  The current early stage in the investigation of the stability of the Kerr
metric is characterized by the study of appropriate model problems.
Particularly interesting is the problem of the stability of the solutions of
the Klein-Gordon equation, describing the propagation of a scalar field in the
background of a rotating (Kerr-) black hole. Results suggest that the stability
of the field depends crucially on its mass $\mu$. Among others, the paper
provides an improved bound for $\mu$ above which the solutions of the reduced,
by separation in the azimuth angle in Boyer-Lindquist coordinates, Klein-Gordon
equation are stable. Finally, it gives new formulations of the reduced
equation, in particular, in form of a time-dependent wave equation that is
governed by a family of unitarily equivalent positive self-adjoint operators.
The latter formulation might turn out useful for further investigation. On the
other hand, it is proved that from the abstract properties of this family alone
it cannot be concluded that the corresponding solutions are stable.
