Levinson-Seaton theorem for potentials with an attractive Coulomb tail

The zero-energy scattering in a particular partial wave by a potential V=${\mathit{V}}_{\mathit{s}}$+${\mathit{V}}_{\mathit{c}}$ that is a superposition of short range and attractive Coulomb components is characterized by the additional phase shift ${\mathrm{\delta }}_{\mathit{s}}$(0), due to ${\mathit{V}}_{\mathit{s}}$. It has been known for many years that ${\mathrm{\delta }}_{\mathit{s}}$(0)(modπ)=μ(∞)π, where μ(n) is the quantum defect of the nth energy level. In analogy with Levinson’s theorem for short-range potentials, one might expect that a more precise statement, based on an absolute definition of the phase shift, would be ${\mathrm{\delta }}_{\mathit{s}}$(0)=μ(∞)π, with the value of the largest integer contained in μ(∞) representing the number of additional bound states due to ${\mathit{V}}_{\mathit{s}}$. A simple derivation of this relation is presented here, based on variational principles for the binding energies and phase shifts, and on the property (fundamental to quantum-defect theory) that appropriately normalized bound-state wave functions for n→∞ merge smoothly into the energy-normalized regular continuum solutions at the continuum threshold.