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Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

  • David Gómez-UllateEmail author
  • Robert Milson
Conference paper
  • 47 Downloads
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

Abstract

These are the lecture notes for a course on exceptional polynomials taught at the AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications that took place in Douala (Cameroon) from October 5–12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past 10 years. In addition, some new results are presented on the construction of rational solutions to Painlevé equation PIV and its higher order generalizations that belong to the \(A_{2n}^{(1)}\)-Painlevé hierarchy. The construction is based on dressing chains of Schrödinger operators with potentials that are rational extensions of the harmonic oscillator. Some of the material presented here (Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crum transformations, etc.) are classical and can be found in many textbooks, while some results (genus, interlacing and cyclic Maya diagrams) are new and presented for the first time in this set of lecture notes.

Keywords

Sturm-Liouville problems Classical polynomials Darboux transformations Exceptional polynomials Painlevé equations Rational solutions Darboux dressing chains Maya diagrams Wronskian determinants 

Mathematics Subject Classification (2000)

Primary 33C45; Secondary 34M55 

Notes

Acknowledgements

The research of DGU has been supported in part by Spanish MINECO-FEDER Grants MTM2015-65888-C4-3 and MTM2015-72907-EXP, and by the ICMAT-Severo Ochoa project SEV-2015-0554. The research of RM was supported in part by NSERC grant RGPIN-228057-2009. DGU would like to thank the Volkswagen Stiftung and the African Institute of Mathematical Sciences for their hospitality during the Workshop on Introduction to Orthogonal Polynomials and Applications, Duala (Cameroon), where these lectures were first taught.

References

  1. 1.
    V. È. Adler, A modification of Crum’s method. Theor. Math. Phys. 101(3), 1381–1386 (1994)MathSciNetCrossRefGoogle Scholar
  2. 2.
    V. È. Adler, Nonlinear chains and Painlevé equations. Phys. D 73(4), 335–351 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    G.E. Andrews, The Theory of Partitions (Cambridge University Press, Cambridge, 1998). MR 1634067Google Scholar
  4. 4.
    G.E. Andrews, K. Eriksson, Integer Partitions (Cambridge University Press, Cambridge, 2004). MR 2122332Google Scholar
  5. 5.
    D. Bermúdez, Complex SUSY transformations and the Painlevé IV equation. SIGMA 8, 069 (2012)zbMATHGoogle Scholar
  6. 6.
    D. Bermúdez, D.J. Fernández, Complex solutions to the Painlevé IV equation through supersymmetric quantum mechanics, in AIP Conference Proceedings, vol. 1420 (AIP, College Park, 2012), pp. 47–51Google Scholar
  7. 7.
    N. Bonneux, A.B.J. Kuijlaars, Exceptional Laguerre polynomials. Stud. Appl. Math. (2018).  https://doi.org/10.1111/sapm.12204
  8. 8.
    P.A. Clarkson, Painlevé equations – nonlinear special functions. J. Comput. Appl. Math. 153(1–2), 127–140 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    P.A. Clarkson, The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44(11), 5350–5374 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    P.A. Clarkson, D. Gómez-Ullate, Y. Grandati, R. Milson, Rational solutions of higher order Painlevé systems I (2018). Preprint. arXiv: 1811.09274Google Scholar
  11. 11.
    S.Yu. Dubov, V.M. Eleonskii, N.E. Kulagin, Equidistant spectra of anharmonic oscillators. Chaos 4(1), 47–53 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A.J. Durán, Exceptional Meixner and Laguerre orthogonal polynomials. J. Approx. Theory 184, 176–208 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A.J. Durán, Exceptional Charlier and Hermite orthogonal polynomials. J. Approx. Theory 182, 29–58 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    A.J. Durán, Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials. Integral Transforms Spec. Funct. 26(5), 357–376 (2015)CrossRefGoogle Scholar
  15. 15.
    A.J. Durán, Exceptional Hahn and Jacobi orthogonal polynomials. J. Approx. Theory 214, 9–48 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A.J. Durán, M. Pérez, Admissibility condition for exceptional Laguerre polynomials. J. Math. Anal. Appl. 424(2), 1042–1053 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Filipuk, P.A. Clarkson, The symmetric fourth Painlevé hierarchy and associated special polynomials. Stud. Appl. Math. 121(2), 157–188 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    P.J. Forrester, N.S. Witte, Application of the τ-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Commun. Math. Phys. 219(2), 357–398 (2001). MR 1833807Google Scholar
  19. 19.
    M. García-Ferrero, D. Gómez-Ullate, Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger’s equation. Lett. Math. Phys. 105(4), 551–573 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. García-Ferrero, D. Gómez-Ullate, R. Milson, A Bochner type characterization theorem for exceptional orthogonal polynomials. J. Math. Anal. Appl. 472, 584–626 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Gómez-Ullate, N. Kamran, R. Milson, Supersymmetry and algebraic Darboux transformations. J. Phys. A 37(43), 10065 (2004)Google Scholar
  22. 22.
    D. Gómez-Ullate, N. Kamran, R. Milson, The Darboux transformation and algebraic deformations of shape-invariant potentials. J. Phys. A 37(5), 1789 (2004)Google Scholar
  23. 23.
    D. Gómez-Ullate, N. Kamran, R. Milson, An extended class of orthogonal polynomials defined by a Sturm–Liouville problem. J. Math. Anal. Appl. 359(1), 352–367 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    D. Gómez-Ullate, N. Kamran, R. Milson, An extension of Bochner’s problem: exceptional invariant subspaces. J. Approx. Theory 162(5), 987–1006 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    D. Gómez-Ullate, N. Kamran, R. Milson, Two-step Darboux transformations and exceptional Laguerre polynomials. J. Math. Anal. Appl. 387(1), 410–418 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    D. Gómez-Ullate, Y. Grandati, R. Milson, Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47(1), 015203 (2013)Google Scholar
  27. 27.
    D. Gómez-Ullate, N. Kamran, R. Milson, A conjecture on exceptional orthogonal polynomials. Found. Comput. Math. 13(4), 615–666 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    D. Gómez-Ullate, Y. Grandati, R. Milson, Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials. J. Phys. A 51(34), 345201 (2018)Google Scholar
  29. 29.
    D. Gómez-Ullate, Y. Grandati, R. Milson, Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials. Stud. Appl. Math. 141(4), 596–625 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    D. Gómez-Ullate, Y. Grandati, S. Lombardo, R. Milson, Rational solutions of dressing chains and higher order Painleve equations (2018). Preprint. arXiv:1811.10186Google Scholar
  31. 31.
    Y. Grandati, Solvable rational extensions of the isotonic oscillator. Ann. Phys. 326(8), 2074–2090 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Y. Grandati, Multistep DBT and regular rational extensions of the isotonic oscillator. Ann. Phys. 327(10), 2411–2431 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    V.I. Gromak, I. Laine, S. Shimomura, Painlevé Differential Equations in the Complex Plane, vol. 28 (Walter de Gruyter, Berlin, 2008)zbMATHGoogle Scholar
  34. 34.
    K. Kajiwara, Y. Ohta, Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37(9), 4693–4704 (1996)MathSciNetCrossRefGoogle Scholar
  35. 35.
    K. Kajiwara, Y. Ohta, Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31(10), 2431 (1998)Google Scholar
  36. 36.
    M.G. Krein, On a continual analogue of a Christoffel formula from the theory of orthogonal polynomials. Dokl. Akad. Nauk SSSR (N.S.) 113, 970–973 (1957). MR 0091396Google Scholar
  37. 37.
    A.B.J. Kuijlaars, R. Milson, Zeros of exceptional Hermite polynomials. J. Approx. Theory 200, 28–39 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    I. Marquette, C. Quesne, New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54(10), 12 pp., 102102 (2013). MR 3134580Google Scholar
  39. 39.
    I. Marquette, C. Quesne, Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators. J. Phys. A 46(15), 155201 (2013)Google Scholar
  40. 40.
    I. Marquette, C. Quesne, Connection between quantum systems involving the fourth Painlevé transcendent and k-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57(5), 15, 052101 (2016). MR 3501792Google Scholar
  41. 41.
    D. Masoero, P. Roffelsen, Poles of Painlevé IV rationals and their distribution. SIGMA 14 (2018), 49, Paper No. 002. MR 3742702Google Scholar
  42. 42.
    T. Masuda, Y. Ohta, K. Kajiwara, A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, 1–25 (2002)MathSciNetCrossRefGoogle Scholar
  43. 43.
    J. Mateo, J. Negro, Third-order differential ladder operators and supersymmetric quantum mechanics. J. Phys. A 41(4), 28, 045204 (2008). MR 2451071Google Scholar
  44. 44.
    K. Matsuda, Rational solutions of the Noumi and Yamada system of type \(A_4^{(1)}\). J. Math. Phys. 53(2), 023504 (2012)Google Scholar
  45. 45.
    Monty Python, And now for something completely different. https://www.imdb.com/title/tt0066765/
  46. 46.
    M. Noumi, Painlevé Equations through Symmetry, vol. 223 (Springer Science & Business, New York, 2004)CrossRefGoogle Scholar
  47. 47.
    M. Noumi, Y. Yamada, Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, 53–86 (1999)MathSciNetCrossRefGoogle Scholar
  48. 48.
    A.A. Oblomkov, Monodromy-free Schrödinger operators with quadratically increasing potentials. Theor. Math. Phys. 121(3), 1574–1584 (1999)CrossRefGoogle Scholar
  49. 49.
    S. Odake, R. Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials. Phys. Lett. B 679(4), 414–417 (2009)MathSciNetCrossRefGoogle Scholar
  50. 50.
    S. Odake, R. Sasaki, Another set of infinitely many exceptional X Laguerre polynomials. Phys. Lett. B 684, 173–176 (2010)MathSciNetCrossRefGoogle Scholar
  51. 51.
    S. Odake, R. Sasaki, Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials. Phys. Lett. B 702(2–3), 164–170 (2011)MathSciNetCrossRefGoogle Scholar
  52. 52.
    S. Odake, R. Sasaki, Extensions of solvable potentials with finitely many discrete eigenstates. J. Phys. A 46(23), 235205 (2013)Google Scholar
  53. 53.
    S. Odake, R. Sasaki, Krein–Adler transformations for shape-invariant potentials and pseudo virtual states. J. Phys. A 46(24), 245201 (2013)Google Scholar
  54. 54.
    K. Okamoto, Studies on the Painlevé equations. III. Second and fourth Painlevé equations, P II and P IV. Math. Ann. 275(2), 221–255 (1986). MR 854008Google Scholar
  55. 55.
    J.B. Olsson, Combinatorics and Representations of Finite Groups. Fachbereich Mathematik [Lecture Notes in Mathematics], vol. 20 (Universität Essen, Essen, 1994)Google Scholar
  56. 56.
    C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. J. Phys. A Math. Theor. 41(39), 392001 (2008)Google Scholar
  57. 57.
    A. Sen, A.N.W. Hone, P.A. Clarkson, Darboux transformations and the symmetric fourth Painlevé equation. J. Phys. A 38(45), 9751–9764 (2005)MathSciNetCrossRefGoogle Scholar
  58. 58.
    K. Takasaki, Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains. Commun. Math. Phys. 241(1), 111–142 (2003)zbMATHGoogle Scholar
  59. 59.
    T. Tsuda, Universal characters, integrable chains and the Painlevé equations. Adv. Math. 197(2), 587–606 (2005)MathSciNetCrossRefGoogle Scholar
  60. 60.
    H. Umemura, Painlevé equations in the past 100 years. Am. Math. Soc. Transl. 204, 81–110 (2001)Google Scholar
  61. 61.
    W. Van Assche, Orthogonal Polynomials and Painlevé Equations. Australian Mathematical Society Lecture Series, vol. 27 (Cambridge University Press, Cambridge, 2018). MR 3729446Google Scholar
  62. 62.
    A.P. Veselov, A.B. Shabat, Dressing chains and the spectral theory of the Schrödinger operator. Funct. Anal. Appl. 27(2), 81–96 (1993)MathSciNetCrossRefGoogle Scholar
  63. 63.
    R. Willox, J. Hietarinta, Painlevé equations from Darboux chains. I. P IIIP V. J. Phys. A 36(42), 10615–10635 (2003). MR 2024916Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Escuela Superior de IngenieríaUniversidad de CádizPuerto RealSpain
  2. 2.Departamento de Física TeóricaUniversidad Complutense de MadridMadridSpain
  3. 3.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

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