Massive vector field perturbations in the Schwarzschild background: stability and unusual quasinormal spectrum

We consider the perturbations of the massive vector field around Schwarzschild black hole, (generally, with non-vanishing $\Lambda$ - term). In the case of a massive field, even spherically symmetrical (monopole) perturbations become radiative. We have proved the stability against these perturbations and investigated the quasinormal spectrum. The quasinormal behaviour is quite unexpected: the fundamental overtone and all higher overtones shows totally different dependence on the mass of the field $m$: as $m$ is increasing, the damping rate of the fundamental overtone is decreasing, what results in appearing of the infinitely long living modes, while, on contrary, damping rate of all higher overtones are increasing, and their real oscillation frequencies gradually go to tiny values. Thereby, for all overetones higher then the lowest one, almost non-oscillatory, damping modes can exist. In the limit of asymptotically high damping, $Re \omega$ goes to $ln3/(8 \pi M)$, while imaginary part shows equdistant behaviour with spacing $Im \omega_{n+1}- Im \omega_{n}=i/4M$.