  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 real, we construct an
invertible antilinear operator \tau such that H is \tau-anti-pseudo-Hermitian,
i.e., H^\dagger=\tau H\tau^{-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.
