A two-parameter family of non-Hermitian Hamiltonians with real spectrum

Abstract

We introduce a Hamiltonian Hα, 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 Hα, β is (i) Hermitian, (ii) non-Hermitian with real spectrum and (iii) non-Hermitian but PT-symmetric. In the case when Hα, 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 Hα, β Hermitian. In a particular case we calculate the Hermitian counterpart of Hα, β.