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 $E_*=α²,β²$ 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=α² 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 -γ² and -δ² 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}