Sklyanin Bracket and Deformation of the Calogero-Moser System

A two-dimensional integrable system being a deformation of the rational Calogero-Moser system is constructed via the symplectic reduction, performed with respect to the Sklyanin algebra action. We explicitly resolve the respective classical equations of motion via the projection method and quantize the system.Comment: 14 pages, no figure

Separation of Variables. New Trends.

The review is based on the author's papers since 1985 in which a new approach to the separation of variables (\SoV) has being developed. It is argued that \SoV, understood generally enough, could be the most universal tool to solve integrable models of the classical and quantum mechanics. It is shown that the standard construction of the action-angle variables from the poles of the Baker-Akhiezer function can be interpreted as a variant of \SoV, and moreover, for many particular models it has a direct quantum counterpart. The list of the models discussed includes XXX and XYZ magnets, Gaudin model, Nonlinear Schr\"odinger equation, $SL(3)$-invariant magnetic chain. New results for the 3-particle quantum Calogero-Moser system are reported.Comment: 33 pages, harvmac, no figure

On the r-matrix structure of the hyperbolic BC(n) Sutherland model

Working in a symplectic reduction framework, we construct a dynamical r-matrix for the classical hyperbolic BC(n) Sutherland model with three independent coupling constants. We also examine the Lax representation of the dynamics and its equivalence with the Hamiltonian equation of motion.Comment: 20 page

Classical Functional Bethe Ansatz for SL(N)SL(N): separation of variables for the magnetic chain

The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which gives separation variables for systems for which an $R$-matrix is known. Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated) $R$-matrices. In this paper I advance Sklyanin's program by giving the FBA for certain systems with $SL(N)$ $R$-matrices. This is achieved by constructing rational functions \A(u) and \B(u) of the matrix elements of $T(u)$, so that, in the generic case, the zeros $x_i$ of \B(u) are the separation coordinates and the P_i=\A(x_i) provide their conjugate momenta. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed.Comment: 14pp LaTex,DAMTP 94-1

Separation of variables for the Ruijsenaars system

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system.Comment: 26 pages, LaTex, no figure

Backlund transformations for the sl(2) Gaudin magnet

Elementary, one- and two-point, Backlund transformations are constructed for the generic case of the sl(2) Gaudin magnet. The spectrality property is used to construct these explicitly given, Poisson integrable maps which are time-discretizations of the continuous flows with any Hamiltonian from the spectral curve of the 2x2 Lax matrix.Comment: 17 pages, LaTeX, refs adde