Study Of The Recurrence Relation For The SPIN-Weighted Spheroidal Function

It’s well known that the study of spin-weighted spheroidal function play a premier role in mathematical physics. They have extensive applications in relativistic physics and signal and digital image processing of practical issues.Recently, the study of spin-weighted spheroidal function are more and more popular, and the research results are also more and more mature. However, the recurrence relations have yet to be studied in depth. This paper studies mainly the recurrence relations for spin-weighted spheroidal function with different spin-weighted. Here, we mainly focus on the recurrence relations caused by changes in the parameters of generalized spin-weighted spheroidal wave equation changes, which are very different from that previously obtained. These content are the first investigation concerning the spin-weighted spheroidal function and very important in theoretical background.The paper is divided into four chapters.In Chapter1, there is a brief introduction of background of the spin-weighted spheroidal function, which contains perturbation of Kerr black holes and derivation of spin-weighted spheroidal function. Also we give short presentation of the present research state of spin-weighted spheroidal functions, its present research, the research significance and research content.In Chapter2, the basic theory of for solving spin-weighted spheroidal function are introduced, including the research methods——supersymmetric quantum mechanics, simple classification of spin-weighted spheroidal function and the base concepts of spin-weighted spherical harmonics are also given. Specifically, these contain the conception of super symmetric Hamiltonian partners, the idea of hierarchy of Hamiltonians,and the property of shape invariant potential, which will be laid a theoretical foundation for the study recurrence relations in chapter3.In Chapter3, our main work of this paper obtained for the recurrence relations for the same parameter m but different s of spin-weighted spheroidal function. First, by constructing hierarchy of Hamiltonians, we take the first three Hamiltonians for relations of their eigenstates. After reviewing the results of the spin-weighted spheroidal wave equation. We apply the theory to them and first obtain the recurrence relations among different spin-weight s for them with the condition β=0, which means spin-weighted spheroidal function reduce to the well known spin-weighted spherical functions. As the recurrence relations of different spin-weight of spin-weighted spheroidal function are existed, so we verify the consistence of the relations of our result with the results in the literatures. In the end, we investigate the recurrence relations for spin-weighted spheroidal function with β≠0.In Chapter4, the work are summarized and prospects for the paper.