Formulas for theδ-shell-plus-Coulomb potential for all partial waves

Abstract

Closed forms are given for a large number of quantities occurring in the theory of potential scattering in an arbitrary partial-wave state $l$ for the cases (i) the $\ensuremath{\delta}$-shell potential $V(r)=\ensuremath{-}\ensuremath{\lambda}{R}^{\ensuremath{-}2}\ensuremath{\delta}(r\ensuremath{-}R)$, and (ii) the $\ensuremath{\delta}$-shell plus Coulomb potential. Furthermore, the trajectories of the poles of the total $T$ operator in the complex $k$ plane, with varying complex $\ensuremath{\lambda}$, are investigated in detail for zero, repulsive, and attractive Coulomb force, respectively. Expressions are given for the effective-range parameters, and the Coulomb-modified effective-range parameters, for all $l$, with application to the $\mathrm{NN}$ system, and the $N\ensuremath{\alpha}$ system, respectively. The connection between Coulomb-level shifts and effective-range parameters is considered. Improvements on the standard small-shift approximation, which is relatively poor, are suggested.NUCLEAR REACTIONS Charged-particle scattering. $\ensuremath{\delta}$-shell plus Coulomb potential. Effective-range parameters for all $l$. Application to $\mathrm{NN}$ and $N\ensuremath{\alpha}$. Coulomb-level shifts.