Electron. J. Diff. Equ., Vol. 2015 (2015), No. 194, pp. 1-10.

Spectral analysis for the exceptional Xm-Jacobi equation

Constanze Liaw, Lance Littlejohn, Jessica Stewart Kelly

We provide the mathematical foundation for the $X_m$-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional $X_m$-Jacobi orthogonal polynomials as eigenfunctions. This proves that those polynomials are indeed eigenfunctions of the self-adjoint operator (rather than just formal eigenfunctions). Further, we prove the completeness of the exceptional $X_m$-Jacobi orthogonal polynomials (of degrees $m, m+1, m+2, \dots$) in the Lebesgue-Hilbert space with the appropriate weight. In particular, the self-adjoint operator has no other spectrum.

Submitted February 6, 2015. Published July 27, 2015.
Math Subject Classifications: 33C45, 34B24, 33C47, 34L05.
Key Words: Exceptional orthogonal polynomial; spectral theory; self-adjoint operator; Darboux transformation.

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Constanze Liaw
Department of Mathematics and CASPER
Baylor University, One Bear Place #97328
Waco, TX 76798-7328, USA
email: Constanze_Liaw@baylor.edu
Lance Littlejohn
Department of Mathematics, Baylor University
One Bear Place #97328
Waco, TX 76798-7328, USA
email: Lance_Littlejohn@baylor.edu
Jessica Stewart Kelly
Department of Mathematics
Christopher Newport University
1 Avenue of the Arts
Newport News, VA 23606, USA
email: Jessica.Stewart@cnu.edu

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