Pith. sign in

REVIEW 4 major objections 4 minor 5 cited by

Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh's approach

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that Keldysh resolvent expansions, built from modes and comodes of a hyperboloidal non-selfadjoint generator, provide a scalar-product-free spectral construction of quasinormal-mode expansions that accurately reproduces…

desk verdict Solid Keldysh formalism and impressive numerics, but the Schwarzschild tail recovery rests on an unproven continuum limit. read the letter →

arxiv 2412.02793 v2 pith:XPOL7367 submitted 2024-12-03 gr-qc physics.optics

Jérémy Besson , José Luis Jaramillo This is my paper · ORCID
classification gr-qcphysics.optics MSC 83C5783C3535P0547A1065M70
keywords quasinormalmodesblackholeperturbationsKeldyshexpansionnon-selfadjointoperatorshyperboloidalfoliationPricelawtailspseudospectrumWeyl
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that quasinormal-mode (QNM) expansions of black hole perturbations are best built not from orthogonality or completeness of modes, but from a Keldysh expansion of the resolvent of the non-selfadjoint time generator in a hyperboloidal slicing. In this setting QNMs are proper eigenvalues with normalisable eigenfunctions, and the expansion coefficients come from dual pairing with the left eigenfunctions, the comodes, so no scalar product is needed. For Gaussian test data the truncated Keldysh series reproduces the boundary time-domain signal from early times dominated by high overtones through late ringdown, with remarkable accuracy in all studied asymptotics. The most striking claim is that applying the same Keldysh prescription to the non-convergent eigenvalues that discretise the Schwarzschild branch cut recovers the power-law late-time tails, including the Price law $tau^{{-(ell+1)}}$. The paper also extracts second-order QNMs, constructs H^p-pseudospectra, identifies H^p transient growths, and confirms the QNM Weyl law N(omega) ~ $omega^{3}$ in flat, de Sitter and anti-de Sitter asymptotics.

What carries the argument

The load-bearing object is the Keldysh expansion of the resolvent of the infinitesimal time generator L in a hyperboloidal compactification. The hyperboloidal foliation and coordinate compactification turn outgoing boundary conditions into regularity requirements, making QNMs proper eigenvalues of a non-selfadjoint operator with normalisable eigenfunctions in a Hilbert or Banach space. The Keldysh identity expresses R_L(omega) as a sum over poles omega_n with residues given by rank-one operators <alpha_n, .> v_n, where alpha_n are comodes solving L^t alpha_n = omega_n alpha_n, plus a holomorphic remainder; this remainder provides the explicit error bound for truncated QNM series. The discrete Chebyshev pseudospectral approximation then replaces L by a matrix, and the same identity, applied to all eigenvalues of the finite-rank approximant, reconstructs the full evolution, which is what makes the branch-cut tail recovery possible.

What would settle it

Take a fixed Schwarzschild case (say ell=2), evaluate the truncated Keldysh sum over the branch-cut eigenvalues at a late time tau, and compare with a direct numerical evaluation of the Bromwich integral around the branch cut for increasing grid sizes N. If the difference does not decrease toward zero as N grows, the recovered Price tail is a finite-rank artifact rather than a property of the true resolvent.

Watch

Extended reading notes

Core claim

The central discovery is that the Keldysh expansion of the resolvent, R_L(omega) = sum_n [<alpha_n, .>/(omega_n - omega)] v_n + H(omega), built from modes v_n and comodes alpha_n of L and its transpose L^t, gives a spectral, scalar-product-free construction of Lax-Phillips resonant expansions. The paper demonstrates numerically, in the Poschl-Teller model and in Schwarzschild, Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter spacetimes, that the resulting QNM series u(tau,x) ~ sum_n $e^{{i omega_n tau}}$ a_n v_n(x), with a_n = <alpha_n,u_0>/<alpha_n,v_n>, matches the hyperboloidal time evolution of Gaussian data at boundaries throughout the signal. Beyond its strict domain of validity, applying the same formula to the discretised branch-cut eigenvalues reproduces the Schwarzschild late-time power-law tails with the correct Price exponent. If this claim survives scrutiny, the Keldysh scheme is a single spectral algorithm that accounts for both the discrete QNM content and the continuous branch-cut content of black hole perturbation dynamics.

Load-bearing premise

The tail recovery rests on the assumption that the non-convergent eigenvalues discretising the Schwarzschild branch cut, each weighted with its Keldysh coefficient, form a valid discrete approximation (like a Riemann sum) of the continuous branch-cut contribution to the inversion integral; the paper does not prove this and notes it only 'could be understood' that way.

Editorial extensions

If this is right

  • Keldysh QNM expansions are independent of the scalar product: only the product a_n v_n(x) is defined, while constant excitation coefficients require an additional norm.
  • The scheme generalises Ansorg-Macedo expansions to arbitrary dimensions and to non-diagonalisable cases, going beyond effective one-dimensional problems.
  • The truncated Keldysh series can be used as a spectral time-domain reconstruction tool: given an acceptable error and a number of modes, the contour plots fix the earliest valid time for the expansion.
  • For Gaussian data in the Poschl-Teller case the bulk QNM series converges for tau > 1/kappa, providing a natural timescale for ringdown fitting.
  • The QNM Weyl law N(omega) ~ omega^3 holds in flat, de Sitter and anti-de Sitter asymptotics.

Reading between the lines

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

  • If the Riemann-sum interpretation of the branch-cut reconstruction is made rigorous with an error estimate, the Keldysh scheme would provide a unified spectral algorithm for ringdown and tails that could be applied to Kerr or higher-dimensional spacetimes.
  • The tau > 1/kappa convergence threshold hints at a first-principles selection criterion for the start time of ringdown fits in gravitational-wave data analysis; testing it on realistic inspiral-merger-ringdown waveforms would be a natural next step.
  • The delta-like H^p transient growth in the p to infinity limit suggests that high-derivative norms can expose an initial loss of regularity; connecting this with Aretakis-type instability at extremal horizons is a testable extension.
  • Since the Keldysh scheme extends to quadratic pencils, it could provide a practical spectral route to second-order QNM excitation coefficients for mode coupling with different angular quantum numbers.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper develops a Keldysh (modes/comodes) expansion of the resolvent for hyperboloidal, non-selfadjoint black-hole perturbation generators, and uses it to construct quasinormal-mode (QNM) expansions. The authors clarify that the dual-pairing form of the expansion does not require a scalar product, while constant excitation coefficients require one, and they apply the scheme to Pöschl-Teller, Schwarzschild, Schwarzschild-dS, and Schwarzschild-AdS testbeds with Gaussian data. They report accurate reconstruction of boundary time series from early times onward, recovery of Schwarzschild late-time power-law tails by applying the Keldysh prescription to discretized branch-cut eigenvalues, studies of early-time convergence and H^p transient growths, H^p pseudospectra consistent with Warnick's band structure, and numerical evidence for a universal Weyl law for QNM counting.

Significance. If the central claims hold, the paper provides an efficient and conceptually clean spectral route to QNM expansions and, strikingly, to late-time tails. The analytic Keldysh formulation in terms of transpose operators is a genuine clarification of the role of scalar products, and Appendix B.3's finite-rank identity is exact and useful. The numerical comparisons between spectral and time-domain signals use no fitted parameters, and the H^p pseudospectra and Weyl-law results are useful exploratory contributions. However, the headline tail-recovery result and the early-time convergence claim rest on heuristic or single-data evidence, and several supporting studies are deferred to future papers by the same authors, so the significance is conditional on those gaps being closed.

major comments (4)
  1. [Sec. 5.1 and App. B.3] The claim that applying the Keldysh scheme to the non-convergent eigenvalues discretizing the Schwarzschild branch cut accurately recovers Price-law tails is load-bearing but not quantitatively supported. Appendix B.3 shows exactly that the finite-rank matrix L_N reconstructs e^{iτL_N}, but this does not establish convergence of the finite-rank branch-cut dynamics to the continuum hyperboloidal evolution. The branch-cut eigenvalues do not converge as N grows (Figs. 2c, 3c), and Fig. 12 only shows that the apparent power-law window extends to later times with increasing N. The paper lacks a fixed-τ error-versus-N comparison against the continuum branch-cut integral, a demonstration that the fitted exponent β_fit converges to −(ℓ+1) as N grows, or an estimate of the discretization error. Without such evidence, the excellent β_fit values in Fig. 11 could be a serendipitous finite-rank effect rather than a property of the true resolvent.
  2. [Sec. 5.2.2, Eqs. (88)–(90), Fig. 16] The convergence result for the bulk QNM series at fixed τ_o depends critically on the estimate C(N_QNM,L) ≲ C e^{N_QNM} in Eq. (89). This estimate is read off from a tangent at the origin in Fig. 16 for a single Gaussian initial data u_0 in Eq. (204), while Eq. (37) states that the error constant should be independent of the initial data. The manuscript does not provide a data-independent bound or even numerical evidence that the growth exponent is stable across initial data. Therefore the conclusion that τ_o > 1/κ implies convergence is, at present, a heuristic observation for a special test case, not a general result. This should be stated explicitly and, ideally, supported by additional data or by a proof of the exponential bound.
  3. [Sec. 5.2.1 and Sec. 6, conclusions ii.3] The conclusions state that Pöschl-Teller and dS asymptotics 'present good, indeed uniform, convergence properties' of the QNM time series, but the body of the paper only reports that Fig. 14 'suggests' pointwise convergence and that the Schwarzschild-dS case is 'more difficult to evaluate.' The contour-line visual inspection is performed for one initial data and is not a proof of uniform convergence. Please align the summary statements with the exploratory status of the numerical evidence, or provide a rigorous or more systematic numerical test of uniform convergence.
  4. [Sec. 3.3, Sec. 6, and abstract] The abstract and conclusions state that the paper 'demonstrates the efficiency and accuracy' of the Keldysh approach, but all demonstrations use a single Gaussian initial data family (Appendix E.3) and are limited by the tolerance of the ODE/DAE solver in several cases (Fig. 7). The paper itself acknowledges that a systematic study of generic initial data is left for future work [56]. For a proof-of-principle article this is acceptable, but the wording should be softened from 'demonstrate' to 'illustrate in proof-of-principle testbeds' so that the claims match the evidence presented.
minor comments (4)
  1. [Footnote 14 and Sec. 6, item i.3] Footnote 14 explicitly states 'We lack a proof of the later statement' regarding uniqueness of the QNM time series at null infinity, yet conclusion i.3 presents this uniqueness as a result. Please add a qualifier such as 'conjectured' or 'supported by the Lax-Phillips framework'.
  2. [Fig. 14 caption and Sec. 5.2.1] The caption of Fig. 14b uses Λ = 0.11, whereas Section 3.2.1 and the surrounding text mainly use Λ = 0.07/M². Please unify the notation so the reader can reproduce the parameter choices.
  3. [Sec. 5.4, Fig. 24] The Weyl-law evidence is based on a linear fit to 'a few dozens of points at the end of the series' for each spacetime. Please report the number of fitted points, the residuals, and the sensitivity of the fitted exponent to the fitting window, since the claim of universality across asymptotics rests on this numerical fit.
  4. [Sec. 5.3, Eq. (102)] The statement |⟨v̂−_p, v̂+_p⟩_{H^p}| ∼ 1 − 1/p^4 is presented without derivation or numerical table; please indicate whether this is an analytic result, a fit, or a numerical observation, and provide the corresponding data or reference.

Circularity Check

1 steps flagged · score 6.0 of 10

Schwarzschild-tail 'recovery' is the finite-rank Keldysh identity: App. B.3's Eq. (139) equals e^{iτL_N} by construction, so the tail agreement is algebraic; the continuum Price law is an unproved limit.

  1. self definitional [Appendix B.3, Eqs. (136)-(139); used in Sec. 5.1]
    "the forthright application of the expressions in the Keldysh scheme to all the eigenvalues of the matrix approximant of L does recover the dynamics from the evolution operator. This fact justifies the presence of polynomial tails when applying the scheme to the appropriate subset of eigenvalues..."

    Eq. (139) is a rewrite of Eq. (136): for the diagonalizable finite-rank approximant L_N, the Keldysh sum over all its eigenvectors is exactly the matrix propagator e^{iτL_N}u0. The branch-cut Keldysh sum is therefore, by construction, the branch-cut component of the same finite-rank evolution that generates the 'time-domain' tail curve (method of lines on L_N, Sec. E.2). The late-time agreement in Figs. 10-11 is an algebraic consistency check, not an independent validation that the continuum Schwarzschild branch cut has been captured. The paper's Price-law claim thus reduces at fixed N to the finite-rank identity; the missing piece is a quantitative convergence proof of the non-convergent branch-cut eigenvalues to the continuum branch-cut integral (Fig. 12 offers only a qualitative trend).

full rationale

This is a largely self-contained spectral-methods paper. The Keldysh resolvent expansion is taken from external literature (Mennicken-Möller, Beyn-Latushkin-Rottmann-Matthes) and rederived in Sec. 2; the QNM coefficients are set by dual pairings and checked against time-domain integrations for four potentials, with no parameter fitted to force agreement. The H^p-pseudospectra and Weyl-law sections are independent numerical analyses. The one genuinely circular element is the advertised 'recovery' of Schwarzschild tails: Appendix B.3 shows that the Keldysh sum over all eigenvalues of the discretized generator L_N is identically the finite-rank propagator e^{iτL_N}. Consequently, the branch-cut part of the Keldysh sum agrees with the late-time part of the same matrix evolution by construction, not by independent physics; the 'Price-law recovery' of Sec. 5.1 is thus a property of the finite-rank model unless one proves convergence of the non-convergent branch-cut eigenvalues to the continuum branch-cut integral, which the paper does not (Fig. 12 is only qualitative). Self-citations [1] and [10] are refinements or background, not load-bearing for the Keldysh formula itself, which comes from external sources. This partial circularity affects one highlighted beyond-validity claim, while the QNM expansion results themselves remain independent.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard operator-theoretic results (Keldysh, Fredholm theory) and on the hyperboloidal framework from [21,22]. The main new assumption is the ad hoc Riemann-sum interpretation of branch-cut eigenvalues for tail recovery, plus an empirically fitted error-constant bound for the convergence claim. No new physical entities are introduced.

free parameters (3)
  • Error-constant growth exponent in convergence bound = C(N_QNM,L) ≤ C·e^{N_QNM}, C unspecified
    Section 5.2.2 estimates C(N_QNM,L) from the tangent at the origin of Fig. 16, using the same Gaussian initial data, to conclude convergence for τ_o > 1/κ. The exponential rate is fitted to numerical data rather than derived.
  • Gaussian initial data parameters = a=8, b=0.1
    Hand-chosen test initial data (Eq. 204); used for all demonstrations. Not fitted to results, but the proof-of-principle conclusions are only directly verified for this family.
  • Cosmological constant and AdS radius choices = Λ=0.07/M^2 for S-dS; α=1 (r_h=R=1) for S-AdS
    Chosen example values in proof-of-principle spirit; authors note dependence on Λ is not systematically studied.
assumptions (6)
  • standard math Keldysh expansion of the resolvent for Fredholm index-0 operator pencils
    Invoked in Section 2.1 (Eqs. 18-22) following [8,35]; forms the basis of the QNM coefficients.
  • domain assumption QNM eigenvalues are simple and isolated in the studied domain
    Section 2.1: 'we assume in addition, for simplicity, that the ω_n's are non-degenerate (simple)'. The text says extension to degenerate cases is straightforward, but it is not carried out.
  • domain assumption Operator F(ω)=L−ωI (or L−ωB) is Fredholm of index 0 on the chosen spaces
    Section 2.1, assumed to guarantee discreteness of spectrum and meromorphy of the resolvent.
  • domain assumption Hyperboloidal compactification turns outgoing conditions into regularity and renders QNMs as eigenvalues of a well-defined non-selfadjoint operator
    Follows [21,22] and is the setting for the whole method; cited as established, not proven in this paper.
  • domain assumption Warnick's H^p band characterization of QNM eigenvalues
    Used in Sections 4.2.2 and 5.3.1 to interpret H^p pseudospectra and transient growths; imported from [21] and cited as theorem.
  • ad hoc to paper Discretized branch-cut eigenvalues provide a Riemann-sum approximation of the continuous branch cut
    Section 5.1 uses this as a heuristic explanation of tail recovery; explicitly not derived or quantified.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh's approach." pith.science (2026). https://pith.science/paper/XPOL7367

@misc{pith2026241202793,
  author       = {Pith},
  title        = {Pith review of: Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh's approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XPOL7367}},
  note         = {Machine review of arXiv:2412.02793}
}
abstract

We study quasinormal mode expansions by adopting a Keldysh scheme for the spectral construction of asymptotic resonant expansions. Quasinormal modes are first cast in terms of a non-selfadjoint problem by adopting, in a black hole perturbation setting, a spacetime hyperboloidal approach. Then the Keldysh expansion of the resolvent, built on bi-orthogonal systems, provides a spectral version of Lax-Phillips expansions on scattering resonances. We clarify the role of scalar product structures in the Keldysh setting, that prove non-necessary to construct the resonant expansions (in particular the quasinormal mode time-series at null infinity), but are required to define the (constant) excitation coefficients in the bulk resonant expansion. We demonstrate the efficiency and accuracy of the Keldysh spectral approach to (non-selfadjoint) dynamics, even beyond its limits of validity, in particular recovering Schwarzschild black hole late power-law tails. We also study early dynamics by exploring i) the existence of an earliest time of validity of the resonant expansion and ii) the interplay between overtones extracted with the Keldysh scheme and regularity. Specifically, we address convergence aspects of the series and, on the other hand, we implement non-modal analysis tools, namely assessing $H^p$-Sobolev dynamical transient growths and constructing $H^p$-pseudospectra. Finally, we apply the Keldysh scheme to calculate ''second-order'' quasinormal modes and complement the qualitative study of overtone distribution by presenting the Weyl law for the counting of quasinormal modes in black holes with different (flat, De Sitter, anti-De Sitter) spacetime asymptotics.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

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

  1. Probing the Unstable Spectrum of Schwarzschild-like Black Holes

    gr-qc 2025-01 conditional novelty 6.0 of 10

    The quasinormal mode spectrum of physically deformed Schwarzschild-like black holes stays unstable, contradicting the expectation that displaced spectra become stable.

  2. The (in)stability of quasinormal modes of Boulware-Deser-Wheeler black hole in the hyperboloidal framework

    gr-qc 2024-12 conditional novelty 6.0 of 10

    For Boulware-Deser-Wheeler black holes, quasinormal-mode spectra are pseudospectrally unstable, yet time-domain waveforms shift only quadratically under small potential bumps.

  3. Exceptional line and pseudospectrum in black hole spectroscopy

    gr-qc 2025-11 conditional novelty 5.0 of 10

    A continuous line of exceptional points exists in the three-parameter space of a Gaussian-bump-perturbed Regge-Wheeler potential, with pseudospectral contour sizes scaling as ε^{1/2} at second-order EPs.

  4. Transients in black hole perturbation theory

    gr-qc 2025-07 conditional novelty 2.0 of 10

    Black hole quasinormal mode sums can display long transient plateaus and transient energy growth due to non-normality of the evolution operator, and this review systematically collects the frequency- and time-domain evidence.

  5. Topical Collection-Hyperboloidal Foliations in the Era of Gravitational-Wave Astronomy: From Mathematical Relativity to Astrophysics

    gr-qc 2025-09 unverdicted novelty 1.0 of 10

    An editorial overview of a topical collection showing that hyperboloidal foliation methods have matured from linear perturbations toward nonlinear evolutions.

Reference graph

Works this paper leans on

99 extracted references · 39 canonical work pages · cited by 5 Pith papers

  1. [56]

    Besson, J., Bizo ´n, P., Jaramillo, J.L., Pook-Kolb, D.: Keldysh quasi-normal mode expan- sions of black hole perturbations: convergence and regularity aspects, in preparation (2025)

    work page 2025

  2. [1]

    Gasperin, E., Jaramillo, J.L.: Energy scales and black hole pseudospectra: the structural role of the scalar product. Class. Quant. Grav.39(11), 115010 (2022) https://doi.org/10. 1088/1361-6382/ac5054 arXiv:2107.12865 [gr-qc]

    arXiv 2022

  3. [2]

    Zeitschrift fur Physik51(3-4), 204– 212 (1928) https://doi.org/10.1007/BF01343196

    Gamow, G.: Zur Quantentheorie des Atomkernes. Zeitschrift fur Physik51(3-4), 204– 212 (1928) https://doi.org/10.1007/BF01343196

    doi:10.1007/bf01343196 1928

  4. [3]

    Pure and Applied Mathematics, vol

    Lax, P.D., Phillips, R.S.: Scattering Theory, Second edition edn. Pure and Applied Mathematics, vol. 26. Academic Press, Boston (1989)

    1989

  5. [4]

    Vainberg, B.R.: Exterior elliptic problems that depend polynomially on the spectral parameter, and the asymptotic behavior for large values of the time of the solutions of 83 nonstationary problems. Mat. Sb. (N.S.)92(134), 224–241 (1973) translated in Math. USSR-Sb. 21 (1973), 221–239

    1973

  6. [5]

    Communications on Pure and Applied Mathematics53(10), 1305–1334 (2000) https://doi.org/10.1002/ 1097-0312(200010)53:10⟨1305::AID-CPA4⟩3.0.CO;2-#

    Tang, S.-H., Zworski, M.: Resonance expansions of scattered waves. Communications on Pure and Applied Mathematics53(10), 1305–1334 (2000) https://doi.org/10.1002/ 1097-0312(200010)53:10⟨1305::AID-CPA4⟩3.0.CO;2-#

    2000

  7. [6]

    Bulletin of Mathematical Sciences7(1), 1–85 (2017)

    Zworski, M.: Mathematical study of scattering resonances. Bulletin of Mathematical Sciences7(1), 1–85 (2017)

    2017

  8. [7]

    Graduate Studies in Mathematics

    Dyatlov, S., Zworski, M.: Mathematical Theory of Scattering Resonances. Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island (2019).https://books.google.fr/books?id=atCuDwAAQBAJ

    2019

Show all 99 references
  1. [8]

    Mennicken, R., M ¨oller, M.: Non-Self-Adjoint Boundary Eigen- value Problems. ISSN. Elsevier Science, Amsterdam (2003). https://books.google.fr/books?id=ze2tvoola9wC

    2003

  2. [9]

    arXiv e-prints, 1210–3952 (2012) arXiv:1210.3952 [math.NA]

    Beyn, W.-J., Latushkin, Y ., Rottmann-Matthes, J.: Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals. arXiv e-prints, 1210–3952 (2012) arXiv:1210.3952 [math.NA]

    2012 arXiv

  3. [10]

    Jaramillo, J.L., Panosso Macedo, R., Al Sheikh, L.: Pseudospectrum and Black Hole Quasinormal Mode Instability. Phys. Rev. X11(3), 031003 (2021) https://doi.org/10. 1103/PhysRevX.11.031003 arXiv:2004.06434 [gr-qc]

    2021 arXiv

  4. [11]

    Nollert, H.-P., Price, R.H.: Quantifying excitations of quasinormal mode systems. J. Math. Phys.40, 980–1010 (1999) https://doi.org/10.1063/1.532698 arXiv:gr- qc/9810074 [gr-qc]

    1999

  5. [12]

    Ching, E.S.C., Leung, P.T., Suen, W.M., Young, K.: Quasinormal mode expansion for linearized waves in gravitational systems. Phys. Rev. Lett.74, 4588–4591 (1995) https: //doi.org/10.1103/PhysRevLett.74.4588

    1995 doi

  6. [13]

    Leung, P.T., Liu, S.Y ., Young, K.: Completeness and orthogonality of quasinormal modes in leaky optical cavities. Phys. Rev. A49, 3057–3067 (1994) https://doi.org/10. 1103/PhysRevA.49.3057

    1994

  7. [14]

    Ching, E.S.C., Leung, P.T., Brink, A., Suen, W.M., Tong, S.S., Young, K.: Quasinormal- mode expansion for waves in open systems. Rev. Mod. Phys.70, 1545–1554 (1998) https://doi.org/10.1103/RevModPhys.70.1545

    1998 doi

  8. [15]

    Laser & Photonics Review12(5), 1700113 (2018) https://doi.org/10.1002/lpor.201700113 arXiv:1705.02433 [physics.optics] 84

    Lalanne, P., Yan, W., Vynck, K., Sauvan, C., Hugonin, J.-P.: Light Interaction with Pho- tonic and Plasmonic Resonances. Laser & Photonics Review12(5), 1700113 (2018) https://doi.org/10.1002/lpor.201700113 arXiv:1705.02433 [physics.optics] 84

    2018 arXiv

  9. [16]

    Beyer, H.R.: On the completeness of the quasinormal modes of the Poschl- Teller potential. Commun. Math. Phys.204, 397–423 (1999) https://doi.org/10.1007/ s002200050651 arXiv:gr-qc/9803034 [gr-qc]

    1999 arXiv

  10. [17]

    Acta Phys

    Warnick, C.: (in)completeness of quasinormal modes. Acta Phys. Pol. B Proc. Suppl. 15, 1-A13 (2022) https://doi.org/10.5506/APhysPolBSupp.15.1-A13

    2022 doi

  11. [18]

    International Journal of Quantum Chemistry14(4), 529–542 (1978)

    Simon, B.: Resonances and complex scaling: A rigorous overview. International Journal of Quantum Chemistry14(4), 529–542 (1978)

    1978

  12. [19]

    Cambridge University Press, Cam- bridge (2011)

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

    2011

  13. [20]

    Melrose, R.B.: Geometric Scattering Theory vol. 1. Cambridge University Press, Cambridge (1995)

    1995

  14. [21]

    Warnick, C.M.: On quasinormal modes of asymptotically anti-de Sitter black holes. Commun. Math. Phys.333(2), 959–1035 (2015) https://doi.org/10.1007/ s00220-014-2171-1 arXiv:1306.5760 [gr-qc]

    2015 arXiv

  15. [22]

    Ansorg, M., Panosso Macedo, R.: Spectral decomposition of black-hole perturbations on hyperboloidal slices. Phys. Rev.D93(12), 124016 (2016) https://doi.org/10.1103/ PhysRevD.93.124016 arXiv:1604.02261 [gr-qc]

    2016 arXiv

  16. [23]

    Panosso Macedo, R., Jaramillo, J.L., Ansorg, M.: Hyperboloidal slicing approach to quasi-normal mode expansions: the Reissner-Nordstr ¨om case. Phys. Rev.D98(12), 124005 (2018) https://doi.org/10.1103/PhysRevD.98.124005 arXiv:1809.02837 [gr-qc]

    2018 arXiv

  17. [24]

    Gajic, D., Warnick, C.: A model problem for quasinormal ringdown of asymptotically flat or extremal black holes. J. Math. Phys.61(10), 102501 (2020) https://doi.org/10. 1063/5.0024699 arXiv:1910.08481 [math.AP]

    2020 arXiv

  18. [25]

    Gajic, D., Warnick, C.: Quasinormal Modes in Extremal Reissner–Nordstr ¨om Space- times. Commun. Math. Phys.385(3), 1395–1498 (2021) https://doi.org/10.1007/ s00220-021-04137-4 arXiv:1910.08479 [gr-qc]

    2021 arXiv

  19. [26]

    Annals of PDE 7, 1–13 (2021)

    Galkowski, J., Zworski, M.: Outgoing solutions via gevrey-2 properties. Annals of PDE 7, 1–13 (2021)

    2021

  20. [27]

    Annales Henri Poincare23(12), 4343–4390 (2022) https://doi.org/10

    Joykutty, J.: Existence of Zero-Damped Quasinormal Frequencies for Nearly Extremal Black Holes. Annales Henri Poincare23(12), 4343–4390 (2022) https://doi.org/10. 1007/s00023-022-01202-z arXiv:2112.05669 [gr-qc]

    2022 arXiv

  21. [28]

    PhD thesis, Depart- ment of Applied Mathematics And Theoretical Physics, Cambridge U

    Joykutty, J.: Quasinormal Modes of Nearly Extremal Black Holes. PhD thesis, Depart- ment of Applied Mathematics And Theoretical Physics, Cambridge U. (2024). https: //doi.org/10.17863/CAM.106739

    2024 doi

  22. [29]

    arXiv preprint 85 arXiv:2407.04098 (2024)

    Gajic, D., Warnick, C.M.: Quasinormal modes on kerr spacetimes. arXiv preprint 85 arXiv:2407.04098 (2024)

    2024 arXiv

  23. [30]

    https://arxiv

    Warnick, C.: (In)stability of de Sitter Quasinormal Mode spectra (2024). https://arxiv. org/abs/2407.19850

    2024 arXiv

  24. [31]

    Panosso Macedo, R., Zenginoglu, A.: Hyperboloidal approach to quasinormal modes. Front. in Phys.12, 1497601 (2024) https://doi.org/10.3389/fphy.2024.1497601 arXiv:2409.11478 [gr-qc]

    2024

  25. [32]

    Zeitschrift f¨ur Analysis und ihre Anwendungen16(1), 9–36 (1997)

    Belger, M., Schimming, R., W ¨unsch, V .: A survey on huygens’ principle. Zeitschrift f¨ur Analysis und ihre Anwendungen16(1), 9–36 (1997)

    1997

  26. [33]

    Baker, B.B., Copson, E.T.: The Mathematical Theory of Huygens’ Principle vol. 329. American Mathematical Soc., ??? (2003)

    2003

  27. [34]

    Green, S.R., Hollands, S., Sberna, L., Toomani, V ., Zimmerman, P.: Conserved currents for a Kerr black hole and orthogonality of quasinormal modes. Phys. Rev. D107(6), 064030 (2023) https://doi.org/10.1103/PhysRevD.107.064030 arXiv:2210.15935 [gr- qc]

    2023 arXiv

  28. [35]

    Linear Algebra and its Applications436(10), 3839–3863 (2012) https://doi.org/10.1016/j.laa

    Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra and its Applications436(10), 3839–3863 (2012) https://doi.org/10.1016/j.laa. 2011.03.030 . Special Issue dedicated to Heinrich V oss’s 65th birthday

    2012 doi

  29. [36]

    European Journal of Mechanics-A/Solids100, 104809 (2023)

    Nicolet, A., Dem ´esy, G., Zolla, F., Campos, C., Roman, J.E., Geuzaine, C.: Physically agnostic quasi normal mode expansion in time dispersive structures: From mechanical vibrations to nanophotonic resonances. European Journal of Mechanics-A/Solids100, 104809 (2023)

    2023

  30. [37]

    Yakubov, S.: Completeness of Root Functions of Regular Differential Operators vol. 71. CRC Press, Boca Raton, Florida (1993)

    1993

  31. [38]

    American Mathematical Soc., 201 Charles Street, Providence (2012)

    Markus, A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils. American Mathematical Soc., 201 Charles Street, Providence (2012)

    2012

  32. [39]

    JHEP09, 131 (2016) https://doi.org/10

    Ammon, M., Grieninger, S., Jimenez-Alba, A., Macedo, R.P., Melgar, L.: Holo- graphic quenches and anomalous transport. JHEP09, 131 (2016) https://doi.org/10. 1007/JHEP09(2016)131 arXiv:1607.06817 [hep-th]

    2016 arXiv

  33. [40]

    Physical Review D68(8), 084014 (2003)

    Zlochower, Y ., Gomez, R., Husa, S., Lehner, L., Winicour, J.: Mode coupling in the nonlinear response of black holes. Physical Review D68(8), 084014 (2003)

    2003

  34. [41]

    Physical Review D—Particles, Fields, Gravitation, and Cosmology76(8), 084007 (2007)

    Nakano, H., Ioka, K.: Second-order quasinormal mode of the schwarzschild black hole. Physical Review D—Particles, Fields, Gravitation, and Cosmology76(8), 084007 (2007)

    2007

  35. [42]

    Physical Review D90(12), 124032 (2014) 86

    London, L., Shoemaker, D., Healy, J.: Modeling ringdown: Beyond the fundamental quasinormal modes. Physical Review D90(12), 124032 (2014) 86

    2014

  36. [43]

    Physical Review D108(10), 104020 (2023)

    Baibhav, V ., Cheung, M.H.-Y ., Berti, E., Cardoso, V ., Carullo, G., Cotesta, R., Del Pozzo, W., Duque, F.: Agnostic black hole spectroscopy: Quasinormal mode content of numerical relativity waveforms and limits of validity of linear perturbation theory. Physical Review D108(...

    2023

  37. [44]

    Physical Review Letters134(6), 061401 (2025)

    Bourg, P., Macedo, R.P., Spiers, A., Leather, B., Bonga, B., Pound, A.: Quadratic quasinormal mode dependence on linear mode parity. Physical Review Letters134(6), 061401 (2025)

    2025

  38. [45]

    Jaramillo, J.L.: Pseudospectrum and binary black hole merger transients. Class. Quant. Grav.39(21), 217002 (2022) https://doi.org/10.1088/1361-6382/ac8ddc arXiv:2206.08025 [gr-qc]

    2022 arXiv

  39. [46]

    Isaacson, R.A.: Gravitational radiation in the limit of high frequency. i. the linear approximation and geometrical optics. Phys. Rev.166, 1263–1271 (1968) https://doi. org/10.1103/PhysRev.166.1263

    1968 doi

  40. [47]

    III - Second order perturbations of collapse with rotation

    Cunningham, C.T., Price, R.H., Moncrief, V .: Radiation from collapsing relativistic stars. III - Second order perturbations of collapse with rotation. Astrophysical Journal 236, 674–692 (1980) https://doi.org/10.1086/157787

    1980 doi

  41. [48]

    Miller, J., Wardell, B., Pound, A.: Second-order perturbation theory: the problem of infinite mode coupling. Phys. Rev. D94(10), 104018 (2016) https://doi.org/10.1103/ PhysRevD.94.104018 arXiv:1608.06783 [gr-qc]

    2016 arXiv

  42. [49]

    Pound, A.: Gauge and motion in perturbation theory. Phys. Rev. D92(4), 044021 (2015) https://doi.org/10.1103/PhysRevD.92.044021 arXiv:1506.02894 [gr-qc]

    2015 arXiv

  43. [50]

    Classical and Quantum Grav- ity25(14), 145002 (2008)

    Zengino ˘glu, A.: Hyperboloidal foliations and scri-fixing. Classical and Quantum Grav- ity25(14), 145002 (2008)

    2008

  44. [51]

    Zenginoglu, A.: A Geometric framework for black hole perturbations. Phys. Rev.D83, 127502 (2011) https://doi.org/10.1103/PhysRevD.83.127502 arXiv:1102.2451 [gr-qc]

    2011 arXiv

  45. [52]

    Panosso Macedo, R.: Hyperboloidal framework for the Kerr spacetime. Class. Quant. Grav.37(6), 065019 (2020) https://doi.org/10.1088/1361-6382/ab6e3e

    2020 doi

  46. [53]

    Philosophical Transactions of the Royal Society A382(2267), 20230046 (2024)

    Panosso Macedo, R.: Hyperboloidal approach for static spherically symmetric space- times: a didactical introduction and applications in black-hole physics. Philosophical Transactions of the Royal Society A382(2267), 20230046 (2024)

    2024

  47. [54]

    arXiv preprint arXiv:2404.01528 (2024)

    Zengino ˘glu, A.: Hyperbolic times in minkowski space. arXiv preprint arXiv:2404.01528 (2024)

    2024 arXiv

  48. [55]

    arXiv preprint arXiv:2409.11478 (2024) 87

    Macedo, R.P., Zenginoglu, A.: Hyperboloidal approach to quasinormal modes. arXiv preprint arXiv:2409.11478 (2024) 87

    2024 arXiv

  49. [57]

    Sarkar, S., Rahman, M., Chakraborty, S.: Perturbing the perturbed: Stability of quasinor- mal modes in presence of a positive cosmological constant. Phys. Rev. D108, 104002 (2023)

    2023

  50. [58]

    Besson, J., Boyanov, V ., Jaramillo, J.L.: Black hole quasi-normal modes as eigenvalues: definition and stability problems, in preparation (2025)

    2025

  51. [59]

    Laser & Photonics Reviews12(5), 1700113 (2018) https://doi.org/10.1002/lpor.201700113 https://onlinelibrary.wiley.com/doi/pdf/10.1002/lpor.201700113

    Lalanne, P., Yan, W., Vynck, K., Sauvan, C., Hugonin, J.: Light inter- action with photonic and plasmonic resonances. Laser & Photonics Reviews12(5), 1700113 (2018) https://doi.org/10.1002/lpor.201700113 https://onlinelibrary.wiley.com/doi/pdf/10.1002/lpor.201700113

    2018 doi

  52. [60]

    Boyanov, V ., Cardoso, V ., Destounis, K., Jaramillo, J.L., Macedo, R.P.: Structural aspects of the anti–de sitter black hole pseudospectrum. Phys. Rev. D109, 064068 (2024) https://doi.org/10.1103/PhysRevD.109.064068

    2024 doi

  53. [61]

    Journal of Computational and Applied Mathematics69(1), 125–142 (1996) https://doi.org/10.1016/0377-0427(95)00021-6

    Driscoll, T.A., Trefethen, L.N.: Pseudospectra for the wave equation with an absorbing boundary. Journal of Computational and Applied Mathematics69(1), 125–142 (1996) https://doi.org/10.1016/0377-0427(95)00021-6

    1996 doi

  54. [62]

    Journal of High Energy Physics2024(10) (2024) https://doi.org/10.1007/jhep10(2024)084

    Carballo, J., Withers, B.: Transient dynamics of quasinormal mode sums. Journal of High Energy Physics2024(10) (2024) https://doi.org/10.1007/jhep10(2024)084

    2024 doi

  55. [63]

    Classical and Quantum Gravity41(23), 235015 (2024) https://doi.org/10.1088/1361-6382/ad89a1

    Chen, J.-N., Wu, L.-B., Guo, Z.-K.: The pseudospectrum and transient of kaluza–klein black holes in einstein–gauss–bonnet gravity. Classical and Quantum Gravity41(23), 235015 (2024) https://doi.org/10.1088/1361-6382/ad89a1

    2024 doi

  56. [64]

    Carballo, J., Pantelidou, C., Withers, B.: Non-modal effects in black hole perturbation theory: Transient Superradiance (2025) arXiv:2503.05871 [gr-qc]

    2025 arXiv

  57. [65]

    Jaramillo, J.L., Panosso Macedo, R., Sheikh, L.A.: Gravitational Wave Signatures of Black Hole Quasinormal Mode Instability. Phys. Rev. Lett.128(21), 211102 (2022) https://doi.org/10.1103/PhysRevLett.128.211102 arXiv:2105.03451 [gr-qc]

    2022 arXiv

  58. [66]

    Destounis, K., Duque, F.: Black-hole spectroscopy: quasinormal modes, ringdown stability and the pseudospectrum, 155–202 (2024)

    2024

  59. [67]

    Nollert, H.-P.: About the significance of quasinormal modes of black holes. Phys. Rev.D53, 4397–4402 (1996) https://doi.org/10.1103/PhysRevD.53.4397 arXiv:gr- qc/9602032 [gr-qc]

    1996

  60. [68]

    Aguirregabiria, J.M., Vishveshwara, C.V .: Scattering by black holes: A Simulated poten- tial approach. Phys. Lett. A210, 251–254 (1996) https://doi.org/10.1016/0375-9601(95) 88 00937-X

    1996 doi

  61. [69]

    In: 18th Conference of the Indian Association for General Relativity and Gravitation, pp

    Vishveshwara, C.V .: On the black hole trail ...: A personal journey. In: 18th Conference of the Indian Association for General Relativity and Gravitation, pp. 11–22 (1996)

    1996

  62. [70]

    Physical Review D109(4), 044010 (2024)

    Zhu, H., Ripley, J.L., C ´ardenas-Avenda˜no, A., Pretorius, F.: Challenges in quasinor- mal mode extraction: Perspectives from numerical solutions to the teukolsky equation. Physical Review D109(4), 044010 (2024)

    2024

  63. [71]

    Jaramillo, J.L., Panosso Macedo, R., Meneses-Rojas, O., Raffaelli, B., Sheikh, L.A.: A Weyl law for black holes. Phys. Rev. D110(10), 104008 (2024) https://doi.org/10.1103/ PhysRevD.110.104008 arXiv:2212.05570 [gr-qc]

    2024

  64. [72]

    Science261(5121), 578–584 (1993) https://doi.org/10.1126/ science.261.5121.578 https://science.sciencemag.org/content/261/5121/578.full.pdf

    Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A.: Hydrodynamic stabil- ity without eigenvalues. Science261(5121), 578–584 (1993) https://doi.org/10.1126/ science.261.5121.578 https://science.sciencemag.org/content/261/5121/578.full.pdf

    1993

  65. [73]

    Princeton University Press, Princeton (2005)

    Trefethen, L.N., Embree, M.: Spectra and Pseudospectra: The Behavior of Non- normal Matrices and Operators. Princeton University Press, Princeton (2005). https://books.google.es/books?id=7gIbT -Y7-AIC

    2005

  66. [74]

    Annual Review of Fluid Mechan- ics39(1), 129–162 (2007) https://doi.org/10.1146/annurev.fluid.38.050304.092139 https://doi.org/10.1146/annurev.fluid.38.050304.092139

    Schmid, P.J.: Nonmodal stability theory. Annual Review of Fluid Mechan- ics39(1), 129–162 (2007) https://doi.org/10.1146/annurev.fluid.38.050304.092139 https://doi.org/10.1146/annurev.fluid.38.050304.092139

    2007

  67. [75]

    In preparation (2022)

    Boyanov, V ., Destounis, K., Jaramillo, J.L., Macedo, R.P., Cardoso, V .: Pseudospec- trum of horizonless compact objects: a bootstrap instability mechanism. In preparation (2022)

    2022

  68. [76]

    Besson, J., Bizo ´n, P., Jaramillo, J.L., Warnick, C.: Non-modal transient growths and regularity in hyperboloidal evolutions, in preparation (2025)

    2025

  69. [77]

    Physics of Fluids 31(8), 2093 (1988)

    Farrell, B.F.: Optimal excitation of perturbations in viscous shear flow. Physics of Fluids 31(8), 2093 (1988)

    1988

  70. [78]

    Oxford University Press, New York, Oxford (1987).http://opac.inria.fr/record=b1081336

    Chandler, D.: Introduction to Modern Statistical Mechanics. Oxford University Press, New York, Oxford (1987).http://opac.inria.fr/record=b1081336

    1987

  71. [79]

    SIAM Rev.39(3), 383–406 (1997)

    Trefethen, L.N.: Pseudospectra of linear operators. SIAM Rev.39(3), 383–406 (1997)

    1997

  72. [80]

    Davies, E.B.: Pseudo-spectra, the harmonic oscillator and complex resonances. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci.455(1982), 585–599 (1999)

    1982

  73. [81]

    Davies, E.B.: Pseudospectra of differential operators. J. Oper. Th43, 243–262 (2000)

    2000

  74. [82]

    Colbrook, M.J., Roman, B., Hansen, A.C.: How to compute spectra with error control. Phys. Rev. Lett.122, 250201 (2019) https://doi.org/10.1103/PhysRevLett.122.250201 89

    2019 doi

  75. [83]

    Hyperboloidal Research Network Seminar, 13 December (2024)

    Besson, J.: Quasi-normal mode expansions of black hole perturbations: a hyperboloidal keldysh’s approach. Hyperboloidal Research Network Seminar, 13 December (2024)

    2024

  76. [84]

    arXiv preprint arXiv:2501.02522 (2025)

    Cai, R.-G., Cao, L.-M., Chen, J.-N., Guo, Z.-K., Wu, L.-B., Zhou, Y .-S.: The pseu- dospectrum for the kerr black hole: spins= 0case. arXiv preprint arXiv:2501.02522 (2025)

    2025 arXiv

  77. [85]

    https://arxiv.org/abs/2406.15924

    Hitrik, M., Zworski, M.: Overdamped QNM for Schwarzschild black holes (2024). https://arxiv.org/abs/2406.15924

    2024 arXiv

  78. [86]

    Journal of High Energy Physics2024(5), 1–39 (2024)

    Cownden, B., Pantelidou, C., Zilh ˜ao, M.: The pseudospectra of black holes in ads. Journal of High Energy Physics2024(5), 1–39 (2024)

    2024

  79. [87]

    Destounis, K., Macedo, R.P., Berti, E., Cardoso, V ., Jaramillo, J.L.: Pseudospectrum of Reissner-Nordstr ¨om black holes: Quasinormal mode instability and universality. Phys. Rev. D104(8), 084091 (2021) https://doi.org/10.1103/PhysRevD.104.084091 arXiv:2107.09673 [gr-qc]

    2021 arXiv

  80. [88]

    Physical Review D103(6), 065014 (2021)

    Albrychiewicz, E., Neiman, Y .: Scattering in the static patch of de sitter space. Physical Review D103(6), 065014 (2021)

    2021

  81. [89]

    Reviews in Mathematical Physics22(04), 431–484 (2010)

    Daud ´e, T., Nicoleau, F.: Inverse scattering in de sitter–reissner–nordstr ¨om black hole spacetimes. Reviews in Mathematical Physics22(04), 431–484 (2010)

    2010

  82. [90]

    Journal of Geometry and Physics3(1), 101–117 (1986)

    Friedrich, H.: Existence and structure of past asymptotically simple solutions of ein- stein’s field equations with positive cosmological constant. Journal of Geometry and Physics3(1), 101–117 (1986)

    1986

  83. [91]

    Philosophical Transactions of the Royal Society A382(2267), 20230037 (2024)

    Hintz, P.: Asymptotically de sitter metrics from scattering data in all dimensions. Philosophical Transactions of the Royal Society A382(2267), 20230037 (2024)

    2024

  84. [92]

    Acta Phys

    Bizo ´n, P., Chmaj, T., Mach, P.: A toy model of hyperboloidal approach to quasinormal modes. Acta Phys. Polon. B51, 1007 (2020) https://doi.org/10.5506/APhysPolB.51. 1007 arXiv:2002.01770 [gr-qc]

    2020 arXiv

  85. [93]

    Theses, Universit ´e Bourgogne Franche- Comt´e (February 2022)

    Al Sheikh, L.: Scattering resonances and Pseudospectrum : stability and completeness aspects in optical and gravitational systems. Theses, Universit ´e Bourgogne Franche- Comt´e (February 2022). https://theses.hal.science/tel-04116011

    2022

  86. [94]

    Destounis, K., Boyanov, V ., Panosso Macedo, R.: Pseudospectrum of de Sitter black holes. Phys. Rev. D109(4), 044023 (2024) https://doi.org/10.1103/PhysRevD.109. 044023 arXiv:2312.11630 [gr-qc]

    2024 arXiv

  87. [95]

    Journal of High Energy Physics2023(12), 1–60 (2023)

    Are ´an, D., Fari ˜na, D.G., Landsteiner, K.: Pseudospectra of holographic quasinormal modes. Journal of High Energy Physics2023(12), 1–60 (2023)

    2023

  88. [96]

    Science China 90 Physics, Mechanics & Astronomy67(10), 100412 (2024)

    Cao, L.-M., Chen, J.-N., Wu, L.-B., Xie, L., Zhou, Y .-S.: The pseudospectrum and spectrum (in) stability of quantum corrected schwarzschild black hole. Science China 90 Physics, Mechanics & Astronomy67(10), 100412 (2024)

    2024

  89. [97]

    arXiv preprint arXiv:2412.21092 (2024)

    Cao, L.-M., Wu, L.-B., Zhou, Y .-S.: The (in) stability of quasinormal modes of boulware-deser-wheeler black hole in the hyperboloidal framework. arXiv preprint arXiv:2412.21092 (2024)

    2024 arXiv

  90. [98]

    Cam- bridge Monographs on Mathematical Physics

    Griffiths, J.B., Podolsk ´y, J.: Exact Space-Times in Einstein’s General Relativity. Cam- bridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009)

    2009

  91. [99]

    Panosso Macedo, R., Ansorg, M.: Axisymmetric fully spectral code for hyperbolic equations. J. Comput. Phys.276, 357–379 (2014) https://doi.org/10.1016/j.jcp.2014.07. 040 arXiv:1402.7343 [physics.comp-ph] 91

    2014 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.