Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion

TL;DR

This paper classifies superintegrable quantum systems with spin that admit second-order tensor and pseudo-tensor integrals of motion, completing the understanding of possible symmetries in such systems.

Contribution

It provides a complete classification of superintegrable systems with spin that have second-order tensor and pseudo-tensor integrals, extending previous scalar and vector results.

Findings

All systems with scalar and vector integrals identified.

Existence of nontrivial pseudo-tensor integrals confirmed.

Complete list of superintegrable systems with second-order integrals provided.

Abstract

We investigate a quantum non-relativistic system describing the interaction of two particles with spin 1/2 and spin 0, respectively. Assuming that the Hamiltonian is rotationally invariant and parity conserving we identify all such systems which allow additional (pseudo)tensor integrals of motion that are second order matrix polynomials in the momenta. Previously we found all the (pseudo)scalar and (axial)vector integrals of motion. No non-obvious tensor integrals exist. However, nontrivial pseudo-tensor integrals do exist. Together with our earlier results we give a complete list of such superintegrable Hamiltonian systems allowing second-order integrals of motion.