Skip to main content
SpringerLink
Log in

The Heun Equation and the Calogero-Moser-Sutherland System IV: The Hermite-Krichever Ansatz

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic ones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Belokolos, E.D., Enolskii, V.Z.: Reduction of Abelian functions and algebraically integrable systems. II. J. Math. Sci. (New York) 108, 295–374 (2002)

    Google Scholar 

  2. Gesztesy, F., Weikard, R.: Treibich-Verdier potentials and the stationary (m)KdV hierarchy. Math. Z. 219, 451–476 (1995)

    Google Scholar 

  3. Hermite, C.: Oeuvres de Charles Hermite, Vol. III, Paris: Gauthier-Villars, 1912

  4. Inozemtsev, V.I.: Lax representation with spectral parameter on a torus for integrable particle systems. Lett. Math. Phys. 17, 11–17 (1989)

    Google Scholar 

  5. Krichever, I.M.: Elliptic solutions of the Kadomcev-Petviasvili equations, and integrable systems of particles. Funct. Anal. Appl. 14, 282–290 (1980)

    Google Scholar 

  6. Maier, R.S.: Lamé polynomials, hyperelliptic reductions and Lamé band structure. http://www.arxiv.org/abs/math-ph/0309005v1, 2003

  7. Ochiai, H., Oshima, T., Sekiguchi, H.: Commuting families of symmetric differential operators. Proc. Japan. Acad. 70, 62–66 (1994)

    MathSciNet  Google Scholar 

  8. Ronveaux, A.(ed.): Heun’s differential equations. Oxford Science Publications, Oxford: Oxford University Press 1995

  9. Smirnov, A.O.: Elliptic solutions of the Korteweg-de Vries equation. Math. Notes 45, 476–481 (1989)

    Google Scholar 

  10. Smirnov, A.O.: Finite-gap elliptic solutions of the KdV equation. Acta Appl. Math. 36, 125–166 (1994)

    Google Scholar 

  11. Smirnov, A.O.: Elliptic solitons and Heun’s equation. In: The Kowalevski property, CRM Proc. Lecture Notes 32, Providence, RI: Amer. Math. Soc., 2002, pp. 287–305

  12. Takemura, K.: The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method. Commun. Math. Phys. 235, 467–494 (2003)

    Google Scholar 

  13. Takemura, K.: The Heun equation and the Calogero-Moser-Sutherland system II: the perturbation and the algebraic solution. Electron. J. Differ. Eqs. 2004(15), 1–30 (2004)

    Google Scholar 

  14. Takemura, K.: The Heun equation and the Calogero-Moser-Sutherland system III: the finite gap property and the monodromy. J. Nonlinear Math. Phys. 11, 21–46 (2004)

    Google Scholar 

  15. Treibich, A., Verdier, J.-L.: Revetements exceptionnels et sommes de 4 nombres triangulaires (French). Duke Math. J. 68, 217–236 (1992)

    Google Scholar 

  16. Weikard, R.: On Hill’s equation with a singular complex-valued potential. Proc. London Math. Soc. 3, 76, 603–633 (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama, 236-0027, Japan

    Kouichi Takemura

Authors
  1. Kouichi Takemura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kouichi Takemura.

Additional information

Communicated by L. Takhtajan

Rights and permissions

Reprints and permissions

About this article

Cite this article

Takemura, K. The Heun Equation and the Calogero-Moser-Sutherland System IV: The Hermite-Krichever Ansatz. Commun. Math. Phys. 258, 367–403 (2005). https://doi.org/10.1007/s00220-005-1359-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1359-9

Keywords

Search

Navigation