Generalized master function approach to one-dimensional quasiexactly solvable models

Abstract

By introducing the generalized master function of order up to four together with corresponding weight function, we have obtained all one-dimensional quasiexactly solvable second order differential equations. It is shown that these differential equations have solutions of polynomial type with factorization properties, that is polynomial solutions Pm(E) can be factorized in terms of polynomial Pn+1(E) for m⩾n+1. All known one-dimensional quasiexactly quantum solvable models can be obtained from these differential equations, where roots of polynomial Pn+1(E) are corresponding eigenvalues.