Adaptive analytical mapping procedure for efficiently solving the radial Schrödinger equation

Abstract

This paper shows that replacing the usual integration variable $r∊[0,\ensuremath{\infty})$ by a reduced radial variable $y\ensuremath{\equiv}y(r;\stackrel{P\vec}{\ensuremath{\alpha}})$ defined analytically on a finite domain $y∊[a,b]$ transforms the conventional radial Schr\"odinger equation into an equivalent form in which treatment of levels lying extremely close to dissociation becomes just as straightforward and routine as treating levels in the lower part of the potential well. Explicit integral expressions for the eigenvalue error due to the use of a finite step size in finite-difference methods of numerical integration are presented and are used to improve calculated eigenvalues as well as to determine optimal values of the mapping parameters $\stackrel{P\vec}{\ensuremath{\alpha}}$. This adaptive mapping procedure is shown to be versatile and efficient for both finite-difference and pseudospectral methods.