Real eigenspectra in non-Hermitian multidimensional Hamiltonians

The Hamiltonian H=((1)/(2))(p2x+p2y)+((1)/(2))(ω2xx2+ω2yy2)-igxNyM, where g is a real parameter and N and M are positive integers, is investigated. We show that energy levels of this Hamiltonian are either real or come in complex conjugate pairs when at least one of N and M is an odd integer, by showing that H is η-pseudo-Hermitian. For even N and M, the spectra was found to be entirely complex. The investigation is general and results are valid for the systems in more than two dimensions. We study, in detail, the cases where H is not PT-symmetric, but η-pseudo-Hermitian