Spectral analysis for the exceptional $X_m$-Jacobi equation

Abstract

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 $m,m+1,m+2,...$) in the Lebesgue--Hilbert space with the appropriate weight. In particular, the self-adjoint operator has no other spectrum.

Submitted. 9 pages

References in corpus (1)

• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Cited by in corpus (4)

• Exceptional Jacobi polynomials
• A Bochner type classification theorem for exceptional orthogonal polynomials
• Moment Representations of Exceptional $X_1$ Orthogonal Polynomials
• Determinantal Formulas for Exceptional Orthogonal Polynomials