Extending PT symmetry from Heisenberg algebra to E2 algebra

TL;DR

This paper explores PT symmetry in the E2 algebra, extending concepts from Heisenberg algebra, and identifies regions of unbroken and broken PT symmetry with real and complex eigenvalues.

Contribution

It introduces PT-symmetric Hamiltonians within the E2 algebra framework and analyzes their spectral properties, revealing phase transition points.

Findings

Identification of unbroken and broken PT symmetry regions

Real eigenvalues in the unbroken phase

Complex eigenvalues in the broken phase

Abstract

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.