  I propose a new version of the Rayleigh - Schr\"{o}dinger perturbation
method. It admits a lower triangular matrix in place of the usual diagonal
propagator. Illustrated on rational anharmonicities
polynomial}(x)/polynomial}(x), treated as perturbations of (quasi-)exact
anharmonic oscilators. In this sense the method works in an
intermediate-coupling regime and bridges the gap between the weak- and
strong-coupling expansions.
