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

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 $H_{SIO}=-\frac{1}{2}\frac{d^2}{d{x}^2}-\frac{1}{2}{x}^2$, 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.

References in corpus (21)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• Inflation and squeezed quantum states
• Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions
• Integrable Systems for Particles with Internal Degrees of Freedom
• Deformed Heisenberg algebra with reflection
• Hidden supersymmetry in quantum bosonic systems
• The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
• Inverted Oscillator
• Bosons, fermions and anyons in the plane, and supersymmetry
• Three aspects of bosonized supersymmetry and linear differential field equation with reflection
• Minimal bosonization of supersymmetry
• Factorization method and new potentials from the inverted oscillator
• Fingerprints of Classical Instability in Open Quantum Dynamics
• Supersymmetric Quantum Mechanics with Reflections
• Operator Methods of the Parabolic Potential Barrier
• Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes
• Statistical fluctuations for the fission process on its decent from saddle to scission
• Bosonized supersymmetry of anyons and supersymmetric exotic particle on the non-commutative plane
• Supersymmetric Quantum Mechanics of Scattering
• Quantum rolling tachyon

Cited by in corpus (2)

• Quantum inverted harmonic potential
• Ladder Operators in Repulsive Harmonic Oscillator with Application to the Schwinger Effect