Journal
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
Volume 47, Issue 1, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/1751-8113/47/1/015203
Keywords
trivial monodromy; rational extensions; exceptional Hermite polynomials; harmonic oscillator; Darboux transformations
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Funding
- Spanish MINECO-FEDER grants [MTM2009-06973, MTM2012-31714]
- Catalan grant [2009SGR-859]
- NSERC [RGPIN-228057-2009]
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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 lambda 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.. Explicit expressions for such recurrence relations are given.
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