Generalized Swanson models and their solutions
We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form $ H = {\cal{A}}^{\dagger} {\cal{A}} + α{\cal{A}} ^2 + β{\cal{A}}^{\dagger 2} $, where $ α, β$ are real constants, with $ α\neq β$, and ${\cal{A}}^{\dagger}$ and ${\cal{A}}$ are generalized creation and annihilation operators. Thus these Hamiltonians may be classified as generalized Swanson models. It is shown that the eigenenergies are real for a certain range of values of the parameters. A similarity transformation $ρ$, mapping the non-Hermitian Hamiltonian $H$ to a Hermitian one $h$, is also obtained. It is shown that $H$ and $h$ share identical energies. As explicit examples, the solutions of a couple of models based on the trigonometric Rosen-Morse I and the hyperbolic Rosen-Morse II type potentials are obtained. We also study the case when the non-Hermitian Hamiltonian is ${\cal{PT}}$ symmetric.