Levinson's Theorem and the Nodes of Zero-Energy Wave Functions for Potentials with Repulsive Coulomb Tails

Levinson's theorem relation phase shifts and bound states, developed for a short-range potential ${V}_{\mathrm{sh}}$, is extended to scattering by ${V}_{\mathrm{sh}}+{V}_{\mathrm{C}}$, where ${V}_{\mathrm{C}}$ is a repulsive Coulomb potential. The subtleties associated with $L=0$, $E=0$ bound states for ${V}_{\mathrm{sh}}$ alone are not present for ${V}_{\mathrm{sh}}+{V}_{\mathrm{C}}$. Information is also obtained on the relation between the nodal structure of the zero-incident-energy wave function and the number of bound states; some extensions to scattering by a compound target are possible.