  We propose a new approach to the Rayleigh-Schr\"{o}dinger perturbation
expansions of bound states in quantum mechanics. We are inspired by the
enormous flexibility of solvable interactions with several (N) discontinuities.
Their standard matching solution is modified and transferred in perturbation
regime. We employ the global renormalization freedom of the local wave
functions and derive a compact N-dimensional matrix formula for corrections. In
applications, our recipe is shown non-numerical for all polynomial
perturbations of any piece-wise constant zero order potential.
