About the Signiicance of Quasinormal Modes of Black Holes

Quasinormal modes have played a prominent role in the discussion of perturbations of black holes, and the question arises whether they are as signiicant as normal modes are for self adjoint systems, such as harmonic oscillators. They can be signiicant in two ways: Individual modes may dominate the time evolution of some perturbation, and a whole set of them could be used to completely describe this time evolution. It is known that quasinormal modes of black holes have the rst property, but not the second. It has recently been suggested that a discontinuity in the underlying system would make the corresponding set of quasinormal modes complete. We therefore turn the Regge-Wheeler potential, which describes perturbations of Schwarzschild black holes, into a series of step potentials, hoping to obtain a set of quasinormal modes which shows both of the above properties. This hope proves to be futile, though: The resulting set of modes appears to be complete, but it does not contain any individual mode any more which is directly obvious in the time evolution of initial data. Even worse: The quasinormal frequencies obtained in this way seem to be extremely sensitive to very small changes in the underlying potential. The question arises whether-and how-it is possible to make any deenite statements about the signiicance of quasinormal modes of black holes at all, and whether it could be possible to obtain a set of quasinormal modes with the desired properties in another way.