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Riesz--Laurent representation of black-hole scattering and sourced response at exceptional points
General Relativity and Quantum Cosmology · Submitted 14 Aug 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Thus two cluster moments provide mode-label-free data from which both scattering and driven responses follow.
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At a black-hole exceptional point (EP), two quasinormal modes coalesce and their separate residues become ill-conditioned. Rather than postulating a near-degenerate modal fit, we derive the response constructively from the complex-scaled Regge--Wheeler--Zerilli resolvent, treating the modes as one isolated rank-two Riesz cluster. Its zeroth and first contour moments determine an exact pair resolvent on both sides of, and at, the EP, without labeling the individual modes or constructing a normalized Jordan chain. At a second-order EP, these moments determine the simple- and double-pole Laurent operators. Although the modal decomposition is singular, fixed-real-frequency transmission and the greybody factor remain real-analytic through the EP, provided that the cluster remains isolated, the complementary resolvent is regular, and no pole reaches the physical axis. Source--observer matrix elements of the Laurent operators define finite, normalization-independent amplitudes and fix both the constant and linear-in-time terms in the causal ringdown. Their equality with the coefficients from the Jost double-zero expansion shows that they are operator-defined coefficients of the specified physical response, rather than fitting parameters. Thus two cluster moments provide mode-label-free data from which both scattering and driven responses follow.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | At a black-hole exceptional point (EP), two quasinormal modes coalesce and their separate residues become ill-conditioned. | background 0.99 | 0.15 | 0.04 | asserted 1.00 |
| s1 | Rather than postulating a near-degenerate modal fit, we derive the response constructively from the complex-scaled Regge--Wheeler--Zerilli resolvent, treating the modes as one isolated rank-two Riesz cluster. | method 0.66 | 0.94 | 0.20 | asserted 1.00 |
| s2 | Its zeroth and first contour moments determine an exact pair resolvent on both sides of, and at, the EP, without labeling the individual modes or constructing a normalized Jordan chain. | method 0.75 | 0.88 | 0.22 | asserted 1.00 |
| s3 | At a second-order EP, these moments determine the simple- and double-pole Laurent operators. | result 0.68 | 0.75 | 0.11 | asserted 1.00 |
| s4 | Although the modal decomposition is singular, fixed-real-frequency transmission and the greybody factor remain real-analytic through the EP, provided that the cluster remains isolated, the complementary resolvent is regular, and no pole reaches the physical axis. | result 0.92 | 0.74 | 0.22 | asserted 0.97 |
| s5 | Source--observer matrix elements of the Laurent operators define finite, normalization-independent amplitudes and fix both the constant and linear-in-time terms in the causal ringdown. | result 0.81 | 0.82 | 0.17 | asserted 1.00 |
| s6 | Their equality with the coefficients from the Jost double-zero expansion shows that they are operator-defined coefficients of the specified physical response, rather than fitting parameters. | interpretation 0.84 | 0.81 | 0.18 | asserted 1.00 |
| s7 | Thus two cluster moments provide mode-label-free data from which both scattering and driven responses follow. | interpretation 0.77 | 0.84 | 0.75 | asserted 1.00 |
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Original abstract
At a black-hole exceptional point (EP), two quasinormal modes coalesce and their separate residues become ill-conditioned. Rather than postulating a near-degenerate modal fit, we derive the response constructively from the complex-scaled Regge--Wheeler--Zerilli resolvent, treating the modes as one isolated rank-two Riesz cluster. Its zeroth and first contour moments determine an exact pair resolvent on both sides of, and at, the EP, without labeling the individual modes or constructing a normalized Jordan chain. At a second-order EP, these moments determine the simple- and double-pole Laurent operators. Although the modal decomposition is singular, fixed-real-frequency transmission and the greybody factor remain real-analytic through the EP, provided that the cluster remains isolated, the complementary resolvent is regular, and no pole reaches the physical axis. Source--observer matrix elements of the Laurent operators define finite, normalization-independent amplitudes and fix both the constant and linear-in-time terms in the causal ringdown. Their equality with the coefficients from the Jost double-zero expansion shows that they are operator-defined coefficients of the specified physical response, rather than fitting parameters. Thus two cluster moments provide mode-label-free data from which both scattering and driven responses follow.
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References
How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 47 sentences without a confident label
Uses the method or tool of 2 papers
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As a concrete two-parameter family containing an exceptional point, we follow the environmental-potential model of Ref. [ref] .
2504.06072 · Gaussian-deformed Regge–Wheeler model setting · 0.90
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The existing complex-scaling implementation of Ref. [ref] and the Julia package [ref] provide the basis and ECS discretizations, while [ref] provides the tortoise map and its inverse.
Complex scaling approach to quasinormal modes of Schwarzschild and Reissner--Nordstr\"om black holes · Numerical realization of the contour-moment construction · 1.00
Compares with 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.
Exceptional points of the Schwarzschild spectrum under local potential perturbation: The Princess and the Pea · Note added · 0.88
Cites as support 1 paper
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In the Kerr EP cascade, the overtone ordering at any fixed parameter point is unambiguous, but adiabatic continuation around the EPs can permute the modes, so that a mode identity continued through parameter space is path dependent [ref] .
Ergodic Hysteresis of the Kerr black hole spectrum · Riesz-cluster construction and main results · 0.99
Cites as background 8 papers
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A second motivation is provided by exceptional points (EPs), at which two QNM eigenvalues and their eigenvectors coalesce [ref] .
Non-Hermitian Hamilton operator in open quantum systems · Exceptional points and the failure of modal coordinates · 0.87
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A second motivation is provided by exceptional points (EPs), at which two QNM eigenvalues and their eigenvectors coalesce [ref] .
The physics of exceptional points · Exceptional points and the failure of modal coordinates · 0.87
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Related notions of QNM excitation coefficients and source-independent excitation factors have long been developed in modal and Green-function analyses [ref] .
Quasinormal ringing of Kerr black holes: The excitation factors · Response to a localized source · 0.98
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