Integrability of difference Calogero–Moser systems

Diejen, J. F. van

doi:10.1063/1.530498

Cited by 134 publications

(146 citation statements)

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“…Therefore, is it natural to expect that the D-operator has a similar relationship with relativistic quantum multiparticle systems. Indeed, this is true for the model proposed by van Diejen [2] and investigated in detail by Komori and Hikami [7]. In the presence of the balancing condition 8 j=1 ε j = p 2 q 2 , the corresponding one-particle Hamiltonian takes the form …”

Section: D(t )U (T)mentioning

confidence: 85%

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Continuous biorthogonality of an elliptic hypergeometric function

Spiridonov

2009

St. Petersburg Math. J.

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Abstract. A family of continuous biorthogonal functions related to an elliptic analog of the Gauss hypergeometric function is constructed. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. The relationship with the Sklyanin algebra and elliptic analogs of the Faddeev modular double are discussed in detail. §1. Introduction Classical and quantum completely integrable systems serve as a rich source of special functions. The complexity of these systems correlates with the complexity of solutions of the Yang-Baxter equation -rational, trigonometric, or elliptic [22]. In the theory of special functions this structural hierarchy is reflected in the existence of the plain and q-hypergeometric functions [1] and their elliptic generalizations [19]. One of our aims in the present paper is to clarify certain questions about this correspondence at the elliptic level.In [17,19], the author introduced an elliptic analog of the Gauss hypergeometric function V (t 1 , . . . , t 8 ) that generalizes many special functions of hypergeometric type obeying "classical" properties [1]. In particular, it was shown that this function exhibits symmetry transformations tied to the exceptional root system E 7 and satisfies the elliptic hypergeometric equation. We start this work with showing in §2 that the V -function satisfies a simple biorthogonality relation corresponding to the continuous values of a spectral parameter. For that we use the elliptic beta integral [16] and an integral analog of the Bailey chains introduced in [18]. Our construction can be viewed as an integral generalization of Rosengren's approach to the elliptic 6j-symbols [11].In §3, we investigate the generalized eigenvalue problem for a second order finite difference operator D (a, b, c, d; p; q). The relationship of such problems with biorthogonality relations was considered in [23]. A problem similar to ours was investigated earlier in [10,11] for the discrete spectrum. Its general solution determines a basis in some space of meromorphic functions. For a simple scalar product, we derive the dual basis defined by solutions of a similar spectral problem for conjugate operators. As a result, the overlap of the basis vectors with their duals coincides with the V -function. In §4, we derive a new contiguous relation for the elliptic hypergeometric function and the biorthogonality condition associated with that.It turns out that the V -function is directly related to the Sklyanin algebra, one of the central objects in the quantum inverse scattering method [14,15]. This relationship is established in §5 on the basis of the observation, due to Rains [9,12], that the D-operator can be represented as a linear combination of all four generators of the Sklyanin algebra.2000 Mathematics Subject Classification. Primary 33C75, 81R12.

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Section: D(t )U (T)mentioning

confidence: 85%

“…The latter equation is known to emerge in the one particle sector of the Inozemtsev integrable model [5]. Our finite difference analog of it turns out to be related to the model introduced by van Diejen [2] for a restricted set of parameters.…”

Section: §1 Introductionmentioning

confidence: 97%

Continuous biorthogonality of an elliptic hypergeometric function

Spiridonov

2009

St. Petersburg Math. J.

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“…On the other hand, the classical equations of motion for this Hamiltonian with the modular parameter τ considered as a time variable leads to the Painlevé VI equation [83]. The Doperator itself appears to be related to the van Diejen model [62,79]. All this and the connection with elliptic hypergeometric functions described below demonstrate some mathematical universality of the operator (8.1).…”

Section: Connection With the Sklyanin Algebramentioning

confidence: 94%

“…A detailed and rigorous analysis of the degeneration procedure is performed in [28,61]. It is necessary to note [58] that the elliptic hypergeometric equation emerges as a particular case of the eigenvalue problem equation for the one particle Hamiltonian of the quantum model proposed by van Diejen [62] and investigated in detail by Komori and Hikami [63]. This Calogero-Sutherland type model represents a generalization of the Ruijsenaars [64] and Inozemtsev [65] systems.…”

Section: The Elliptic Hypergeometric Equationmentioning

confidence: 99%

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Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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Abstract. We give a brief review of the main results of the theory of elliptic hypergeometric functions -a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

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Theta Functions Associated with Affine Root Systems and the Elliptic Ruijsenaars Operators

Komori

2000

Physical Combinatorics

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Integrability of difference Calogero–Moser systems

Cited by 134 publications

References 9 publications

Continuous biorthogonality of an elliptic hypergeometric function

Continuous biorthogonality of an elliptic hypergeometric function

Essays on the theory of elliptic hypergeometric functions

Theta Functions Associated with Affine Root Systems and the Elliptic Ruijsenaars Operators

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