The hyperbolic step potential: Antibound states, SUSY partners and Wigner time delays

TL;DR

This paper analyzes the scattering properties of a hyperbolic step potential, exploring anti-bound states, SUSY partner potentials, and time delays, revealing insights into quantum and classical scattering similarities.

Contribution

It introduces the supersymmetric partners of the hyperbolic step potential using anti-bound states and compares quantum and classical time delays.

Findings

More bound states in SUSY partners lead to smaller time delays.

Quantum and classical time delays show striking similarities.

The model exhibits no resonances but has an infinite number of anti-bound states.

Abstract

We study the scattering produced by a one dimensional hyperbolic step potential, which is exactly solvable and shows an unusual interest because of its asymmetric character. The analytic continuation of the scattering matrix in the momentum representation has a branch cut and an infinite number of simple poles on the negative imaginary axis which are related with the so called anti-bound states. This model does not show resonances. Using the wave functions of the anti-bound states, we obtain supersymmetric (SUSY) partners which are the series of Rosen Morse II potentials. We have computed the Wigner reflection and transmission time delays for the hyperbolic step and such SUSY partners.Our results show that the more bound states a partner Hamiltonian has the smaller is the time delay. We also have evaluated time delays for the hyperbolic step potential in the classical case and have obtained striking similitudes with the quantum case.