Continued fraction method for high overtone quasinormal modes in effective potentials with discontinuity

TL;DR

This paper extends Leaver's continued fraction method to accurately compute high overtone quasinormal modes in black hole systems with discontinuous potentials, revealing universal spectral features and potential observational implications.

Contribution

It introduces a novel algorithm that expands wavefunctions at discontinuities and incorporates junction conditions, enabling high-precision calculation of high overtone QNMs in such systems.

Findings

High overtones are significantly deformed due to potential discontinuities.

The method achieves excellent agreement with existing techniques for low-lying modes.

Universal features in the asymptotic QNM spectrum are observed across different potentials.

Abstract

In this study, we extend Leaver's continued fraction method to evaluate black hole quasinormal modes (QNMs) in systems where the effective potential exhibits a discontinuity. Besides the low-lying modes, we particularly focus on high overtones, which are physically pertinent due to the substantial deformation of the QNM spectrum triggered by spectral instability. In our algorithm, we expand the wavefunction at the point of discontinuity, instead of the black hole horizon, and incorporate the Israel-Lanczos-Sen junction conditions. %As the wavefunction convergence condition becomes irrelevant, our proposed algorithm generalizes the original method by expanding the wavefunctions at the point of discontinuity, and the associated difficulty is mitigated by rectifying the recurrence relations between the expansion coefficients to incorporate the Israel-Lanczos-Sen junction conditions. We apply this algorithm to compute the QNMs of the modified Regge-Wheeler potential up to $2000$ modes with high precision. For the low-lying modes, the numerical results show excellent agreement with those obtained using the matrix and Prony methods. The high overtones are significantly deformed, owing to the presence of echoes due to the discontinuity. This deformation in the asymptotic QNM spectrum reveals universal features that are largely independent of the specific form of the discontinuity in the potential, seemingly coinciding with those observed in the modified P\"oschl-Teller effective potential. We speculate on whether the collective effect of the high overtones has an observational impact on gravitational wave signals.