# Coulomb- quantum oscillator correspondence in two dimension, pure gauge field and half-quantized vortex

## TL;DR

This paper explores the partial correspondence between 2D Coulomb and oscillator quantum problems, highlighting gauge transformations, vortex structures, and implications for particle physics.

## Contribution

It provides a comprehensive analysis of the Coulomb-oscillator correspondence in 2D, including wavefunction transformations and vortex phenomena, with implications for QCD and hadron spectra.

## Key findings

- Partial wavefunction correspondence established
- Emergence of half-quantized vortex with spin-half
- Discussion on vortex structure and charge origin

## Abstract

Isotropic oscillator and Coulomb problems are known to have interesting correspondence. We focus on 2D quantum problems and present complete treatment on the correspondence including the Schroedinger equation, eigenfunctions and eigenvalues, and the integrals of motion. We find only partial equivalence. The wavefunction correspondence is examined introducing local gauge transformation and the emergence of half-quantized vortex with the associated spin-half is established. Vortex structure of the electron proposed by us and the origin of charge are discussed in the present context. Outlook on the implications for QCD and hadron spectrum is outlined.