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arXiv:nlin/0007041 · Journal of Physics A Mathematical and General

Bäcklund transformations for thesl(2) Gaudin magnet

Andrew N. W. Hone, V B Kuznetsov, O. Ragnisco

Published Mar 14, 2001• 24 citations
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Abstract

Elementary, one- and two-point, Bäcklund 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 2 × 2 Lax matrix.

DOI
10.1088/0305-4470/34/11/336

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Elementary, one- and two-point, Bäcklund 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 2 × 2 Lax matrix.
Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Baecklund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable mu for the parameter lambda of the BT B_lambda such that mu belongs to the spectrum of the Lax operator L(lambda). As a consequence, the generating function of the composition of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by mu is introduced.
We construct integrable cases of generalized classical and quantum Gaudin spin chains in an external magnetic field. For this purpose, we generalize the 'shift of argument method' onto the case of classical and quantum integrable systems governed by an arbitrary -valued, non-dynamical classical r -matrix with spectral parameters. We consider several examples of the obtained construction for the cases of skew-symmetric, 'twisted' non-skew-symmetric and 'anisotropic' non-skew-symmetric classical r -matrices. We show, in particular, that in a general case in order for the Gaudin system in a magnetic field to be integrable, the corresponding magnetic field should be non-homogeneous.
We construct generalized Gaudin systems in an external magnetic field corresponding to arbitrary so(3)-valued non-skew-symmetric r-matrices with spectral parameters and non-homogeneous external magnetic fields. In the case of r-matrices diagonal in the sl(2) basis we calculate the spectrum and the eigen-values of the corresponding generalized Gaudin hamiltonians using the algebraic Bethe ansatz. We explicitly consider several one-parametric families of non-skew-symmetric classical r-matrices and the corresponding generalized Gaudin systems in a magnetic field. We apply these results to fermionic systems and obtain a wide class of new integrable fermionic BCS-type hamiltonians.