The Role of Commuting Operators in Quantum Superintegrable Systems

We discuss the role of commuting operators for quantum superintegrable systems, showing how they are used to build eigenfunctions. These ideas are illustrated in the context of resonant harmonic oscillators, the Krall-Sheffer operators, with polynomial eigenfunctions, and the Calogero-Moser system with additional harmonic potential. The construction is purely algebraic, avoiding the use of separation of variables and differential equation theory.