VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Confluent Darboux transformations and new exceptional orthogonal polynomials

María Ángeles Garcia-Ferrero (Heidelberg University)

Wednesday 26th May 13:30-14:30 Zoom seminar hosted by ICMS

Abstract

Commutation methods allow to modify the discrete spectrum of Schrödinger operators in one dimension. In this talk, we will apply these methods to related Sturm-Liouville problems and we will interpret the double commutation method as a pair of confluent Darboux transformations at the same energy level, focusing on the case of isospectral transformations. Our aim is to apply this construction to exceptional orthogonal polynomial systems, which are complete polynomial systems arising as eigenfunctions of Sturm-Liouville problems and generalizing the classical families of Hermite, Laguerre and Jacobi.  We show that confluent Darboux transformations lead to new classes of exceptional orthogonal polynomials, which fall outside of the current classification scheme.  One of the most remarkable properties of these new exceptional polynomials is the fact that they contain an arbitrary number of continuous parameters. This is a joint work with David Gómez-Ullate and Robert Milson.

 

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