Superintegrable systems of two interacting spin-$\frac12$ particles with first-order vector integrals of motion

Abstract

We study quantum superintegrability for two interacting non-relativistic spin-$\frac12$ particles in three-dimensional Euclidean space. The Hamiltonian contains a central potential together with spin-orbit, spin-spin, tensor, and quadratic spin-orbit interaction terms, all depending only on the relative distance. We restrict the classification to the $V_4=0$ case, so that the spin-momentum interaction term is not included. We determine all such systems admitting non-trivial first-order vector integrals of motion. For this purpose, we construct the most general Hermitian first-order vector operator built from the relative position, momentum, orbital angular momentum, and the two spin vectors. The commutativity condition with the Hamiltonian leads to an overdetermined system of radial determining equations, whose solution gives the complete list of admissible potentials and the corresponding vector integrals within this class. The results extend the previous classifications of scalar and pseudo-scalar first-order integrals for two particles with spin. We also discuss selected symmetry algebras generated by the vector integrals and show, in one representative case, how a scalar reduction leads to exact Coulomb- and oscillator-type solutions.