Riccati equations and perturbation expansions in quantum mechanics

Abstract

General perturbation expansions, which allow corrections to any order to be written in quadrature, are presented for Riccati and other nonlinear first-order equations. These results are valid for eigenfunctions which are free of poles and zeros. A Riccati equation suitable for a Schrödinger or Klein–Gordon particle in a central field is expanded for a general state, with corrections to all orders expressed in quadrature. A general Riccati equation for a meromorphic eigenfunction leads to a similar expansion with corrections to all orders, including corrections to the zeros and simple poles, expressed in quadrature. This form is suitable for a Dirac particle in a central field but is more general. The general results are applied to specific examples from the literature.