LEVINSON-SEATON THEOREM FOR POTENTIALS WITH AN ATTRACTIVE COULOMB TAIL

The zero-energy scattering in a particular partial wave by a potential V=V-s+V-c that is a superposition of short range and attractive Coulomb components is characterized by the additional phase shift delta(s)(0), due to V-s. It has been known for many years that delta(s)(0)(mod pi)=mu(infinity)pi, where mu(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 delta(s)(0)=mu(infinity)pi, with the value of the largest integer contained in mu(infinity) representing the number of additional bound states due to V-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-->(infinity) merge smoothly into the energy-normalized regular continuum solutions at the continuum threshold.