Complete total-transmission modes of Kerr black holes

Abstract

We construct the complete spectrum of gravitational total-transmission modes (TTMs) of Kerr black holes and find four globally continuous families, $n_\infty=1,2,3,4$, each with two complex-conjugate frequency branches. Compared with the three-family classification of Cook and Lu, our global continuation resolves the mirror-related sectors of their $n=2$ family into separate families without introducing additional symmetry-unrelated roots. This organization distinguishes complex conjugation at fixed $m$ from the mirror symmetry connecting the $m$ and $-m$ spectra. The $n_\infty=3$ family approaches the Schwarzschild algebraically special frequency, whereas the $n_\infty=1,2,$ and $4$ families diverge as $ω\propto a^{-4/3}$ along lower-half-plane directions $-150^\circ$, $-90^\circ$, and $-30^\circ$, respectively. High-precision data up to $\ell=32$ confirm the Cook--Lu small-spin asymptotics and reveal how the divergent branches are embedded in the global four-family spectrum, with a family-dependent relation between the spherical-limit angular labels and the large-$|aω|$ ordering. For axisymmetric perturbations, the $n_\infty=2$ and $n_\infty=3$ branches coalesce at genuine exceptional points, where the scattering spectral function develops a second-order zero and the TTM excitation factors are strongly enhanced, offering a different interpretation of the structure previously described by Cook and Lu as an overtone-multiplet splitting. Finally, we identify a previously unreported anomalous proximity between the $n_\infty=3$ TTM family and an unconventional Kerr quasinormal-mode sequence, with frequency separations reaching order $10^{-8}$ in the representative case studied.