Superintegrable systems with spin induced by coalgebra 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 ${\mathbb{E}}_3$ with the following properties. It describes a rotationally invariant interaction between a particle of spin $\frac{1}{2}$ and one of spin 0. Its Hamiltonian commutes with total angular momentum $\vec{\mathcal{J}}$ and with additional vector integrals of motion $\vec{X}$, $\vec{Y}$ 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.

12 pages

References in corpus (5)

• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Integrable and superintegrable systems with spin
• SUSY approach to Pauli Hamiltonians with an axial symmetry
• (Super)integrability from coalgebra symmetry: formalism and applications
• Quantum superintegrable system for arbitrary spin

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