Some integrable models in quantized spaces

TL;DR

This paper explores integrable models like Coulomb and harmonic oscillator in a quantized space governed by the B2 algebra, revealing spectrum similarities with constant curvature spaces but with quantum number limitations.

Contribution

It demonstrates that Coulomb and harmonic oscillator spectra in B2 quantized space match those in constant curvature spaces, with specific quantum number restrictions.

Findings

Coulomb spectrum matches Schrödinger's in curved space

Harmonic oscillator spectrum is similarly derived

Quantum number limits are identified in the quantized space

Abstract

It is shown that in a quantized space determined by the $B_2\quad (O(5)=Sp(4))$ algebra with three dimensional parameters of the length $L^2$, momentum $(Mc)^2$, and action $S$, the spectrum of the Coulomb problem with conserving Runge-Lenz vector coincides with the spectrum found by Schr\"odinger for the space of constant curvature but with the values of the principal quantum number limited from the side of higher values. The same problem is solved for the spectrum of a harmonic oscillator.