Spectral analysis of the exceptional Laguerre and Jacobi equations.

It was believed that Bochner's characterization of all sequences of polynomials {Ƥ_n}∞_(n=0), with deg Ƥ_n=n≥0, that are eigenfunctions of a second-order differential equation and are orthogonal with respect to a positive Borel measure on the real line having finite moments of all orders, was the only classification result of its kind. This result has been generalized, most notably in 2009 by Gómez-Ullate, Kamran, and Milson who characterized all sequences of polynomials {Ƥ_n}∞_(n=1), with deg Ƥ_n=n≥1 which have the remaining properties as those polynomial systems in Bochner's result. Up to a complex linear change of variable, the only such sequences are the exceptional X₁-Laguerre and the X₁-Jacobi polynomials. Additionally, their result was later extended to include exceptional X_m polynomial sequences; that is sequences which omit m polynomials from the standard sequence {Ƥ_n}∞_(n=0), but still satisfy the remaining properties as the polynomial systems from Bochner's result. In fact, there are two existing families of generalized X_m-Laguerre polynomials, Type I and Type II, and we show the existence of a Type III family. The X₁ and generalized X_m families are excellent examples on which to apply the classical Glazman, Krein, Naimark theory as it pertains to the study of spectral analysis. The full spectral analysis for each of these families of exceptional polynomials as well as the analysis for extreme parameter choices is given in this dissertation.Ph.D