Resonant states in the presence of Coulomb interactions

Abstract

continuum. However, as is always the case, the derivation relied on the strict finite range of the potential. In such a situation, the analytic structure of the Green's function in the complex-energy plane can readily be established. When a long-range force is present, such as the Coulomb potential, the Green's function no longer satisfies the usual outgoingwave boundary conditions and the arguments leading to the eigenfunction expansion become invalid. In this note we impose a suitable modification of the boundary conditions for the radial wave functions in order to include a Coulomb potential in the Schrgdinger equation. In addition, we show that the arguments leading to the eigenfunction expansion of the Green's function in terms of its poles can be recovered, once the essential singularity at the origin of the complex k-plane is properly taken into account. For simplicity, we restrict ourselves to a particle of zero angular momentum and choose units in which h = 2m= 1. The extension to higher angular momentum is straighforward. The Schr5dinger equation that' we consider is then