Pseudo supersymmetric partners for the generalized Swanson model

Abstract

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., \( H_- = {\cal{A}}^{\dagger} {\cal{A}} + \alpha {\cal{A}} ^2 + \beta {\cal{A}}^{\dagger 2} \), where \( \alpha \beta \) are real constants, with \( \alpha \neq \beta \), and \({\cal{A}}^{\dagger}\) and \({\cal{A}}\) are generalized creation and annihilation operators. It is shown that the initial Hamiltonian \(H_-\), and its partner \(H_+\), are related by pseudo supersymmetry, and they share all the eigen energies except for the ground state. This pseudo supersymmetric extension enlarges the class of non Hermitian Hamiltonians \(H_{\pm}\), related to their respective Hermitian counterparts \(h_{\pm}\), through the same similarity transformation operator \(\rho\) : \( H_{\pm} = \rho ^{-1} h_{\pm} \rho \). The formalism is applied to the entire class of shape-invariant models.