E. K. Sklyanin

Boundary conditions for integrable quantum systems

A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.

YANG-BAXTER EQUATION AND REPRESENTATION THEORY: I

The problem of constructing the GL(N,ℂ) solutions to the Yang-Baxter equation (factorizedS-matrices) is considered. In caseN=2 all the solutions for arbitrarily finite-dimensional irreducible representations of GL(2,ℂ) are obtained and their eigenvalues are calculated. Some results for the caseN>2 are also presented.

Separation of Variables New Trends

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, Nonlin

Solutions of the Yang-Baxter equation

We give the basic definitions connected with the Yang-Baxter equation (factorization condition for a multiparticle S-matrix) and formulate the problem of classifying its solutions. We list the known methods of solution of the Y-B equation, and also various applications of this equation to the theory of completely integrable quantum and classical systems. A generalization of the Y-B equation to the case ofZ2-graduation is obtained, a possible connection with the theory of representations is noted. The supplement contains about 20 explicit solutions.

Separation of variables in the Gaudin model

We separate variables for the Gaudin model (degenerate case of an integrable quantum magnet SU(2) -chain) by means of an explicit change of coordinates. We get a description of the space of states in the language of ideals in polynomial rings. The structure of the eigenfunctions is studied.

Quantum inverse scattering method and the Heisenberg ferromagnet

Abstract The quantum version of the inverse scattering method is formulated for the Heisenberg ferromagnet. The elements of the transition matrix for the corresponding auxiliary linear problem are the quantum analogue of the action-angle variables. In a continuous limit the model is shown to yield the quantum nonlinear Schrodinger equation.

ALGEBRAIC BETHE ANSATZ FOR THE XYZ GAUDIN MODEL

Abstract The eigenvectors of the Hamiltonians of the XYZ Gaudin model are constructed by means of the algebraic Bethe ansatz. The construction is based on the quasiclassical limit of the corresponding results for the inhomogeneous higher spin eight vertex model.

SEPARATION OF VARIABLES FOR THE RUIJSENAARS SYSTEM

Abstract:We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of the r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking the nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system.

O(N)-invariant nonlinear Schrödinger equation — A new completely integrable system

Abstract A class of reductions of the matrix nonlinear Schrodinger equation is considered which yields a new completely integrable equation. This equation describes an O(N)-invariant interaction of the vector multiplet of nonrelativistic fields and can be solved exactly both in the classical and quantum cases.

Separation of Variables in the Elliptic Gaudin Model

Abstract:For the elliptic Gaudin model (a degenerate case of the 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 the separating integral operator is constructed. The simplest one-degree-of-freedom case is studied in detail.