Abstract
We investigate a charged, massive scalar field around a static, spherically symmetric black hole immersed into an external asymptotically uniform magnetic field $B$. It is shown that for given multipole number $\ensuremath{\ell}$ there are $2\ensuremath{\ell}+1$ numbers of modes due to the Zeeman effect appearing by an interaction of the external magnetic and charged scalar fields introducing an effective mass of the scalar field ${\ensuremath{\mu}}_{\mathrm{eff}}=\sqrt{{\ensuremath{\mu}}^{2}\ensuremath{-}mqB}$ where $m$ is the azimuthal number and $q$ is the charge coupling constant. We calculate threshold value of effective mass in which quasinormal modes are arbitrarily long lived and beyond that value quasinormal modes vanish. In the case of $mqB<0$ quasinormal modes are longer lived with larger oscillation frequencies. Whenever, magnetic and massive scalar fields satisfies condition ${\ensuremath{\mu}}_{\mathrm{eff}}^{2}<0$, an instability appears, i.e., if $qB>0$ or $qB<0$ there is an instability for the values of azimuthal number $m>{\ensuremath{\mu}}^{2}/qB$ or $m<{\ensuremath{\mu}}^{2}/qB$, respectively.