Supersymmetry, PT-symmetry and Spectral Bifurcation

TL;DR

This paper explores PT-symmetric complex potentials, revealing how different superpotentials influence spectral properties, including real spectra, shape invariance, and spectral bifurcations, with implications for supersymmetry in quantum systems.

Contribution

It uncovers the existence of two distinct superpotentials in PT-symmetric potentials and their roles in spectral bifurcation and shape invariance, advancing understanding of supersymmetry in non-Hermitian systems.

Findings

Real spectra associated with unique superpotentials and shape invariance.

Broken PT-symmetry leads to spectral bifurcation and disjoint Hilbert space sectors.

Complex parametric shifts arise in the shape invariance of broken PT-symmetric potentials.

Abstract

We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by \textit{real} units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces \textit{complex} parametric shifts.