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 $\mathcal{P}\mathcal{T}$ -symmetric and non-Hermitian Hamiltonian H=J 2+igu, where again g is real. As in the case of $\mathcal{P}\mathcal{T}$ -symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this $\mathcal{P}\mathcal{T}$ -symmetric Hamiltonian, a region of unbroken $\mathcal{P}\mathcal{T}$ symmetry in which all the eigenvalues are real and a region of broken $\mathcal{P}\mathcal{T}$ symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.
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