Pseudo-Hermiticity versus PT -Symmetry II: A complete characterization of non-Hermitian Hamiltonians with a real spectrum

We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors. Recently, we have explored in [1] the basic mathematical structure underlying the spectral properties of PT -symmetric Hamiltonians [2]. In particular, we have shown that these properties are associated with a class of more general (not necessarily Hermitian) Hamiltonians H satisfying H = η H η, (1) where † denotes the adjoint of the corresponding operator and η is a Hermitian invertible linear operator. We have termed such a Hamiltonian ‘η-pseudo-Hermitian.’ Hermitian and the PT -symmetric Hamiltonians that admit a complete set of biorthonormal eigenvectors constitute subsets of the set of pseudo-Hermitian Hamiltonians. For a PT -symmetric Hamiltonian, the exactness of PT -symmetry ensures the reality of the energy spectrum. The purpose of this article is to provide a complete characterization of the Hamiltonians that ∗E-mail address: [email protected]