Non-hermitian quantum mechanics in non-commutative space

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.