An asymptotic series solution to the Schrodinger equation for a model potential

Using the method of Frobenius, the attempt to find a closed-form solution to the Schrodinger equation for a model potential V(r) = -Ze(2)/root r(2) + alpha(2) is described. Inspection of the series solution in the asymptotic region suggests a substitution of a four term recurrence relation by a two term relation in order to extract explicit solutions that approach to those obtained for the hydrogen atom as alpha --> 0 smoothly. The resulting approximate solutions are compared with numerical ones and shown to provide a good basis set for high Rydberg states.