Quasilinear Approximation and WKB

Quasilinear solutions of the radial Schrödinger equation for different potentials are compared with corresponding WKB solutions. For this study, the Schrödinger equation is first cast into a nonlinear Riccati form. While the WKB method generates an expansion in powers of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hbar$\end{document}, the quasi-linearization method (QLM) approaches the solution of the Riccati equation by approximating its nonlinear terms by a sequence of linear iterates. Although iterative, the QLM is not perturbative and does not rely on the existence of any kind of smallness parameters. If the initial QLM guess is properly chosen, the usual QLM solution, unlike the WKB, displays no unphysical turning-point singularities. The first QLM iteration is given by an analytic expression. This allows one to estimate analytically the role of different parameters, and the influence of their variation on the boundedness or unboundedness of a critically stable quantum system, with much more precision than provided by the WKB approximation, which often fails miserably for systems on the border of stability. It is therefore demonstrated that the QLM method is preferable over the usual WKB method.