One-body overlap functions, equations of motion, and phenomenological potentials

One-body overlap functions play an important role for the description of nuclear structure and nuclear reactions. Equations of motion for the one-body overlaps, based on particle-only, hole-only, and particle-hole approaches, are studied. A given overlap function is shown to satisfy four different Schr\"odinger-like equations, all of which can be derived in the framework of the Feshbach projection operator formalism. Approximating the relevant potential by a local potential is only valid in the particle-hole approach. Previously proposed one-body functions, which can be derived from the overlap functions, are also considered. It is argued that the latter do not satisfy a Schr\"odinger-like equation with an approximately local potential.