A New Integral Equation for the Spheroidal equations in case of m equal 1

The spheroidal wave functions are investigated in the case m=1. The integral equation is obtained for them. For the two kinds of eigenvalues in the differential and corresponding integral equations, the relation between them are given explicitly. Though there are already some integral equations for the spheroidal equations, the relation between their two kinds of eigenvalues is not known till now. This is the great advantage of our integral equation, which will provide useful information through the study of the integral equation. Also an example is given for the special case, which shows another way to study the eigenvalue problem.


I. INTRODUCTION OF THE SPHEROIDAL FUNCTIONS
The spheroidal wave equations are extension of the ordinary spherical wave equations. There are many fields where spheroidal functions play important roles just as the spherical functions do. So far, in comparison to simpler spherical special functions (the associated Lengdre's functions ) their properties still are difficult for study than their counterpart[1]- [3].
Their differential equations are This is a kind of the Sturm-Liouville eigenvalue problem with the natural conditions that Θ is finite at the boundaries x = ±1.
Under the condition β = 0, they reduce to the Spherical equation and the solutions to the Sturm-Liouville eigenvalue problem are the associated Legendrefunctions P m n (x)(the spherical functions) with the eigenvalues E n = n(n + 1), n = m + 1, m + 2, . . . . They only have one more term β 2 x 2 than the spherical ones (the associated Lendgre's equations). However, the extra term presents many mathematical difficulties for one to treat the equations.
Though the spheroidal wave equations are extension of the ordinary spherical wave functions equations, the difference between this two kinds of wave funtions are far greater than their similarity [1].
Usual way to study the spheroidal equations is the perturbation one resulting in the continued fraction to determine the eigenvalues and eigenfunctions. Recently, new * Electronic address: tgh-2000@263.net, tgh20080827@gmail.com † Electronic address: shuqzhong@gmail.com methods are used to re-investigate the problems again. The new methods mainly include the perturbation one in supersymmetry quantum mechanics, which gives rise to many nice results. Some of the results are the extension of the recurrence relation of the spherical functions to the spheroidal functions, which makes one could obtain the excited spheroidal functions from the ground one. Other results might give new method in their numerical calculation [14]. There are also the integral equations, which provides another way to numerically study the spheroidal functions. In Ref. [15], the integral equations are extended to the spin-weighted spheroidal case.
For example, the integral equation for the prolate spheroidal wave equation is [1]- [14] where the kernel K(x, y) is withβ = iβ. There are two eigenvalues appear in the differential and the integral equations, that is, the quantities E, λ . However, the relation between the eigenvalues E, λ is unclear [1]- [14]. In this letter, we will report a new integral equation for the spheroidal equation in the case of m = 1.Because the integral equation is derived from the Green function of the equation, the advantage of the new integral equation shows the concise relation between the eigenvalues E, λ. From eq.(1) and by the transformation we could obtain the following The above equation becomes very simple when m = 1, that is, where It is easy to find the Green functions for the equations (6), that is The Green function G(x, ξ) satisfies the following and the boundary conditions Hence the the Sturm-Liouville eigenvalue problem turns into the integral equation form: (13) The great advantage lies in that the relation between the integral eigenvalues λ sin 2β and E of the differential equations for the spheroidal is given explicitly by Though the green function G(x, ξ) is symmetry with respect to the variables x, ξ, the kernal in Eq. (13) is not symmetrical at all. Nevertheless, it is easy to make the kernal be symmetry. That is, changing as desired by our requirement. It is well-known that one could easily to study the integral equations if their kernels are symmetry. Hence, the usual method to solve the integral equations could be used to treat the problem here too. We will stop here.
The Green function G(x, ξ) for the spheroidal equations in m = 1 includes all cases of the parameter β as complex number. When β is pure imaginary, the corresponding equation is the prolate spheroidal equation and the Green function turns out as whereβ = iβ is real. If one supposes the parameter λ = E + β 2 = E −β 2 = 0, the parameterβ will stand in the position of the eigenvalues in the Sturm-Liouville eigenvalue problem. Of course, the parameter β orβ is no longer a fixed quantity in this case. Notice that the case is special because the parameterβ is not a fixed quantity in contrasting with the usual cases. The original equation becomes The Green function in Eq.(16) will give much information about the eigenvalues and eigenfunctions in this special case. Now the Green function could be regarded as the functions of the parameterβ. One could expands this Green function in the form The eigenvalues are determined by the poles of the Green functions, that is sin 2β = 0.
[1], though they are derived from the different way. As stated in Ref.
[1], the function is one of the eigenfunctions for the fixed parameterβ = nπ 2 , n = 2, 4, 6, · · · of the original equation (1) in case m = 1, so does for the fixed parameterβ = nπ 2 , n = 1, 3, 5, · · · . The above example just provides some clues on the connection between the Green function and the solutions to the corresponding the Sturm-Liouville eigenvalue problem. If the Green function is the one corresponding with the parameter λ = 0, they will more useful than just giving the integral equation. However, one could not obtain directly the information on the eigenvalues and eigenfunctions from the the Green function corresponding with the parameter λ = 0 . In this situation, the useful information could be obtained through the study on the integral equation. Here the Green function G(x, ξ) satisfies Eq.(10), rather than the following This Green functionḠ(x, ξ) is connected with the eigenfunctions Ψ n (x) bȳ Our Green function G(x, ξ) related toḠ(x, ξ) by Of course,Ḡ(x, ξ) contains much more useful information on the eigenvalues and eigenfunctions than that of G(x, ξ), but it is much harder to obtain. Even it is inferior toḠ(x, ξ), G(x, ξ) still could provide useful information through the integral equation, which will be our further study.