Self-similar features around spectral singularity in complex barrier potential

The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential $iV$, $V \in R^{+}$. We show that these loci are elliptical and self-similar to each other in $E-V$ plane where $E$ is energy of the wave. The orientations of these ellipses for reflection ($R$) and transmission ($T$) amplitude around a given SS are same. This is surprising due to the different mathematical forms of $R$ and $T$. In the vicinity of SS, $R \approx T$.