Behavior of eigenvalues in a region of broken PT symmetry

PT-symmetric quantum mechanics began with a study of the Hamiltonian H=p2+x2(ix)ϵ. When ϵ≥0, the eigenvalues of this non-Hermitian Hamiltonian are discrete, real, and positive. This portion of parameter space is known as the region of unbroken PT symmetry. In the region of broken PT symmetry, ϵ<0, only a finite number of eigenvalues are real and the remaining eigenvalues appear as complex-conjugate pairs. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry has thus far been unexplored. This paper presents a detailed numerical and analytical examination of the behavior of the eigenvalues for -4<ϵ<0. In particular, it reports the discovery of an infinite-order exceptional point at ϵ=-1, a transition from a discrete spectrum to a partially continuous spectrum at ϵ=-2, a transition at the Coulomb value ϵ=-3, and the behavior of the eigenvalues as ϵ approaches the conformal limit ϵ=-4.