An essential singularity of the cotangent of the Coulomb-nuclear phase shift, and a finite limit of the nuclear part of the effective-range function derived at zero energy

Abstract

The Coulomb-nuclear phase shift ${\delta }_l^{(cs)}$, $\cot {\delta }_l^{(cs)}$ and a finite limit of the nuclear part ${\Delta }_l(k)$ of the effective-range function (ERF) are derived for an arbitrary orbital momentum $l$ when energy $E\to 0$. It is proved that $\cot {\delta }_l^{(cs)}$ has an essential singularity at zero energy, but ${\Delta }_l(k)$ does not. The explicit finite limit of ${\Delta }_l(0)$ is found. The property of ${\Delta }_l(k)$ as a meromorphic function makes possible the analytical continuation of a re-normalized scattering amplitude from the physical energy region to a bound state pole. Then the asymptotic normalization coefficients (ANC) can be deduced from experimental phase-shift data and applied to radiative capture processes which are important in nuclear astrophysics for new elements creation. Our results are in agreement with the results published for $S$ wave scattering.