Superintegrable systems with spin induced by co-algebra symmetry

Abstract

A method for deriving superintegrable Hamiltonians with a spin-orbital interaction is presented. The method is applied to obtain a new superintegrable system in Euclidean space with the following properties. It describes a rotationally invariant interaction between a particle of spin 1/2 and one of spin 0. Its Hamiltonian commutes with total angular momentum and with additional vector integrals of motion , with components that are third-order differential operators. The integrals of motion form a polynomial algebra under commutation. The system is exactly solvable (in terms of Laguerre polynomials) and the bound state energy levels are degenerate and described by a Balmer type formula. When the spin-orbital potential is switched off the system reduces to a hydrogen atom.