A Conjecture on Exceptional Orthogonal Polynomials
- 535 Downloads
Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm–Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Cariñena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPSs. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.
KeywordsExceptional orthogonal polynomials Sturm–Liouville problems Darboux–Crum transformation Bochner theorem
Mathematics Subject Classification33E99 33C4 34A30 34A05
The research of DGU was supported in part by MICINN-FEDER grant MTM2009-06973 and CUR-DIUE grant 2009SGR859. The research of NK was supported in part by NSERC grant RGPIN 105490-2011. The research of RM was supported in part by NSERC grant RGPIN-228057-2009.
- 10.L.E. Gendenshtein, Derivation of exact spectra of the Schroedinger equation by means of supersymmetry, JETP Lett. 38, 356 (1983). Google Scholar
- 14.D. Gómez-Ullate, N. Kamran, R. Milson, Quasi-exact solvability in a general polynomial setting, Inverse Probl. 23, 2007 (1915–1942). Google Scholar
- 19.D. Gómez-Ullate, N. Kamran, R. Milson, On orthogonal polynomials spanning a non-standard flag, in Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, ed. by P. Acosta-Humánez et al. Contemp. Math., vol. 563 (2012), pp. 51–72. CrossRefGoogle Scholar
- 21.Y. Grandati, Multistep DBT and regular rational extensions of the isotonic oscillator, arXiv:1108.4503.
- 23.F.A. Grünbaum, L. Haine, Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation, in Symmetries and Integrability of Differential Equations. CRM Proc. Lecture Notes, vol. 9 (Am. Math. Soc., Providence, 1996), pp. 143–154. Google Scholar
- 39.G. Szegő, Orthogonal Polynomials, 4th edn. Am. Math. Soc. Colloq. Publ., vol. 23, (Am. Math. Soc., Providence, 1975). Google Scholar