Orthogonal polynomials and separation of variables

Abstract

These are the notes for an introductory course into the theory of quantum. integrability and separability, a rather incomplete status of which dictates a special way of presentation. Therefore, I have chosen to work out a few very basic examples which shall help to exemplify the most important notions and the general methodology of the separation approach to solving quantum integrable systems. All bases considered (and factorized) in these notes appear to be special cases of the celebrated Jack polynomials which were the object of our study in the work [15] joint with Vladimir Mangazeev and Evgeny Sklyanin. These notes are designed to be a good preparation for a more dedicated reading into the subject.