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Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

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dc.contributor.authorGómez-Ullate Otaiza, David
dc.contributor.authorGrandati, Yves
dc.contributor.authorMilson, Robert
dc.date.accessioned2023-06-19T14:54:29Z
dc.date.available2023-06-19T14:54:29Z
dc.date.issued2014-01-10
dc.description© 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.
dc.description.abstractWe 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.
dc.description.departmentDepto. de Física Teórica
dc.description.facultyFac. de Ciencias Físicas
dc.description.refereedTRUE
dc.description.sponsorshipSpanish MINECO-FEDER
dc.description.sponsorshipCatalan
dc.description.sponsorshipNSER
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/30746
dc.identifier.doi10.1088/1751-8113/47/1/015203
dc.identifier.issn1751-8113
dc.identifier.officialurlhttp://dx.doi.org/10.1088/1751-8113/47/1/015203
dc.identifier.relatedurlhttp://iopscience.iop.org
dc.identifier.relatedurlhttp://arxiv.org/abs/1306.5143
dc.identifier.urihttps://hdl.handle.net/20.500.14352/34707
dc.issue.number1
dc.journal.titleJournal of physics A: Mathematical and theoretical
dc.language.isoeng
dc.publisherIOP Publishing Ltd
dc.relation.projectIDMTM2009-06973
dc.relation.projectIDMTM2012-31714
dc.relation.projectID2009SGR-859
dc.relation.projectIDRGPIN-228057-2009
dc.rights.accessRightsopen access
dc.subject.cdu51-73
dc.subject.keywordShape-invariant potentials
dc.subject.keywordQuasi-exact solvability
dc.subject.keywordOrthogonal polynomials
dc.subject.keywordDarboux transformations
dc.subject.keywordLaguerre-polynomials
dc.subject.keywordMechanics
dc.subject.keywordEquation
dc.subject.keywordFormula
dc.subject.ucmFísica-Modelos matemáticos
dc.subject.ucmFísica matemática
dc.titleRational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
dc.typejournal article
dc.volume.number47
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