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30
The Heun equation and the CalogeroMoserSutherland system I: The Bethe ANSATZ METHOD
, 2002
"... We propose and develop the Bethe Ansatz method for the Heun equation. As an application, holomorphy of the perturbation for the BC1 Inozemtsev model from the trigonometric model is proved. ..."
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Cited by 46 (16 self)
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We propose and develop the Bethe Ansatz method for the Heun equation. As an application, holomorphy of the perturbation for the BC1 Inozemtsev model from the trigonometric model is proved.
A.Varchenko, Algebraic Bethe ansatz for the elliptic quantum group
 Eτ,η(sl2), Nuclear Physics B 480
, 1996
"... Abstract. To each representation of the elliptic quantum group Eτ,η(sl2) is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eightvertex mode ..."
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Cited by 38 (6 self)
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Abstract. To each representation of the elliptic quantum group Eτ,η(sl2) is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eightvertex model and for the twobody Ruijsenaars operator. The latter is a qdeformation of Hermite’s solution of the Lamé equation. 1.
Duality for KnizhnikZamolodchikov and dynamical equations
 ACTA APPL. MATH
, 2001
"... We consider the KnizhnikZamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the (gl k, gl n) duality. We show that the KZ and dynamical equations naturally exchange under the duality. ..."
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Cited by 31 (9 self)
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We consider the KnizhnikZamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the (gl k, gl n) duality. We show that the KZ and dynamical equations naturally exchange under the duality.
Elliptic quantum groups and Ruijsenaars models qalg 9704005 26
 Frenkel E., Feigin B. Quantum Walgebras and elliptic algebras Comm. Math. Phys. A3
, 1996
"... Abstract. We construct symmetric and exterior powers of the vector representation of the elliptic quantum groups Eτ,η(glN). The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases ..."
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Cited by 19 (0 self)
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Abstract. We construct symmetric and exterior powers of the vector representation of the elliptic quantum groups Eτ,η(glN). The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators. 1.
Twisted WessZuminoWitten models on elliptic curves
, 1996
"... Abstract. We construct a Gaudin type lattice model as the WessZuminoWitten model on elliptic curves at the critical level. Bethe eigenvectors are obtained by the bosonisation technique. The goal of this article is to construct a lattice model which is a variant of the Gaudin model with the help of ..."
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Cited by 17 (4 self)
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Abstract. We construct a Gaudin type lattice model as the WessZuminoWitten model on elliptic curves at the critical level. Bethe eigenvectors are obtained by the bosonisation technique. The goal of this article is to construct a lattice model which is a variant of the Gaudin model with the help of the WessZuminoWitten (WZW) model on elliptic curves at the critical level and to find its eigenvectors by means of the bosonisation of the WZW model.
Etingof P., Oblomkov A., Generalized Lamé operators
 Comm. Math. Phys
"... Abstract. We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero–Moser operator, others are related to the root systems and their deformations. We conjecture that th ..."
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Cited by 14 (2 self)
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Abstract. We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero–Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finitegap property of the Lamé operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh–Veselov conjecture for the elliptic Calogero–Moser problem for all root systems. We also establish algebraic integrability in all known twodimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to B2 case, another one is a certain deformation of the A2 case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald–Ruijsenaars type. 1.
The Perturbation Of The Quantum CalogeroMoserSutherland System And Related Results
 COMMUN. MATH. PHYS
, 2002
"... The Hamiltonian of the trigonometric CalogeroSutherland model coincides with some limit of the Hamiltonian of the elliptic CalogeroMoser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential selfadjointness of the Ha ..."
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Cited by 12 (2 self)
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The Hamiltonian of the trigonometric CalogeroSutherland model coincides with some limit of the Hamiltonian of the elliptic CalogeroMoser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential selfadjointness of the Hamiltonian of the elliptic CalogeroMoser model and the regularity (convergence) of the perturbation for the arbitrary root system. We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x 1 ; : : : ; xN ) for the AN01 case. As a result, the algebraic calculation of the perturbation is justified.
Quantum integrable systems and differential galois theory
 Transformation Groups
, 1997
"... Abstract. This paper is devoted to a systematic study of quantum completely integrable systems (i.e. complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the corresponding Dmodule when the eige ..."
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Cited by 11 (0 self)
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Abstract. This paper is devoted to a systematic study of quantum completely integrable systems (i.e. complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the corresponding Dmodule when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e. its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finitezone potentials and the elliptic CalogeroMoser system. In the second example, we obtain a proof of the ChalyhVeselov conjecture that the CalogeroMoser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko. 0.1. Let us recall that in classical mechanics an integrable Hamiltonian system on a manifold X of dimension n is a collection of functions I1,..., In on the cotangent
An explicit solution of the (quantum) elliptic Calogero–Sutherland model
"... Dedicated to the memory of Ludwig Pittner We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic CalogeroSutherland (eCS) model as formal power series to all orders, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explici ..."
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Cited by 10 (4 self)
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Dedicated to the memory of Ludwig Pittner We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic CalogeroSutherland (eCS) model as formal power series to all orders, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explicit formulas for an elliptic deformation of the Jack polynomials. 1
Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator
"... Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in te ..."
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Cited by 8 (5 self)
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Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl2 type.