Wave propagation in gravitational systems: Completeness of quasinormal modes

Gravitational waves emitted from relativistic stars and black holes are often studied in terms of the quasinormal modes (QNM's) of the Klein-Gordon (KG) equation with different effective potentials $V\left(x\right)\mathrm{}$ describing the scattering of waves by the geometry of spacetime. The characteristics of these QNM's that may be observed by gravitational wave detectors in the next decade will provide important information about the spacetime structure of various gravitational systems. In this paper we present a systematic study of the relation between the QNM's of the KG equation and the form of $V\left(x\right)\mathrm{}$. In particular, we find the conditions on $V\left(x\right)\mathrm{}$ in order for the QNM's to form a complete set for the expansion of outgoing waves. Even for relativistic systems whose QNM's do not form a complete set, a complete set of QNM's can often be obtained by introducing a very small change in the effective potential.