On the Pseudo-Hermiticity of General PT-Symmetric Standard Hamiltonians in One Dimension

For a given standard Hamiltonian H with arbitrary complex scalar and vector potentials in one-dimension, we construct an invertible antilinear operator τ such that H is τ -anti-pseudo-Hermitian, i.e., H = τHτ−1. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT symmetric Hamiltonian with a real degree of freedom is pseudo-Hermitian. In a recent series of papers [1, 2, 3] we have revealed the basic mathematical structure responsible for the intriguing spectral properties of PT -symmetric systems [4]. The main ingredient leading to the results of [1, 2, 3] and their ramifications [5, 6, 7, 8, 9] is the concept of a pseudo-Hermitian operator. By definition, a linear operator H acting in a Hilbert space is pseudo-Hermitian if there is a linear Hermitian invertible operator η satisfying H = ηHη, (1) ∗E-mail address: amostafazadeh@ku.edu.tr