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

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 $\delta$-shell potential $V\left(r\right)=-\lambda R^{-2\mathrm{}}\delta \mathrm{}(r-R)\mathrm{}$, and (ii) the $\delta$-shell plus Coulomb potential. Furthermore, the trajectories of the poles of the total $T$ operator in the complex $k$ plane, with varying complex $\lambda$, 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 $N\alpha$ 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.