Any realistic central-spike repulsion superimposed with a complex-spike screening seems hard to solve.
Even for its most simple model
V(r) = A r^2 + B/r^2 + C/r^4 + D/r^6 + F/(1+G r^2)
the elementary, quasi-exact doubets of bound states did not seem to exist at all (Znojil and Leach, 1992).
Using the s. c. Groebner bases we succeeded in improving the underlying method.
By solving the coupled polynomial algebraic equations we clarified the whole puzzle.
You may click on the ps formatted full text of this result as published in
Czechoslovak J. Phys. 48 (1998) 1 - 8.