Cylindrical first order superintegrability with complex magnetic fields

TL;DR

This paper explores superintegrable quantum systems with complex magnetic fields in three-dimensional space, identifying new systems and extensions of known ones through complex coupling constants, despite challenges posed by non-Hermitian settings.

Contribution

It introduces the study of superintegrable systems with complex electromagnetic fields, extending known systems with complex couplings and discovering a new multiseparable system.

Findings

Known systems can be extended with complex coupling constants.

A new superintegrable system with complex constants is identified.

Systems are multiseparable despite non-Hermitian challenges.

Abstract

This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space $\mathbb{E}_3$ in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [X. Peng et al., Phys. Rev. Lett. 114 (2015)], they were so far not addressed in the growing literature on superintegrability. Here we venture into this field by searching for additional first order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.