Cesàro convergence of the high-order WKB method and its applications to black-hole overtones and long-lived modes

Abstract: We develop a fully automatic Mathematica implementation of the black-hole WKB method at very high orders based on the Bender-Wu algorithm, which in principle is limited only by memory and computational time, and show that when pushed to sufficiently high order and improved by diagonal Padé approximants the method becomes efficient for two regimes which are usually regarded as difficult for the standard low-order WKB treatment: the first several overtones with n>l and the very long-lived quasinormal modes of massive fields. At the same time, we show that this efficiency has a nontrivial limitation: for black-hole metrics belonging to the non-moderate class, especially when higher coefficients of the near-horizon parametrization become large, the WKB sequence may exhibit an apparent convergence to values which are nevertheless far from the accurate quasinormal frequencies. Thus, numerical stabilization of the WKB output alone is not always a sufficient criterion of correctness. However, we observe that although the WKB method with diagonal or near-diagonal Padé approximants does not exhibit monotonic convergence order by order, the corresponding Cesàro means become monotonically convergent once a sufficiently high WKB order is reached. This behavior may serve as an internal WKB criterion for the convergence of the method.

• The paper establishes an automated high-order WKB method using Cesàro averaging and Padé resummation to ensure accurate black-hole overtone computations.
• It demonstrates that genuine convergence can be distinguished from apparent stabilization even in deformed near-horizon geometries.
• Numerical comparisons with the Frobenius-Leaver method highlight efficiency gains and broadened application in spectral analysis of black-hole metrics.

Cesàro Convergence of High-Order WKB: Applications to Black-Hole Overtones and Long-Lived Modes

Overview

This paper establishes an automated symbolic implementation of the black-hole WKB method at arbitrarily high order, leveraging a Mathematica-based recursion via the Bender-Wu algorithm. The approach, paired with diagonal and near-diagonal Padé resummation, demonstrates effectiveness for two regimes traditionally challenging for standard low-order WKB: the computation of higher overtones ($n>\ell$), and quasi-resonant modes of massive fields exhibiting extremely low damping. The authors identify nontrivial limitations in WKB efficiency, particularly for non-moderate metrics where near-horizon deformations induce large coefficients in continued-fraction parametrization, resulting in apparent numerical convergence that does not guarantee correctness. Cesàro summation emerges as a robust internal criterion for convergence at sufficiently high WKB order, offering a monotonic stabilization of Padé-improved sequences.

Parametrized Black-Hole Geometry and Spectral Sensitivity

The paper uses a continued-fraction parametrization of asymptotically flat, spherically symmetric black-hole metrics (Rezzolla et al., 2014), permitting systematic separation between weak-field and near-horizon physics. This allows the study of how overtone frequencies respond to localized geometric deformations, isolating three reference classes: Schwarzschild (no deformation), moderately deformed (smooth variations), and non-moderately deformed metrics (sharp near-horizon changes driven by large higher-order coefficients). The latter class is particularly probed, as it can leave the fundamental mode and optical observables nearly invariant while causing dramatic spectral shifts in the overtone sector (Konoplya et al., 2022).

High-Order WKB Expansion, Padé Resummation, and Borel Summation

The authors detail the WKB expansion as a formal asymptotic series, recast via a perturbative eigenvalue analogy with the quantum anharmonic oscillator [43.605], enabling recursive coefficient generation using the Bender-Wu algorithm [7.1620]. For practical convergence diagnostics, Padé transformation of the WKB series ($$\omega^2 \approx \mathcal{P}^{\tilde{m}}_{\tilde{n}}(1)$$) is employed—diagonal and near-diagonal approximants reveal stabilization of the sequence, overcoming non-monotonic oscillations typical in raw WKB truncations (Matyjasek et al., 2019). For cases with slow order-by-order convergence, Borel and Borel-Padé summation are invoked to extract physically meaningful results from asymptotic series (Hatsuda, 2019).

Numerical Results: Overtones and Quasi-Resonant Modes

Strong numerical evidence is provided for the efficacy of very-high-order WKB combined with Padé resummation in reproducing precise overtone frequencies and quasi-resonances, even in the presence of three-turning-point potentials, as long as the effective barrier maximum is maintained. For moderately deformed backgrounds, convergence is rapid; for nonmoderate backgrounds (large near-horizon coefficients), convergence is considerably slower and prone to apparent stabilization at incorrect values unless sufficiently high WKB order is reached. The Cesàro mean of the Padé sequence, $\sigma_N = \frac{1}{N+1}\sum_{k=0}^{N}\omega_k$, is shown to monotonically stabilize and provide a robust internal convergence criterion, albeit at the cost of slower practical convergence compared to Padé clustering.

Numerical comparisons against the Frobenius-Leaver continued-fraction method are provided. The latter remains the "gold standard" for precise quasinormal frequency computation but is computationally expensive and requires rational metric forms [402.285]. The WKB-Padé approach, by contrast, is massively parallelizable and does not necessitate an initial frequency guess, granting efficiency for large-scale parameter scans.

Apparent vs Genuine Convergence

An important conclusion is that numerical stabilization of the high-order Padé sequence can be misleading in sharply deformed near-horizon geometries: clustering of Padé approximants may persist for many orders around incorrect values. The authors systematically distinguish genuine convergence (verified against Frobenius reference) from apparent or false stabilization. Cesàro averaging, while conservative, can help diagnose the attainable asymptotic regime, especially in sectors where the WKB order required for accuracy grows rapidly with overtone number or deformation strength.

Theoretical and Practical Implications

The formalism is extensible to parametrized metrics for black holes, wormholes, and higher-dimensional generalizations (Konoplya et al., 2020, Bronnikov et al., 2021). The practical computational methodology established here enables the study of overtone sensitivity and the extraction of event-horizon information from spectral data (Konoplya et al., 2022). From a theoretical standpoint, the high-order WKB+Padé paradigm reinforces the importance of resummation for asymptotic spectral problems, suggesting that much more information can be extracted than from low-order expansions.

The possibility of extracting near-horizon geometry from overtones ("sound of the event horizon") (Konoplya, 2023) points to implications for gravitational-wave ringdown analysis and the testing of strong-field gravity via quasinormal mode spectroscopy. Automated implementation via symbolic recursion further unlocks mass parameter regimes (quasi-resonances, long-lived modes) previously inaccessible to semi-analytic methods.

Future Directions

Potential future developments include generalization to axisymmetric and time-dependent backgrounds, integration of Borel-Padé resummation into automated workflows, and systematic exploration of the threshold for genuine convergence as a function of parametrization coefficients and overtone number. The extension of Cesàro convergence diagnostics to other asymptotic spectral problems in gravitational physics appears promising.

Conclusion

The paper establishes the utility and limitations of very-high-order WKB, with Padé and Cesàro convergence analysis, for quasinormal mode calculation in parametrized black-hole geometries. While numerical stabilization is not always a sufficient correctness criterion, the combined use of Padé resummation and Cesàro means provides a rigorous internal diagnostic for the convergence of WKB expansions. The methodology significantly expands the practical parameter space for automatic spectral computations and deepens understanding of overtone sensitivity to near-horizon structures. Theoretical implications for black-hole spectroscopy and gravitational-wave data interpretation are substantial, and further extension to more general backgrounds appears tractable.