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 d dx. Also, by changing the derivative d dx by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted harmonic oscillator HSIO = -1 2 d2 dx2 -1 2x2, 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.