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

SIGMA 14 (2018), 061, 16 pages      arXiv:1712.09564      https://doi.org/10.3842/SIGMA.2018.061
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

On $q$-Deformations of the Heun Equation

Kouichi Takemura
Department of Mathematics, Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan

Received January 18, 2018, in final form May 29, 2018; Published online June 18, 2018

Abstract
The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the $q$-Heun equation and its variants. Namely we investigate finite-dimensional subspaces which are invariant under the action of the $q$-Heun operator or variants of the $q$-Heun operator.

Key words: Heun equation; $q$-deformation; regular singularity; quasi-exact solvability; degeneration.

pdf (360 kb)   tex (17 kb)

References

  1. Adams C.R., On the linear ordinary $q$-difference equation, Ann. of Math. 30 (1928), 195-205.
  2. van Diejen J.F., Integrability of difference Calogero-Moser systems, J. Math. Phys. 35 (1994), 2983-3004.
  3. Finkel F., Gómez-Ullate D., González-López A., Rodríguez M.A., Zhdanov R., New spin Calogero-Sutherland models related to $B_N$-type Dunkl operators, Nuclear Phys. B 613 (2001), 472-496, hep-th/0103190.
  4. Hahn W., On linear geometric difference equations with accessory parameters, Funkcial. Ekvac. 14 (1971), 73-78.
  5. Jimbo M., Sakai H., A $q$-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154, chao-dyn/9507010.
  6. Komori Y., Ruijsenaars' commuting difference operators and invariant subspace spanned by theta functions, J. Math. Phys. 42 (2001), 4503-4522.
  7. Ruijsenaars S.N.M., Integrable $BC_N$ analytic difference operators: hidden parameter symmetries and eigenfunctions, in New Trends in Integrability and Partial Solvability, NATO Sci. Ser. II Math. Phys. Chem., Vol. 132, Kluwer Acad. Publ., Dordrecht, 2004, 217-261.
  8. Takemura K., Quasi-exact solvability of Inozemtsev models, J. Phys. A: Math. Gen. 35 (2002), 8867-8881, math.QA/0205274.
  9. Takemura K., Heun's differential equation, in Selected Papers on Analysis and Differential Equations, Amer. Math. Soc. Transl. Ser. 2, Vol. 230, Amer. Math. Soc., Providence, RI, 2010, 45-68.
  10. Takemura K., Degenerations of Ruijsenaars-van Diejen operator and $q$-Painlevé equations, J. Integrable Syst. 2 (2017), xyx008, 27 pages, arXiv:1608.07265.
  11. Turbiner A.V., Quasi-exactly-solvable problems and ${\rm sl}(2)$ algebra, Comm. Math. Phys. 118 (1988), 467-474.
  12. Yamada Y., Lax formalism for $q$-Painlevé equations with affine Weyl group symmetry of type $E^{(1)}_n$, Int. Math. Res. Not. 2011 (2011), 3823-3838, arXiv:1004.1687.

Previous article  Next article   Contents of Volume 14 (2018)