Accurate WKB wave functions for weakly attractive inverse-square potentials

Abstract

For a weakly attractive inverse-square potential, $V(x)=\ensuremath{-}g{\ensuremath{\Elzxh}}^{2}{/(2\mathrm{mx}}^{2})$ with $0<g<~1/4,$ the standard WKB wave function shows unphysical divergence near the origin. Introducing an appropriate nonvanishing reference point and a related phase yields WKB wave functions whose deviation from the regular solution of the Schr\"odinger equation decreases asymptotically as ${1/(\mathrm{kx})}^{3}.$ This is two orders better than the alternative technique involving the Langer modification of the potential. The performance of the correspondingly modified quantization conditions is demonstrated for the bound states of vanishing angular momentum in the two-dimensional circle billiard and in a two-dimensional Woods-Saxon well.