Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

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