Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, $R^{l\mathrm{/}r}(k)$ and $T(k)$, of general ${\cal PT}$-symmetric scattering potentials. We use these identities to give a general proof of the relations, $\mathrm{\mid }T(-k)\mathrm{\mid }=\mathrm{\mid }T(k)\mathrm{\mid }$ and $\mathrm{\mid }{R}^r(-k)\mathrm{\mid }=\mathrm{\mid }{R}^l(k)\mathrm{\mid }$, conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: $R^{l\mathrm{/}r}(k)R^{l\mathrm{/}r}(-k)+\mathrm{\mid }T(k){\mathrm{\mid }}^2=1$, and show that it is a common property of both real and complex $\mathcal{P}\mathcal{T}$-symmetric potentials. The same holds for $T(-k)=T(k{)}^{\ast }$ and $\mathrm{\mid }{R}^r(-k)\mathrm{\mid }=\mathrm{\mid }{R}^l(k)\mathrm{\mid }$.