Electromagnetic wave propagation in an active medium and the equivalent Schrödinger equation with an energy-dependent complex potential

We study the massless limit of the Klein-Gordon (K-G) equation in $1+1$ dimensions with static complex potentials in order to provide an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in an active medium. In the case of a dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy dependent. We study also the nonrelativistic Schr\"odinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of the solutions of this Schr\"odinger equation is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. In particular, both equations exhibit a discrepancy between the time-dependent and stationary results; our study attributes this discrepancy to the appearance of time-growing bound eigenstates corresponding to poles in the transmission and reflection amplitudes located in the upper half of the wave-number plane. The omission of these bound states in the expansion in stationary states leads to the observed discrepancy. Furthermore, it was demonstrated that there is a frequency- (energy) -and-size-dependent gain threshold above which this discrepancy appears. This threshold corresponds to the value of the gain at which the first pole crosses the real axis.