QNM–algebraically special frequency puzzle and anomalous multiplet splitting

Determine the precise relationship between quasinormal modes and algebraically special frequencies in Schwarzschild and Kerr black holes, including whether QNMs occur exactly at algebraically special frequencies and explaining the observed anomalous doublet/multiplet splitting for Kerr with ℓ=2 and m≥0 that appears to emerge from the Schwarzschild algebraically special frequency while no splitting is seen for m<0; clarify the physical meaning and roles of QNMs versus total transmission modes in this regime.

Background

Numerical studies have repeatedly found QNMs close to, but not precisely at, the algebraically special (AS) frequencies. In Kerr, a doublet of modes for m≥0 appears to emerge from the Schwarzschild AS frequency, while no splitting is seen for m<0, raising questions about the underlying mechanism.

Understanding whether and how QNMs coincide with or relate to TTMs (total transmission modes) at AS frequencies is central for interpreting the spectrum and resolving conflicting numerical and analytical indications.

References

Quasinormal mode (QNM) spectra of black holes exhibit two open problems [Conf. Proc. C {0405132}, 145 (2004); CQG {26}, 163001 (2009)]: (i) the discontinuity in highly damped QNMs between Schwarzschild and Kerr solutions as $a \to 0$, and (ii) the unexplained spectral proximity between QNMs and algebraically special (AS) frequencies, particularly the anomalous multiplet splitting for Kerr $\ell=2$, $m \geq 0$ modes.

— Complete quasinormal modes of Type-D black holes (2506.14635 - Chen et al., 17 Jun 2025) in Abstract