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Two-step rational extensions of the harmonic oscillator: Exceptional orthogonal polynomials and ladder operators

par Marquette, Ian;Quesne, Christiane
Référence Journal of Physics A: Mathematical and Theoretical, 46, 15, 155201
Publication Publié, 2013-04

Article révisé par les pairs

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Résumé :

The type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one (with m1 even and m 2 odd such that m2 > m1) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order m2 - m1 + 1, which may alternatively be interpreted in terms of a special type of (m2 - m1 + 2)th-order shape invariance property. © 2013 IOP Publishing Ltd.