On the Pseudo-Hermiticity of a Class of PT-Symmetric Hamiltonians in One Dimension

For a given standard Hamiltonian H = [p − A(x)]2/(2m) + V (x) with arbitrary complex scalar potential V and vector potential A, with x ∈ R, 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. Our results do not make use of the assumption that H is diagonalizable or that its spectrum is discrete. In a recent series of papers [1, 2, 3] we have revealed the basic mathematical structure responsible for the intriguing spectral properties of certain 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: [email protected]