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SIGMA 3 (2007), 028, 9 pages math.QA/0702676
https://doi.org/10.3842/SIGMA.2007.028
Contribution to the Vadim Kuznetsov Memorial Issue
Bethe Ansatz for the Ruijsenaars Model of BC1-Type
Oleg Chalykh
School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Received December 14, 2006, in final form February
06, 2007; Published online February 22, 2007
Abstract
We consider one-dimensional elliptic Ruijsenaars model
of type BC1. It is given by a three-term difference
Schrödinger operator L containing 8 coupling constants. We
show that when all coupling constants are integers, L has
meromorphic eigenfunctions expressed by a variant of Bethe ansatz.
This result generalizes the Bethe ansatz formulas known in the
A1-case.
Key words:
Heun equation; three-term difference operator; Bloch eigenfunction; spectral curve.
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References
- Askey R., Wilson J., Some basic hypergeometric orthogonal
polynomials that generalize Jacobi polynomials, Mem. Amer.
Math. Soc. 319 (1985), 1-53.
- Chalykh O.,
Macdonald polynomials and algebraic integrability, Adv.
Math. 166 (2002), 193-259,
math.QA/0212313.
- Chalykh O., Etingof P., Oblomkov A., Generalized Lamé operators,
Comm. Math. Phys. 239 (2003), 115-153,
math.QA/0212029.
- van Diejen J.F., Integrability of difference Calogero-Moser
systems, J. Math. Phys. 35 (1994), 2983-3004.
- van Diejen J.F., Difference Calogero-Moser systems and finite
Toda chains, J. Math. Phys. 36 (1995), 1299-1323.
- Dubrovin B.A., Krichever I.M., Novikov S.P., Integrable systems I,
Dynamical systems IV, Encyclopaedia Math. Sci., Vol. 4,
Springer, Berlin, 2001, 177-332.
- Felder G., Varchenko A., Algebraic integrability of the two-body
Ruijsenaars operator, Funktsional Anal. i Prilozhen.
32 (1998), 8-25,
q-alg/9610024.
- Felder G., Varchenko A., Algebraic Bethe ansatz for the elliptic
quantum group Et,h(sl2), Nuclear
Phys. B 480 (1996), 485-503,
q-alg/9605024.
- Inozemtsev V.I., Lax representation with spectral parameter on a
torus for integrable particle systems, Lett. Math. Phys.
17 (1989), 11-17.
- Komori Y., Hikami K., Quantum integrability of the generalized
elliptic Ruijsenaars models, J. Phys. A: Math. Gen. 30
(1997), 4341-4364.
- Komori Y., Hikami K., Conserved operators of the generalized
elliptic Ruijsenaars models, J. Math. Phys. 39 (1998),
6175-6190.
- Koornwinder T.H., Askey-Wilson polynomials for root systems of
type BC, in Hypergeometric Functions on Domains of
Positivity, Jack Polynomials, and Applications, Editor D.St.P.
Richards, Contemp. Math., 138 (1992), 189-204.
- Krichever I., Zabrodin A., Spin generalization of the
Ruijsenaars-Schneider model, the nonabelian 2D Toda lattice, and
representations of the Sklyanin algebra, Uspekhi Mat. Nauk
50 (1995), 3-56,
hep-th/9505039.
- Ruijsenaars S.N.M., Complete integrability of relativistic
Calogero-Moser systems and elliptic functions identities,
Comm. Math. Phys. 110 (1987), 191-213.
- Sklyanin E.K., Some algebraic structures connected with the
Yang-Baxter equation (Russian), Funktsional. Anal. i
Prilozhen. 16 (1982), no. 4, 27-34.
- Sklyanin E.K., Some algebraic structures connected with the
Yang-Baxter equation, Representations of a quantum algebra
(Russian), Funktsional. Anal. i Prilozhen. 17 (1983), no. 4, 34-48.
- Smirnov A.O., Elliptic solitons and Heun's equation, in The
Kowalevski Property (Leeds, 2000), CRM Proc. Lecture Notes,
Vol. 32, Editor V.B. Kuznetsov, American Mathematical Society,
Providence, RI, 2002, 287-305,
math.CA/0109149.
- Takemura K., The Heun equation and the Calogero-Moser-Sutherland
system I, The Bethe ansatz method, Comm. Math. Phys
235 (2003), 467-494,
math.CA/0103077.
- Treibich A., Verdier J.-L., Solitons elliptiques, in The
Grothendieck Festschrift, Vol. III, Editots P. Cartier et al.,
Progr. Math., Vol. 88, Birkhäuser, Boston, 1990.
- Whittaker E.T., Watson G.N., A course of modern analysis,
Cambridge University Press, New York, 1962.
- Zabrodin A., On the spectral curve of the difference Lamé
operator, Int. Math. Res. Not. (1999), no. 11, 589-614,
math.QA/9812161.
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