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 quantum Integrable system for example a) Heisenberg model (quantum), b) quantum Nonlinear Schrödinger equation (also known as Lieb-Liniger Model or Tonks–Girardeau gas) and c) Hubbard model... Theory of correlation functions was developed: determinant representations, description by differential equations and Riemann-Hilbert problem. Asymptotic of correlation functions (even for space, time and temperature dependent) was evaluated in 1991. Explicit expression for higher conservation laws was obtained in 1989. In mathematics quantum inverse scattering method led to formulation of quantum groups. Especially interesting is Yangian, the center of the Yangian is given by quantum determinant. Essential progress was achieved in study of Ice-type model: the bulk free energy of six vertex model depends on boundary conditions even 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