Superradiant stability of the Kerr black holes

We study the superradiant stability of the system of a Kerr black hole and a massive scalar perturbation. It was proved previously that this system is superradiantly stable when μ≥2mΩH, where μ is the proper mass of the scalar, m is the azimuthal number of the scalar mode, and ΩH is the angular velocity of the Kerr black hole horizon. Our study is a complementary work of this result. We analytically prove that in the complementary parameter region μ<2mΩH, when the parameters of scalar perturbation and Kerr black hole satisfy two simple inequalities, ω<μ2, r−r+<0.802, the system is also superradiantly stable.