Perturbed Bound-State Poles in Potential Scattering. II

The problem of maintaining the locality of the bound-state wave function is discussed in the $S$-matrix approach to the first-order energy shift. It is shown first that the nearby-singularity prescription in the Dashen-Frautschi formalism leads to the inclusion of unphysical terms coming from the nonlocal component of the wave function. It is pointed out that the infrared divergence of Dashen and Frautschi comes from this unphysical term. We then present an alternative method based on the Gelfand-Levitan formalism on the inverse-scattering problem. We discuss here an approximate solution of the Gelfand-Levitan equation that leads to a localized bound-state wave function. It is pointed out that this wave function gives the first-order energy shift without the infrared difficulty. Using soluble exponential potentials, we discuss numerical accuracies of this approach.