Non-Hermitian oscillators with TdTd symmetry

• PT-symmetric oscillators exhibit real eigenvalues.
• Not all space–time symmetries lead to real eigenvalues.
• Some Hamiltonians are invariant under unitary transformations.
• Point-group symmetry greatly simplifies the calculation of eigenvalues and eigenfunctions.
• Group theory and perturbation theory enable one to predict the occurrence of real eigenvalues.

We analyse some PT-symmetric oscillators with TdTd symmetry that depend on a potential parameter gg. We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of gg. Pairs of eigenvalues coalesce at exceptional points gcgc; 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 gg 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 gg.