Pseudo-Hermitian Systems with PT-Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT -symmetry (P-2 = 1, T-2 = 1) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems (P-2 = 1, T-2 = -1) which is appropriate to the fermions (Scolarici and Solombrino, Phys. Lett. A 303, 239 2002; Jones-Smith and Mathur, Phys. Rev. A 82, 042101 2010). We establish that the pseudo-Hermitian Hamiltonians with even PT -symmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator.s which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken PT-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even PT-symmetry.