Scattering by a separable potential in one dimension

Abstract

The scattering by a separable potential in one dimension is described by means of explicit expressions for the reflectance coefficient, phase shift, and excess density. Resonances, bound and virtual states are analyzed in the light of the position of the poles of the resolvent, and a physical consequence of the spurious poles of the S matrix is indicated. Deviations from the usual partial wave analysis in three dimensions, especially a modification of Levinson's theorem, are also studied. It is shown that at very low energies the slope of the phase shift is not always a valid indicator of the excess density in the interaction region.