Non-Hermitian Hamiltonians of Lie algebraic type

We analyse a class of non-Hermitian Hamiltonians, which can be expressed in terms of bilinear combinations of generators in the sl2(R)-Lie algebra or their isomorphic su(1, 1)-counterparts. The Hamiltonians are prototypes for solvable models of Lie algebraic type. Demanding a real spectrum and the existence of a well-defined metric, we systematically investigate the constraints these requirements impose on the coupling constants of the model and the parameters in the metric operator. We compute isospectral Hermitian counterparts for some of the original non-Hermitian Hamiltonians. Alternatively, we employ a generalized Bogoliubov transformation, which allows us to compute explicitly real-energy eigenvalue spectra for these type of Hamiltonians, together with their eigenstates. We compare the two approaches