Quasinormal modes of magnetized black hole

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 ℓ there are 2 ℓ+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 μ_eff=√{μ²-m q B } 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 m q B <0 quasinormal modes are longer lived with larger oscillation frequencies. Whenever, magnetic and massive scalar fields satisfies condition μ_eff²<0 , an instability appears, i.e., if q B >0 or q B <0 there is an instability for the values of azimuthal number m >μ²/q B or m <μ²/q B , respectively.