  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
$\alpha\equiv M\mu\ll1$ (here $\mu$ is the mass of the scalar field), these
quasi-stationary scalar resonances are characterized by the familiar hydrogenic
oscillation spectrum: $\omega_{\text{R}}/\mu=1-\alpha^2/2{\bar n}^2_0$, where
the integer $\bar n_0(l,n;\alpha\to0)=l+n+1$ 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 ($\alpha\ll1$) 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 $\alpha=O(1)$. In
particular, we derive the non-hydrogenic (and, in general, non-degenerate)
resonance oscillation spectrum
${{\omega_{\text{R}}}/{\mu}}=\sqrt{1-(\alpha/{\bar n})^2}$, where $\bar
n(l,n;\alpha)=\sqrt{(l+1/2)^2-2m\alpha+2\alpha^2}+1/2+n$ 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.
