A Spectral Study of the Second-Order Exceptional $X_1$-Jacobi Differential Expression and a Related Non-classical Jacobi Differential Expression

Abstract

The exceptional $X_1$-Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by Gómez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions ${\left\{{\hat{P}}_n^{(\alpha ,\beta )}\right\}}_{n=1}^{\infty }$ called the exceptional $X_1$-Jacobi polynomials. There is no exceptional $X_1$-Jacobi polynomial of degree zero. These polynomials form a complete orthogonal set in the weighted Hilbert space $L^2((-1,1);{\hat{w}}_{\alpha ,\beta }),$ where ${\hat{w}}_{\alpha ,\beta }$ is a positive rational weight function related to the classical Jacobi weight. Among other conditions placed on the parameters $\alpha$ and $\beta ,$ it is required that $\alpha ,\beta >0.$ In this paper, we develop the spectral theory of this expression in $L^2((-1,1);{\hat{w}}_{\alpha ,\beta })$. We also consider the spectral analysis of the `extreme' non-exceptional case, namely when $\alpha =0$. In this case, the polynomial solutions are the non-classical Jacobi polynomials ${\left\{P_n^{(-2,\beta )}\right\}}_{n=2}^{\infty }.$ We study the corresponding Jacobi differential expression in several Hilbert spaces, including their natural $L^2$ setting and a certain Sobolev space $S$ where the full sequence ${\left\{P_n^{(-2,\beta )}\right\}}_{n=0}^{\infty }$ is studied and a careful spectral analysis of the Jacobi expression is carried out.

26 pages

References in corpus (3)

• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Exceptional orthogonal polynomials and the Darboux transformation
• Note on the X_(1)-Jacobi orthogonal polynomials