Scattering from a truncated von Neumann–Wigner potential

Abstract

In 1929, von Neumann and Wigner [Phys. Z. 30, 465 (1929)] demonstrated the existence of local potentials which support a bound state with energy embedded in the continuous spectrum. In this paper, we study the $s$-wave scattering produced by a class of von Neumann–Wigner potentials which are perturbed by truncating them at some large radius $r=a.\mathrm{}$ For the class of potentials considered, we obtain an analytic expression for the Jost function. The original continuum bound state is replaced by a well isolated complex zero of the Jost function (equivalent to a pole of the $S$ matrix), but this does not produce a conventional Breit-Wigner resonance. Instead, we find twin peaks in the cross section, separated by a very narrow gap associated with the Jost function zero. Our study continues with an account of bound states and virtual states supported by the potential, and with an investigation of the form of asymptotically large zeros of the Jost function. We compare the results of this paper with the significantly different results we obtained using a different perturbation [Phys. Rev. A 52, 3932 (1995)].