A note on quantum Bohlin transformation

TL;DR

This paper explores how certain quantum oscillator systems can be reduced via symmetry group actions to simpler systems like hydrogen atoms and magnetic vortex systems, revealing new relationships and potential applications.

Contribution

It introduces a novel reduction method for quantum oscillators using $Z_N$-group actions, linking them to systems with different potentials and spins.

Findings

Reduction of quantum oscillator to hydrogen atom and magnetic vortex systems.

Generalization to $Z_N$-reductions leading to multiple bound systems.

Identification of spin and potential relationships in reduced systems.

Abstract

It is shown, that the reduction of the circular quantum oscillator by the $Z_2$-group action results to the two systems: a two-dimensional hydrogen atom, and a ``charge - charged magnetic vortex" one, with the spin $\frac 12$. Analogously, the $Z_N$-reduction of the two-dimensional system with the central potential $r^{2(N-1)}$ results into $N$ bound ``charge - magnetic vertex" systems with the interaction potential $r^{2(1/N-1)}$ and spins $\sigma=\frac kN$, $k =0,1,..., (N-1)$.