Abstract
A recent investigation of the possibility of having a \(\mathcal{PT}\) -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a \(\mathcal{PT}\) -symmetric deformation of this space. Specifically, a \(\mathcal{PT}\) -symmetric harmonic oscillator together with an iC(x 1+x 2) interaction are discussed in this space, and solutions are obtained. We show that in the \(\mathcal{PT}\) deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x 1+x 2) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not \(\mathcal{PT}\) -symmetric. A complex interacting anisotropic oscillator system also is discussed.
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