A Spectral Study of the Second-Order Exceptional $X_1$-Jacobi Differential Expression and a Related Non-classical Jacobi Differential Expression (1404.1882v1)

Abstract: The exceptional $X_{1}$-Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by G\'{o}mez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions $\left{\widehat{P} {n}^{(\alpha,\beta)}\right}{n=1}^{\infty}$ 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 $L^{2}((-1,1);\widehat{w}_{\alpha,\beta}),$ where $\widehat{w}{\alpha,\beta}$ 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 $L^{2}((-1,1);\widehat{w}{\alpha,\beta})$. 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.