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