Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

Abstract

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, $R^{l/r}(k)$ and $T(k)$, of general $\mathcal{P}\mathcal{T}$-symmetric scattering potentials. We use these identities to give a general proof of the relations, $|T(-k)|=|T(k)|$ and $|R^r(-k)|=|R^l(k)|$, conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: $R^{l/r}(k)R^{l/r}(-k)+|T(k){|}^2=1$, 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 $|R^r(-k)|=|R^l(k)|$.

arXiv comment6 pages

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