618 research outputs found
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, -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 : 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 -matrix is known.
Previously the FBA was only known for and (and associated)
-matrices. In this paper I advance Sklyanin's program by giving the FBA for
certain systems with -matrices. This is achieved by constructing
rational functions \A(u) and \B(u) of the matrix elements of , so
that, in the generic case, the zeros 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
Backlund transformations for the elliptic Gaudin model and a Clebsch system
A two-parameters family of Backlund transformations for the classical
elliptic Gaudin model is constructed. The maps are explicit, symplectic,
preserve the same integrals as for the continuous flows and are a time
discretization of each of these flows. The transformations can map real
variables into real variables, sending physical solutions of the equations of
motion into physical solutions. The starting point of the analysis is the
integrability structure of the model. It is shown how the analogue
transformations for the rational and trigonometric Gaudin model are a limiting
case of this one. An application to a particular case of the Clebsch system is
given.Comment: 18 pages; the following article has been submitted to J. Math. Phys.
After it is published, it will be found at http://jmp.aip.org
Separation of Variables in BC-type Gaudin Magnet
The integrable system is introduced based on the Poisson -matrix
structure. This is a generalization of the Gaudin magnet, and in SL(2) case
isomorphic to the generalized Neumann model. The separation of variables is
discussed for both classical and quantum case.Comment: 11 pages, macros from ftp.ioppublishing.co
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