Pseudo-Hermitian Systems with $\mathcal {P}\mathcal {T}$-Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even \(\mathcal {P}\mathcal {T}\)-symmetry \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=1)\) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd \(\mathcal {P}\mathcal {T}\)-symmetric systems \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=-1)\) 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 \(\mathcal {P}\mathcal {T}\)-symmetry admit a degeneracy structure if the operator \(\mathcal {PT}\) anticommutes with the metric operator η _σ 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 \(\mathcal {P}\mathcal {T}\)-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even \(\mathcal {P}\mathcal {T}\) -symmetry.