Asymptotic spectroscopy of rotating black holes

We calculate analytically the transmission and reflection amplitudes for waves incident on a rotating black hole in $d=4$, analytically continued to asymptotically large, nearly imaginary frequency. These amplitudes determine the asymptotic resonant frequencies of the black hole, including quasinormal modes, total-transmission modes and total-reflection modes. We identify these modes with semiclassical bound states of a one-dimensional Schrödinger equation, localized along contours in the complexified $r$-plane which connect turning points of corresponding null geodesics. Each family of modes has a characteristic temperature and chemical potential. The relations between them provide hints about the microscopic description of the black hole in this asymptotic regime.