The two dimensional harmonic oscillator on a noncommutative space with minimal uncertainties

TL;DR

This paper explores a two-dimensional harmonic oscillator on a noncommutative space, analyzing its eigenvalue spectrum and potential PT-symmetry breaking, revealing new insights into minimal uncertainty relations and spectral properties.

Contribution

It provides a novel representation of the algebra in noncommutative space and investigates the spectral behavior, including the possibility of exceptional points and PT-symmetry breaking.

Findings

Eigenvalue spectrum analyzed with perturbation theory

Potential existence of an exceptional point where spectrum becomes complex

Indications of spontaneous PT-symmetry breaking

Abstract

The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken.