The perturbed ladder operator method. Perturbed eigenvalues and eigenfunctions from finite difference considerations

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Abstract
The finite difference aspect of the perturbed ladder operator method is reinvestigated. By the use of finite difference calculus, resolution of the factorisability condition is achieved, at any order of perturbation, without assuming for the ladder and factorisation functions any particular dependence on the quantum number. A novel procedure of obtaining perturbed eigenfunctions in terms of the unperturbed functions is described. The method, which holds for any type of factorisation (types A to F), is applied to resolution of the type A wave equation. The perturbed type A problem, which has not been previously treated, contains, as particular cases, the perturbation of the spherical harmonics Ylm (or generalised Yl, gamma m) functions of the symmetric top functions Dmki and, more generally, of the hypergeometric functions F( alpha , beta ; gamma ;x).