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. In addition to 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 wave function at the point of discontinuity, instead of the black hole horizon, and 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 Pöschl-Teller effective potential. We speculate on whether the collective effect of the high overtones has an observational impact on gravitational-wave signals.

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• Classical black holes
• Perturbative methods