Pseudo-reality and pseudo-adjointness of Hamiltonians

Abstract

We define pseudo-reality and pseudo-adjointness of a Hamiltonian, $H$, as $ρH ρ^{-1}=H^\ast$ and $μH μ^{-1}=H^\prime$, respectively. We prove that the former yields the {\it necessary} condition for spectrum to be real whereas the latter helps in fixing a definition for inner-product of the eigenstates. Here we separate out adjointness of an operator from its Hermitian-adjointness. It turns out that a Hamiltonian possessing real spectrum is first pseudo-real, further it could be Hermitian, PT-symmetric or pseudo-Hermitian.