Aviso: para depositar documentos, por favor, inicia sesión e identifícate con tu cuenta de correo institucional de la UCM con el botón MI CUENTA UCM. No emplees la opción AUTENTICACIÓN CON CONTRASEÑA
 

Two-step Darboux transformations and exceptional Laguerre polynomials

Simple item page
dc.contributor.authorGómez-Ullate Otaiza, David
dc.contributor.authorKamran, Niky
dc.contributor.authorMilson, Robert
dc.date.accessioned2023-06-20T03:53:36Z
dc.date.available2023-06-20T03:53:36Z
dc.date.issued2012-03-01
dc.description© 2011 Elsevier Inc. All rights reserved. The research of DGU was supported in part by MICINN-FEDER grant MTM2009- 06973 and CUR-DIUE grant 2009SGR859. The research of NK was supported in part by NSERC grant RGPIN 105490-2004. The research of RM was supported in part by NSERC grant RGPIN-228057-2004.
dc.description.abstractIt has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these parameters, the classical Laguerre and the type II X(l)-Laguerre polynomials are recovered.
dc.description.departmentDepto. de Física Teórica
dc.description.facultyFac. de Ciencias Físicas
dc.description.refereedTRUE
dc.description.sponsorshipMICINN
dc.description.sponsorshipCUR-DIUE
dc.description.sponsorshipNSERC
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/30775
dc.identifier.doi10.1016/j.jmaa.2011.09.014
dc.identifier.issn0022-247X
dc.identifier.officialurlhttp://dx.doi.org/10.1016/j.jmaa.2011.09.014
dc.identifier.relatedurlhttp://www.sciencedirect.com/
dc.identifier.relatedurlhttp://arxiv.org/abs/1103.5724
dc.identifier.urihttps://hdl.handle.net/20.500.14352/44627
dc.issue.number1
dc.journal.titleJournal of mathematical analysis and applications
dc.language.isoeng
dc.page.final418
dc.page.initial410
dc.publisherElsevier
dc.relation.projectIDMTM2009-06973
dc.relation.projectID2009SGR859
dc.relation.projectIDRGPIN 105490-2004
dc.relation.projectIDRGPIN-228057-2004
dc.rights.accessRightsopen access
dc.subject.cdu51-73
dc.subject.keywordShape-invariant potentials
dc.subject.keywordOrthogonal polynomials
dc.subject.ucmFísica-Modelos matemáticos
dc.subject.ucmFísica matemática
dc.titleTwo-step Darboux transformations and exceptional Laguerre polynomials
dc.typejournal article
dc.volume.number387
dcterms.references[1] V. E. Adler, A modification of Crum’s method, Theor. Math. Phys. 101 (1994) 1381–1386. [2] S. Bochner, Über Strum-Liouvillsche Polynomsysteme, Math. Z. 29 (1929), 730-736. [3] M. M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. 6 (1955) 121. [4] G. Darboux Théorie Générale des Surfaces vol 2, Gauthier-Villars, Paris, 1888. [5] D. Dutta and P. Roy, Conditionally exactly solvable potentials and exceptional orthogonal polynomials, J. Math. Phys. 51 (2010) 042101. [6] A. Erd´elyi et al., Higher Transcendental Functions vol 1, McGraw-Hill, New York, 1953. [7] D. Gómez-Ullate, N. Kamran and R. Milson, Quasi-exact solvability and the direct approach to invariant subspaces. J. Phys. A 38 (2005) 2005–2019. [8] D. Gómez-Ullate, N. Kamran and R. Milson, Quasi-exact solvability in a general polynomial setting, Inverse Problems, 23 (2007) 1915. [9] D. Gómez-Ullate, N. Kamran and R. Milson, The Darboux transformation and algebraic deformations of shape-invariant potentials, J. Phys. A 37 (2004) 1789–1804. [10] D. Gómez-Ullate, N. Kamran and R. Milson, Supersymmetry and algebraic Darboux transformations, J. Phys. A 37 (2004) 10065–10078. [11] D. Gómez-Ullate, N. Kamran, and R. Milson, An extension of Bochner’s problem: exceptional invariant subspaces, J. Approximation Theory 162 (2010) 897–1006. [12] D. Gómez-Ullate, N. Kamran, and R. Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl. 359 (2009) 352–367. [13] D. Gómez-Ullate, N. Kamran, and R. Milson, Exceptional orthogonal polynomials and the Darboux transformation, J. Phys. A 43 (2010) 434016. [14] D. Gómez-Ullate, N. Kamran, and R. Milson, On orthogonal polynomials spanning a nonstandard flag, Contemp. Math. (in press) arXiv:1101.5584. [15] Y. Grandati, Solvable rational extensions of the isotonic oscillator , Ann Phys. 326 (2011) 2074–2090. [16] C-L. Ho, Dirac(-Pauli), FokkerPlanck equations and exceptional Laguerre polynomials, Ann Phys. 326 (2011) 797807. [17] P. Lesky, Die Charakterisierung der klassischen orthogonalen Polynome durch SturmLiouvillesche Differentialgleichungen, Arch. Rat. Mech. Anal. 10 (1962), 341–352. [18] B. Midya and B. Roy, Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schr¨odinger Hamiltonians, Phys. Lett. A 373(45) (2009) 4117–4122. [19] C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A 41 (2008) 392001–392007. [20] C. Quesne, Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics, SIGMA 5 (2009) 084. [21] S. Odake and R. Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679 (2009) 414. [22] S. Odake and R. Sasaki, Another set of infinitely many exceptional (X_ℓ) Laguerre polynomials, Phys. Lett. B684 (2010) 173–176. [23] S. Odake and R. Sasaki, Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials, J. Math. Phys. 51 (2010) 053513. [24] R. Sasaki, S. Tsujimoto, and A. Zhedanov, Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations, J. Phys. A 43 (2010) 315204. [25] G. Szegö, Orthogonal polynomials, Colloquium Publications 23, American Mathematical Society, Providence, 1939. [26] T. Tanaka, N-fold Supersymmetry and quasi-solvability associated with X2-Laguerre polynomials, J. Math. Phys. 51 (2010) 032101
dspace.entity.typePublication

Download

Original bundle

Now showing 1 - 1 of 1
Thumbnail Image
Name:
gomez-ullate09preprint.pdf
Size:
170.34 KB
Format:
Adobe Portable Document Format
downoload Download

Collections

Artículos