Classification of exceptional Jacobi polynomials

TL;DR

This paper presents a comprehensive classification of exceptional Jacobi polynomials and operators, introducing spectral diagrams and explicit construction formulas, and explores the full range of rational Darboux transformations involved.

Contribution

It provides a complete classification scheme for exceptional Jacobi operators using spectral diagrams and explicit construction formulas, including degenerate cases with multiple parameters.

Findings

Six degeneracy classes based on parameter integrality

Spectral diagrams encode eigenfunction asymptotics

Explicit formulas for constructing exceptional operators

Abstract

We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi operators are in one-to-one correspondence with spectral diagrams, a combinatorial object that describes the number and asymptotic behaviour at the endpoints of $(-1,1)$ of all quasi-rational eigenfunctions of the operator. With a convenient indexing scheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptional operators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class $\alpha,\beta\in\mathbb N_0$ there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classification result is achieved by a careful description of all possible rational Darboux transformations that can be performed on exceptional Jacobi operators.