Bound-state-induced persistent oscillations in the transient behavior of the probability density for the attractive δ potential

An exact analytical solution to the time-dependent Schroedinger equation for an attractive δ potential using cutoff plane-wave initial conditions is derived to investigate the effect of a bound state on the transmission transients. We find that at short distances from the potential the probability density behaves harmonically at all times. This unexpected behavior, which corresponds to the trapping of a fraction of the transmitted wave, decreases exponentially with distance, so that at large distances the solution goes into the stationary solution. We find also that this behavior has no effect on the phase time.