On the equations governing the axisymmetric perturbations of the Kerr black hole

It is shown how Teukolsky's equation, governing the perturbations of the Kerr black hole, can be reduced, in the axisymmetric case, to a one-dimensional wave equation with four possible potentials. The potentials are implicitly dependent on the frequency; and besides, depending on circumstances, they can be complex. In all cases (i.e. whether or not the potentials are real or complex), the problem of the reflexion and the transmission of gravitational waves by the potential barriers can be formulated, consistently, with the known conservation laws. It is further shown that all four potentials lead to the same reflexion and transmission coefficients.