A Tentative Study Of The Spheroidal Wave Function And Master Equation

There are two parts in this thesis. The first one is derivation about the eigenvalues and the eigenfunctions of the spheroidal functions in the case of s=2 and s=-2. The second one is for deriving the coefficients of master equation, given the Drude special density, which is one of the important concepts in stochastic gravity theory. They are described in three chapters as following:In Chapter 1, the base concepts, research significance and the situation of Spin-weighted Spheroidal wave function and the master equation are introduced, including the formation of the Spin-weighted Spheroidal wave function and the applications etc; At the same time, there is the related knowledge about stochastic gravity theory and the research significance and the present research of master equation.In Chapter 2, we will try to derive the formations of the spin-weighted spheroidal wave function in the case of s=2 and s=-2, using the method of super-symmetric quantum mechanics. First of all, we use this method to calculate super-potential and eigenvalues of spheroidal functions in the first several terms. Further more, we derive the ground eigenfunction in the first order terms. At last, we will utilize the properties of the shape invariant of the potential energy to build the recurrence relation between the ground state wave function and the wave functions of the excited states and obtain the solution of these wave functions of the excited states. This super-symmetric quantum mechanics method and the formation of the ground state wave function are very useful for applying of the spheroidal wave function probably. In Chapter 3, we will attempt to derive the coefficients of the master equation in the special case: Drude spectral density, via the Laplace transformation, at low-temperature approximation. First, we utilize the Laplace transformation method to solve the elementary functionsu,(i= 1,2), which could constructed the coefficients of Drude's master equation, and then we try to calculate the Green functions Gi(i= 1,2), which are the compound functions of the elementary functions ui;(i= 1,2); At the same time, we could obtain the noise kernel according to Drude spectral density. At last, we will derive the coefficients of Drude's master equation, which are propitious to solution of the master equation.