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

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