| In this thesis we mainly study the spin-weighted spheroidal equation, which appeared first in stable perturbation equations of the kerr black hole. Through the separation of variables, two equations:radial wave equation and angle equation have been obtained by Teukolsky in1973. What we study is the angle equation, spin-weighted spheroidal equation.The thesis is divided into four chapters. In the first chapter, we briefly introduced the kerr black hole perturbation theory, especially the N-P formula for gravitational radiation perturbation equation. In the end of the chaperter the origin form of the of Teukolsky equations are given.After that, we introduce the principle of the super-symmtric quantum mechanics(SUSYQM), including the properties of hamiltonian operator, superpotential and the invariance of the potential. Later, what we mainly give the purpose of the research on the spin-weighted spheroidal equation and their research status.The main work in my thesis is mainly divided into the following two parts in the second chapter and the third chapter. In the second chapter we introduce a new expression of the superpotential to obtain the excited solutionsfor the spin-weighted spheroidal equation with the help of the property of shape-invariance in SUSYQM. Through the result we obtain, we can get a more thorough understanding of the spin-weighted spheroidal functions’poperties, especially the integrability. In the third chapter, we carry out the main work, studying the convergence of the solutions we have obtained. Because of the infinite power series in the expressions of the superpotential and power function, we depict the curve by means of numerical methods, making the research more thorough. Analyzing the curves we have obtained can give us some important information about the convergence of the solutions. This work can contribute to the equation and improve theory.The fourth Chapter has carried on the review, summary and outlook. |
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