The problem becomes an eigenvalue problem. The quasi-normal modes are of damping type in time, although these waves diverge in space as r ∗ ∗ → → ± ± ∞ ∞ (this is due to the implicit assumption that the perturbation in quasi-normal modes is 'infinite' in the remote past). Again because of the relation mentioned between the two problem, the spectrum of Z + and Z − − are identical and thus it enough to consider the spectrum of Z − − . The problem is simplified by introducing