Truncated Radial Oscillators with a Bound State in the Continuum via Darboux Transformations

Abstract

Abstract The radial oscillator with zero angular momentum is used to construct a short-range model by cutting-off the potential at a given radius r = b, and by substituting it with a constant potential for r > b. The new potential, called truncated radial oscillator, admits both bound and scattering states. It is shown that the appropriate Darboux transformation leads to new exactly solvable models that have the entire energy spectrum of the truncated radial oscillator plus a new discrete energy eigenvalue. The latter defines a square-integrable wave function for the new system although it is embedded in the scattering regime of the energy spectrum. The new potentials are radial and such that their asymptotic behavior coincides with the profile predicted by von Neumann and Wigner for a potential to admit an eigenvalue in the continuum.