%0 Journal Article
%A Gómez-Ullate Otaiza, David
%A Grandati, Yves
%A Milson, Robert
%T Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
%D 2014
%@ 1751-8113
%U https://hdl.handle.net/20.500.14352/34707
%X 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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