An unknown branch of the total-transmission modes for the Kerr-geometry

The gravitational modes of the Kerr geometry include both quasinormal modes and total-transmission modes. Sequences of these modes are parameterized by the angular momentum of the black hole. The quasinormal and total-transmission modes are usually distinct, having mode frequencies that are different at any given value of the angular momentum. But a discrete and countably infinite subset of the left-total-transmission modes are simultaneously quasinormal modes. Most of these special modes exist along previously unknown branches of the gravitational total-transmission modes. In this paper, we give detailed plots of the total-transmission modes for harmonic indices $\ell=[2,7]$, with special emphasis given to the $m=0$ modes which all contain previously unknown branches. All of these unknown branches have purely imaginary mode frequencies. We find that as we approach the Schwarzschild limit along these new branches, the mode frequencies approach $-i\infty$ in stark contrast to the finite mode frequency obtained in the Schwarzschild limit along the previously known branches. We explain when and why, at certain frequencies, the left-total-transmission modes are simultaneously quasinormal modes. At these same frequencies, the right-total-transmission modes are missing. We also derive analytic expressions for the asymptotic behavior of the total-transmission mode frequencies, and for the values of the angular momentum at which the modes are simultaneously quasinormal modes.