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Quasibound states of massive scalar around the Kerr black hole

In this work, we obtain quasibound states of a massive scalar around the Kerr black hole. A massive scalar propagating around the Kerr black hole induces the far-region potential with either a trapping well or a tiny well, which produces a peak of resonance or a bound state in the far-region. The former represents superradiant instability, while the latter could accommodate superradiant instability via a positive $ω_I$. If the scalar's energy is really quantized around a tiny well, an ultralight boson could be expressed in terms of a hydrogen-like bound state, leading to a gravitational atom when incorporating superradiant instability. If there are no wells outside the outer horizon, their far-region wave functions represent purely bound state without node. Finally, we discriminate between superradiant instability and stability.