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Method used to solve integrable many-body quantum systems
In quantum physics, the quantum inverse scattering method (QISM), similar to the closely related algebraic Bethe ansatz, is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979.[1]
It can be viewed as a quantized version of the classical inverse scattering method pioneered by Norman Zabusky and Martin Kruskal[2] used to investigate the Korteweg–de Vries equation and later other integrable partial differential equations. In both, a Lax matrix features heavily and scattering data is used to construct solutions to the original system.
While the classical inverse scattering method is used to solve integrable partial differential equations which model continuous media (for example, the KdV equation models shallow water waves), the QISM is used to solve many-body quantum systems, sometimes known as spin chains, of which the Heisenberg spin chain is the best-studied and most famous example. These are typically discrete systems, with particles fixed at different points of a lattice, but limits of results obtained by the QISM can give predictions even for field theories defined on a continuum, such as the quantum sine-Gordon model.
^Takhtadzhan, L A; Faddeev, Lyudvig D (31 October 1979). "The Quantum Method of the Inverse Problem and the Heisenberg Xyz Model". Russian Mathematical Surveys. 34 (5): 11–68. Bibcode:1979RuMaS..34...11T. doi:10.1070/RM1979v034n05ABEH003909.
^Zabusky, N. J.; Kruskal, M. D. (9 August 1965). "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States". Physical Review Letters. 15 (6): 240–243. Bibcode:1965PhRvL..15..240Z. doi:10.1103/PhysRevLett.15.240.
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