Quantum Toda Chain
**NOTICE:**
The lecture originally scheduled for Wednesday, 30 April 2025, has been rescheduled to Tuesday, 29 April 2025. The time remains the same (9:50 AM), but the Zoom link has been updated:
🔗 New Zoom Link:
https://us02web.zoom.us/i/8157628413?pwd=S1JRUGhgUzdrcXhXTX30iBpZU84Zz09
All other lecture times and Zoom links remain unchanged.
In the first part of the course in octobet-december we have considered the following topics
1) Quantum Toda open chain. Hamiltonian and family of commuting operators.
2) Lax operator, monodromy matrix and Yang-Baxter commutation relations.
3) Iterative construction of common eigenfunctions of the family of commuting operators.
4) Ladder-operators. Iterative construction of eigenfunction in coordinate space -- Gauss-Givental representation.
5) Iterative construction of eigenfunction in spectral-space -- Mellin-Barnes representation.
6) Calculation of the scalar product: Orthogonality and the Sklyanin measure. Feynman diagrams. Proof of the main relations.
7) Equivalence of the Gauss-Givental and the Mellin-Barnes representations.
8) Gustafson integrals and completeness.
We have concentrated mainly on the exact integral representations for the eigenfunctions and the proofs of the main integral identities.
The maing goal is to demonstrate in details how everything works and to show the main technical tricks.
In the second part of the course in april-may we will concentrate on the operator solution of the Yang-Baxter equation and the role of these integral R-operators in construction of eigenfunctions.
We are going to consider the following topics.
1) Solution of the Yang-Baxter equation and integral R-operators.
2) Q-operator. Definition, kernel identity. Baxter equation.
3) Sklyanin representation of separated variables. Periodic Toda chain.
4) Inverse problem -- solution by O.Babelon and E.Sklyanin.
5) Quasiclassical approximation. Classical Baecklund transformation. Solution of the classical open Toda chain. Solution of the scattering problem for the classical open Toda chain.
6) Reflection equations. BC-Toda chain. Construction of eigenfunctions.
The lecture originally scheduled for Wednesday, 30 April 2025, has been rescheduled to Tuesday, 29 April 2025. The time remains the same (9:50 AM), but the Zoom link has been updated:
🔗 New Zoom Link:
https://us02web.zoom.us/i/8157628413?pwd=S1JRUGhgUzdrcXhXTX30iBpZU84Zz09
All other lecture times and Zoom links remain unchanged.
In the first part of the course in octobet-december we have considered the following topics
1) Quantum Toda open chain. Hamiltonian and family of commuting operators.
2) Lax operator, monodromy matrix and Yang-Baxter commutation relations.
3) Iterative construction of common eigenfunctions of the family of commuting operators.
4) Ladder-operators. Iterative construction of eigenfunction in coordinate space -- Gauss-Givental representation.
5) Iterative construction of eigenfunction in spectral-space -- Mellin-Barnes representation.
6) Calculation of the scalar product: Orthogonality and the Sklyanin measure. Feynman diagrams. Proof of the main relations.
7) Equivalence of the Gauss-Givental and the Mellin-Barnes representations.
8) Gustafson integrals and completeness.
We have concentrated mainly on the exact integral representations for the eigenfunctions and the proofs of the main integral identities.
The maing goal is to demonstrate in details how everything works and to show the main technical tricks.
In the second part of the course in april-may we will concentrate on the operator solution of the Yang-Baxter equation and the role of these integral R-operators in construction of eigenfunctions.
We are going to consider the following topics.
1) Solution of the Yang-Baxter equation and integral R-operators.
2) Q-operator. Definition, kernel identity. Baxter equation.
3) Sklyanin representation of separated variables. Periodic Toda chain.
4) Inverse problem -- solution by O.Babelon and E.Sklyanin.
5) Quasiclassical approximation. Classical Baecklund transformation. Solution of the classical open Toda chain. Solution of the scattering problem for the classical open Toda chain.
6) Reflection equations. BC-Toda chain. Construction of eigenfunctions.
Lecturer
Sergey Derkachov
Date
7th April ~ 21st May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 09:50 - 11:25 | A7-307 | ZOOM 06 | 537 192 5549 | BIMSA |
Reference
1) The Quantum Toda Chain, E.K. Sklyanin, Lect.Notes Phys. 226 (1985) 196-233
2) How algebraic Bethe ansatz works for integrable model, L.D. Faddeev, e-Print: hep-th/9605187 [hep-th]
3) Integral representation for the eigenfunctions of quantum periodic Toda chain, S. Kharchev, D. Lebedev, Lett.Math.Phys. 50 (1999) 53-77, e-Print: hep-th/9910265 [hep-th]
4) Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism, S. Kharchev, D. Lebedev, J.Phys.A 34(2001) 2247-2258, e-Print: hep-th/0007040 [hep-th]
5) The periodic Toda chain and a matrix generalization of the Bessel function's recursion relations, M. Gaudin, V. Pasquier, J.Phys.A 25 (1992) 5243-5252
6) Bispectrality for the quantum open Toda chain, E.Sklyanin, J. Phys. A: Math & Theor, 46:38 (2013) 382001, arXiv:1306.0454
7) Baecklund transformations and Baxter's Q-operator, E. K. Sklyanin, arXiv:nlin/0009009
2) How algebraic Bethe ansatz works for integrable model, L.D. Faddeev, e-Print: hep-th/9605187 [hep-th]
3) Integral representation for the eigenfunctions of quantum periodic Toda chain, S. Kharchev, D. Lebedev, Lett.Math.Phys. 50 (1999) 53-77, e-Print: hep-th/9910265 [hep-th]
4) Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism, S. Kharchev, D. Lebedev, J.Phys.A 34(2001) 2247-2258, e-Print: hep-th/0007040 [hep-th]
5) The periodic Toda chain and a matrix generalization of the Bessel function's recursion relations, M. Gaudin, V. Pasquier, J.Phys.A 25 (1992) 5243-5252
6) Bispectrality for the quantum open Toda chain, E.Sklyanin, J. Phys. A: Math & Theor, 46:38 (2013) 382001, arXiv:1306.0454
7) Baecklund transformations and Baxter's Q-operator, E. K. Sklyanin, arXiv:nlin/0009009
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Sergey Derkachov is leading researcher of the Laboratory of Mathematical Problems of Physics in St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences. The research activity in the last period of time is concentrated on application of the methods of integrable systems to the problems of the quatum field theory, for example, the multiloop calculations.