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 PT-symmetric scattering potentials. We use these identities to give a general proof of the relations, vertical bar T(-k)vertical bar=vertical bar T(k)vertical bar and vertical bar R-r(- k)vertical bar=vertical bar R-l(k)vertical bar, conjectured in Ahmed (2012 J. Phys. A: Math. Theor. 45 032004), and the generalized unitarity relation: R-l/r(k) R-l/r(-k)+vertical bar T(k)vertical bar 2=1, and show that it is a common property of both real and complex PT-symmetric potentials. The same holds for T(-k)=T(k)* and vertical bar R-r(- k)vertical bar=vertical bar R-l(k)vertical bar.