Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

Ali Mostafazadeh

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 \(|R^r(-k)|=|R^l(k)|\).
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

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