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Spectral analysis for the exceptional $X_m$-Jacobi equation

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.