An update on coherent scattering from complex non-PT-symmetric Scarf II potential with new analytic forms

The versatile and exactly solvable Scarf II has been predicting, confirming and demonstrating interesting phenomena in complex PT-symmetric sector, most impressively. However, for the non-PT-symmetric sector it has gone underutilized. Here, we present most simple analytic forms for the scattering coefficients $(T(k),R(k),|\det S(k)|)$. On one hand, these forms demonstrate earlier effects and confirm the recent ones. On the other hand they make new predictions - all simply and analytically. We show the possibilities of both self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT sectors (potentials). The former one is not accompanied by time-reversed coherent perfect absorption (CPA) and gives rise to the parametrically controlled splitting of SS in to a finite number of complex conjugate pairs of eigenvalues (CCPEs). The latter ones (NSDSS) behave just oppositely: CPA but no splitting of SS. We demonstrate a one-sided reflectionlessness without invisibility. Most importantly, we bring out a surprising co-existence of both real discrete spectrum and a single SS in a fixed potential. Nevertheless, the complex Scarf II is not known to be pseudo-Hermitian ($\eta^{-1} H\eta=H^\dagger$) under a metric of the type $\eta(x)$, so far.