PT-symmetry broken by point-group symmetry

TL;DR

This paper investigates PT-symmetry breaking in a quantum system with complex eigenvalues, showing how point-group symmetry can predict when eigenvalues become complex, supported by perturbation theory and numerical analysis.

Contribution

It demonstrates how point-group symmetry can be used to predict PT-symmetry breaking in specific Hamiltonians, combining perturbation theory with numerical diagonalization.

Findings

Complex eigenvalues appear for small coupling in one potential

Point-group symmetry helps predict PT-symmetry breaking

Different potentials exhibit distinct eigenvalue behaviors

Abstract

We discuss a PT-symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schr\"{o}dinger equation for a particle in a square box with the PT-symmetric potential $V(x,y)=iaxy$. Perturbation theory clearly shows that some of the eigenvalues are complex for sufficiently small values of $|a|$. Point-group symmetry proves useful to guess if some of the eigenvalues may already be complex for all values of the coupling constant. We confirm those conclusions by means of an accurate numerical calculation based on the diagonalization method. On the other hand, the Schr\"odinger equation with the potential $V(x,y)=iaxy^{2}$ exhibits real eigenvalues for sufficiently small values of $|a|$. Point group symmetry suggests that PT-symmetry may be broken in the former case and unbroken in the latter one.