51 research outputs found
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
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
Baxter's Q-operator for the homogeneous XXX spin chain
Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator
for the homogeneous XXX model as integral operator in standard representation
of SL(2). The connection between Q-operator and local Hamiltonians is
discussed. It is shown that operator of Lipatov's duality symmetry arises
naturally as leading term of the asymptotic expansion of Q-operator for large
values of spectral parameter.Comment: 23 pages, Late
New boundary conditions for integrable lattices
New boundary conditions for integrable nonlinear lattices of the XXX type,
such as the Heisenberg chain and the Toda lattice are presented. These
integrable extensions are formulated in terms of a generic XXX Heisenberg
magnet interacting with two additional spins at each end of the chain. The
construction uses the most general rank 1 ansatz for the 2x2 L-operator
satisfying the reflection equation algebra with rational r-matrix. The
associated quadratic algebra is shown to be the one of dynamical symmetry for
the A1 and BC2 Calogero-Moser problems. Other physical realizations of our
quadratic algebra are also considered.Comment: 22 pages, latex, no figure
Dynamical boundary conditions for integrable lattices
Some special solutions to the reflection equation are considered. These
boundary matrices are defined on the common quantum space with the other
operators in the chain. The relations with the Drinfeld twist are discussed.Comment: LaTeX, 12page
Separation of variables for A2 Ruijsenaars model and new integral representation for A2 Macdonald polynomials
Using the Baker-Akhiezer function technique we construct a separation of
variables for the classical trigonometric 3-particle Ruijsenaars model
(relativistic generalization of Calogero-Moser-Sutherland model). In the
quantum case, an integral operator M is constructed from the Askey-Wilson
contour integral. The operator M transforms the eigenfunctions of the commuting
Hamiltonians (Macdonald polynomials for the root sytem A2) into the factorized
form S(y1)S(y2) where S(y) is a Laurent polynomial of one variable expressed in
terms of the 3phi2(y) basic hypergeometric series. The inversion of M produces
a new integral representation for the A2 Macdonald polynomials. We also present
some results and conjectures for general n-particle case.Comment: 31 pages, latex, no figures, Proposition 12 correcte
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
Exact Diagonalisation of The XY-Hamiltonian of Open Linear Chains with Periodic Coupling Constants and Its Application to Dynamics of One-Dimensional Spin Systems
A new method of diagonalisation of the XY-Hamiltonian of inhomogeneous open
linear chains with periodic (in space) changing Larmor frequencies and coupling
constants is developed. As an example of application, multiple quantum dynamics
of an inhomogeneous chain consisting of 1001 spins is investigated. Intensities
of multiple quantum coherences are calculated for arbitrary inhomogeneous
chains in the approximation of the next nearest interactions.
{\it Key words:} linear spin chain, nearest--neighbour approximation,
three--diagonal matrices, diagonalisation, fermions, multiple--quantum NMR,
multiple--quantum coherence, intensities of multiple--quantum coherences.
{\it PACS numbers:} 05.30.-d; 76.20.+qComment: 21 pages + 1 figure (to download separately via ps-format
- …