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Exceptional Points in Scalar Quasinormal Modes of Rastall String-Fluid Black Holes

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Ming Zhang, Xiao-Mei Wang, Ya-Nan Wang, Miao Wang, Lei Zhang, Xufen Zhang, De-Cheng Zou

General Relativity and Quantum Cosmology · Submitted 8 Nov 2026 · 61 references

Pith's verdict: A careful numerical EP claim with honest limitations; the A=1 result is solid, but the phenomenological bump keeps it a mathematical demonstration rather than a physical prediction.

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arXiv ID2611.11366 (version 1)
Pith IDGQSED4T6, Pith's permanent record of this paper and its review
CitationDe-Cheng Zou, Lei Zhang, Miao Wang, Ming Zhang, Xiao-Mei Wang, Xufen Zhang, Ya-Nan Wang. "Exceptional Points in Scalar Quasinormal Modes of Rastall String-Fluid Black Holes." arXiv:2611.11366 (2026). https://pith.science/paper/GQSED4T6
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The paper

The paper claims

A resolution-stable second-order exceptional point is found in the scalar quasinormal modes of a Rastall string-fluid black hole, confirmed by eigenvector coalescence, Jordan-chain residuals, square-root splitting, and closed-loop mode…

The reading

This paper claims that by adding a localized Pöschl–Teller bump to the scalar potential of a Rastall string-fluid black hole, one can create a second-order exceptional point (EP) in the quasinormal-mode spectrum. The bump’s position and amplitude provide the two independent controls needed to satisfy the codimension-two condition for an isolated EP. For the regular $A=1$ branch, a two-parameter search using hyperboloidal spectral methods yields a candidate whose coordinates stabilize from $N=24$ to $N=60$, and multiple diagnostics — eigenvector overlap, self-orthogonality, Jordan-chain construction, square-root splitting, and closed-loop exchange — confirm defectiveness. The time-domain response extracted via a Riesz projector shows the characteristic linear-in-time term $(C+Dt)e^{i\omega_{\rm EP}t}$ without fitting. For the logarithmic $A=2$ branch, finite-dimensional degeneracies appear in every truncation but fail to converge, so no continuum EP is claimed. The bump is phenomenological and not a solution of the field equations, so the result demonstrates a spectral mechanism rather than an astrophysical prediction.

What carries the argument

The central machinery is the combination of a hyperboloidal compactification (mapping the black-hole exterior to a finite interval) with two-domain Chebyshev–Lobatto collocation, allowing quasinormal modes to be computed without artificial outer boundaries. A localized Pöschl–Teller bump $V_{\rm bump}(r_*) = \epsilon_b r_h^{-2} \operatorname{sech}^2(r_*/r_h - a)$ is added to the scalar potential, providing two independent control parameters $(a,\epsilon_b)$. Exceptional points are located by solving the two real equations $\operatorname{Re} D = \operatorname{Im} D = 0$ for the complex discriminant $D = (\omega_1-\omega_2)^2$ using arbitrary-precision Newton iteration. Defectiveness is tested via eigenvector overlap $C_R$, left–right self-orthogonality $R$, Jordan-chain residuals $(r_0,r_1,\chi_0)$, closed-loop topological exchange, and the local square-root splitting exponent.

Extended reading notes

Core claim

For the $A=1$, $\epsilon_R=-1$ branch of the Rastall string-fluid black hole with $M=L=1$, $Q=1.2$, and $\ell=2$, the two-parameter search in bump position $a$ and amplitude $\epsilon_b$ locates a second-order exceptional point at $a_{\rm EP}=9.372247727781(9)$, $\epsilon_{b,\rm EP}=0.0082296235574(3)$, with centre frequency $\omega_{\rm EP}=0.907881134084(2)+0.212872971497(1)i$. The defectiveness is confirmed by eigenvector coalescence ($1-C_R$ below $10^{-18}$ at $N=60$), vanishing left–right self-orthogonality, a length-two Jordan chain with residual below $10^{-29}$, square-root splitting exponent $0.4999999$ on both sides, and one-loop mode exchange with zero swapped distance. The constrained time evolution projects onto the EP subspace and yields the Jordan response $(C+Dt)e^{i\omega_{\rm EP}t}$ directly from the nilpotent part of the compressed generator, without fitting.

Load-bearing premise

The central result depends on the assumption that a Pöschl–Teller bump added by hand to the scalar potential captures the essential physics of an environmental perturbation; the authors emphatically state that the bump is used only as a 'phenomenological local environmental correction' and is not a solution of the Rastall field equations.

Editorial extensions

If this is right

  • If confirmed in more realistic models, the existence of an exceptional point in black-hole quasinormal spectra implies that ringdown waveforms can contain a linear-in-time Jordan term $(C+Dt)e^{i\omega_{\rm EP}t}$, beyond the usual superposition of damped sinusoids.
  • The two-parameter search methodology can be applied to other black-hole backgrounds (Kerr, Kerr–Newman, hairy solutions) to systematically locate exceptional points and exceptional lines.
  • For the $A=2$ logarithmic branch, the failure of coordinate convergence points to a need for discretizations adapted to endpoint nonanalyticities (e.g., $\sigma^3\log\sigma$) before a continuum EP can be claimed.
  • The condition-number gap between the two-exponential fit (2QNM) and the Jordan fit (EP total) suggests that near an EP, the Jordan parameterization is numerically better conditioned, which may improve ringdown parameter estimation.
  • Future work can extend the analysis to higher mode pairs, electromagnetic and gravitational perturbations, and study the organization of exceptional points into lines or surfaces in parameter space.

Reading between the lines

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

  • The paper’s reliance on a single phenomenological bump shape leaves open whether other types of environmental perturbations (e.g., power-law tails, non-local interactions) also produce exceptional points. A systematic study with different bump profiles would test the universality of the mechanism.
  • The fact that the $A=2$ branch fails to converge numerically despite local defectiveness suggests that spectral methods lose exponential convergence when the potential has endpoint branch points. This is a caution for any QNM calculation on backgrounds with non-analytic asymptotic behavior.
  • The numerical approach — hyperboloidal compactification, Chebyshev collocation, two-parameter Newton search, and multipronged defectiveness diagnostics — could serve as a template for rigorous EP searches in other open systems described by non-Hermitian spectral problems.
Original abstract

In this paper, we study exceptional points (EPs) in scalar quasinormal modes(QNMs) of Rastall string-fluid black holes by adding a localized P\"oschl--Teller perturbation to the scalar potential, with the bump position $a$ and amplitude $\eb$ serving as independent controls. We solve the hyperboloidal problem with two-domain Chebyshev collocation and arbitrary-precision linear algebra, and use the complex discriminant, eigenvector coalescence, Jordan-chain residuals, square-root splitting and closed-loop mode exchange as complementary tests of defectiveness. For the regular $A=1$, $\eR=-1$ branch, the two-parameter search locates a resolution-stable second-order EP from $N=24$ to $N=60$, while constrained semidiscrete evolution gives the Jordan response $(C+Dt)e^{\ii\omega_{\rm EP}t}$. For the logarithmic $A=2$ branch, defective points occur in each finite truncation but their coordinates do not converge, so we make no continuum claim for this branch.

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, 2026-11-10: 1 request, 3,972 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.64 · compares with prior results 0.18 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

In this paper, we study exceptional points (EPs) in scalar quasinormal modes(QNMs) of Rastall string-fluid black holes by adding a localized P\"oschl--Teller perturbation to the scalar potential, with the bump position $a$ and amplitude $\eb$ serving as independent controls. We solve the hyperboloidal problem with two-domain Chebyshev collocation and arbitrary-precision linear algebra, and use the complex discriminant, eigenvector coalescence, Jordan-chain residuals, square-root splitting and closed-loop mode exchange as complementary tests of defectiveness. For the regular $A=1$, $\eR=-1$ branch, the two-parameter search locates a resolution-stable second-order EP from $N=24$ to $N=60$, while constrained semidiscrete evolution gives the Jordan response $(C+Dt)e^{\ii\omega_{\rm EP}t}$. For the logarithmic $A=2$ branch, defective points occur in each finite truncation but their coordinates do not converge, so we make no continuum claim for this branch.

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 In this paper, we study exceptional points (EPs) in scalar quasinormal modes(QNMs) of Rastall string-fluid black holes by adding a localized P\"oschl--Teller perturbation to the scalar potential, with the bump position $a$ and amplitude $\eb$ serving as independent controls. contribution 0.98 0.92 0.07 asserted 0.91
s1 We solve the hyperboloidal problem with two-domain Chebyshev collocation and arbitrary-precision linear algebra, and use the complex discriminant, eigenvector coalescence, Jordan-chain residuals, square-root splitting and closed-loop mode exchange as complementary tests of defectiveness. method 1.00 0.95 0.04 asserted 0.99
s2 For the regular $A=1$, $\eR=-1$ branch, the two-parameter search locates a resolution-stable second-order EP from $N=24$ to $N=60$, while constrained semidiscrete evolution gives the Jordan response $(C+Dt)e^{\ii\omega_{\rm EP}t}$. result 1.00 0.96 0.40 asserted 1.00
s3 For the logarithmic $A=2$ branch, defective points occur in each finite truncation but their coordinates do not converge, so we make no continuum claim for this branch. limitation 0.92 0.94 0.11 asserted 0.98

What kind of paper the classifier reports this is: simulation study 0.88 · evidence from simulation or computation 1.00.

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
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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. Background geometry and localized perturbation for the regular
Figure 2. tracks four low-lying branches at N = 30 while a varies at fixed amplitude. Labels follow continuity between neighboring points. Two branches approach ω ≃ 0.908 + 0.213i and form the interacting pair. The other two remain separated. The coarse step ∆a = 0.2 misses the rapid motion near the degeneracy, so we refine the search locally. This scan identifies the pair but cannot locate the EP without the second control parameter. 0 0.2 0.4 0.6 0.8 1 1.2 0.5 1 Re ω Im ω (a) Global QNM migration branch 0 branch 1 branch 2 branch 3 continuum EP 0.902 0.904 0.906 0.908 0.91 0.912 0.914 0.21…
Figure 3. Two-parameter localization near the N = 42 EP of the A = 1 branch, with fixed M = L = 1, Q = 1.2, εR = −1 and ℓ = 2. The color represents log10 |ω1 −ω2| in the (a, ϵb) plane; the axes are scaled by 103 and 106 , respectively. The star marks the EP center relocated with arbitrary-precision arithmetic, and the isolated minimum identifies the interacting pair. For each N, we solve Re D = Im D = 0 independently and seed Newton’s iteration with the previous resolution
Figure 4. Radial-eigenfunction coalescence at the N = 42 candidate of the A = 1 branch, with fixed M = L = 1, Q = 1.2, εR = −1 and ℓ = 2. (a) Real parts and (b) imaginary parts of the two unit-L 2 -normalized, phase-aligned eigenfunctions; the solid and dashed curves denote ϕ1 and ϕ2, respectively. (c) Their distance at fixed ϵb = ϵ (42) b,EP; the star marks the EP and the dashed line indicates the distance 1.660 × 10−5 at that point
Figure 5. Resolution convergence and EP diagnostics for the
Figure 6. Closed-loop transport of the A = 1 branch at N = 54, with fixed M = L = 1, Q = 1.2, εR = −1 and ℓ = 2. (a) The loop in the (a, ϵb) plane. (b) The continuously tracked complex￾frequency trajectories. Arrows indicate the transport direction, and the star marks the EP center. One circuit exchanges the two modes. 4 A = 1 branch: time-domain response 4.1 Constraint elimination and a stable initial-value problem The interface conditions replace two rows of the discrete system. Thus B is singular and cannot be inverted directly. We eliminate the constraints explicitly. Let Φ and Ψ denote…
Figure 7. Time-domain waveform of the A = 1 branch at N = 42, with fixed M = L = 1, Q = 1.2, εR = −1, ℓ = 2 and the EP coordinates in Eqs. (30)–(32). The black solid curve is the initial-value evolution in the stable Schur subspace, the blue dashed curve is the EP two￾dimensional contribution, and the red dotted curve is the Jordan reconstruction; the lower panel shows the absolute differences from the EP-pair propagation. 13
Figure 8. Jordan response of the A = 1 branch at N = 42, with fixed M = L = 1, Q = 1.2, εR = −1 and ℓ = 2. The curves show the constant term 2 Re(Cesct ), the linear term 2 Re(Dtesct ), and their sum, which contains the characteristic linear-in-time contribution. 14
Figure 9. Fits of the same A = 1 EP two-mode signal with 2QNM, EP linear and EP total, for fixed M = L = 1, Q = 1.2, εR = −1, ℓ = 2 and 12 ≤ t/rh ≤ 35. The upper panel shows the waveforms and the lower panel the absolute residuals. 5 10 15 10−10 10−7 10−4 10−1 fit start t0/rh NRMS (a) Normalized RMS mismatch 5 10 15 100 101 102 103 104 105 fit start t0/rh κ2(X) (b) Linear design conditioning 2QNM EP linear EP total
Figure 10. Dependence of the three A = 1 time-domain models on the fitting start time t0, with fixed M = L = 1, Q = 1.2, εR = −1, ℓ = 2 and t1 = 35rh. The upper panel shows the normalized residual and the lower panel the design-matrix condition number. 16
Figure 11. Time-domain resolution comparison for the
Figure 12. Local frequency structure of the A = 2 branch at N = 42, with fixed M = L = λR = Q = 1, εR = +1 and ℓ = 2. (a) The local coalescing pair of continuously tracked complex￾frequency branches; the star marks the center frequency in Eq. (50). (b) Two-parameter local￾ization near the EP, with color denoting log10 |ω1 − ω2|; the star marks the finite-dimensional EP center relocated with 128-bit arithmetic, and the heat map is interpolated from a 5 × 5 search grid. 19
Figure 13. Defectiveness and Jordan-chain diagnostics of the
Figure 14. Closed-loop transport of the A = 2 branch at N = 42, with fixed M = L = λR = Q = 1, εR = +1 and ℓ = 2. (a) The loop in the (a, ϵb) plane, with coordinates normalized by the semiaxes 0.002 and 5 × 10−6 . (b) The continuously tracked complex-frequency trajectories. Arrows indicate the transport direction, and the star marks the finite-dimensional EP center. One circuit exchanges the two modes. 20 30 40 50 60 70 10.097001 10.097501 10.098001 N a A = 2 EP coordinate 20 30 40 50 60 70 0.009434 0.009436 0.009438 0.00944 N ϵb A = 2 bump amplitude 20 30 40 50 60 70 0.7310301 0.7310401…
Figure 15. Resolution dependence of independently relocated coordinates for the
Figure 16. Time-domain waveform of the A = 2 branch at N = 42, with fixed M = L = λR = Q = 1, εR = +1, ℓ = 2 and the finite-dimensional EP coordinates in Eq. (50). The black solid curve is the initial-value evolution in the stable Schur subspace, the blue dashed curve is the EP two-dimensional contribution, and the red dotted curve is the Jordan reconstruction; the lower panel shows the absolute differences from the EP-pair propagation. The corresponding real response is ΦJ(t) = 2 Re (C + Dt)e sct
Figure 17. Jordan response of the A = 2 branch at N = 42, with fixed M = L = λR = Q = 1, εR = +1 and ℓ = 2. The curves show the constant term 2 Re(Cesct ), the linear term 2 Re(Dtesct ), and their sum, which contains the characteristic linear-in-time contribution
Figure 18. Fits of the same A = 2 EP two-mode signal at N = 42 with 2QNM, EP linear and EP total, for fixed M = L = λR = Q = 1, εR = +1, ℓ = 2 and 12 ≤ t/rh ≤ 35. The upper panel shows the waveform and the lower panel the absolute residual. 5 10 15 10−10 10−7 10−4 10−1 fit start t0/rh NRMS (a) Normalized RMS mismatch 5 10 15 100 101 102 103 104 105 fit start t0/rh κ2(X) (b) Linear design conditioning 2QNM EP linear EP total
Figure 19. Dependence of the three A = 2 time-domain models on the fitting start time t0, with fixed M = L = λR = Q = 1, εR = +1, ℓ = 2 and t1 = 35rh. The upper panel shows the normalized residual and the lower panel the design-matrix condition number. 25
Figure 20. Time-domain resolution comparison for the

Figures reproduced from arXiv:2611.11366 by the authors.

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