### Abstract

We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero-Moser system. We start with the description of Treibich-Verdier two-gap elliptic potentials. The explicit formulae for the covers, wavefunctions and Lame polynomials are derived, together with a new Lax representation for the particle dynamics on the locus. We then consider more general potentials within the Weierstrass reduction theory of theta functions to lower genera. The reduction conditions in the moduli space of the genus-2 algebraic curves are given. This is a subvariety of the Humbert surface, which can be singled out by the condition of the vanishing of some theta constants.

Original language | English |
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Article number | 028 |

Pages (from-to) | 1069-1088 |

Number of pages | 20 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 28 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1995 |

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*Journal of Physics A: Mathematical and General*,

*28*(4), 1069-1088. [028]. https://doi.org/10.1088/0305-4470/28/4/028

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*Journal of Physics A: Mathematical and General*, vol. 28, no. 4, 028, pp. 1069-1088. https://doi.org/10.1088/0305-4470/28/4/028

**On the two-gap locus for the elliptic Calogero-Moser model.** / Enolskii, V. Z.; Eilbeck, J. C.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the two-gap locus for the elliptic Calogero-Moser model

AU - Enolskii, V. Z.

AU - Eilbeck, J. C.

PY - 1995

Y1 - 1995

N2 - We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero-Moser system. We start with the description of Treibich-Verdier two-gap elliptic potentials. The explicit formulae for the covers, wavefunctions and Lame polynomials are derived, together with a new Lax representation for the particle dynamics on the locus. We then consider more general potentials within the Weierstrass reduction theory of theta functions to lower genera. The reduction conditions in the moduli space of the genus-2 algebraic curves are given. This is a subvariety of the Humbert surface, which can be singled out by the condition of the vanishing of some theta constants.

AB - We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero-Moser system. We start with the description of Treibich-Verdier two-gap elliptic potentials. The explicit formulae for the covers, wavefunctions and Lame polynomials are derived, together with a new Lax representation for the particle dynamics on the locus. We then consider more general potentials within the Weierstrass reduction theory of theta functions to lower genera. The reduction conditions in the moduli space of the genus-2 algebraic curves are given. This is a subvariety of the Humbert surface, which can be singled out by the condition of the vanishing of some theta constants.

UR - http://www.scopus.com/inward/record.url?scp=21844497180&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/28/4/028

DO - 10.1088/0305-4470/28/4/028

M3 - Article

VL - 28

SP - 1069

EP - 1088

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

SN - 0305-4470

IS - 4

M1 - 028

ER -