Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 085, 18 pages      arXiv:1207.5302      http://dx.doi.org/10.3842/SIGMA.2012.085
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity

Ryu Sasaki a and Kouichi Takemura b
a) Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
b) Department of Mathematics, Faculty of Science and Technology, Chuo University, 1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan

Received July 24, 2012, in final form November 07, 2012; Published online November 15, 2012

Abstract
Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent −2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

Key words: multi-indexed orthogonal polynomials; solvable systems; Fuchsian differential equations; Heun's equation; apparent singularities; high characteristic exponents.

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References

  1. Adler V.È., On a modification of Crum's method, Theoret. and Math. Phys. 101 (1994), 1381-1386.
  2. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  3. Crum M.M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121-127, physics/9908019.
  4. Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. II, Gauthier-Villars, Paris, 1989.
  5. Duistermaat J.J., Grünbaum F.A., Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240.
  6. García-Gutiérrez L., Odake S., Sasaki R., Modification of Crum's theorem for `discrete' quantum mechanics, Progr. Theoret. Phys. 124 (2010), 1-26, arXiv:1004.0289.
  7. Gendenshtein L.E., Derivation of exact spectra of the Schrödinger equation by means of supersymmetry, JETP Lett. 38 (1983), 356-359.
  8. Gibbons J., Veselov A.P., On the rational monodromy-free potentials with sextic growth, J. Math. Phys. 50 (2009), 013513, 25 pages, arXiv:0807.3502.
  9. Gómez-Ullate D., Kamran N., Milson R., A conjecture on exceptional orthogonal polynomials, Found. Comput. Math., to appear, arXiv:1203.6857.
  10. Gómez-Ullate D., Kamran N., Milson R., Two-step Darboux transformations and exceptional Laguerre polynomials, J. Math. Anal. Appl. 387 (2012), 410-418, arXiv:1103.5724.
  11. Hiroe K., Oshima T., A classification of roots of symmetric Kac-Moody root systems and its application, available at http://akagi.ms.u-tokyo.ac.jp/~oshima/index.html.
  12. Ho C.-L., Sasaki R., Takemura K., Confluence of apparent singularities in multi-indexed orthogonal polynomials: the Jacobi case, arXiv:1210.0207.
  13. Infeld L., Hull T.E., The factorization method, Rev. Modern Physics 23 (1951), 21-68.
  14. Krein M.G., On a continual analogue of a Christoffel formula from the theory of orthogonal polynomials, Dokl. Akad. Nauk SSSR 113 (1957), 970-973.
  15. Oblomkov A.A., Monodromy-free Schrödinger operators with quadratically increasing potential, Theoret. and Math. Phys. 121 (1999), 1574-1584.
  16. Odake S., Sasaki R., Crum's theorem for 'discrete' quantum mechanics, Progr. Theoret. Phys. 122 (2009), 1067-1079, arXiv:0902.2593.
  17. Odake S., Sasaki R., Exact solutions in the Heisenberg picture and annihilation-creation operators, Phys. Lett. B 641 (2006), 112-117, quant-ph/0605221.
  18. Odake S., Sasaki R., Exactly solvable 'discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states, Progr. Theoret. Phys. 119 (2008), 663-700, arXiv:0802.1075.
  19. Odake S., Sasaki R., Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials, Phys. Lett. B 702 (2011), 164-170, arXiv:1105.0508.
  20. Odake S., Sasaki R., Exceptional Askey-Wilson-type polynomials through Darboux-Crum transformations, J. Phys. A: Math. Theor. 43 (2010), 335201, 18 pages, arXiv:1004.0544.
  21. Odake S., Sasaki R., Exceptional (Xl) (q)-Racah polynomials, Progr. Theoret. Phys. 125 (2011), 851-870, arXiv:1102.0813.
  22. Odake S., Sasaki R., Discrete quantum mechanics, J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages, arXiv:1104.0473.
  23. Odake S., Sasaki R., Dual Christoffel transformations, Progr. Theoret. Phys. 126 (2011), 1-34, arXiv:1101.5468.
  24. Odake S., Sasaki R., Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials, Phys. Lett. B 682 (2009), 130-136, arXiv:0909.3668.
  25. Odake S., Sasaki R., Multi-indexed (q-)Racah polynomials, J. Phys. A: Math. Theor. 45 (2012), 385201, 21 pages, arXiv:1203.5868.
  26. Odake S., Sasaki R., Multi-indexed Wilson and Askey-Wilson polynomials, arXiv:1207.5584.
  27. Odake S., Sasaki R., Orthogonal polynomials from Hermitian matrices, J. Math. Phys. 49 (2008), 053503, 43 pages, arXiv:0712.4106.
  28. Odake S., Sasaki R., Unified theory of annihilation-creation operators for solvable ("discrete") quantum mechanics, J. Math. Phys. 47 (2006), 102102, 33 pages, quant-ph/0605215.
  29. Oshima T., Classification of Fuchsian systems and their connection problem, arXiv:0811.2916.

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