Abstract
The Schr\"odinger-Infeld-Hull factorization method is extended within the perturbation scheme in order to treat nonfactorizable Sturm-Liouville eigenequations in the same way as factorizable ones. It is shown that, provided suitable choices of the expansion basis set for the perturbing potential and for the associated perturbed ladder function are made, the solution of the factorizability condition associated with the perturbed eigenequation can be achieved by using an elementary finite difference calculus. An algebraic manufacturing process allowing the determination of the perturbed ladder and factorization functions, capable of handling any order of the perturbation and any type of factorization (Infeld-Hull types A to E), is given. This procedure, well adapted for computer algebra, allows an analytical determination of the perturbed eigenvalues and eigenfunctions without calculation of either the excited unperturbed eigenfunctions or any matrix element. This extension of the exact factorization method within the perturbation scheme can be applied to many model equations of current interest in quantum physics. Special attention is paid to perturbed factorizations that correspond to unperturbed ladder operators that are linear functions of the quantum number (types A to D). Illustrative applications are given. Particularly, the perturbed harmonic-oscillator ladder operators and eigenenergies are obtained in closed form.
Published Version
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