Massive scalar perturbations in Kerr Black Holes: near extremal analysis

We study quasinormal modes of massive scalar perturbations in Kerr black holes using the isomonodromic method. For arbitrary scalar masses $M\mu$ and black hole spins $a\mathrm{/}M$, we numerically determine the quasinormal frequencies for various orbital $\mathrm{\ell }$, azimuthal $m$, and overtone $n$ numbers. In particular, we derive an analytic expression for frequencies of the zero-damping modes near the extremal limit $a\mathrm{/}M\to 1$. For $\mathrm{\ell }=m=1$, we reveal that the fundamental mode becomes a damped mode (rather than a zero-damping mode) if the scalar field is sufficiently heavy. By exploring the parameter space, we find numerical evidence for level-crossing between the longest-living mode and the first overtone at an exceptional point $(M\mu {)}_c\simeq 0.3704981$ and $(a\mathrm{/}M{)}_c\simeq 0.9994660$.