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NATO ASI series. Series B : Physics

Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

Igor Moiseevich Krichever, Ovidiu Lipan, Paul B. Wiegmann, Anton Vladimirovich Zabrodin

Published Jan 1, 1997• 70 citations
Publisher
DOI
10.1007/978-1-4899-1919-9_16

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We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. namely, the eigenvalues of the quantum transfer matrix and the scattering S-matrix itself are identified with a certain τ-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems.1
The solution of the Yang–Baxter equation for integrable systems is shown to be equivalent to the existence of a differential identity. Quantum integration formulas for the calculation of commutators of monodromy matrices are given. Based on the integration formulas and the systematic use of differential identities, the Yang–Baxter equations for the nonlinear Schrödinger model for the quantum case of both bosons and fermions are derived. The case for discrete models is also included. The parallelism between the classical and quantum case and the classical limiting process from the latter to the former are discussed.
The generalized elliptic Ruijsenaars models, which are regarded as a difference analog of the Calogero–Sutherland–Moser models associated with the classical root systems are studied. The integrability and the duality using the fusion procedure of operator-valued solutions of the Yang–Baxter equation and the reflection equation are shown. In particular a new integrable difference operator of type-D is proposed. The trigonometric models are also considered in terms of the representation of the affine Hecke algebra.