The canonical functions method and singular potentials (Q805174)

The aim of this paper is to show that the canonical functions method of the first author is a powerful alternative to the treatment outlined by \textit{L. Gr. Ixaru} [ibid. 72, 270-274 (1987; Zbl 0621.65099)] for the numerical integration of the radial Schrödinger equation \[ (1)\quad d^ 2y/dr^ 2+(E-U(r)-\ell (\ell +1)/r^ 2)y(r)=0. \] In this method the computation of the wave function y(r) (implying an initial value problem) is replaced by that of the canonical functions \(\alpha (r_ 0;r)\) and \(\beta (r_ 0;r)\) which are particular solutions of (1) with the initial values \(\alpha (r_ 0;r_ 0)=1,\quad \alpha '(r_ 0;r_ 0)=0,\quad \beta (r_ 0;r_ 0)=0,\quad \beta '(r_ 0;r_ 0)=1,\) \(r_ 0\) being an ``arbitrary'' origin, \(0<r_ 0<\infty\). To show the validity of the present method, the authors consider the Coulomb potential \(U(r)=- 2/r\) having the exact eigenvalues \(E=-1/(\ell +1)^ 2\).