Polynomial oscillators as perturbations of multiple square wells

We propose an extension of applicability of the Rayleigh-Schrödinger perturbation expansions. Our innovated prescription employs an N−parametric re-normalization of wavefunctions and is shown to be still able to define the corrections via an N by N matrix inversion. For the first few positive integers N , the piecewise constant forces with N discontinuities are recommended as a promising and new unperturbed model. The trigonometric form of its bound states facilitates a consequent perturbative treatement of an arbitrary polynomial potential: Constructively, we demonstrate that the necessary N +N “input” matrix elements are obtainable non-numerically in all orders.