A conjecture on Exceptional Orthogonal Polynomials

Abstract

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. 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-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.

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• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Two-step Darboux transformations and exceptional Laguerre polynomials
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• Parametric symmetries in exactly solvable real and P T symmetric complex potentials
• Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
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• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
• Exceptional Legendre Polynomials and Confluent Darboux Transformations
• Shape Invariant Rational Extensions And Potentials Related to Exceptional Polynomials
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• Symmetric Tops Subject to Combined Electric Fields: Conditional Quasi-Solvability via the Quantum Hamilton-Jacobi Theory
• A Spectral Study of the Second-Order Exceptional $X_1$-Jacobi Differential Expression and a Related Non-classical Jacobi Differential Expression
• A unified treatment of polynomial sectors and constraint polynomials of the Rabi models
• Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials
• Moment Representations of Exceptional $X_1$ Orthogonal Polynomials
• General solution of the exceptional Hermite differential equation and its minimal surface representation