One-parameter potential from Darboux theorem

We consider the stationary one-dimensional Schrödinger equation with potential u(x; i) = ∑_j=-2 ² f_j(i)x^j, where the coefficients f_j(i) are functions of a discrete parameter i. We establish the most general form of the coefficients f_j(i) and obtain the ladder operators for the solution of Schrödinger equation by a Darboux transform. Generally speaking, the Darboux transform is obtained through a so-called superpotential W(x), which is derived from a Riccati equation. We first propose a convenient ansatz for the function W(x) and then yield a set of nine difference equations for the coefficients f_j(i). This set of difference equations establishes the explicit form of the coefficients f_j(i), in the potential u(x; i). Our results are consistent with some well-known quantum potentials in special cases.