Non-Hermitian oscillators with Td symmetry

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. (C) 2014 Elsevier Inc. All rights reserved.