The noncommutative harmonic oscillator in more than one dimensions

TL;DR

This paper explores the properties of the noncommutative harmonic oscillator in multiple dimensions, analyzing its eigenvalue problems, symmetries, and specific examples like the Landau problem and a 3D oscillator.

Contribution

It provides a detailed decomposition of the noncommutative harmonic oscillator problem into separate equations for each dimension and investigates angular momentum and specific asymmetric cases.

Findings

Eigenvalues depend on energy differences between dimensions.

Angular momentum eigenvalues are equivalent to the usual problem.

Solutions for symmetric and asymmetric noncommutative oscillators are derived.

Abstract

The noncommutative harmonic oscillator in arbitrary dimension is examined. It is shown that the $\star$-genvalue problem can be decomposed into separate harmonic oscillator equations for each dimension. The noncommutative plane is investigated in greater detail. The constraints for rotationally symmetric solutions and the corresponding two-dimensional harmonic oscillator are solved. The angular momentum operator is derived and its $\star$-genvalue problem is shown to be equivalent to the usual eigenvalue problem. The $\star$-genvalues for the angular momentum are found to depend on the energy difference of the oscillations in each dimension. Furthermore two examples of assymetric noncommutative harmonic oscillator are analysed. The first is the noncommutative two-dimensional Landau problem and the second is the three-dimensional harmonic oscillator with symmetrically noncommuting coordinates and momenta.