Behavior of eigenvalues in a region of broken-PT symmetry

Abstract

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$. When $\epsilon \ge 0$, 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 $\epsilon <0$ 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 $-4<\epsilon <0$. In particular, it reports the discovery of an infinite-order exceptional point at $\epsilon =-1$, a transition from a discrete spectrum to a partially continuous spectrum at $\epsilon =-2$, a transition at the Coulomb value $\epsilon =-3$, and the behavior of the eigenvalues as $\epsilon$ approaches the conformal limit $\epsilon =-4$.

14 pages, 19 figures

References in corpus (5)

• Making Sense of Non-Hermitian Hamiltonians
• The physics of exceptional points
• The ODE/IM Correspondence
• Bifurcation diagram and pattern formation in superconducting wires with electric currents
• Scattering in the PT-symmetric Coulomb potential

Cited by in corpus (7)

• Dynamics of finite dimensional non-hermitian systems with indefinite metric
• $\mathcal{P}\mathcal{T}$-Symmetry in Hartree-Fock Theory
• PT-symmetric eigenvalues for homogeneous potentials
• Analytic approximation for eigenvalues of a class of $\mathcal{P}\mathcal{T}$ symmetric Hamiltonians
• Necessary condition for information transfer under simulated parity-time-symmetric evolution
• Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics
• Exponential asymptotics for the eigenvalues in the broken PT-symmetric region