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Exceptional points of the Schwarzschild spectrum under local potential perturbation: The Princess and the Pea
General Relativity and Quantum Cosmology · Submitted 17 Aug 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Finally, we study the response to initial data and find that, although a weak perturbation can substantially reconfigure the QNM spectrum, the corresponding time-domain response remains perturbative.
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Black-hole quasinormal modes (QNMs) may coalesce at exceptional points (EPs) in the frequency spectrum. We study EPs and their phenomenology for gravitational perturbations of a Schwarzschild black hole whose Regge-Wheeler potential is perturbed by a localized delta-function of amplitude $\epsilon$ at radius $r_0$. For real $\epsilon$, we find infinite families of discrete EPs. Each family approaches a distinct unperturbed QNM overtone frequency as $r_0$ grows large. Under perturbation at fixed $r_0$, we find that neighboring QNM overtones migrate in the complex plane to meet at these EPs, with the higher overtone moving further. We derive large-$r_0$ asymptotics, finding that EPs have (asymptotically) equal spacing in the tortoise coordinate, and that EP amplitudes $\epsilon$ can be exceedingly small, decaying exponentially according to a rule governed by the complex frequency of the unperturbed QNM. Using quadratic approximations around EPs and high-precision numerical methods, we investigate a range of phenomena in the model: square-root splittings, avoided crossings and hyperbolae, hysteresis and geometric phases generated by encircling an EP, adiabatic invariants, deformed Cassini ovals and cusp-like trajectories, as well as the lemniscate geometry of the excitation factors. We then replace the delta-function perturbation by a finite-width Gaussian, verify the persistence of the EP structure, and show that varying the width extends isolated EPs into exceptional lines in parameter space. Finally, we study the response to initial data and find that, although a weak perturbation can substantially reconfigure the QNM spectrum, the corresponding time-domain response remains perturbative. This contrast is captured by a re-interpretation of the tale of The Princess and the Pea.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Black-hole quasinormal modes (QNMs) may coalesce at exceptional points (EPs) in the frequency spectrum. | background 1.00 | 0.08 | 0.12 | hedged possibility 1.00 |
| s1 | We study EPs and their phenomenology for gravitational perturbations of a Schwarzschild black hole whose Regge-Wheeler potential is perturbed by a localized delta-function of amplitude $\epsilon$ at radius $r_0$. | contribution 0.87 | 0.90 | 0.05 | asserted 0.93 |
| s2 | For real $\epsilon$, we find infinite families of discrete EPs. | result 1.00 | 0.96 | 0.17 | asserted 1.00 |
| s3 | Each family approaches a distinct unperturbed QNM overtone frequency as $r_0$ grows large. | result 1.00 | 0.91 | 0.12 | asserted 1.00 |
| s4 | Under perturbation at fixed $r_0$, we find that neighboring QNM overtones migrate in the complex plane to meet at these EPs, with the higher overtone moving further. | result 1.00 | 0.95 | 0.12 | asserted 1.00 |
| s5 | We derive large-$r_0$ asymptotics, finding that EPs have (asymptotically) equal spacing in the tortoise coordinate, and that EP amplitudes $\epsilon$ can be exceedingly small, decaying exponentially according to a rule governed by the complex frequency of the unperturbed QNM. | result 0.99 | 0.97 | 0.18 | asserted 1.00 |
| s6 | Using quadratic approximations around EPs and high-precision numerical methods, we investigate a range of phenomena in the model: square-root splittings, avoided crossings and hyperbolae, hysteresis and geometric phases generated by encircling an EP, adiabatic invariants, deformed Cassini ovals and cusp-like trajectories, as well as the lemniscate geometry of the excitation factors. | contribution 0.66 | 0.96 | 0.09 | asserted 1.00 |
| s7 | We then replace the delta-function perturbation by a finite-width Gaussian, verify the persistence of the EP structure, and show that varying the width extends isolated EPs into exceptional lines in parameter space. | result 0.50 | 0.96 | 0.08 | asserted 1.00 |
| s8 | Finally, we study the response to initial data and find that, although a weak perturbation can substantially reconfigure the QNM spectrum, the corresponding time-domain response remains perturbative. | result 0.99 | 0.96 | 0.27 | asserted 0.99 |
| s9 | This contrast is captured by a re-interpretation of the tale of The Princess and the Pea. | interpretation 1.00 | 0.46 | 0.10 | asserted 0.98 |
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Original abstract
Black-hole quasinormal modes (QNMs) may coalesce at exceptional points (EPs) in the frequency spectrum. We study EPs and their phenomenology for gravitational perturbations of a Schwarzschild black hole whose Regge-Wheeler potential is perturbed by a localized delta-function of amplitude $\epsilon$ at radius $r_0$. For real $\epsilon$, we find infinite families of discrete EPs. Each family approaches a distinct unperturbed QNM overtone frequency as $r_0$ grows large. Under perturbation at fixed $r_0$, we find that neighboring QNM overtones migrate in the complex plane to meet at these EPs, with the higher overtone moving further. We derive large-$r_0$ asymptotics, finding that EPs have (asymptotically) equal spacing in the tortoise coordinate, and that EP amplitudes $\epsilon$ can be exceedingly small, decaying exponentially according to a rule governed by the complex frequency of the unperturbed QNM. Using quadratic approximations around EPs and high-precision numerical methods, we investigate a range of phenomena in the model: square-root splittings, avoided crossings and hyperbolae, hysteresis and geometric phases generated by encircling an EP, adiabatic invariants, deformed Cassini ovals and cusp-like trajectories, as well as the lemniscate geometry of the excitation factors. We then replace the delta-function perturbation by a finite-width Gaussian, verify the persistence of the EP structure, and show that varying the width extends isolated EPs into exceptional lines in parameter space. Finally, we study the response to initial data and find that, although a weak perturbation can substantially reconfigure the QNM spectrum, the corresponding time-domain response remains perturbative. This contrast is captured by a re-interpretation of the tale of The Princess and the Pea.
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References
How the literature uses it
1 arXiv paper cite this in 6 sentences
Papers citing it · 2 sentences without a confident label
Compares with it 1 paper
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Likewise, Ref. [ref] shows that the divergent difference of the two excitation factors is compensated by the vanishing frequency separation, so that their product has a finite and sheet-independent exceptional-point limit.
Riesz--Laurent representation of black-hole scattering and sourced response at exceptional points · Note added · 0.88
What it cites · 79 sentences without a confident label
Builds on 1 paper
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In the current work, building on Ref. [ref] , we focus on a simple two-parameter model: a delta-function perturbation $(ϵ, r_0)$ that is essentially equivalent to a Gaussian bump of vanishing width ($σ→0$).
Dynamical system approach to the spectral (in)stability of black holes under localised potential perturbations · Introduction · 0.95
Uses the method or tool of 1 paper
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In Ref. [ref] , to find QNM trajectories for fixed $r_0$ and increasing $ϵ$, this was cast as a simple dynamical system: $dω/dϵ= 1/F_ω(ω; r_0)$ (where $F_ω= ∂_ωF$).
Dynamical system approach to the spectral (in)stability of black holes under localised potential perturbations · Introduction · 0.87
Cites as background 33 papers
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The Exceptional Points of QNM spectra have only recently been studied, through pioneering work by Motohashi [ref] and, in parallel, Cavalcante, Richartz and da Cunha [ref] .
Massive scalar perturbations in Kerr Black Holes: near extremal analysis · Introduction · 0.81
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The Exceptional Points of QNM spectra have only recently been studied, through pioneering work by Motohashi [ref] and, in parallel, Cavalcante, Richartz and da Cunha [ref] .
Resonant Excitation of Quasinormal Modes of Black Holes · Introduction · 0.81
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The Exceptional Points of QNM spectra have only recently been studied, through pioneering work by Motohashi [ref] and, in parallel, Cavalcante, Richartz and da Cunha [ref] .
Exceptional point and hysteresis in perturbations of Kerr black holes · Introduction · 0.81
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