Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry

Abstract

Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian Hamiltonian $H_{-}=ω(ξ^† ξ+\1/2)+αξ^{2}+βξ^{†2}$, where $\alpha \ne \beta$ and $\xi$ is a first order differential operator, to obtain the partner potentials $V_{+}(x)$ and $V_{-}(x)$ which are new isotonic and isotonic nonlinear oscillators, respectively, as the Hermitian equivalents of the non-Hermitian partner Hamiltonians $H_{\pm }$. We have provided an algebraic way to obtain the spectrum and wavefunctions of a nonlinear isotonic oscillator. The solutions of $V_{-}(x)$ which are Hermitian counterparts of Swanson Hamiltonian are obtained under some parameter restrictions that are found. Also, we have checked that if the intertwining operator satisfies ${\eta }_1{H}_{-}=H_{+}{\eta }_1$, where ${\eta }_1={\rho }^{-1}\mathcal{A}\rho$ and $\mathcal{A}$ is the first order differential operator, which factorizes Hermitian equivalents of $H_{\pm }$.

11 pages

References in corpus (8)

• Solvable rational extensions of the isotonic oscillator
• Non-Hermitian Hamiltonians of Lie algebraic type
• The generalized quantum isotonic oscillator
• The structure of supersymmetry in $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics
• A generalized non-Hermitian oscillator Hamiltonian, N-fold supersymmetry and position-dependent mass models
• Generalized quantum isotonic nonlinear oscillator in d dimensions
• Pseudo supersymmetric partners for the generalized Swanson model
• Quasi-Hermitian supersymmetric extensions of a non-Hermitian oscillator Hamiltonian and of its generalizations

Cited by in corpus (8)

• Non-Hermitian anisotropic XY model with intrinsic rotation-time reversal symmetry
• A Quantum Quasi-Harmonic Nonlinear Oscillator with an Isotonic Term
• Quantifying the nonlinearity of a quantum oscillator
• $\mathcal{P}\mathcal{T}$ Symmetric Hamiltonian Model and Dirac Equation in 1+1 dimensions
• A generalized Swanson Hamiltonian in a second-derivative pseudo-supersymmetric framework
• Metric Operator For The Non-Hermitian Hamiltonian Model and Pseudo-Supersymmetry
• Swanson Hamiltonian: non-PT-symmetry phase
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry