1,204 research outputs found
Algebraic Bethe Ansatz for XYZ Gaudin model
The eigenvectors of the Hamiltionians of the XYZ Gaudin model are constructed
by means of the algebraic Bethe Ansatz. The construction is based on the
quasi-classical limit of the corresponding results for the inhomogeneous higher
spin eight vertex model.Comment: 11 pages, Latex file; minor correction
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
Contractions of quantum algebraic structures
A general framework for obtaining certain types of contracted and centrally
extended algebras is presented. The whole process relies on the existence of
quadratic algebras, which appear in the context of boundary integrable models.Comment: 6 pages, Latex. Proceedings contribution to the "9th Hellenic School
on Elementary Particle Physics and Gravity" Corfu, September 2009. Based on a
talk given by A.
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
B\"acklund Transformation for the BC-Type Toda Lattice
We study an integrable case of n-particle Toda lattice: open chain with
boundary terms containing 4 parameters. For this model we construct a
B\"acklund transformation and prove its basic properties: canonicity,
commutativity and spectrality. The B\"acklund transformation can be also viewed
as a discretized time dynamics. Two Lax matrices are used: of order 2 and of
order 2n+2, which are mutually dual, sharing the same spectral curve.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables
We analyze a completely integrable two-dimensional quantum-mechanical model
that emerged in the recent studies of the compound gluonic states in
multi-color QCD at high energy. The model represents a generalization of the
well-known homogenous Heisenberg spin magnet to infinite-dimensional
representations of the SL(2,C) group and can be reformulated within the Quantum
Inverse Scattering Method. Solving the Yang-Baxter equation, we obtain the
R-matrix for the SL(2,C) representations of the principal series and discuss
its properties. We explicitly construct the Baxter Q-operator for this model
and show how it can be used to determine the energy spectrum. We apply
Sklyanin's method of the Separated Variables to obtain an integral
representation for the eigenfunctions of the Hamiltonian. We demonstrate that
the language of Feynman diagrams supplemented with the method of uniqueness
provide a powerful technique for analyzing the properties of the model.Comment: 61 pages, 19 figures; version to appear in Nucl.Phys.
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