Supersymmetric quantum mechanics on non-commutative space

We construct a supersymmetric quantum mechanics in terms of two real supercharges on non-commutative space in arbitrary dimensions. We obtain the exact eigenspectra of the two- and three-dimensional non-commutative superoscillators. We further show that a reduction in the phase space occurs for a critical surface in the space of parameters. At this critical surface, the energy spectrum of the bosonic sector is infinitely degenerate, while the degeneracy in the spectrum of the fermionic sector gets enhanced by a factor of two for each pair of reduced canonical coordinates. For the two-dimensional non-commutative "inverted superoscillator", we find exact eigenspectra with a well-defined ground state for certain regions in the parameter space, which have no smooth limit to the ordinary commutative space.