Recent studies have shown that a secondary potential barrier, forming a potential well outside the event horizon, can destabilize the Quasinormal Mode (QNM) spectrum of black holes. We find that spectral instability may persist even after the potential well vanishes, giving rise to a distinct family of spectrally unstable QNMs that differ from the spectrally stable modes localized near the potential peak and associated with the photon sphere. Nevertheless, time-domain simulations reveal that early-time ringdown waveforms remain dominated by stable modes, while unstable modes have only a subdominant contribution. These results highlight the robustness of black hole spectroscopy, as the observable ringdown signal is primarily governed by the most stable QNMs.
Wang et al. (Tue,) studied this question.