Quantum cylindrical integrability in magnetic fields

TL;DR

This paper classifies quantum cylindrical integrable systems with magnetic fields, identifying cases with and without quantum corrections, and highlights systems that are not separable, laying groundwork for future superintegrability studies.

Contribution

It provides a classification of quantum cylindrical integrable systems with magnetic fields, including quantum corrections to scalar potentials, extending classical results.

Findings

Identified 4 quantum systems with cylindrical magnetic fields, 2 with no correction, 2 with quantum correction.

Found 2 systems with quadratic integrals that are not separable.

Established a basis for future superintegrability research.

Abstract

We present the classification of quadratically integrable systems of the cylindrical type with magnetic fields in quantum mechanics. Following the direct method used in classical mechanics by [F Fournier et al 2020 J. Phys. A: Math. Theor. 53 085203] to facilitate the comparison, the cases which may a priori differ yield 2 systems without any correction and 2 with it. In all of them, the magnetic field $B$ coincides with the classical one, only the scalar potential $W$ may contain a $\hbar^2$-dependent correction. Two of the systems have both cylindrical integrals quadratic in momenta and are therefore not separable. These results form a basis for a prospective study of superintegrability.