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

4

E.G. KALNINS, V.B. KUZNETSOV AND WILLARD MILLER, JR.

The changing of variables x; p for the new variables ui; vi is the procedure of separation of variables. The matrix elements of the L-operator can be expressed in terms of these variables according to the formulas?? A(u)=iB (u) 2 cosh u+ n=11 uj? n=1 e vn j (2.5)+n?1 X

?2vj=1 sinh(u? uj ) k=j sinh(uj? uk ) 6 j=1n?1 X

n

3 sinh(uj? e ) 5 e?un: 3 sinh(uj? e ) 5:

The entry C (u) can be computed by using the formula?? p=x e?un 2 cosh n=11 uj? 6= e vn j (2.6)+n

?2vj=1 sinh(e? uj ) k=j sinh(uj? uk ) 6 j=1

This gives the relation between the coordinates x; p and ui; vi where vn= H0. The equation for the eigenvalue curve?: det(L(u)? I )= 0, has the form 2? A(u)2? B (u)C (u)= 0: If we put u= uj; j= 1;:::; n?

1 into this equation then= vj . Thus variables uj and vj (j= 1;:::; n? 1) lie on the curve?: (2.7)

vj?2

n X

=1

2 2 H coth(uj? e )? H0 vj+ det L(uj )= 0:

Equations (2.7) are the separation equations forP degrees of freedom connected the with the values of the integrals H . (Note that n=1 H= 0.) For illustrative reasons it is more transparent to use the variables Ai= eei and Ui= eui . Then many of the expressions given have algebraic form. For example, the nearest object we have to a Hamiltonian in the case of XXZ r-matrix algebra P is H= n=1 A2Hi which has the form i i

H=(2.8) Note that

n sinh(ui? e )=1 j=i sinh(ui? uj ) 6 i=1 n?1? n 2 2 X 2 2 k Ui pUi+ H0 U 2 k=1(AU 2?? U) ):= (4 n=1Aj )?1(?1)n j 2 6 i j=i ( i j i=1

n?1 X

e?

Pj6=i uj?? uie2

2 vi2+ H0

n U1 Un?1 X A x2: A1 An=1 If we adopt the standard procedure and write pUi=@W=@UiP the equation then? H= E admits separation of variables via the usual ansatz W= n=11 Wi (Ui ): The i separation equations can be written in the alternate form n?1 2 n (U 2? A2 )@Wj= H0 U 2n?2+ H0 (?1)n? k A2 U?2+ X k U 2k?2; k j k=1 j k j j@Uj k=1

eun=