New features of scattering from a one-dimensional non-Hermitian (complex) potential
Zafar Ahmed
For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time reversal: R ( − k ) ≠ R ( k ) and T ( − k ) ≠ 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 1. We present a new non-Hermitian parametrization of the 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 obtain 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 real 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. We find them as E + M = −(γ − M ) 2 and E − N = −(δ − N ) 2 ; M ( N ) = 0, 1, 2, ... with 0 ⩽ M ( N ) < γ(δ).
arXiv:1110.4485•Journal of Physics A Mathematical and Theoretical•Dec 19, 2011•49 citations