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

TL;DR

This paper explores the pseudo-supersymmetric relationship between non-Hermitian and Hermitian isotonic oscillators, deriving their spectra and wavefunctions algebraically, and establishing connections to the Swanson Hamiltonian.

Contribution

It introduces new isotonic and nonlinear oscillators as Hermitian counterparts of non-Hermitian Hamiltonians using pseudo-supersymmetry, and provides an algebraic method to find their spectra and wavefunctions.

Findings

Derived partner potentials as Hermitian equivalents of non-Hermitian Hamiltonians.

Obtained solutions for the nonlinear isotonic oscillator related to Swanson Hamiltonian.

Established conditions for intertwining operators to connect the Hamiltonians.

Abstract

Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian Hamiltonian $H_{-}=\omega(\xi^{\dag} \xi+\1/2)+\alpha \xi^{2}+\beta \xi^{\dag 2}$, where $\alpha \neq \beta$ 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 $H_{\pm}$.