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

TL;DR

This paper critiques a previous comment on PT-symmetry in complex potentials, clarifying conditions for symmetry breaking and emphasizing the independence of spectral bifurcation from $sl(2)$ symmetry in PT-symmetric models.

Contribution

It corrects the argument regarding PT-symmetry breaking conditions and highlights that spectral bifurcation is independent of $sl(2)$ symmetry considerations.

Findings

Corrects the PT-symmetry condition in broken regime

Shows spectral bifurcation is independent of $sl(2)$ symmetry

Clarifies the nature of PT-symmetry in the models

Abstract

In "Comment on Supersymmetry, PT-symmetry and spectral bifurcation" \cite{BQ1}, Bagchi and Quesne correctly show the presence of a class of states for the complex Scarf-II potential in the unbroken PT-symmetry regime, which were absent in \cite{AP}. However, in the spontaneously broken PT-symmetry case, their argument is incorrect since it fails to implement the condition for the potential to be PT-symmetric: $C^{PT}[2(A-B)+\alpha]=0$. It needs to be emphasized that in the models considered in \cite{AP}, PT is spontaneously broken, implying that the potential is PT- symmetric, whereas the ground state is not. Furthermore, our supersymmetry (SUSY)-based 'spectral bifurcation' holds \textit{independent} of the $sl(2)$ symmetry consideration for a large class of PT-symmetric potentials.