Perturbation method for non-square Hamiltonians and its application to polynomial oscillators

A remarkable extension of Rayleigh-Schrödinger perturbation method is found and described. Its (N+q)×(N+1)-dimensional Hamiltonians are assumed emerging during quasi-exact constructions of bound states. At all q>1, the role of the traditional single eigenvalue is taken over by an energy/coupling q-plet. In a way circumventing both the non-linearity and non-Hermiticity difficulties, the corrections are defined by compact, q-dimensional matrix-inversion formulae.