Scattering from a shifted von Neumann–Wigner potential

Abstract

We study the s-wave scattering produced by perturbing a family of potentials that support a single bound s state with energy embedded in the continuous spectrum. Such bound states were first studied by von Neumann and Wigner [Z. Phys. 20, 465 (1929)], and we refer to the class of potentials that we perturb as von Neumann–Wigner potentials. We solve the radial Schrödinger equation exactly for von Neumann–Wigner potentials which have been perturbed by a radial displacement. We analyze the Jost function and its zeros, including how the latter moves as the perturbation is changed. Depending on the amount of the displacement, the perturbation may produce a sharp resonance at an energy shifted from that of the unperturbed continuum bound state, a virtual state, or a conventional bound state with negative energy. In all cases, no physically acceptable solution of the perturbed Schrödinger equation has the energy of the unperturbed continuum bound state, and the scattering phase shift that is defined from the Jost function has a discontinuity of magnitude π at that energy. The continuum bound state is recovered in the limit as the perturbation goes to zero, but the limit is nonuniform, and we demonstrate a number of anomalies resulting from this.