QuantumThe quantum inverse scattering method relates two different approaches:
1) The [[InverseBethe scattering transformansatz ]] is, a method of solving classical integrable differentialquantum equationsmodels ofin one space and one evolutionarytime type.dimension;
2) the [[Inverse scattering transform]], a method of solving classical integrable differential equations of the evolutionary type.
Important concept is [[Lax representation]].
QuantumAn important concept in the [[Inverse scattering transform]] is the [[Lax representation]]; the quantum inverse scattering method starts by the quantization of Lax representation and reproducereproduces the results of the Bethe ansatz.▼
2) [[Bethe ansatz ]] is a method of solving quantum models in one space and one time dimension.
ActuallyIn fact it permits toallows rewritethe Bethe ansatz to be written in a new form: the algebraic Bethe ansatz. This led to further progress in the understanding of quantum [[Integrable systemsystems]] for example: a) the [[Heisenberg model (quantum)]],▼
▲Quantum inverse scattering method starts by quantization of Lax representation and reproduce results of Bethe ansatz.
b) the quantum [[Nonlinear Schrödinger equation ]] (also known as the [[Lieb-Liniger Model]] or the [[Tonks–Girardeau gas]]) and c) the [[Hubbard model]]. The
▲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]]... Theorytheory of correlation functions was developed: determinant representations, descriptiondescriptions by differential equations and the [[Riemann-Hilbert problem]]. Asymptotic Asymptotics of correlation functions (even for space, time and temperature dependentdependence) waswere evaluated in 1991. Explicit expressionexpressions for the higher [[conservation laws]] was of the integrable models were obtained in 1989. In mathematics the quantum inverse scattering method led to the formulation of [[quantum groups]]. Especially interesting is the [[Yangian]], and the center of the Yangian is given by the quantum determinant. Essential progress was achieved in study of [[Ice-type modelmodels]]: the bulk free energy of the six vertex model depends on boundary conditions even in the [[thermodynamic limit]].
In mathematics, the '''quantum inverse scattering method''' is a method for solving [[integrable model]]s in 1+1 dimensions introduced by [[L. D. Faddeev]] in about 1979.
