Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case

TL;DR

This paper constructs new three-dimensional quantum systems with magnetic fields in Cartesian coordinates, featuring pairs of commuting integrals of motion, many of which are unrelated to variable separation methods.

Contribution

It introduces novel integrable quantum systems with magnetic fields in Cartesian coordinates, expanding the class of known integrable models.

Findings

Most systems are new and not related to separation of variables.

Constructed systems admit pairs of commuting second-order integrals.

The work broadens understanding of quantum integrability in magnetic fields.

Abstract

In this paper we construct integrable three-dimensional quantum-mechanical systems with magnetic fields, admitting pairs of commuting second-order integrals of motion. The case of Cartesian coordinates is considered. Most of the systems obtained are new and not related to the separation of variables in the corresponding Schr\"odinger equation.