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.
Alberto Parmeggiani (2014). Non-Commutative Harmonic Oscillators and Related Problems. MILAN JOURNAL OF MATHEMATICS, 82(2), 343-387 [10.1007/s00032-014-0220-z].
Non-Commutative Harmonic Oscillators and Related Problems
PARMEGGIANI, ALBERTO
2014
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.