Nonlocal potentials, isolated states, and Levinson’s theorem

Abstract

A compact expression for the calculation of phase shifts is derived for a potential which is the sum of local and nonlocal parts. Nonlocal potentials can support positive energy bound states, that is, states embedded in the continuous energy spectrum. These states, sometimes referred to as “isolated” states, are not associated with any poles of the S matrix. Some controversy exists in the literature on how such bound states are included in Levinson’s theorem; it is found that the phase shift should be taken continuous at the energy of the bound state rather than taken to have a discontinuity of π. For simplicity, the analysis is restricted to the radial s wave Schrödinger equation and separable nonlocal potentials.