Pseudo-Hermitian systems with PT-symmetry: Degeneracy and Krein space

TL;DR

This paper demonstrates that pseudo-Hermitian Hamiltonians with even PT-symmetry exhibit degeneracy structures, and utilizes Krein space to analyze unbroken PT-symmetry, extending understanding beyond traditional odd PT-symmetric systems.

Contribution

It establishes conditions for degeneracy in even PT-symmetric pseudo-Hermitian systems and highlights Krein space as an effective framework for their analysis.

Findings

Degeneracy occurs when PT anticommutes with the indefinite metric operator.

Krein space provides a suitable setting for unbroken PT-symmetry analysis.

Illustrated with a detailed four-level pseudo-Hermitian Hamiltonian example.

Abstract

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT-symmetry admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems which is appropriate to the fermions as shown by Jones-Smith and Mathur [1] who extended PT-symmetric quantum mechanics to the case of odd time-reversal symmetry. We establish that the pseudo-Hermitian Hamiltonians with even PT-symmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator {\eta} 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.