A spectral study of the second-order exceptional X-1-Jacobi differential expression and a related non-classical Jacobi differential expression

作者：Liaw Constanze; Littlejohn Lance L^*; Stewart Jessica; Wicks Quinn

来源：Journal of Mathematical Analysis and Applications, 2015, 422(1): 212-239.

DOI：10.1016/j.jmaa.2014.08.016

• Orthogonal polynomials
• Spectral theory
• Glazman-Krein-Naimark theory
• X-1-Jacobi polynomials
• Left-definite theory

The exceptional X-1-Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by Gomez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions {(P) over cap ((alpha,beta))(n)}(n=1)(infinity), 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); (omega) over cap (alpha,beta)), where (omega) over cap (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); (omega) over cap (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 {P-n((-2,beta))}(n=2)(infinity). 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 {P-n((-2,beta))}(n=0)(infinity) is studied and a careful spectral analysis of the Jacobi expression is carried out.

• 出版日期2015-2-1