SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET(4)

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

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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=