Metrics and isospectral partners for the most generic cubic PT - symmetric non - Hermitian Hamiltonian Paulo

We investigate properties of the most general PT -symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic potential, the so-called Swanson Hamiltonian or the transformed version of the from below unbounded quartic −x4-potential. In addition, it also reduces to various models not considered in the present context, namely the single site lattice Reggeon model and a transformed version of the massive sextic ±x6-potential, which plays an important role as a toy model to identify theories with vanishing cosmological constant.