Cylindrical first-order superintegrability with complex magnetic fields

Abstract

This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space E3 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 [Peng et al., Phys. Rev. Lett. 114, 010601 (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.

Non-Hermitian superintegrable systems

“…Quantum non-Hermitian Hamiltonians in terms of symmetry generators (or creation and annihilation operators) give rise to the Swanson model [52] and its extensions [53] including supersymmetric ones [54]. Recently, complex magnetic fields have also attracted the attention from the point of view of superintegrability, see for instance [55] and in graphene systems with non-Hermitian Dirac-Weyl Hamiltonians [56]. So far there are several approaches to introduce non-Hermitian deformations, one way considers the complexification of coordinates, field or parameters, while another form modifies the underlying symmetry structure via the generators or the root systems.…”