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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

## Abstract

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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Additional Information: | © IOP Publishing Ltd. |

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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