Generalized Swanson Models and their solutions

We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form H = A†A + αA2 + βA† 2, where α , β are real constants, with α 6= β, and A† and 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 RosenMorse I and the hyperbolic Rosen-Morse II type potentials are obtained. We also study the case when the non-Hermitian Hamiltonian is PT symmetric.