Time evolution of decay for purely absorptive potentials: The effect of spectral singularities

We consider an analytical approach that involves the complex poles of the propagator to investigate the effect due to the lack of time-reversal invariance on the time evolution of decay for purely absorptive potentials. We find that the choice of the initial state may change the exponential decaying regime into a nonexponential oscillatory behavior at time scales of the order of a lifetime of the system. We illustrate this effect for a spectral singularity corresponding to a purely imaginary δ-shell potential.