Speaker: Professor Guihua Tian, Department of Physics, Beijing University of Posts and Telecommunications

Title: Research on the recurrence relations for the spin-weighted spheroidal harmonics

Abstract: The spin-weighted spheroidal harmonics (SWSHs) are first defined by Teukolsky in the study of the perturbation of the Kerr black holes. They are the extension of the spin-weighted spherical harmonics and important for the study of the perturbation of the Kerr blackhole. The spin-weight $s$ of SWSHs can be $0, \pm 1/2, \pm 1, \pm 3/2, \pm 2$, and the corresponding perturbation fields to Kerr blackhole are the scalar, neutrino, electromagnetic, Rarita-schwinger, and gravitational ones, respectively. The spheroidal harmonics are SWSHs when the spin-weight is zero. The spheroidal harmonics are easier to study than the counterparts of the spin-weight $s \neq 0$. Through the methods in super-symmetric quantum mechanics, we study the recurrence relations for SWSHs with different spin weight. These relations can be applied to derive SWSHs of $s = \pm 1, \pm 2$ from the spheroidal functions, that is SWSHs of $s = 0$. They also give SWSHs of $s = -1/2, \pm3/2$ from that of $s = 1/2$. These recurrence relations are first investigated and are very important both in theoretical background and the astrophysical applications.

Keywords: spin-weighted spheroidal harmonics, recurrence relation, super-symmetric quantum mechanics, shape-invariance

PACS numbers: 02.30.Gp, 03.65.Ge, 11.30.Pb