Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

  • I. Krichever
  • O. Lipan
  • P. Wiegmann
  • A. Zabrodin
Part of the NATO ASI Series book series (NSSB, volume 361)


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
$$ E = \sum\limits_i {\varepsilon \left( {{u_i}} \right)} $$
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
$$ {e^{ - 4\eta \nu }}\frac{{\phi \left( {{u_j}} \right)}}{{\phi \left( {{u_j} - 2} \right)}} = - \mathop \Pi \limits_k \frac{{\sigma \left( {\eta \left( {{u_j} - {u_k} + 2} \right)} \right)}}{{\sigma \left( {\eta \left( {{u_j} - {u_k} - 2} \right)} \right)}} $$
where σ(x) is the Weierstrass σ-function and
$$ \phi \left( u \right) = \mathop \Pi \limits_{k = 1}^N \sigma \left( {\eta \left( {u - {y_k}} \right)} \right) $$


Transfer Matrix Bethe Equation Auxiliary Space Determinant Formula Quantum Integrable System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • I. Krichever
    • 1
    • 2
  • O. Lipan
    • 3
  • P. Wiegmann
    • 4
    • 5
  • A. Zabrodin
    • 6
    • 7
  1. 1.Department of Mathematics of Columbia UniversityMoscowRussia
  2. 2.Landau Institute for Theoretical PhysicsMoscowRussia
  3. 3.James Franck Institute of the University of ChicagoChicagoUSA
  4. 4.James Franck and Enrico Fermi InstitutesThe University of ChicagoChicagoUSA
  5. 5.Landau Institute for Theoretical PhysicsRussia
  6. 6.Joint Institute of Chemical PhysicsMoscowRussia
  7. 7.ITEPMoscowRussia

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