Solve spheroidal wave functions by SUSY method ∗

In the paper, we study the spin-weighted spheroidal wave functions in the case of s = m = 0. Their eigenvalue problem is investigated by the perturbation method in supersymmetric quantum mechanics (SUSYQM). For the ground state, the first three terms of ground eigenvalue and eigenfunction in parameter α = aw are obtained. The obtained ground eigenfunction is elegantly in closed forms. Due to the good form of the first term in the superpotential and its shape-invariant property, we also obtain the eigenvalues and eigenfunctions of excited state in first term , esp the first term eigenfunctions are in closed form. These results are new and very useful for solving relative physical problems. PACs:04.25Nx; 04.70-s; 04.70Bw It is well-known that the spin-weighted spheroidal harmonics are indispensable in many physical process, such as gravitational wave detection, quantum field theory in curved spacetime, black hole stable problem; nuclear modeling; spheroidal cavity problem; spheroidal electromagnetic diffraction, scattering and similar problems in acoustic science, etc[1]-[3]. Spin-weighted spheroidal harmonics first appeared in the theory of Kerr black hole’s perturbation. In 1973, Teukolsky first obtained separable equations of scalar, electromagnetic and gravitational fields’ perturbation to the Kerr black-hole[4]. The separated angular equations are called the spin-weighted spheroidal wave equations. Though they are extension of the ordinary spherical harmonics equations. So far, in comparison to simpler spherical special functions, their properties still are difficult for study than their counterpart[1]-[14]. The dynamical wave equations for the perturbation field ψs of the Kerr black-hole of mass M , angular momentum aM are as following[4]: