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Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials



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Gómez-Ullate Otaiza, David and Grandati, Yves and Milson, Robert (2014) Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. Journal of physics A: Mathematical and theoretical, 47 (1). ISSN 1751-8113

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Official URL: http://dx.doi.org/10.1088/1751-8113/47/1/015203


We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2l + 3 recurrence relation where l is the length of the partition λ. Explicit expressions for such recurrence relations are given.

Item Type:Article
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© IOP Publishing Ltd.
The research of the first author (DGU) has been supported in part by Spanish MINECO-FEDER grants MTM2009-06973, MTM2012-31714, and the Catalan grant 2009SGR-859. The research of the third author (RM) was supported in part by NSERC grant RGPIN-228057-2009.

Uncontrolled Keywords:Shape-invariant potentials; Quasi-exact solvability; Orthogonal polynomials; Darboux transformations; Laguerre-polynomials; Mechanics; Equation; Formula
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:30746
Deposited On:10 Jun 2015 09:28
Last Modified:10 Dec 2018 15:09

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