Spectral Parameter Power Series Analysis of Supersymmetric Schrödinger Equations with Balanced Gain and Loss Potentials

In this work the Zakharov-Shabat system is addressed to obtain a pair of supersymmetric Schrödinger equations. The scattering and resonance states of these equations are investigated. Explicit solutions for the equations are obtained in the form of power series of the spectral parameter. In the case of the scattering states, we obtain expressions for the transmission and reflection coefficients. In the case of the resonance states we obtain the characteristic equation that defines their complex energies. We show that finding approximate complex energies of the resonance states reduces to calculating polynomial roots from certain characteristic polynomial. Some cases of interest are numerically implemented.