  Quasinormal modes have played a prominent role in the discussion of
perturbations of black holes, and the question arises whether they are as
significant as normal modes are for self adjoint systems, such as harmonic
oscillators. They can be significant 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 first 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
definite statements about the significance 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.
