Quantum inverse scattering method: Difference between revisions

Revision as of 15:52, 27 September 2011

Revision as of 16:05, 27 September 2011

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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]] likefor example a) [[Heisenberg model (quantum)]],

b) quantum [[Nonlinear Schrödinger equation ]] (also known as [[Lieb-Liniger Model]] or [[BoseTonks–Girardeau gas]]) withand delta interactionc) and [[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 asymptotictemperature dependent) was evaluated in 1991. Explicit expression for higher [[Conservation law]] was obtained in 1989. In mathematics itquantum inverse scattering method led to formulation of [[quantum groups]]. Especially interesting one 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]].