Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

Abstract

In spite of the diversity of solvable models of quantum field theory and the vast variety of methods, the final results display dramatic unification: the spectrum of an integrable theory with a local interaction is given by a sum of elementary energies

$$ E = \sum\limits_i {\varepsilon \left( {{u_i}} \right)} $$

((1.1))

where u _i obey a system of algebraic or transcendental equations known as Bethe equations [1], [2]. The major ingredients of Bethe equations are determined by the algebraic structure of the problem. A typical example of a system of Bethe equations (related to A _i -type models with elliptic R-matrix) is

$$ {e^{ - 4\eta \nu }}\frac{{\phi \left( {{u_j}} \right)}}{{\phi \left( {{u_j} - 2} \right)}} = - \mathop \Pi \limits_k \frac{{\sigma \left( {\eta \left( {{u_j} - {u_k} + 2} \right)} \right)}}{{\sigma \left( {\eta \left( {{u_j} - {u_k} - 2} \right)} \right)}} $$

((1.2))

where σ(x) is the Weierstrass σ-function and

$$ \phi \left( u \right) = \mathop \Pi \limits_{k = 1}^N \sigma \left( {\eta \left( {u - {y_k}} \right)} \right) $$

((1.3))

.