733 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
The Nonlinear Schrodinger Equation on the Half Line
The nonlinear Schrodinger equation on the half line with mixed boundary
condition is investigated. After a brief introduction to the corresponding
classical boundary value problem, the exact second quantized solution of the
system is constructed. The construction is based on a new algebraic structure,
which is called in what follows boundary algebra and which substitutes, in the
presence of boundaries, the familiar Zamolodchikov-Faddeev algebra. The
fundamental quantum field theory properties of the solution are established and
discussed in detail. The relative scattering operator is derived in the
Haag-Ruelle framework, suitably generalized to the case of broken translation
invariance in space.Comment: Tex file, no figures, 32 page
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
Quantum group symmetry of integrable systems with or without boundary
We present a construction of integrable hierarchies without or with boundary,
starting from a single R-matrix, or equivalently from a ZF algebra. We give
explicit expressions for the Hamiltonians and the integrals of motion of the
hierarchy in term of the ZF algebra. In the case without boundary, the
integrals of motion form a quantum group, while in the case with boundary they
form a Hopf coideal subalgebra of the quantum group.Comment: 14 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
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