Intertwining of the equations of black-hole perturbations

Abstract

For both Schwarzschild and Reissner-Nordstr\"om black holes, equations have been given in which the dynamics of perturbations are governed by effective potentials. These potentials have different form for odd- and for even-parity perturbations, yet they are equivalent. In particular, they give rise to the same spectrum of quasinormal frequencies. Though all the potentials are rational functions of the radial coordinate, the odd-parity potentials are markedly simpler and are necessary for certain semianalytic approaches to numerical studies. We investigate here whether there can be yet simpler equivalent potentials which could further simplify numerical work. With the "intertwining operator" viewpoint we show that no such further simplification is possible. This viewpoint also gives added insights into the relationship of the equivalent effective potentials.