Classification of the conditionally observable spectra exhibiting central symmetry

The emergence of upside-down symmetry of the bound-state energies E1 = −EN , E2 = −EN−1, . . . has been observed, in strong-coupling regime, in several pseudoHermitian N−state quantum systems. We show that such a symmetry assumption also simplifies a combinatorial classification of these systems since non-equivalent hierarchies of complexifications beyond the strong-coupling boundaries induce nonequivalent pairwise links between the energy levels.