2021)

    work page 2021

  13. [13]

    Non-Hermitian topology and exceptional-point geometries,

    K. Ding, C. Fang, and G. Ma, “Non-Hermitian topology and exceptional-point geometries,” Nature Rev. Phys.4, 745 (2022), arXiv:2204.11601 [quant-ph]

    work page arXiv 2022

  14. [14]

    Rabi Oscillations at Exceptional Points in Microwave Billiards

    B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscilla- tions at exceptional points in microwave billiards,” Phys. Rev. E75, 027201 (2007), arXiv:cond-mat/0612547

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  15. [15]

    Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems

    F. Verstraete, V. Murg, and J. Cirac, “Matrix prod- uct states, projected entangled pair states, and vari- ational renormalization group methods for quantum spin systems,” Advances in Physics57, 143 (2008), arXiv:0907.2796 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  16. [16]

    Time behaviour near to spectral singularities

    W. D. Heiss, “Time behaviour near to spectral singular- ities,” The European Physical Journal D60, 257 (2010), arXiv:1009.5780 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  17. [17]

    Pseudospectrum and black hole quasi-normal mode (in)stability

    J. L. Jaramillo, R. Panosso Macedo, and L. Al Sheikh, “Pseudospectrum and Black Hole Quasinormal Mode Instability,” Phys. Rev. X11, 031003 (2021), arXiv:2004.06434 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  18. [18]

    Pseudospectrum and binary black hole merger transients,

    J. L. Jaramillo, “Pseudospectrum and binary black hole merger transients,” Class. Quant. Grav.39, 217002 (2022), arXiv:2206.08025 [gr-qc]

    work page arXiv 2022

  19. [19]

    Resonant Excitation of Quasinormal Modes of Black Holes,

    H. Motohashi, “Resonant Excitation of Quasinormal Modes of Black Holes,” Phys. Rev. Lett.134, 141401 (2025), arXiv:2407.15191 [gr-qc]

    work page arXiv 2025

  20. [20]

    Exceptional Point and Hysteresis in Perturbations of Kerr Black Holes,

    J. P. Cavalcante, M. Richartz, and B. C. da Cunha, “Exceptional Point and Hysteresis in Perturbations of Kerr Black Holes,” Phys. Rev. Lett.133, 261401 (2024), arXiv:2407.20850 [gr-qc]

    work page arXiv 2024

  21. [21]

    Massive scalar perturbations in Kerr black holes: Near extremal analysis,

    J. P. Cavalcante, M. Richartz, and B. C. da Cunha, “Massive scalar perturbations in Kerr black holes: Near extremal analysis,” Phys. Rev. D110, 124064 (2024), arXiv:2408.13964 [gr-qc]

    work page arXiv 2024

  22. [22]

    Black hole spectroscopy: from theory to experiment

    E. Bertiet al., “Black hole spectroscopy: from theory to experiment,” (2025), arXiv:2505.23895 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  23. [23]

    Unstable Chords and Destructive Resonant Excitation of Black Hole Quasinormal Modes,

    N. Oshita, E. Berti, and V. Cardoso, “Unstable Chords and Destructive Resonant Excitation of Black Hole Quasinormal Modes,” Phys. Rev. Lett.135, 031401 (2025), arXiv:2503.21276 [gr-qc]

    work page arXiv 2025

  24. [24]

    Quasinormal modes and excitation factors of Kerr black holes,

    R. K. L. Lo, L. Sabani, and V. Cardoso, “Quasinormal modes and excitation factors of Kerr black holes,” Phys. Rev. D111, 124002 (2025), arXiv:2504.00084 [gr-qc]

    work page arXiv 2025

  25. [25]

    Black Hole Quasi- 6 normal Mode Resonances,

    Y. Yang, E. Berti, and N. Franchini, “Black Hole Quasi- 6 normal Mode Resonances,” Phys. Rev. Lett.135, 201401 (2025), arXiv:2504.06072 [gr-qc]

    work page arXiv 2025

  26. [26]

    Resonance of black hole quasinormal modes in cou- pled systems,

    T. Takahashi, H. Motohashi, and K. Takahashi, “Resonance of black hole quasinormal modes in cou- pled systems,” Phys. Rev. D112, 064006 (2025), arXiv:2505.03883 [gr-qc]

    work page arXiv 2025

  27. [27]

    Complete quasinormal modes of type-D black holes,

    C. Chen, J. Jing, Z. Cao, and M. Wang, “Complete quasinormal modes of type-D black holes,” Phys. Rev. D 112, 103036 (2025), arXiv:2506.14635 [gr-qc]

    work page arXiv 2025

  28. [28]

    Resonance in black hole ringdown: Benchmarking quasinormal mode excitation and extraction,

    K.-i. Kubota and H. Motohashi, “Resonance in black hole ringdown: Benchmarking quasinormal mode excitation and extraction,” (2025), arXiv:2509.06411 [gr-qc]

    work page arXiv 2025

  29. [29]

    Ergodic Hysteresis of the Kerr black hole spectrum,

    J. P. Cavalcante, M. Richartz, and B. C. da Cunha, “Ergodic Hysteresis of the Kerr black hole spectrum,” (2025), arXiv:2511.16640 [gr-qc]

    work page arXiv 2025

  30. [30]

    Ex- ceptional line and pseudospectrum in black hole spec- troscopy,

    L.-M. Cao, M.-F. Ji, L.-B. Wu, and Y.-S. Zhou, “Ex- ceptional line and pseudospectrum in black hole spec- troscopy,” (2025), arXiv:2511.17067 [gr-qc]

    work page arXiv 2025

  31. [31]

    Spectral decomposition of the perturba- tion response of the Schwarzschild geometry,

    E. W. Leaver, “Spectral decomposition of the perturba- tion response of the Schwarzschild geometry,” Phys. Rev. D34, 384 (1986)

    work page 1986

  32. [32]

    A detailed study of quasinormal frequencies of the Kerr black hole

    H. Onozawa, “A Detailed study of quasinormal frequen- ciesoftheKerrblackhole,” Phys.Rev.D55,3593(1997), arXiv:gr-qc/9610048

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  33. [33]

    Quasinormal modes of nearly extremal Kerr spacetimes: spectrum bifurcation and power-law ringdown

    H. Yang, A. Zimmerman, A. Zenginoğlu, F. Zhang, E. Berti, and Y. Chen, “Quasinormal modes of nearly extremal Kerr spacetimes: spectrum bifurcation and power-law ringdown,” Phys. Rev. D88, 044047 (2013), arXiv:1307.8086 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  34. [34]

    Gravitational perturbations of the Kerr geometry: High-accuracy study

    G. B. Cook and M. Zalutskiy, “Gravitational perturba- tions of the Kerr geometry: High-accuracy study,” Phys. Rev. D90, 124021 (2014), arXiv:1410.7698 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  35. [35]

    Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for the numerical computation of quasinormal modes

    A. Jansen, “Overdamped modes in Schwarzschild-de Sit- ter and a Mathematica package for the numerical com- putation of quasinormal modes,” Eur. Phys. J. Plus132, 546 (2017), arXiv:1709.09178 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  36. [36]

    Eigenvalue repulsions in the quasinormal spectra of the Kerr-Newman black hole,

    O. J. C. Dias, M. Godazgar, J. E. Santos, G. Carullo, W. Del Pozzo, and D. Laghi, “Eigenvalue repulsions in the quasinormal spectra of the Kerr-Newman black hole,” Phys. Rev. D105, 084044 (2022), arXiv:2109.13949 [gr- qc]

    work page arXiv 2022

  37. [37]

    StrongCosmicCensorshipandeigenvaluerepulsions for rotating de Sitter black holes in higher-dimensions,

    A. Davey, O. J. C. Dias, P. Rodgers, and J. E. San- tos,“StrongCosmicCensorshipandeigenvaluerepulsions for rotating de Sitter black holes in higher-dimensions,” JHEP07, 086 (2022), arXiv:2203.13830 [gr-qc]

    work page arXiv 2022

  38. [38]

    Eigen- value repulsions and quasinormal mode spectra of Kerr- Newman: an extended study,

    O. J. C. Dias, M. Godazgar, and J. E. Santos, “Eigen- value repulsions and quasinormal mode spectra of Kerr- Newman: an extended study,” JHEP07, 076 (2022), arXiv:2205.13072 [gr-qc]

    work page arXiv 2022

  39. [39]

    Quasinormal mode spectrum of the AdS black hole with the Robin boundary condition,

    S. Kinoshita, T. Kozuka, K. Murata, and K. Sugawara, “Quasinormal mode spectrum of the AdS black hole with the Robin boundary condition,” Class. Quant. Grav.41, 055010 (2024), arXiv:2305.17942 [gr-qc]

    work page arXiv 2024

  40. [40]

    Chandrasekhar,The mathematical theory of black holes(Oxford University Press, 1985)

    S. Chandrasekhar,The mathematical theory of black holes(Oxford University Press, 1985)

    work page 1985

  41. [41]

    Destabilizing the Fundamental Mode of Black Holes: The Elephant and the Flea,

    M. H.-Y. Cheung, K. Destounis, R. P. Macedo, E. Berti, and V. Cardoso, “Destabilizing the Fundamental Mode of Black Holes: The Elephant and the Flea,” Phys. Rev. Lett.128, 111103 (2022), arXiv:2111.05415 [gr-qc]

    work page arXiv 2022

  42. [42]

    A geometric framework for black hole perturbations

    A. Zenginoglu, “A Geometric framework for black hole perturbations,” Phys. Rev. D83, 127502 (2011), arXiv:1102.2451 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  43. [43]

    Hyperboloidal approach for static spherically symmetric spacetimes: a didactical intro- ductionand applications in black-hole physics,

    R. Panosso Macedo, “Hyperboloidal approach for static spherically symmetric spacetimes: a didactical intro- ductionand applications in black-hole physics,” Phil. Trans. Roy. Soc. Lond. A382, 20230046 (2024), arXiv:2307.15735 [gr-qc]

    work page arXiv 2024

  44. [44]

    Hyperboloidal approach to quasinormal modes,

    R. Panosso Macedo and A. Zenginoglu, “Hyperboloidal approach to quasinormal modes,” Front. in Phys.12, 1497601 (2024), arXiv:2409.11478 [gr-qc]

    work page arXiv 2024

  45. [45]

    Spectral decomposition of black-hole perturbations on hyperboloidal slices

    M. Ansorg and R. Panosso Macedo, “Spectral decomposi- tion of black-hole perturbations on hyperboloidal slices,” Phys. Rev. D93, 124016 (2016), arXiv:1604.02261 [gr- qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  46. [46]

    Axisymmetric fully spectral code for hyperbolic equations

    R. Panosso Macedo and M. Ansorg, “Axisymmetric fully spectralcodeforhyperbolicequations,” J.Comput.Phys. 276, 357 (2014), arXiv:1402.7343 [physics.comp-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  47. [47]

    Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime,

    P. Bourg, R. Panosso Macedo, A. Spiers, B. Leather, B. Béatrice, and A. Pound, “Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime,” Phys. Rev. D112, 044049 (2025), arXiv:2503.07432 [gr- qc]

    work page arXiv 2025

  48. [48]

    Hyperboloidal method for frequency-domain self-force calculations,

    R. Panosso Macedo, B. Leather, N. Warburton, B. Wardell, and A. Zenginoğlu, “Hyperboloidal method for frequency-domain self-force calculations,” Phys. Rev. D105, 104033 (2022), arXiv:2202.01794 [gr-qc]

    work page arXiv 2022

  49. [49]

    Limiting geome- try and spectral instability in Schwarzschild–de Sit- ter spacetimes,

    Y. Zhou and R. Panosso Macedo, “Limiting geome- try and spectral instability in Schwarzschild–de Sit- ter spacetimes,” Phys. Rev. D112, 084063 (2025), arXiv:2507.05370 [gr-qc]

    work page arXiv 2025

  50. [50]

    Stability of the fundamental quasinormal mode in time-domain observations against small perturbations,

    E. Berti, V. Cardoso, M. H.-Y. Cheung, F. Di Filippo, F. Duque, P. Martens, and S. Mukohyama, “Stability of the fundamental quasinormal mode in time-domain observations against small perturbations,” Phys. Rev. D 106, 084011 (2022), arXiv:2205.08547 [gr-qc]

    work page arXiv 2022

  51. [51]

    Quasinor- mal modes of Schwarzschild black holes on the real axis,

    K. Kyutoku, H. Motohashi, and T. Tanaka, “Quasinor- mal modes of Schwarzschild black holes on the real axis,” Phys. Rev. D107, 044012 (2023), arXiv:2206.00671 [gr- qc]

    work page arXiv 2023

  52. [52]

    [30] also comments on a similar result

    While finishing this work, Ref. [30] also comments on a similar result

    work page

  53. [53]

    Quasinormal modes on Kerr spacetimes,

    D. Gajic and C. M. Warnick, “Quasinormal modes on Kerr spacetimes,” (2024), arXiv:2407.04098 [gr-qc]

    work page arXiv 2024