Analytic treatment of black-hole gravitational waves at the algebraically special frequency

We study the Regge-Wheeler (RWE) and Zerilli equations (ZE) at the ``algebraically special frequency'' Ω, where these equations admit an exact solution (elaborated here), generating the supersymmetry (SUSY) relationship between them. The physical significance of the SUSY generator and of the solutions at Ω in general is elucidated as follows. The RWE has no (quasinormal or total-transmission) modes at all; however, Ω is nonetheless ``special'' in that (a) for the outgoing wave into the horizon one has a ``miraculous'' cancellation of a divergence expected due to the exponential potential tail, and (b) the branch-cut discontinuity at ω=Ω vanishes in the outgoing wave to infinity. Moreover, (a) and (b) are related. For the ZE, its only mode is the---inverse---SUSY generator, which is at the same time a quasinormal mode and a total-transmission mode propagating to infinity. The subtlety of these findings (of general relevance for future study of the equations on or near the negative imaginary {ω}-axis) may help explain why the situation has sometimes been controversial. For finite black-hole rotation, the algebraically special modes are shown to be totally transmitting, and the implied singular nature of the Schwarzschild limit is clarified. The analysis draws on a recent detailed investigation of SUSY in open systems [math-ph/9909030].