Structure Theory for Extended Kepler–coulomb 3d Quantum Superintegrable Systems

A quantum superintegrable system is an integrable n-dimensional Hamiltonian system with potential: H = ∆n + V that admits 2n − 1 algebraically independent partial differential operators commuting with the Hamiltonian, the maximum number possible. The system is of order ` if the maximum order of the symmetry operators, other than H, is `. Typically, the algebra generated by the symmetry operators and their commutators has been proven to close polynomially. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (2nd order superintegrable) appeared to be an exception as Kalnins et al. (2007) showed that it didn’t close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, Tanoudis and Daskaloyannis (2011) showed it closed polynomially. We consider an infinite class of quantum extended KeplerCoulomb systems that we show to be superintegrable of arbitrarily high order, compute the structure algebras and demonstrate that algebraic closure is the norm, whereas polynomial closure requires extra symmetry. This is a report on joint work with Ernie Kalnins (University of Waikato) and Jonathan Kress (University of new South Wales).