Title of lectures
Bäcklund transformations, Baxter's Q-operators and separation of
variables
Lecturer:
Evgueni SKLYANIN, University of St. Petersburg, Russia
Outline:
This course of 5 lectures will contain a detailed comment on the recently
discovered (Gaudin- Pasquier, 1992) connection between Bäcklund
transformations in the theory of classical integrable systems on one hand,
and Baxter's Q-operator for quantum integrable systems, on the other hand.
We shall restrict our attention to the the systems with finite number of
degrees of freedom. Our main illustrative examples will be the periodic Toda
chain, Heisenberg spin chains, Calogero- Moser model. We shall discuss
applications of BT and Q-operators to the separation of variables and theory
of special functions.
Lecture 1. Hamiltonian integrable systems and r-matrix formalism.
Examples of Bäcklund transformation (Toda lattice, DST model, Heisenberg
magnet, Calogero-Moser model). Canonicity of BT.
Lecture 2. Axiomatics of BT.
Dual BT. BT for the relativistic Ruijsenaars model.
Lecture 3. Quantization.
Quantum Toda chain. Baxter's
Q-operator. Quantum DST model. Applications to special functions.
Lecture 4. BT and separation of variables.
SoV for the Ruisenaars model.
Lecture 5. SoV for A_2 Macdonald polynomials.
References:
[1] Baxter, R. I., Exactly Solved Models in Statistical Mechanics, London:
Academic, 1982.
[2] Faddeev, L.D. and Takhtajan, L.A., Hamiltonian Methods in the Theory of Solitons,
Springer, 1987.
[3] Toda, M., Theory of Nonlinear Lattices, Springer, Berlin, 1981.
[4] Pasquier, V. and Gaudin, M., The periodic Toda chain and a matrix
generalization of the Bessel function recursion relations, J. Phys. A: Math.
Gen. 25 (1992), 5243-5252.
[5] Kuznetsov, V. B. and Sklyanin, E. K., On Bäcklund transformation for
many-body systems. J. Phys. A: Math. Gen. 31 (1998), 2241-2251.
[6] Kuznetsov, V. B. and Sklyanin, E. K., Separation of variables for the A_2
Ruijsenaars model and a new integral representation for the A_2 Macdonald
polynomials, J. Phys. A: Math. Gen. 29 (1998), 2779-2804.
[7] Kuznetsov, V.B., Nijhoff, F.W., and Sklyanin, E.K., Separation of variables
for the Ruijsenaars system. Commun. Math. Phys. 189 (1997), 855-877.