SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET(3)

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

3

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: