Complete quasinormal modes of Type-D black holes. II. Extremal Kerr black holes

Abstract

We develop a double-confluent Heun method that extends our previous study of complete quasinormal-mode (QNM) spectra of Type-D black holes to extremal Kerr black holes. Radial boundary conditions and angular regularity determine the gravitational frequencies at extremality. Compared with earlier spectra restricted to the conventional Riemann sheet, analytic continuation across frequency branch cuts reveals additional QNMs on unconventional Riemann sheets. Spin continuation follows our Schwarzschild mode families toward extremality, with the survey extending through the 41st overtone on specified angular branches and Riemann sheets. We show for the first time that unconventional Class I modes with $m<0$, continued from unconventional Schwarzschild modes, form zero-damping sequences approaching the superradiant bound $Mω=m/2$ in the extremal limit. Within a fixed angular sector, finite-damping and zero-damping sequences are interspersed in the Schwarzschild overtone ordering, so a single overtone cutoff cannot separate them. Matching near the superradiant bound distinguishes nonextremal sequences accumulating at the critical frequency from isolated QNMs at exact extremality. Scattering-amplitude calculations verify representative roots and their sheet assignments. The expanded spectrum clarifies how Schwarzschild mode families reorganize at extremality, with completeness established by contour counting within a bounded frequency domain on a specified angular branch and Riemann sheet.