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arXiv:solv-int/9504001 · Progress of Theoretical Physics Supplement

Separation of Variables

Evgueni K. Sklyanin

Published Jan 1, 1995• 581 citations
arXivHTMLPDF

Abstract

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, $SL(3)$-invariant magnetic chain. New results for the 3-particle quantum Calogero-Moser system are reported.

DOI
10.1143/ptps.118.35

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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.
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of gl(n)\mathfrak{gl}(n)gl(n) in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for gl(n)\mathfrak{gl}(n)gl(n) spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
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
. 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...
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.
We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix.We show that for the case of such twist matrices one can interchange the role of classical separating functions A(u) and B(u) and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin.We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.