Quantum inverse scattering method: Difference between revisions

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#the [[Inverse scattering transform]], a method of solving classical integrable differential equations of the evolutionary type.

An important concept in the [[Inverse scattering transform]] is the [[Lax pair|Lax representation]]; the quantum inverse scattering method starts by the [[quantization (physics)|quantization]] of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the ''algebraic Bethe ansatz''.<ref>cf. e.g. the lectures by N.A. Slavnov, {{arXiv|1804.07350}}</ref>. This led to further progress in the understanding of quantum [[Integrable system]]s, for example:

a) the [[Heisenberg model (quantum)]],

b) the quantum [[Nonlinear Schrödinger equation ]] (also known as the [[Lieb–Liniger model]] or the [[Tonks–Girardeau gas]]) and

c) the [[Hubbard model]].

The theory of [[correlation function]]s was developed {{when|date=November 2015}}: determinant representations, descriptions by differential equations and the [[Riemann–Hilbert problem]]. Asymptotics of correlation functions (even for space, time and temperature dependence) were evaluated in 1991.

Explicit expressions for the higher [[conservation law]]s of the integrable models were obtained in 1989.

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==References==

{{Reflist}}

*{{Citation | last1=Faddeev | first1=L. | title=Instructive history of the quantum inverse scattering method |doi=10.1007/BF00994626 | mr = 1329554 | year=1995 | journal=Acta Applicandae Mathematicae | volume=39 | issue=1 | pages=69–84}}

*{{Citation | last1=Korepin | first1=V. E. | last2=Bogoliubov | first2=N. M. | last3=Izergin | first3=A. G. | title=Quantum inverse scattering method and correlation functions | publisher=[[Cambridge University Press]] | series=Cambridge Monographs on Mathematical Physics | isbn=978-0-521-37320-3 | mr =1245942 | year=1993}}

[[Category:Exactly solvable models]]

[[Category:Quantum mechanics]]