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We find quasinormal spectrum of the massive scalar field in the background ofthe Kerr black holes. We show that all found modes are damped under thequasinormal modes boundary conditions when $\mu M$ is not large, therebyimplying stability of the massive scalar field. This complements the region ofstability determined by the Beyer inequality for large masses of the field. Weshow that, similar to the case of a non-rotating black holes, the massive termof the scalar field does not contribute in the regime of high damping. Thereby,the high damping asymptotic should be the same as for the massless scalarfield.Comment: 3 figures, RevTe

Stability and quasinormal modes of the massive scalar field around Kerr black holes