Non-Hermitian quantum mechanics in non-commutative space

TL;DR

This paper explores non-Hermitian quantum systems within noncommutative and cPTe-symmetric spaces, revealing conditions for real eigenvalues and analyzing specific oscillator models with complex interactions.

Contribution

It introduces the study of non-Hermitian quantum mechanics in noncommutative spaces, including cPTe-symmetric deformations and their spectral properties, which is a novel extension.

Findings

cPTe-deformed noncommutative space can yield real or complex eigenvalues depending on parameters

In standard noncommutative space, certain complex interactions produce real eigenvalues without cPTe-symmetry

Solutions for cPTe-symmetric harmonic oscillators are obtained and analyzed

Abstract

We study non Hermitian quantum systems in noncommutative space as well as a \cal{PT}-symmetric deformation of this space. Specifically, a \mathcal{PT}-symmetric harmonic oscillator together with iC(x_1+x_2) interaction is discussed in this space and solutions are obtained. It is shown that in the \cal{PT} deformed noncommutative space the Hamiltonian may or may not possess real eigenvalues depending on the choice of the noncommutative parameters. However, it is shown that in standard noncommutative 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 has also been discussed.