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GómezUllate 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 17518113

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Official URL: http://dx.doi.org/10.1088/17518113/47/1/015203
URL  URL Type 

http://iopscience.iop.org  Publisher 
http://arxiv.org/abs/1306.5143  Organisation 
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 statedeleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a DarbouxCrum 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 

Additional Information:  © IOP Publishing Ltd. 
Uncontrolled Keywords:  Shapeinvariant potentials; Quasiexact solvability; Orthogonal polynomials; Darboux transformations; Laguerrepolynomials; Mechanics; Equation; Formula 
Subjects:  Sciences > Physics > PhysicsMathematical 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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