Raising and lowering operators of spin-weighted spheroidal harmonics

Abstract

Differential operators for raising and lowering angular momentum for spherical harmonics are used widely in many branches of physics. Less well known are raising and lowering operators for both spin and the azimuthal component of angular momentum (Goldberg et al. in J Math Phys 8:2155, 1967). In this paper we generalize the spin-raising and lowering operators of spin-weighted spherical harmonics to operators linear-in-γ for spin-weighted spheroidal harmonics, where γ is an additional parameter present in the second order ordinary differential equation governing these harmonics. Constructing these operators has required using all the ℓ-, s- and m-raising and lowering operators (and various combinations of them) for spin-weighted spherical harmonics, which have been calculated and shown explicitly in this paper. Following a well-defined procedure, the operators given could be generalized to higher powers in γ.