A new symmetry theory for non-Hermitian Hamiltonians

TL;DR

This paper introduces a new symmetry operator, {ta}, expanding the pseudo PT symmetry framework to non-Hermitian Hamiltonians, enabling them to have real spectra under broader conditions.

Contribution

It proposes a novel {ta} pseudo PT symmetry theory that generalizes existing symmetry concepts for non-Hermitian Hamiltonians with real spectra.

Findings

The {ta} operator commutes with certain non-Hermitian Hamiltonians.

The theory applies to coupled non-Hermitian harmonic oscillators.

Real eigenvalues are achieved under the new symmetry conditions.

Abstract

The {\eta} pseudo PT symmetry theory, denoted by the symbol {\eta}, explores the conditions under which non-Hermitian Hamiltonians can possess real spectra despite the violation of PT symmetry, that is the adjoint of H, denoted H^{{\dag}} is expressed as H^{{\dag}}=PTHPT. This theory introduces a new symmetry operator, {\eta}=PT{\eta}, which acts on the Hilbert space. The {\eta} pseudo PT symmetry condition requires the Hamiltonian to commute with the {\eta} operator, leading to real eigenvalues. We discuss some general implications of our results for the coupled non hermitian harmonic oscillator.