We introduce a Hamiltonian H_(α,β) depending on the two complex parameters α and β and develop a through analysis of the spectral properties of this Hamiltonian and its adjoint. We determine the regions in the space of the complex parameters α and β, where H _(α,β) is (i) Hermitian, (ii) non-Hermitian with real spectrum and (iii) non-Hermitian but PT-symmetric. In the case when H _(α,β) is non-Hermitian having real spectrum, we derive a closed formula for a family of the metric operators, depending on two arbitrary real positive parameters, which render the Hamiltonian H _(α,β) Hermitian. In a particular case we calculate the Hermitian counterpart of H_(α,β).
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