Superintegrability in the interaction of two particles with spin: First-order pseudo-scalar integrals of motion

Abstract

In recent work, we initiated a research program aimed at the systematic investigation of quantum superintegrable systems describing the interaction of two non-relativistic spin-$1/2$ particles in three-dimensional Euclidean space. In that study, we classified all such superintegrable systems admitting additional first-order scalar integrals of motion. In the present paper, we continue this program by focusing on systems that admit additional pseudo-scalar integrals of motion. Starting from the most general rotationally invariant Hamiltonian for two interacting spin-$1/2$ particles, we construct the most general first-order pseudo-scalar operator in the form of a matrix polynomial in the momenta. Imposing the commutativity of this operator with the Hamiltonian leads to a system of determining equations. By solving these equations, we obtain a complete classification of such superintegrable systems and determine the corresponding pseudo-scalar integrals of motion. The resulting classification provides new families of superintegrable systems with spin-dependent interactions. These systems enrich the class of integrable models relevant to nucleon--nucleon interactions and contribute to the broader program of classifying superintegrable quantum systems with spin. For selected cases, we further construct the associated polynomial symmetry algebras generated by the integrals of motion, providing additional insight into the algebraic structure of the systems.