Non-Hermitian oscillators with $T_{d}$ symmetry

TL;DR

This paper investigates PT-symmetric oscillators with $T_{d}$ symmetry, analyzing eigenvalues and eigenfunctions, and exploring the conditions under which real eigenvalues occur, especially near exceptional points and potential phase transitions.

Contribution

It provides a detailed analysis of eigenvalues and eigenfunctions of $T_{d}$ symmetric oscillators, combining group theory and perturbation theory to predict real eigenvalues for small potential parameters.

Findings

Eigenvalues coalesce at exceptional points $g_c$

Difficulty in confirming phase transitions at nonzero $g$

Group and perturbation theory predict real eigenvalues for small $g$

Abstract

We analyse some PT-symmetric oscillators with $T_{d}$ symmetry that depend on a potential parameter $g$. We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of $g$. Pairs of eigenvalues coalesce at exceptional points $g_c$; their magnitude roughly decreasing with the magnitude of the eigenvalues. It is difficult to estimate whether there is a phase transition at a nonzero value of $g$ as conjectured in earlier papers. Group theory and perturbation theory enable one to predict whether a given space-time symmetry leads to real eigenvalues for sufficiently small nonzero values of $g$.