Non-Hermitian Hamiltonians with unitary and antiunitary symmetries

• PT-symmetric Hamiltonians exhibit real eigenvalues when PT symmetry is unbroken.
• PT-symmetric multidimensional oscillators appear to show PT phase transitions.
• This transition was conjectured to be a high-energy phenomenon.
• We show that point group symmetry is useful for predicting broken PT symmetry in multidimensional oscillators.
• PT-symmetric oscillators with C2vC2v symmetry exhibit phase transitions at the trivial Hermitian limit.

We analyse several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. It enables us to predict the degeneracy of the energy levels and to reduce the dimension of the matrices necessary for the diagonalization of the Hamiltonian in a given basis set. We can also classify the solutions according to the irreducible representations of the point group and thus analyse their properties separately. One of the main results of this paper is that some PT-symmetric Hamiltonians with point-group symmetry C2vC2v exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit which suggests that the phenomenon is not robust. Point-group symmetry enables us to explain such anomalous behaviour and to choose a suitable antiunitary operator for the PT symmetry.