Non-Hermitian oscillators with Td symmetry

Abstract

We analyse some PT-symmetric oscillators with Td 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 gc; 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.