  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.
