Behavior of eigenvalues in a region of broken-PT symmetry - Bender, Carl M.; Klevansky, Sandra Pamela (Arxiv, 2017)
PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^ε$. 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 $ε=-4$.