I'm trying to solve this differential equation in elliptic coordinate system:
My real problem is inserting the physical boundary condition that makes the function a well behavior one, and obtaining the analytical solution in Wolfram Mathematica. Using DSolve I got a differential root witch means that the solution should be numerical but I'm sure this is not my case. I'm asking you how can I solve this kind of differential equation in Mathematica with an analytical solution.
Here is the present status of the problem. The Schrödinger equation for the hydrogen molecule ion (within the Born-Oppenheimer approximation) can be separated in prolate spheroidal coordinates (xi, eta, phi) into 3 ODEs. No analytic solutions have as yet been found for the xi and eta differential equations, although very accurate asymptotic expansions can be derived. The energy eigenvalues, which are related to the separation constant, can be expressed as an infinite series or an infinite continued fraction, which can be evaluated to arbitrary accuracy. A good update is given in http://arxiv.org/pdf/physics/0607081v1.pdf
Have you seen The hydrogen molecule ion? The exact solution is possible following it. What has been done so far?