A new symmetry theory for non-Hermitian Hamiltonians

The η pseudo PT symmetry theory, denoted by the symbol η, 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^{†} is expressed as H^{†}=PTHPT. This theory introduces a new symmetry operator, η=PTη, which acts on the Hilbert space. The η pseudo PT symmetry condition requires the Hamiltonian to commute with the η operator, leading to real eigenvalues. We discuss some general implications of our results for the coupled non hermitian harmonic oscillator.