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Quasi-bound states of massive scalar fields in the Kerr black-hole spacetime: Beyond the hydrogenic approximation

Rotating black holes can support quasi-stationary (unstable) bound-state resonances of massive scalar fields in their exterior regions. These spatially regular scalar configurations are characterized by instability timescales which are much longer than the timescale $M$ set by the geometric size (mass) of the central black hole. It is well-known that, in the small-mass limit $α\equiv Mμ\ll1$ (here $μ$ is the mass of the scalar field), these quasi-stationary scalar resonances are characterized by the familiar hydrogenic oscillation spectrum: $ω_{\text{R}}/μ=1-α^2/2{\bar n}^2_0$, where the integer $\bar n_0(l,n;α\to0)=l+n+1$ is the principal quantum number of the bound-state resonance (here the integers $l=1,2,3,...$ and $n=0,1,2,...$ are the spheroidal harmonic index and the resonance parameter of the field mode, respectively). As it depends only on the principal resonance parameter $\bar n_0$, this small-mass ($α\ll1$) hydrogenic spectrum is obviously degenerate. In this paper we go beyond the small-mass approximation and analyze the quasi-stationary bound-state resonances of massive scalar fields in rapidly-spinning Kerr black-hole spacetimes in the regime $α=O(1)$. In particular, we derive the non-hydrogenic (and, in general, non-degenerate) resonance oscillation spectrum ${ω_{\text{R}}/μ}=\sqrt{1-(α/{\bar n})^2}$, where $\bar n(l,n;α)=\sqrt{(l+1/2)^2-2mα+2α^2}+1/2+n$ is the generalized principal quantum number of the quasi-stationary resonances. This analytically derived formula for the characteristic oscillation frequencies of the composed black-hole-massive-scalar-field system is shown to agree with direct numerical computations of the quasi-stationary bound-state resonances.