Group Analysis of Differential Equations and Integrable Systems − 2018
(Czech Technical University in Prague, Czech Republic)
Superintegrable 3D systems in a magnetic field and separation of variables
This is a joint work with L. Šnobl and P. Winternitz. We study the problem of the classification of three dimensional superintegrable systems in a magnetic field in the case they admit integrals polynomial in the momenta, two of them in involution and at most of second order (besides the Hamiltonian). Both the classical and quantum case are considered, as it is known that already in two dimensions, when a magnetic field is present, classical and quantum integrable systems do not necessarily coincide. We start by considering second order integrable systems that would separate in subgroup-type coordinates in the limit when the magnetic field vanishes. We look for additional integrals which make these systems minimally or maximally superintegrable. We show that the leading structure terms of the second order integrals responsible for integrability should be considered in a more general form than for the case without magnetic field.