Non-Hermitian inverted Harmonic Oscillator-Type Hamiltonians Generated from Supersymmetry with Reflections

TL;DR

This paper introduces four novel classes of non-Hermitian inverted harmonic oscillator Hamiltonians derived through supersymmetry modifications, expanding the understanding of their spectral properties and solutions.

Contribution

It develops four new sets of non-Hermitian Hamiltonians for the inverted harmonic oscillator using supersymmetry with reflections, Dunkl derivatives, and complex parameters.

Findings

Derived four distinct Hamiltonian sets for inverted harmonic oscillators.

Provided eigensolutions with definite parity for each Hamiltonian set.

Extended supersymmetric quantum mechanics to non-Hermitian cases.

Abstract

By modifying and generalizing known supersymmetric models we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of $\frac{d}{dx}$. Also, by changing the derivative $\frac{d}{dx}$ by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted Harmonic oscillator $H_{SIO}=-\frac{1}{2}\frac{d^2}{dx^2}-\frac{1}{2}x^2$, we generate the third set of Hamiltonians. The fourth set of Hamiltonians emerges by allowing a parameter of the supersymmetric two-body Calogero-type model to take imaginary values. The eigensolutions of definite parity for each set of Hamiltonians are given.