SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET(7)

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