Publication: Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
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2014-01-10
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IOP Publishing Ltd
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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.
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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.
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