Spectral analysis for the exceptional Xm-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 $m, m+1, m+2, \dots$) in the Lebesgue-Hilbert space with the appropriate weight. In particular, the self-adjoint operator has no other spectrum.

Keywords

• Exceptional orthogonal polynomial
• spectral theory
• self-adjoint operator
• Darboux transformation