Quantum counterpart of spontaneously broken classical PT symmetry

The classical trajectories of a particle governed by the PT-symmetric Hamiltonian $H=p^2+x^2(ix)^ε$ ($ε\geq0$) have been studied in depth. It is known that almost all trajectories that begin at a classical turning point oscillate periodically between this turning point and the corresponding PT-symmetric turning point. It is also known that there are regions in $ε$ for which the periods of these orbits vary rapidly as functions of $ε$ and that in these regions there are isolated values of $ε$ for which the classical trajectories exhibit spontaneously broken PT symmetry. The current paper examines the corresponding quantum-mechanical systems. The eigenvalues of these quantum systems exhibit characteristic behaviors that are correlated with those of the associated classical system.

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