Quantum inverse scattering method

Quantum inverse scattering method relates two different approaches: 1)Inverse scattering transform is a method of solving classical integrable differential equations of evolutionary type. Important concept is Lax representation. 2) Bethe ansatz is a method of solving quantum models in one space and one time dimension. Quantum inverse scattering method starts by quantization of Lax representation and reproduce results of Bethe ansatz. Actually it permits to rewrite Bethe ansatz in a new form: algebraic Bethe ansatz. This led to further progress in understanding of Heisenberg model (quantum), quantum Nonlinear Schrödinger equation (also known as Lieb-Liniger Model or Bose gas with delta interaction) and Hubbard model. Theory of correlation functions was developed: determinant representations, description by differential equations and Riemann-Hilbert problem and asymptotic. Explicit expression for higher conservation laws. In mathematics it led to formulation of quantum groups. Especially interesting one is Yangian. Essential progress was achieved in study of 6 vertex model: the bulk free energy depends on boundary conditions in thermodynamic limit.

In mathematics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions introduced by L. D. Faddeev in about 1979.

References

• Faddeev, L. (1995), "Instructive history of the quantum inverse scattering method", Acta Applicandae Mathematicae, 39 (1): 69–84, doi:10.1007/BF00994626, MR 1329554
• Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1993), Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, ISBN 978-0-521-37320-3, MR 1245942