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.
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Belokolos, E.D., Enolskii, V.Z.: Reduction of Abelian functions and algebraically integrable systems. II. J. Math. Sci. (New York) 108, 295–374 (2002)
Gesztesy, F., Weikard, R.: Treibich-Verdier potentials and the stationary (m)KdV hierarchy. Math. Z. 219, 451–476 (1995)
Hermite, C.: Oeuvres de Charles Hermite, Vol. III, Paris: Gauthier-Villars, 1912
Inozemtsev, V.I.: Lax representation with spectral parameter on a torus for integrable particle systems. Lett. Math. Phys. 17, 11–17 (1989)
Krichever, I.M.: Elliptic solutions of the Kadomcev-Petviasvili equations, and integrable systems of particles. Funct. Anal. Appl. 14, 282–290 (1980)
Maier, R.S.: Lamé polynomials, hyperelliptic reductions and Lamé band structure. http://www.arxiv.org/abs/math-ph/0309005v1, 2003
Ochiai, H., Oshima, T., Sekiguchi, H.: Commuting families of symmetric differential operators. Proc. Japan. Acad. 70, 62–66 (1994)
Ronveaux, A.(ed.): Heun’s differential equations. Oxford Science Publications, Oxford: Oxford University Press 1995
Smirnov, A.O.: Elliptic solutions of the Korteweg-de Vries equation. Math. Notes 45, 476–481 (1989)
Smirnov, A.O.: Finite-gap elliptic solutions of the KdV equation. Acta Appl. Math. 36, 125–166 (1994)
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
Takemura, K.: The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method. Commun. Math. Phys. 235, 467–494 (2003)
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)
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)
Treibich, A., Verdier, J.-L.: Revetements exceptionnels et sommes de 4 nombres triangulaires (French). Duke Math. J. 68, 217–236 (1992)
Weikard, R.: On Hill’s equation with a singular complex-valued potential. Proc. London Math. Soc. 3, 76, 603–633 (1998)
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Communicated by L. Takhtajan
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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
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DOI: https://doi.org/10.1007/s00220-005-1359-9