Research On The Analytic Solutions And Numerical Solutions Of The Spin-Weighted Spheroidal Harmonics

Linear perturbations of black holes (BH) play a major role in physics and BH perturbation theory is a useful tool to investigate issues in astrophysics and high-energy physics. The spin-weighted spheroidal harmonics were first defined by Teukolsky with the context of BH perturbation theory. They result from the separation of angular variables in the equations describing the propagation of a spin-s field in a rotating Kerr black hole background. They have two parameters s and β, the parameter s is the helicity of the fields and β is the angular frequency of the perturbation. The parameter s=0,±1/2,±1,±3/2,±2correspond to the scalar, neutrino, electromagnetic, Rarita-Schwinger’s or gravitational perturbations, respectively. And in our method β is also the perturbation parameter of series expansion.This dissertation is devoted to analytic and numerical calculations of the eigenvalues and eigen-functions for the spin-weighted spheroidal equations, and we present numerical results for the range of β. The main results of this dissertation consist of three parts:1. The analytic and numerical solutions of ground states for the spin-weighted spheroidal equationsSuper-symmetric quantum mechanics (SUSYQM) is employed to obtain the analytic solutions for the spin-weighted spheroidal harmonics. Here we take the spin5=1as an example (the author obtained the analytic solutions for cases s=1and s=2, and other members in my tutor’s group got the cases s=1/2and s=1/3). Then we get the general formula of the n-th order super-potential and the correctness of general formula is proved by inductive method. Furthermore, we evaluate the coefficients of super-potential numerically and analyze the convergence of the super-potential for the cases s=1/2,1,3/2, The analytic solutions of eigen-values and the diagrams of the corresponding eigen-functions are obtained, and we present how the parameters m and β influence the images of the eigen-functions. Meanwhile, the eigen-values and eigen-functions of the ground states for the cases s=0,β≠0, s≠0,β=0and s=0,β=0have also been re-evaluated and figured out.2. The range of perturbation parameterβWorking in a perturbative scheme, we give the series expansion of the eigenvalues and super-potentials for small β. And it is necessary to analyze the distribution of singularities. The range of the perturbation parameter β≤3.3has been analyzed numerically, and the range of β is related to m. This is a meaningful work.3. The analytic and numerical solutions of excited states for the spin-weighted spheroidal equationsIn this part we still take the spin s=1as an example to study the excited states. Based on the shape-invariant property of potential, the analytic recurrence relations between ground and excited states of the spin-weighted spheroidal equations are obtained. So the perturbation parameter β of the excited states for the spin-weighted spheroidal equations has the same range as well as the ground states. Similarly, the excited eigen-values and eigen-functions for cases s=1/2,3/2,2are given numerically and graphically. Meanwhile, the eigen-values and eigen-functions of the excited states for the cases s≠0,β=0and s=0,/3=0have also been re-analyzed by our method. By comparing the excited states of cases s≠0, β=0and s≠0, β≠0, we analyzed the eigen-functions with different β.