Abstract. In this work we generalise previous results connecting (rational) Gaudin magnet models and classical separation of variables. It is shown that the connection persists for the case of linear r-matrix algebra which corresponds to the trigonometric

SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET

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i.e., that det L(u) is a generating function of the constants of the motion. In particular we have (1.8) n X C 2 2? det L(u)= A (u)+ B (u)C (u)=+ H coth(u? e )+ H0; 2 sinh (u? e )=1 where C= (S3 )2+ S+ S? are the Casimir elements of the algebra generated P by elements S+; S? and S3 . Furthermore H0= S3 and (1.9) H=

X6=

2S3 S3 coth(e? e )+ sinh(e1? e ) (S+ S?+ S+ S? ):

With the following realization of the algebra in terms of the canonical coordinates x and p, fp; x g=?:

i 1 i S+= 2 x2; S?= 2 p2; S3=? 2 x p; the constants (1.9) have the form X?x2 p2+ x2 p2? 2x x p p cosh(e? e );?1 (1.12) H= 1 4 6= sinh(e? e )(1.10) . Notice that all C= 0 in such a representation. 2. Variable Separation for the XXZ Magnet. Proceeding as in 4,7,8], we choose separable coordinates such that B (u)= 0, i.e., u= uj; j= 1;:::; n? 1. P x2=sinh(u? e )= 0 for u= u;:::; u, which in turn implies This implies 1 n?1 that we choose coordinates according to (2.1)n X

and H0=? 1 2

P xp

x

2

= eun

n?1 sinh(uj? e ) j=1; 6= sinh(e? e )

motivated by the general formula (2.2)n?1 x2 un j=1 sinh(u? uj ): sinh(u? e )= e n=1 sinh(u? e )=1

For each uj we can de ne the canonically conjugate coordinate vj as follows:n 1 X coth(u? e ) x p; 1 j n? 1; (2.3) vj= A(uj )=? 2 j=1

v n= H0:

The coordinates ui; vj (i; j= 1;:::; n) satisfy the canonical bracket relations (2.4)

fui; uj g= 0; fvi; uj g=

ij;

fvi; vj g= 0: