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We study the scattering produced by a one dimensional hyperbolic steppotential, which is exactly solvable and shows an unusual interest because ofits asymmetric character. The analytic continuation of the scattering matrix inthe momentum representation has a branch cut and an infinite number of simplepoles on the negative imaginary axis which are related with the so calledanti-bound states. This model does not show resonances. Using the wavefunctions of the anti-bound states, we obtain supersymmetric (SUSY) partnerswhich are the series of Rosen Morse II potentials. We have computed the Wignerreflection and transmission time delays for the hyperbolic step and such SUSYpartners.Our results show that the more bound states a partner Hamiltonian hasthe smaller is the time delay. We also have evaluated time delays for thehyperbolic step potential in the classical case and have obtained strikingsimilitudes with the quantum case.

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