Comment on `Supersymmetry, PT-symmetry and spectral bifurcation'

TL;DR

This paper critiques a previous study on PT-symmetric potentials, emphasizing the importance of parameter invariance and clarifying the existence of two real energy eigenvalue series for unbroken PT-symmetry.

Contribution

The authors highlight a key oversight in prior work by demonstrating the invariance under parameter exchange and clarifying the spectral structure of the complexified Scarf II potential.

Findings

The potential has two series of real eigenvalues for unbroken PT-symmetry.

Invariance under parameter exchange is crucial for correct spectral analysis.

Previous work missed the dual superpotential structure due to symmetry oversight.

Abstract

We demonstrate that the recent paper by Abhinav and Panigrahi entitled `Supersymmetry, PT-symmetry and spectral bifurcation' [Ann.\ Phys.\ 325 (2010) 1198], which considers two different types of superpotentials for the PT-symmetric complexified Scarf II potential, fails to take into account the invariance under the exchange of its coupling parameters. As a result, they miss the important point that for unbroken PT-symmetry this potential indeed has two series of real energy eigenvalues, to which one can associate two different superpotentials. This fact was first pointed out by the present authors during the study of complex potentials having a complex $sl(2)$ potential algebra.