SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET(9)

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