Advertisement

Foundations of Computational Mathematics

, Volume 13, Issue 4, pp 615–666 | Cite as

A Conjecture on Exceptional Orthogonal Polynomials

  • David Gómez-Ullate
  • Niky KamranEmail author
  • Robert Milson
Article

Abstract

Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm–Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Cariñena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPSs. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.

Keywords

Exceptional orthogonal polynomials Sturm–Liouville problems Darboux–Crum transformation Bochner theorem 

Mathematics Subject Classification

33E99 33C4 34A30 34A05 

Notes

Acknowledgements

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 105490-2011. The research of RM was supported in part by NSERC grant RGPIN-228057-2009.

References

  1. 1.
    V.E. Adler, A modification of Crum’s method, Theor. Math. Phys. 101, 1381–1386 (1994). zbMATHCrossRefGoogle Scholar
  2. 2.
    S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29, 730–736 (1929). MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    J.F. Cariñena, A.M. Perelomov, M.F. Rañada, M. Santander, A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator, J. Phys. A 41, 085301 (2008). MathSciNetCrossRefGoogle Scholar
  4. 4.
    M.M. Crum, Associated Sturm–Liouville systems, Q. J. Math. Oxf. Ser. (2) 6, 121 (1955). MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S.Y. Dubov, V.M. Eleonskii, N.E. Kulagin, Equidistant spectra of anharmonic oscillators, Sov. Phys. JETP 75, 446–451 (1992). Chaos 4, 47–53 (1994). MathSciNetGoogle Scholar
  6. 6.
    D. Dutta, P. Roy, Conditionally exactly solvable potentials and exceptional orthogonal polynomials, J. Math. Phys. 51, 042101 (2010). MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Dutta, P. Roy, Information entropy of conditionally exactly solvable potentials, J. Math. Phys. 52, 032104 (2011). MathSciNetCrossRefGoogle Scholar
  8. 8.
    W.N. Everitt, L.L. Littlejohn, R. Wellman, The Sobolev orthogonality and spectral analysis of the Laguerre polynomials \({L^{-k}_{n}}\) for positive integers k, J. Comput. Appl. Math. 171, 199–234 (2004). MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    J.M. Fellows, R.A. Smith, Factorization solution of a family of quantum nonlinear oscillators, J. Phys. A 42, 335303 (2009). MathSciNetCrossRefGoogle Scholar
  10. 10.
    L.E. Gendenshtein, Derivation of exact spectra of the Schroedinger equation by means of supersymmetry, JETP Lett. 38, 356 (1983). Google Scholar
  11. 11.
    D. Gómez-Ullate, N. Kamran, R. Milson, Quasi-exact solvability and the direct approach to invariant subspaces, J. Phys. A 38(9), 2005–2019 (2005). MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    D. Gómez-Ullate, N. Kamran, R. Milson, The Darboux transformation and algebraic deformations of shape-invariant potentials, J. Phys. A 37, 1789–1804 (2004). MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    D. Gómez-Ullate, N. Kamran, R. Milson, Supersymmetry and algebraic Darboux transformations, J. Phys. A 37, 10065–10078 (2004). MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    D. Gómez-Ullate, N. Kamran, R. Milson, Quasi-exact solvability in a general polynomial setting, Inverse Probl. 23, 2007 (1915–1942). Google Scholar
  15. 15.
    D. Gómez-Ullate, N. Kamran, R. Milson, An extension of Bochner’s problem: exceptional invariant subspaces, J. Approx. Theory 162, 987–1006 (2010). MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    D. Gómez-Ullate, N. Kamran, R. Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl. 359, 352–367 (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    D. Gómez-Ullate, N. Kamran, R. Milson, Exceptional orthogonal polynomials and the Darboux transformation, J. Phys. A 43, 434016 (2010). MathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Gómez-Ullate, N. Kamran, R. Milson, Two-step Darboux transformations and exceptional Laguerre polynomials, J. Math. Anal. Appl. 387, 410–418 (2012). MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    D. Gómez-Ullate, N. Kamran, R. Milson, On orthogonal polynomials spanning a non-standard flag, in Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, ed. by P. Acosta-Humánez et al. Contemp. Math., vol. 563 (2012), pp. 51–72. CrossRefGoogle Scholar
  20. 20.
    A. González-López, N. Kamran, P.J. Olver, Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators, Commun. Math. Phys. 153(1), 117–146 (1993). zbMATHCrossRefGoogle Scholar
  21. 21.
    Y. Grandati, Multistep DBT and regular rational extensions of the isotonic oscillator, arXiv:1108.4503.
  22. 22.
    Y. Grandati, Solvable rational extensions of the isotonic oscillator, Ann. Phys. 326, 2074–2090 (2011). MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    F.A. Grünbaum, L. Haine, Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation, in Symmetries and Integrability of Differential Equations. CRM Proc. Lecture Notes, vol. 9 (Am. Math. Soc., Providence, 1996), pp. 143–154. Google Scholar
  24. 24.
    C.-L. Ho, Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials, Ann. Phys. 326, 797–807 (2011). zbMATHCrossRefGoogle Scholar
  25. 25.
    C.-L. Ho, S. Odake, R. Sasaki, Properties of the exceptional (X) Laguerre and Jacobi polynomials, SIGMA 7, 107 (2011). MathSciNetGoogle Scholar
  26. 26.
    N. Kamran, P.J. Olver, Lie algebras of differential operators and Lie-algebraic potentials, J. Math. Anal. Appl. 145(2), 342–356 (1990). MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M.G. Krein, Dokl. Akad. Nauk SSSR 113, 970–973 (1957). MathSciNetzbMATHGoogle Scholar
  28. 28.
    P. Lesky, Die Charakterisierung der klassischen orthogonalen Polynome durch Sturm-Liouvillesche Differentialgleichungen, Arch. Ration. Mech. Anal. 10, 341–352 (1962). MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    B. Midya, B. Roy, Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schrödinger Hamiltonians, Phys. Lett. A 373(45), 4117–4122 (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A 41, 392001 (2008). MathSciNetCrossRefGoogle Scholar
  31. 31.
    C. Quesne, Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics, SIGMA 5, 084 (2009). MathSciNetGoogle Scholar
  32. 32.
    C. Quesne, Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials, Mod. Phys. Lett. A 26, 1843–1852 (2011). MathSciNetCrossRefGoogle Scholar
  33. 33.
    A. Ronveaux, Sur l’équation différentielle du second ordre satisfaite par une classe de polynômes orthogonaux semi-classiques, C. R. Acad. Sci. Paris Sér. I Math. 305(5), 163–166 (1987). MathSciNetzbMATHGoogle Scholar
  34. 34.
    A. Ronveaux, F. Marcellán, Differential equation for classical-type orthogonal polynomials, Can. Math. Bull. 32(4), 404–411 (1989). zbMATHCrossRefGoogle Scholar
  35. 35.
    S. Odake, R. Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679, 414–417 (2009). MathSciNetCrossRefGoogle Scholar
  36. 36.
    S. Odake, R. Sasaki, Another set of infinitely many exceptional (Xm) Laguerre polynomials, Phys. Lett. B 684, 173–176 (2010). MathSciNetCrossRefGoogle Scholar
  37. 37.
    S. Odake, R. Sasaki, Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials, Phys. Lett. B 702, 164–170 (2011). MathSciNetCrossRefGoogle Scholar
  38. 38.
    R. Sasaki, S. Tsujimoto, A. Zhedanov, Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux–Crum transformations, J. Phys. A 43, 315204 (2010). MathSciNetCrossRefGoogle Scholar
  39. 39.
    G. Szegő, Orthogonal Polynomials, 4th edn. Am. Math. Soc. Colloq. Publ., vol. 23, (Am. Math. Soc., Providence, 1975). Google Scholar
  40. 40.
    A. Turbiner, Quasi-exactly-solvable problems and sl(2) algebra, Commun. Math. Phys. 118(3), 467–474 (1988). MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    V.B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions, USSR Comput. Math. Math. Phys. 9, 25–36 (1969). CrossRefGoogle Scholar

Copyright information

© SFoCM 2012

Authors and Affiliations

  • David Gómez-Ullate
    • 1
  • Niky Kamran
    • 2
    Email author
  • Robert Milson
    • 3
  1. 1.Departamento de Física Teórica IIUniversidad Complutense de MadridMadridSpain
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

Personalised recommendations