Abstract:
This paper is a developed and consecutive account of a quantum version of the method of the inverse scattering problem on the example of the nonlinear Schrödinger equation. The method of $R$-matrices developed by the author is given basic consideration. The generating functions of quantum integrals of motion and action-angle variables for the quantum nonlinear Schrцdinger equation are obtained. There is also described a classical version of the method of $R$-matrices.
Citation:
E. K. Sklyanin, “Quantum version of the method of inverse scattering problem”, Differential geometry, Lie groups and mechanics. Part III, Zap. Nauchn. Sem. LOMI, 95, "Nauka", Leningrad. Otdel., Leningrad, 1980, 55–128; J. Soviet Math., 19:5 (1982), 1546–1596
\Bibitem{Skl80}
\by E.~K.~Sklyanin
\paper Quantum version of the method of inverse scattering problem
\inbook Differential geometry, Lie groups and mechanics. Part~III
\serial Zap. Nauchn. Sem. LOMI
\yr 1980
\vol 95
\pages 55--128
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl3802}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=606022}
\zmath{https://zbmath.org/?q=an:0464.35071|0497.35072}
\transl
\jour J. Soviet Math.
\yr 1982
\vol 19
\issue 5
\pages 1546--1596
\crossref{https://doi.org/10.1007/BF01091462}
Linking options:
https://www.mathnet.ru/eng/znsl3802
https://www.mathnet.ru/eng/znsl/v95/p55
This publication is cited in the following 148 articles:
F Delduc, B Hoare, M Magro, “Towards a quadratic Poisson algebra for the subtracted classical monodromy of symmetric space sine-Gordon theories”, J. Phys. A: Math. Theor., 57:6 (2024), 065401
Roberto Ruiz, Alejandro Sopena, Max Hunter Gordon, Germán Sierra, Esperanza López, “The Bethe Ansatz as a Quantum Circuit”, Quantum, 8 (2024), 1356
Nafiz Ishtiaque, Seyed Faroogh Moosavian, Yehao Zhou, “Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence”, SIGMA, 20 (2024), 099, 95 pp.
A. V. Pribytok, “Novel integrability in string theory from automorphic symmetries”, Theoret. and Math. Phys., 217:3 (2023), 1914–1937
Yi 艺 Qiao 乔, Junpeng 俊鹏 Cao 曹, Wen-Li 文力 Yang 杨, Kangjie 康杰 Shi 石, Yupeng 玉鹏 Wang 王, “Off-diagonal approach to the exact solution of quantum integrable systems”, Chinese Phys. B, 32:11 (2023), 117504
Xavier Poncini, Jørgen Rasmussen, “Integrability of planar-algebraic models”, J. Stat. Mech., 2023:7 (2023), 073101
Riccardo Borsato, Christian Ferko, Alessandro Sfondrini, “Classical integrability of root-
TT¯
flows”, Phys. Rev. D, 107:8 (2023)
Wei Wang, Yi Qiao, Rong-Hua Liu, Wu-Ming Liu, Junpeng Cao, “Elementary excitations in an integrable twisted
J1-J2
spin chain in the thermodynamic limit”, Phys. Rev. D, 107:5 (2023)
Qian Tang, Xiaomeng Xu, “Stokes phenomenon and Poisson brackets in r-matrix form”, Journal of Geometry and Physics, 188 (2023), 104809
Anatolij K. Prykarpatski, “Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems”, Universe, 8:5 (2022), 288
Alessandro Torrielli, “A study of integrable form factors in massless relativistic AdS
3”, J. Phys. A: Math. Theor., 55:17 (2022), 175401
G Niccoli, V Terras, “Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields”, J. Phys. A: Math. Theor., 55:40 (2022), 405203
N. Belousov, “Bäcklund Transformation for the Nonlinear Schrödinger Equation”, J Math Sci, 264:3 (2022), 203
Vincent Caudrelier, Matteo Stoppato, “Multiform description of the AKNS hierarchy and classical r-matrix”, J. Phys. A: Math. Theor., 54:23 (2021), 235204
R S Vieira, A Lima-Santos, “Solutions of the Yang–Baxter equation for (n + 1) (2n + 1)-vertex models using a differential approach”, J. Stat. Mech., 2021:5 (2021), 053103
Citation in format AMSBIB
Contact us: