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

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{d²}{dx²}-frac{1}{2}x², 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.