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New features of scattering from a one-dimensional non-Hermitian (complex) potential

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: $R(-k)\ne R(k)$ and $T(-k) \ne T(k)$, unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that $R_{left}(-k)=R_{right}(k)$ and $T(-k)=T(k)$. So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ($E_*=α^2,β^2$) either in $T(k)$ or in $T(-k)$, when $αβ>0$. Thirdly, when $αβ<0$ it possesses one SS in $T(k)$ and the other in $T(-k)$. Fourthly, when the potential becomes PT-symmetric $[(α+β)=0]$, we get $T(k)=T(-k)$, it possesses a unique SS at $E=α^2$ in both $T(-k)$ and $T(k)$. Lastly, for completeness, when $α=iγ$ and $β=iδ$, there are no SS, instead we get two negative energies $-γ^2$ and $-δ^2$ of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound state eigenvalues and no spectral singularity exists in this case. We find them as $E^{+}_{M}=-(γ-M)^2$ and $E^{-}_{N}=-(δ-N)^2$; $M(N)=0,1,2,...$ with $0 \le M (N)< γ(δ)$. {PACS: 03.65.Nk,11.30.Er,42.25.Bs}