Quantum counterpart of spontaneously broken classical PT symmetry

TL;DR

This paper explores the quantum analogs of classical PT-symmetric systems, revealing how quantum eigenvalues reflect classical behaviors, including symmetry breaking and trajectory dynamics, for Hamiltonians of the form $H=p^2+x^2(ix)^psilon$.

Contribution

It provides a detailed analysis of the quantum eigenvalues corresponding to classical PT-symmetric Hamiltonians, highlighting the relationship between classical trajectory behaviors and quantum spectral properties.

Findings

Quantum eigenvalues show characteristic behaviors linked to classical PT symmetry breaking.

Classical trajectories exhibit regions of rapid period variation and spontaneous PT symmetry breaking.

Quantum systems reflect classical dynamics, indicating a deep connection between classical and quantum PT symmetry phenomena.

Abstract

The classical trajectories of a particle governed by the PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) 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 $\epsilon$ for which the periods of these orbits vary rapidly as functions of $\epsilon$ and that in these regions there are isolated values of $\epsilon$ 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.