All Hermitian Hamiltonians have parity

It is shown that if a Hamiltonian H is Hermitian, then there always exists an operator P having the following properties: (i) P is linear and Hermitian; (ii) P commutes with H; (iii) P(2) = 1; (iv) the nth eigenstate of H is also an eigenstate of P with eigenvalue (-1)(n). Given these properties, it is appropriate to refer to P as the parity operator and to say that H has parity symmetry, even though P may not refer to spatial reflection. Thus, if the Hamiltonian has the form H = P(2) + V (x), where V (x) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has PT symmetry. This shows that PT symmetry is a generalization of Hermiticity: all Hermitian Hamiltonians of the form H = P(2) + V(x) have PT symmetry, but not all PT-symmetric Hamiltonians of this form are Hermitian.