1,322 research outputs found
QKZ-Ruijsenaars correspondence revisited
We discuss the Matsuo-Cherednik type correspondence between the quantum
Knizhnik-Zamolodchikov equations associated with and the -particle
quantum Ruijsenaars model, with being not necessarily equal to . The
quasiclassical limit of this construction yields the quantum-classical
correspondence between the quantum spin chains and the classical Ruijsenaars
models.Comment: 14 pages, minor correction
Self-dual form of Ruijsenaars-Schneider models and ILW equation with discrete Laplacian
We discuss a self-dual form or the B\"acklund transformations for the
continuous (in time variable) Ruijsenaars-Schneider model. It is
based on the first order equations in complex variables which include
positions of particles and dual variables. The latter satisfy equations of
motion of the Ruijsenaars-Schneider model. In the elliptic case it
holds while for the rational and trigonometric models is not
necessarily equal to . Our consideration is similar to the previously
obtained results for the Calogero-Moser models which are recovered in the
non-relativistic limit. We also show that the self-dual description of the
Ruijsenaars-Schneider models can be derived from complexified intermediate long
wave equation with discrete Laplacian be means of the simple pole ansatz
likewise the Calogero-Moser models arise from ordinary intermediate long wave
and Benjamin-Ono equations.Comment: 16 pages, references adde
R-matrix-valued Lax pairs and long-range spin chains
In this paper we discuss -matrix-valued Lax pairs for
Calogero-Moser model and their relation to integrable quantum long-range spin
chains of the Haldane-Shastry-Inozemtsev type. First, we construct the
-matrix-valued Lax pairs for the third flow of the classical Calogero-Moser
model. Then we notice that the scalar parts (in the auxiliary space) of the
-matrices corresponding to the second and third flows have form of special
spin exchange operators. The freezing trick restricts them to quantum
Hamiltonians of long-range spin chains. We show that for a special choice of
the -matrix these Hamiltonians reproduce those for the Inozemtsev chain. In
the general case related to the Baxter's elliptic -matrix we obtain a
natural anisotropic extension of the Inozemtsev chain. Commutativity of the
Hamiltonians is verified numerically. Trigonometric limits lead to the
Haldane-Shastry chains and their anisotropic generalizations.Comment: 12 pages, Introduction added, minor correction
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