Coherent scattering from semi-infinite non-Hermitian potentials

When two identical (coherent) beams are injected at a semi-infinite non-Hermitian medium from left and right, we show that both reflection $(r_L,r_R)$ and transmission $(t_L,t_R)$ amplitudes are non-reciprocal. In a parametric domain, there exists Spectral Singularity (SS) at a real energy $E=E_*$ and the determinant of the time-reversed two port S-matrix i.e., $|\det(S)|=|t_L t_R-r_L r_R|$ vanishes sharply at $E=E_*$ displaying the phenomenon of Coherent Perfect Absorption (CPA). In the complimentary parametric domain, the potential becomes either left or right reflectionless at $E=E_z$. But we rule out the existence of Invisibility despite $r_R(E_i)=0$ and $t_R(E_i)=1$ in these new models. We present two simple exactly solvable models where the expressions for $E_*$, $E_z$, $E_i$ and the parametric conditions on the potential have been obtained in explicit and simple forms. Earlier, the novel phenomena of SS and CPA have been found to occur only in the scattering complex potentials which are spatially localized (vanish asymptotically) and having $t_L=t_R$.