"Hirota-Miwa difference equations"의 두 판 사이의 차이
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| + | ==introduction== | ||
| + | * how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's <math>Q</math>-operators, etc.) with elements of classical nonlinear integrable difference equations (<math>\tau</math>-functions, Baker-Akhiezer functions, etc.). | ||
| + | |||
| + | |||
| + | ==dictionary== | ||
| + | * The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with the classical Hirota bilinear difference equation. | ||
| + | * This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. | ||
| + | * Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. | ||
| + | * The elliptic solutions relevant to the Bethe ansatz are studied. | ||
| + | * The nested Bethe ansatz equations for <math>A_{k−1}</math>-type models appear as discrete time equations of motions for zeros of classical <math>\tau</math>-functions and Baker-Akhiezer functions. | ||
| + | * Determinant representations of the general solution to the bilinear discrete Hirota equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. | ||
| + | ===Baxter <math>Q</math>-operator=== | ||
| + | * Eigenvalues of Baxter's <math>Q</math>-operator are solutions to the auxiliary linear problems for the classical Hirota equation. | ||
| + | * Difference equations for eigenvalues of the <math>Q</math>-operators which generalize Baxter's three-term <math>T-Q</math>-relation are derived | ||
| + | |||
| + | |||
| + | |||
==related items== | ==related items== | ||
* [[Hirota bilinear method]] | * [[Hirota bilinear method]] | ||
* [[T-system]] | * [[T-system]] | ||
| + | * [[Determinant solutions of T-systems]] | ||
* [[Octahedral recurrence]] | * [[Octahedral recurrence]] | ||
| + | * [[Difference L-operators and TT-relations]] | ||
==expositions== | ==expositions== | ||
| + | * Zabrodin, A. 2012. “Bethe Ansatz and Hirota Equation in Integrable Models.” arXiv:1211.4428 [hep-Th, Physics:math-Ph] (November 19). http://arxiv.org/abs/1211.4428. | ||
| + | * Zabrodin, A. V. 1998. “Hirota Equation and Bethe Ansatz.” Theoretical and Mathematical Physics 116 (1) (July 1): 782–819. doi:10.1007/BF02557123. | ||
| + | * Wiegmann, P. 1997. “Bethe Ansatz and Classical Hirota Equation.” International Journal of Modern Physics B 11 (01n02) (January 20): 75–89. doi:10.1142/S0217979297000101. | ||
* Zabrodin, A. V. 1997. “Hirota’s Difference Equations.” Theoretical and Mathematical Physics 113 (2) (November 1): 1347–1392. doi:10.1007/BF02634165. http://arxiv.org/abs/solv-int/9704001 | * Zabrodin, A. V. 1997. “Hirota’s Difference Equations.” Theoretical and Mathematical Physics 113 (2) (November 1): 1347–1392. doi:10.1007/BF02634165. http://arxiv.org/abs/solv-int/9704001 | ||
| − | + | * Zabrodin, A. V. 1997. “Bethe Ansatz and Classical Hirota Equations.” In Proceedings of the Second International A. D. Sakharov Conference on Physics (Moscow, 1996), 627–632. World Sci. Publ., River Edge, NJ. http://arxiv.org/abs/hep-th/9607162. | |
| + | |||
| + | |||
==articles== | ==articles== | ||
| − | + | * Krichever, I., O. Lipan, P. Wiegmann, and A. Zabrodin. 1997. “Quantum Integrable Models and Discrete Classical Hirota Equations.” Communications in Mathematical Physics 188 (2): 267–304. doi:10.1007/s002200050165. http://arxiv.org/abs/hep-th/9604080. | |
| − | * Krichever, I., O. Lipan, P. Wiegmann, and A. Zabrodin. | ||
* T. Miwa, Proc. Japan. Acad. 58, 9 (1982). | * T. Miwa, Proc. Japan. Acad. 58, 9 (1982). | ||
* R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn. 50, 3785 (1981). | * R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn. 50, 3785 (1981). | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
| + | [[분류:Integrable systems]] | ||
| + | [[분류:migrate]] | ||
2020년 12월 28일 (월) 05:25 기준 최신판
introduction
- how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's \(Q\)-operators, etc.) with elements of classical nonlinear integrable difference equations (\(\tau\)-functions, Baker-Akhiezer functions, etc.).
dictionary
- The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with the classical Hirota bilinear difference equation.
- This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries.
- Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices.
- The elliptic solutions relevant to the Bethe ansatz are studied.
- The nested Bethe ansatz equations for \(A_{k−1}\)-type models appear as discrete time equations of motions for zeros of classical \(\tau\)-functions and Baker-Akhiezer functions.
- Determinant representations of the general solution to the bilinear discrete Hirota equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained.
Baxter \(Q\)-operator
- Eigenvalues of Baxter's \(Q\)-operator are solutions to the auxiliary linear problems for the classical Hirota equation.
- Difference equations for eigenvalues of the \(Q\)-operators which generalize Baxter's three-term \(T-Q\)-relation are derived
- Hirota bilinear method
- T-system
- Determinant solutions of T-systems
- Octahedral recurrence
- Difference L-operators and TT-relations
expositions
- Zabrodin, A. 2012. “Bethe Ansatz and Hirota Equation in Integrable Models.” arXiv:1211.4428 [hep-Th, Physics:math-Ph] (November 19). http://arxiv.org/abs/1211.4428.
- Zabrodin, A. V. 1998. “Hirota Equation and Bethe Ansatz.” Theoretical and Mathematical Physics 116 (1) (July 1): 782–819. doi:10.1007/BF02557123.
- Wiegmann, P. 1997. “Bethe Ansatz and Classical Hirota Equation.” International Journal of Modern Physics B 11 (01n02) (January 20): 75–89. doi:10.1142/S0217979297000101.
- Zabrodin, A. V. 1997. “Hirota’s Difference Equations.” Theoretical and Mathematical Physics 113 (2) (November 1): 1347–1392. doi:10.1007/BF02634165. http://arxiv.org/abs/solv-int/9704001
- Zabrodin, A. V. 1997. “Bethe Ansatz and Classical Hirota Equations.” In Proceedings of the Second International A. D. Sakharov Conference on Physics (Moscow, 1996), 627–632. World Sci. Publ., River Edge, NJ. http://arxiv.org/abs/hep-th/9607162.
articles
- Krichever, I., O. Lipan, P. Wiegmann, and A. Zabrodin. 1997. “Quantum Integrable Models and Discrete Classical Hirota Equations.” Communications in Mathematical Physics 188 (2): 267–304. doi:10.1007/s002200050165. http://arxiv.org/abs/hep-th/9604080.
- T. Miwa, Proc. Japan. Acad. 58, 9 (1982).
- R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn. 50, 3785 (1981).