Generalized Swanson models and their solutions

We analyse a class of non-Hermitian quadratic Hamiltonians, which are of the form H = A(dagger) A + alpha A(2) + beta A(dagger) (2), where alpha, beta are real constants, with a alpha not equal beta, and A(dagger) and A are generalized creation and annihilation operators respectively. 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., 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 where the non-Hermitian Hamiltonian is PT symmetric.