On the spectrum of oscillations of a Schwarzschild black hole

The spectral properties of the resonant oscillations of a Schwarzschild black hole are studied. The analogy with the Coulomb wave equation is developed, and it is shown that the high-overtone quasi-normal modes are intimately connected with the long-range nature of the gravitational interaction. This is followed by an explicit construction of the quasi-normal eigenfunctions in analogy with the bound states of the inverted black-hole potentials. The special gravitational quasi-normal mode is discussed and its relationship with the algebraically special solution of the gravitational perturbations is clarified.