Approximation for the Phase Shifts Produced by Repulsive Potentials Strongly Singular in the Origin

Abstract

An approximate expression for the partial-wave scattering phase shifts produced by a potential strongly divergent at the origin is introduced. This approximation is applicable over the whole energy range; it yields the correct threshold behavior of the phase shifts and becomes exact at high energy. The high-energy behavior of the phase shifts ${\delta }_l$ is ascertained; we find, as $k$ diverges ($l$ fixed), ${\delta }_l=-A{k}^{1-\frac{2}{m}}$, where $m$ characterizes the behavior of the potential at the origin, $V\left(r\right)\mathrm{}\sim r^{-m}$, $m>\mathrm{}2\mathrm{}$, and $A$ is a constant independent of $l$ which is evaluated explicitly. Numerical tests of the accuracy of the approximation are given, including comparisons with the WKB approximation.