Perturbed harmonic oscillator ladder operators: eigenenergies and eigenfunctions for the X2+ λX2/(1+gX2) interaction

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Abstract
The perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of Hermite polynomials Hk, when V(X)=b2X2+ Sigma kCkHk(b12/X). It is found that the use of a Hermite polynomials basis, together with the use of binomial coefficient functions in the quantum number, greatly simplifies the determination of the perturbed ladder and factorisation functions. Thus, one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation, without increasing intricacy. Thorough calculation has been given for a perturbing potential V(X) function even in X. As an illustrative application of the procedure, the resolution of the Schrodinger equation with a potential function V(X)=X2+ lambda X2/(1+gX2), g>0 is reinvestigated.