Is space-time symmetry a suitable generalization of parity-time symmetry?

• Space-time symmetry is a generalization of PT symmetry.
• The eigenvalues of a space-time Hamiltonian are either real or appear as pairs of complex conjugate numbers.
• In some cases all the eigenvalues are real for some values of a potential-strength parameter gg.
• At some value of gg space-time symmetry is broken and complex eigenvalues appear.
• Some multidimensional oscillators exhibit broken space-time symmetry for all values of gg.

We discuss space-time symmetric Hamiltonian operators of the form H=H0+igH′H=H0+igH′, where H0H0 is Hermitian and gg real. H0H0 is invariant under the unitary operations of a point group GG while H′H′ is invariant under transformation by elements of a subgroup G′G′ of GG. If GG exhibits irreducible representations of dimension greater than unity, then it is possible that HH has complex eigenvalues for sufficiently small nonzero values of gg. In the particular case that HH is parity-time symmetric then it appears to exhibit real eigenvalues for all 00g>0. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.