Superradiant instability for black holes immersed in a magnetic field

We consider the bound states of the massive scalar field around a rotating black hole immersed in the asymptotically uniform magnetic field. In the regime of slow black hole rotation, the Klein-Gordon equation allows separation of variables. We show that the growth rate of the instability can be amplified a few times by the magnetic field. The effect occurs because the magnetic field adds the "effective mass" term $B |m|$ to the scalar field potential for a Kerr black hole. In addition, and as a by-product, we discuss the behavior of the quasinormal modes for the magnetized rotating black holes.