Start from the Lanczos tridiagonalization of your uperturbed Hamiltonian T.
In zero order, consider just the "QES" cases which terminate
(illustration: V = r^2 + a r^2/(1 + g r^2)).
Recall the modified Rayleigh-Schrödinger perturbation recipe where
the wavefunctions become easily defined (by recurrences) in all the higher orders, and
the current diagonalization of H_0 is replaced by the (much simpler) recurrent inversion.
Published in Phys. Lett. A 155 (1991) 87 - 93.