Non-Commutative Harmonic Oscillators and Related Problems

This paper is mainly meant to be a survey on the state-of-the-art of the understanding we have of a family of systems with polynomial coefficients, called non-commutative harmonic oscillators (NCHOs), that has shown itself to be very rich in structure. The study of this family has required, and is requiring, the study of problems arising from different parts of Mathematics, from spectral theory to the theory of modular forms, just to mention a few of them. On the side of new results, they will be concerned with the creation-annihilation relations for NCHOs and with Fredholm properties of operators belonging to certain global Weyl-Hörmander classes, of which NCHOs are a particular case.