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 introduced by Cariñena et al., [3]. 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 (X 2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X 2-OPS. The classification includes all cases known to date plus some new examples of X 2-Laguerre and X 2-Jacobi polynomials.

The electrostatic properties of zeros of exceptional Laguerre and Jacobi polynomials and stable interpolation

“…The exceptional Laguerre and Jacobi polynomials discussed below are reachable from the classical polynomials by 1-step Darboux transformations, but there are much general classes of exceptional orthogonal polynomials. More precisely as it was conjectured in [11] and proved for Hermite subclass in [7], "every X m orthogonal polynomial system for any codimension m can be obtained by applying a sequence of at most m Darboux transformations to a classical orthogonal polynomial system".…”

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

“…After the introduction of the first families of exceptional orthogonal polynomials (EOP) in the context of Sturm-Liouville theory [11,12], the realization of their usefulness in constructing new SI extensions of ES potentials in quantum mechanics [13,14,15], and the rapid developments that followed in this area [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], it soon appeared that only some of the well-known SI potentials led to rational extensions connected with EOP. In this category, one finds the radial oscillator [13,15,16,17,18,22,23,24], the Scarf I (also called trigonometric Pöschl-Teller or Pöschl-Teller I) [13,15,16,17,22,24], and the generalized Pöschl-Teller (also termed hyperbolic Pöschl-Teller or Pöschl-Teller II) [14,16,17].…”

Multi-indexed Jacobi polynomials and Maya diagrams

“…Recently systems of orthogonal polynomials which are out of range of Bochner's theorem are studied actively (see [4,7,5,8] and references therein). A typical example of them is the multi-indexed Jacobi polynomials, which is a generalizations of exceptional Jacobi polynomials.…”

“…is given by φ n (x; g, h) = (sin x) g (cos x) h P (g−1/2,h−1/2) n (η(x)), η(x) = cos(2x), (1.2) where P (α,β) n (η) is the Jacobi polynomial in the variable η defined by Then the eigenstate is square-integrable π/2 0 φ n (x; g, h) 2 dx < +∞ in the case g > −1/2, h > −1/2. To define the multi-indexed Jacobi polynomials ( [9,4]), we introduce three types of seed polynomial solutions indexed by v ∈ Z ≥0 : φ I v (x; g, h) = (sin x) g (cos x) 1−h P (g− They are solutions of the Schrödinger equation (1.1) with the eigenvaluesẼ I v (g, h) = −4(g + v + 1/2)(h − v − 1/2),Ẽ II v (g, h) =Ẽ I −(v+1) (g, h),Ẽ III v (g, h) = E −(v+1) (g, h) respectively, which are not square-integrable in the case g ≥ 3/2, h ≥ 3/2. In this paper we assume that g±h ∈ Z and g, h ∈ Z+1/2 under which the distinct eigenstates and seed solutions are linearly independent.…”