New investigation for the spheroidal wave functions ∗

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study the spheroidal wave functions’ eigenvalue problem. Expanded by the parameter α, the first order term of ground eigen-value and the eigen-function are gotten. In virtue of the good form of the first term in the superpotential and its shape-invariant property in the first order, we also obtain the eigenvalues and eigenfunctions of excited state in first order term from that of ground state. In the paper, the very good results are that all the first term eigenfunctions obtained are in closed form. They are advantageous to investigating for involved physical problems of spheroidal harmonics. PACs:04.25Nx; 04.70-s; 04.70Bw During the last five decades, spheroidal wave functions play a premier role in mathematical physics. Nowadays, they have made strong contributions to extensively theoretical and practical applications in science and engineering, such as gravitational wave detection, quantum field theory in curved space-time, black hole stable problem; spheroidal antenna analysis and design in 3G mobile and broad band satellite telecommunication; steady flow of a viscous fluid past a spheroidal, spheroidal shapes calculations of rockets, aircraft noses and guided missiles; spheroidal cavity problem; spheroidal electromagnetic diffraction, scattering and similar problems in acoustic science, etc[1]-[5]. The spheroidal wave equations 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]-[3]. Basically, the methods of evaluating the eigenvalues and eigenfunctions of spheroidal harmonics mainly rely on the three-term recurrence relation: one could solve the transcendental equation in continued fraction form or its equivalent or by power series expansion etc[1][3],[6]-[13]. For the details of these methods and their advantage and disadvantage, one could see the reference [13]. These methods mainly work for the numerical purpose, and also rely heavily on the numerical method. However, all previous work concentrates on the side of numerical calculations of the eigenvalues, and particularly emphasizes the small parameter and large parameter limits form of the eigenvalues. But, less effort has been devoted to the ∗E-mail: [email protected], [email protected], [email protected], [email protected] 1 related eigenfunctions, especially the analytic property. This mainly because the difficulty for it. Even in the approximation by power series expansion, whether from the continued fraction or quantum mechanics method, all eigenfunctions come into this kind of form Fkm = ∑∞ n=0 α fn,km with the subscript k being indication of the kth state eigenfunction Fkm. The nth term fn,km in α could be written as fn,km = ∑∞ q=0 cnq,kmP m m+q with P q m+q being the associated-Legendre functions. Obviously, the nth term fn,km is only in the series form, this actually does not reveal much about the eigen-functions at all. In the paper, we give a completely new method to calculate the ground eigenvalues and eigenfunctions of spheroidal harmonics in the first term of the parameter α. The good result is that the ground eigenfunction of the first order is in closed form. Furthermore, using the perturbation in supersymmetric quantum mechanics (SUSYQM), we find the closed form f1,k0 due to the properties of their potential’s shape-invariance in the first term in the parameter. Just like in the associated-Lendgre functions’ case, the shape-invariance property makes it easy to calculate the f1,km from the f1,0m. Unfortunate, the higher order term of the potentials have no such a good property, so we only get the ground function for higher order terms in parameter α. In the following, we first give brief review of supersymmetric quantum mechanics (SUSYQM) in solving the eigenvalue-problem, then using it solving our problem, finally we give some discussion and conclusion.