Generalized Swanson models and their solutions

Abstract

We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form $H={\mathcal{A}}^{†}\mathcal{A}+\alpha {\mathcal{A}}^2+\beta {\mathcal{A}}^{†2}$, where $\alpha ,\beta$ are real constants, with $\alpha \ne \beta$, and ${\mathcal{A}}^{†}$ and $\mathcal{A}$ 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 $\rho$, 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 $\mathcal{P}\mathcal{T}$ symmetric.

17 pages

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• The Quantum Effective Mass Hamilton-Jacobi Problem
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