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Separation of Variables in the Elliptic Gaudin Model

Evgeny Konstantinovich Sklyanin, St. Petersburg Branch, Takashi Takebe

Published Jan 1, 1998• 21 citations
OpenAlex

Abstract

. For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of separating integral operator is constructed. The simplest onedegree -of-freedom case is studied in detail. 1 Introduction The quantum elliptic (or XYZ) Gaudin model was introduced in [1], see also [2], as a limiting case of the integrable XYZ spin chain [3]. The commuting Hamiltonians H n of the model are expressed as quadratic combinations of sl 2 spin operators. Determining the spectrum of H n turned out to be a difficult problem like the original XYZ spin chain. Let us list the known facts related to this problem. ffl A solution by means of Algebraic Bethe Ansatz has been obtained only recently [4]. See also [5] ffl As shown in [6], in the SU(2...

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arXiv:hep-th/9403030•Journal of Mathematical Physics•Nov 1, 1994•38 citations
. For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of separating integral operator is constructed. The simplest onedegree -of-freedom case is studied in detail. 1 Introduction The quantum elliptic (or XYZ) Gaudin model was introduced in [1], see also [2], as a limiting case of the integrable XYZ spin chain [3]. The commuting Hamiltonians H n of the model are expressed as quadratic combinations of sl 2 spin operators. Determining the spectrum of H n turned out to be a difficult problem like the original XYZ spin chain. Let us list the known facts related to this problem. ffl A solution by means of Algebraic Bethe Ansatz has been obtained only recently [4]. See also [5] ffl As shown in [6], in the SU(2...
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
This chapter introduces an alternative framework for solving spectral problems in classical and quantum integrable systems via separation of variables, grounded in the Hamilton–Jacobi theory. The method is first outlined in the classical context and then extended to quantum integrable spin chains with gl ell symmetry and associated Yangians. A key objective is to construct separated representations for the spectrum, reducing multidimensional eigenvalue problems to a set of one-dimensional ones solvable through algebraic and representation-theoretic techniques. This approach proves especially effective in cases where the completeness of the Bethe ansatz is unclear, offering complementary insights. Focusing on the classical gl ell magnets, the chapter details the construction of separated variables using the Baker–Akhiezer function and its role in identifying canonical coordinates on the spectral curve. In the quantum case, complete diagonalization of the gl 2 spin chain is achieved through separation of variables. For higher-rank models and generic local representations of gl ell , the full construction remains an open and active area of research. The chapter closes with an overview of current developments and challenges in advancing this method.
The review is based on the author's papers since 1985 in which a new approach to the separation of variables (\SoV) has being developed. It is argued that \SoV, understood generally enough, could be the most universal tool to solve integrable models of the classical and quantum mechanics. It is shown that the standard construction of the action-angle variables from the poles of the Baker-Akhiezer function can be interpreted as a variant of \SoV, and moreover, for many particular models it has a direct quantum counterpart. The list of the models discussed includes XXX and XYZ magnets, Gaudin model, Nonlinear Schrödinger equation, SL(3)SL(3)SL(3)-invariant magnetic chain. New results for the 3-particle quantum Calogero-Moser system are reported.
In this thesis we give accounts on the solution of the open XXZ and XYZ quantum spin-1/2 chains with the most generic integrable boundary terms. By using the the Separation of Variables method (SoV), due to Sklyanin, we are able, in the inhomogeneous case, to build the complete set of eigenstates and the associated eigenvalues. The characterization of these quantities is made through a maximal system of N quadratic equations, where N is the size of the chain. Different methods, like the Algebraic Bethe ansatz (ABA) or other generalized Bethe ansatz techniques, have been used, in the past, in order to tackle these problems. None of them resulted effective in the reproduction of the full set of eigenstates and eigenvalues in the case of most general boundary conditions. A Vandermonde determinant formula for the scalar products of SoV states is obtained as well. The scalar product formula represents a first step towards the calculation of form factors and correlation functions.
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.