Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

TL;DR

This paper derives identities for reflection and transmission amplitudes in PT-symmetric scattering potentials, proving conjectured relations and establishing a generalized unitarity relation applicable to both real and complex PT-symmetric potentials.

Contribution

It provides a general proof of reflection and transmission amplitude relations and introduces a unified unitarity relation for PT-symmetric scattering potentials.

Findings

Proved the relations |T(-k)|=|T(k)| and |R^r(-k)|=|R^l(k)|.

Established the generalized unitarity relation R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1.

Showed these properties hold for both real and complex PT-symmetric potentials.

Abstract

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, $R^{l/r}(k)$ 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 $|R^r(-k)|=|R^l(k)|$, 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 ${\cal PT}$-symmetric potentials. The same holds for $T(-k)=T(k)^*$ and $|R^r(-k)|=|R^l(k)|$.