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

9

Here the S, S3 obey the same commutation relations as (1.4). We choose the L(u) operator to be (3.3) )S; sn(u? e ) 3=1 n X 1 B (u)= 2sn(u? e ) (1+ dn(u? e ))S?+ (1? dn(u? e ))S+];=1 n X C (u)= 2sn(u1? e ) (1+ dn(u? e ))S++ (1? dn(u? e ))S?]:=1

A(u)=

n X cn(u? e

The determinant of L(u) is once again a generator of the constants of the motion. It has the form (3.4) where det L(u)=n X

H E (u? e+ iK 0 )+ H0

=1

sn(e? e ) S1 S1

+ dn(e? e )S2 S2+ cn(e? e )S3 S3]; X E( H0= 2k2 0 sn(e? e ) S1 S1+ dn(e? e )S2 S2+ cn(e? e )S3 S3] e?e ); (3.5)

H= 2k2

X6=

1

?

X0;

k cn(e? e )S2 S2+ dn(e? e )S3 S3]?2

n X

2 2 k2S2+ S3]:

Here E (z)= z dn2(u)du is Jacobi's epsilon function. The same is now true as for the case of XXZ r-matrix algebras: if we subject the e 's and the Si 's to the transformations given by (2.11), then we arrive at the generating function for the constants of motion for a root structure having the signature N1; N2;::::; Np. The expression for this function is (3.6) det L(u)=J? X NX3 1

R

=1

k=1

r=0

!2@ r f (u? J e )(Z J ); 1 NJ?r k@ J e1 k

where f1 (z)= 1=sn(z), f2(z)= dn(z)=sn(z), and f3(z)= cn(z)=sn(z). As an example, the generating function corresponding to signature 2,1 is det L(u)= H1E (u? e1+ iK 0 )+ H2E (u? e3+ iK 0 )+ H3+ sn4 (u1? e ) H4 1 cn(u? e1 )dn(u? e1 ) H+ 1 1+ 5 sn3(u? e ) sn2(u? e ) H6+ sn2(u? e ) H7;1 1 3