Pseudo supersymmetric partners for the generalized Swanson model

Abstract

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., $H_{-}={\mathcal{A}}^{†}\mathcal{A}+\alpha {\mathcal{A}}^2+\beta {\mathcal{A}}^{†2}$, where $\alpha \beta$ are real constants, with $\alpha \ne \beta$, and ${\mathcal{A}}^{†}$ and $\mathcal{A}$ are generalized creation and annihilation operators. 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.

22 pages, to be published in J. Phys. A

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