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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For the case of signature N1; N2;:::; Np the coordinates are given by the relationsj X i=1

xJ xJ+1?i= i j

@ r n=1 ((AJ )2? Ui2): 1 i@AJ AJ L6=J (2AL)NL 1 1 1 This gives a complete description of the separation of variables procedure for the signature N1; N2;:::; Np case. We illustrate these ideas with two examples. A. The case of signature 2,1 and dimension 3. In this case the generating function assumes the form?(Z 1) (Z 2) 1 1 (Z 1:Z 1+ Z 1:Z 1)? 1 det L(u)= 1 1 2 2 2 sinh(e1? e3 ) 1 2 1 2 sinh (u? e1 ) 2 1 2 1 2+(Z1 )1 (Z1 )1+ cosh(e1? e3 )(Z1 )3(Z1 )3g 1 1? 1 2 1 2+ Z1:Z1 coth(u? e1 ) sinh(e1? e ) (Z2 )1(Z1 )1+ (Z2 )2(Z1 )2 4 2 sinh (u? e1 ) 1 3?(Z 1) (Z 2) 1 1 2+ cosh(e1? e3 )(Z2 )3 (Z1 )3? 2 sinh (e1? e3 ) 1 3 1 3 1 2 1 2+ cosh(e1? e3 )((Z1 )2(Z1 )2+ (Z1 )1(Z1 )1?? 2 1 2 1? coth(u? e3 ) sinh(e1? e )? (Z1 )2(Z2 )2+ (Z1 )1(Z2 )1 1 3?(Z 2) (Z 1) 1 2 1+ cosh(e1? e3 )(Z1 )3 (Z2 )3+ sinh2(e1? e3 ) 1 3 1 3? 2 1 2 1+ cosh(e1? e3 ) (Z1 )1 (Z2 )1+ (Z1 )2 (Z2 )2: The constants of the motion can be deduced from the coe cents of independent functions of u. In the coordinate representation these

constants have the form? H1= x2p2+ p2 x2? 2x1 x2p1 p2+ sinh(e1? e ) x2p2+ x2 p2? 2x1 x3p1 p3; 1 2 1 2 1 3 3 1 1 3? H2= sinh(e2? e ) x1 x2p2+ p1 p2x2? (x1x3 p2 p3+ p1 p3 x2x3 ) cosh(e1? e3 ) 3 3 1 3? 21 (x1p3+ p1 x3)2; sinh (e1? e3 ) where we have used the notation x1= x1; x2= x1 and x3= x2, with similar 1 2 1 relations for the pi 's. The coordinates are given by the formulas? e1 ) x2=? sinh(u1sinh(e sinh(u2? e1 ); 1 1? e3 ) 2x1 x2=? sinh(u1? e1 ) sinh(u2? e1 ) cosh(e1? e3 )? sinh(e1? e ) sinh2(e1? e3 ) 1 3 (sinh(u1? e1 ) cosh(u2? e1 )+ sinh(u1? e3 ) cosh(u2? e3 )); x2= sinh(u1? e3 ) sinh(u2? e3 ): 3 sinh2 (e1? e3 )

r=1 L )NL] L6=J (A1 (2NJ )AJ r! 1

n Uk] k=1

(X j?2

(NJ? r? 3) r?2(NJ+ q? 2)] q=0