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arXiv:hep-th/9403030 · Journal of Mathematical Physics

Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain

D. R. D. Scott

Published Nov 1, 1994• 38 citations

Abstract

It is desirable for one to be able to construct separation variables for systems with SL(N) R matrices or Lax pairs. This article relates and extends the two known methods of construction. The Sklyanin’s functional Bethe ansatz (FBA) [Commun. Math. Phys. 150, 181–191 (1992)] was previously only known for SL(2) and SL(3) (and associated) R matrices. The algebraic geometric method of Adams, Harnad, and Hurtubise [Commun. Math. Phys. 155, 385–413 (1993)] has been shown to work for SL(N) but up until now only for linear Poisson brackets. In this article Sklyanin’s program is advanced by giving the FBA for certain systems with SL(N) R matrices. This is achieved by constructing rational functions 𝒜(u) and ℬ(u) of the matrix elements of T(u), so that, in the generic case, the zeros xi of ℬ(u) are the separation coordinates and the Pi=𝒜(xi) provide their conjugate momenta. It is shown that the separation variables thus defined are the same as those given (as the simultaneous solutions of several equations in two variables) by the other method. The crucial calculation of Adams et al. of the commutation relations is also adapted to the case of quadratic Poisson brackets. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed. No knowledge of algebraic geometry is required.

DOI: 10.1063/1.530712

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Abstract. In this work we generalise previous results connecting (rational) Gaudin magnet models and classical separation of variables. It is shown that the connection persists for the case of linear r-matrix algebra which corresponds to the trigonometric 4 × 4 r-matrix (of the XXZ type). We comment also on the corresponding problem for the elliptic (XYZ) r-matrix. A prescription for obtaining integrable systems associated with multiple poles of an L-operator is given. 1. Introduction. Separation of variables for the Hamilton-Jacobi and Schrödinger equations have long been known as methods for explicit solution of these equations in appropriate circumstances. The technical requirements for this method of solution have quite fully developed in recent years (see [1–8]). In particular the relationship between the separable systems and the Gaudin magnet [4,9] integrable

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In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off-shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.