Generalized Swanson models and their solutions

TL;DR

This paper investigates a class of non-Hermitian quadratic Hamiltonians called generalized Swanson models, deriving conditions for real eigenvalues, constructing similarity transformations to Hermitian forms, and providing explicit solutions for specific potential models.

Contribution

The paper extends the analysis of Swanson models to a generalized form with explicit solutions and similarity transformations, including PT-symmetric cases.

Findings

Eigenvalues are real within certain parameter ranges.

A similarity transformation maps non-Hermitian Hamiltonians to Hermitian counterparts.

Explicit solutions are provided for Rosen-Morse type potential models.

Abstract

We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form $ H = {\cal{A}}^{\dagger} {\cal{A}} + \alpha {\cal{A}} ^2 + \beta {\cal{A}}^{\dagger 2} $, where $ \alpha, \beta $ are real constants, with $ \alpha \neq \beta $, 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 $\rho$, 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 ${\cal{PT}}$ symmetric.