Classical Functional Bethe Ansatz for $SL(N)$: separation of variables for the magnetic chain

Details Classical Functional Bethe Ansatz for $SL(N)$: separation of variables for the magnetic chain Classical Functional Bethe Ansatz for $SL(N)$: separation of variables for the magnetic chain

1994-03-04

arxiv.org/abs/hep-th/9403030v1

J.Math.Phys. 35 (1994) 5831-5843

Physics

High Energy Physics

High Energy Physics - Theory

14pp LaTex,DAMTP 94-17

Scientific paper

10.1063/1.530712

The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which gives separation variables for systems for which an $R$-matrix is known. Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated) $R$-matrices. In this paper I advance Sklyanin's program by giving the FBA for certain systems with $SL(N)$ $R$-matrices. This is achieved by constructing rational functions $\A(u)$ and $\B(u)$ of the matrix elements of $T(u)$, so that, in the generic case, the zeros $x_i$ 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.

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