Pseudo supersymmetric partners for the generalized Swanson model

TL;DR

This paper constructs new non-Hermitian Hamiltonians related to the generalized Swanson model, demonstrating pseudo supersymmetry and extending the class of shape-invariant models with shared spectra.

Contribution

It introduces pseudo supersymmetric partners for the generalized Swanson model, expanding the framework to include non-Hermitian Hamiltonians related via similarity transformations.

Findings

Hamiltonians are related by pseudo supersymmetry

Shared eigenvalues except for the ground state

Extension to shape-invariant models

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.