How to solve Schroedinger's equation with Kratzer potential?

Question

Kratzer potential is defined by

$$V(r)={\frac{\alpha}{r}+\frac{\beta}{r^2}}.$$

I read that the Schroedinger equation for this potential has an analytical solution in terms of hypergeometric functions. Most papers use SUSYQM or Asymptotic Iteration Methods, involving different approximation schemes. I believe there is some more direct way to express the solution in terms of hypergeometric functions,without any approximations as in the case or other similar potentials like Morse, Eckart, .etc, by some substitutions/ansatz, etc.? How can I proceed? Can anyone provide me some references for the same?

Answer 1

If you are okay with approximations and some nasty follow-up calculations, you can do the following not-so-delightful recipe:

1- Expand the $V(r)$ in its analytical form using Maclaurin Seires. (Hint, use $r=\zeta + 1$)

2- Stop when you reach the order that you feel appropriate for you. (e.g. 3)

3- Solve the Schrödinger Equation for each term after expansion independently.

4- Apply Perturbation Theory to find the energies and states produced by combining the terms back together.

Answer 2

I found the solution in---

Siegfried Flugge's Springer book "Practical Quantum Mechanics",2nd reprint(1994),page-178 , in addition to solutions of problems involvin a bunch of other similar potentials,which are treated in relatively less books.