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

Abstract

We investigate a quantum non-relativistic system describing the interaction of two particles with spin and spin 0, respectively. Assuming that the Hamiltonian is rotationally invariant and parity conserving we identify all such systems which allow additional tensor and pseudo-tensor integrals of motion that are second-order matrix polynomials in the momenta. Previously we found all the scalar, pseudo-scalar, vector 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.

Symmetries of Schrödinger–Pauli equations for charged particles and quasirelativistic Schrödinger equations

“…Indeed, the correct QRSE should include both of them. We remind that the superintegrability aspects of systems with spin-orbit coupling (but without Pauli one) and Pauli coupling (but without spin-orbit one) are relatively well studied, refer to [13][14][15] and [18][19][20]. Thus the next step is to investigate the systems including both these couplings.…”

“…In particular this system includes two matrix equations ( 15) and (16). Collecting in (15) the terms proportional to the unit matrix we obtain:…”

“…It was Winternitz again [13][14][15] who starts and continued the study of superintegrable SEs generalized by the presence of the spin-orbit coupling term. The principally new feature of such equations is that they include matrix potentials with generate essential complications of the related classification procedure.…”

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

“…However, in general, there is no guarantee that integrals of degree 2 would close to form a three-generator quadratic algebra when the systems contain magnetic fields or spins. See for instance, [42,43]. So the determination of symmetry algebras of superintegrable systems rely on explicit differential operator realizations of the integrals of motion and require case-by-case computations.…”

Doubly Exotic Nth-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

“…Superintegrability with magnetic field was first explored in the articles [5,10] and much recently developed in the articles [4,43,44]. The systematic investigation of integrability and superintegrability for systems involving particles with spin was initiated in the reference [68] and subsequently all the rotationally invariant superintegrable systems in E 3 were classified in the articles [9,69,70]. On the other hand, spin dependent superintegrable systems were studied in the works [53,54] for matrix potentials simulating charged or neutral fermions with non-trivial dipole moment in the presence of an electric field.…”