SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET

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

E.G. KALNINS, V.B. KUZNETSOV and WILLARD MILLER, Jr.Department of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand Department of Mathematics y x University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands School of Mathematics z and Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455, USA.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 4 r-matrix (of the XXZ type). We comment also on the corresponding problem for the elliptic (XYZ) r-matrix. A prescription for obtaining integrable systems associated with multiple poles of an L-operator is given.

1. Introduction. Separation of variables for the Hamilton-Jacobi and Schrodinger

equations have long been known as methods for explicit solution of these equations in appropriate circumstances. The technical requirements for this method of solution have quite fully developed in recent years (see 1{8]). In particular the relationship between the separable systems and the Gaudin magnet 4,9] integrable systems models has been established via r-matrix algebra methods, where the rmatrix corresponds to the rational or so called XXX case, 4{8]. This relationship works very clearly with separable coordinate systems on spaces of constant curvature. The question we answer here is how these notions can be extended to include the so-called trigonometric r-matrix algebra in the XXZ case. To do this let usy Work supported by the Netherlands Organisation for Scienti c Research (NWO) z Work supported in part by the National Science Foundation under grant DMS 94{00533 x On leave from Department of Mathematical and Computational Physics, Institute of Physics,Typeset by AMS-TEX

St. Petersburg University, St. Petersburg 198904, Russia

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

2

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

recall the fundamental ideas of the r-matrix formalism (see 10,11] and references in there). For a classical mechanical system the basic (linear) r-matrix algebra is (1.1)

fL(u) I; I L(v)g= r(u? v); L(u) I+ I L(v)];L(u)11=?L(u)22= A(u); L(u)12= B (u); L(u)21= C (u);

where f; g is the Poisson bracket and;] the matrix commutator bracket. The operator L(u) is taken to be the 2 2 matrix and r(u) is a suitable 4 4 matrix of scalars solving the classical Yang-Baxter equation 10,11]; u being arbitrary constant is called the spectral parameter. In the case of the XXZ r-matrix algebra the non zero elements of r can be taken to be 1 (1.2) r(u)11= r(u)44= coth(u); r(u)23= r(u)32= sinh(u): In component form, the r-matrix algebra relations are (1.3)

1 fA(u); C (v)g= sinh(u? v) (? cosh(u? v)C (v)+ C (u));? fB (u); C (v)g= sinh(u2? v) (A(u)? A(v)): If we now ma

ke the ansatz A(u)= coth(u)S3; B (u)= (1= sinh(u))S+ and C (u)= (1= sinh(u))S? these relations imply (1.4)n X

fA(u); A(v)g= fB (u); B (v)g= fC (u); C (v)g= 0; 1 fA(u); B (v)g= sinh(u? v) (cosh(u? v)B (v)? B (u));

fS3; S g= S;

fS+; S?g= 2S3:n X

To relate this observation to the separation of variables methods, we form the L(u) operator with elements 1 B (u)= sinh(u? e ) S+;=1

C (u)=

(1.5) where (1.6) (1.7)

1 sinh(u? e ) S?;=1

A(u)=

n X=1

coth(u? e ) S3;

fS3; S g=

S;

fS+; S? g= 2 S3:

The r-matrix algebra relations, (1.1) or (1.3), imply

fdet L(u); det L(v)g= 0;

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:

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=

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

5

where the k are related to the Hj via

P2n?4 (u)k= H01

2u?2

n (u2? A2 ) k=1 k

X<:::< n?k

n X HiAi u?A;2

i=1

2

2

i

(?1)n?k A2 1

1 A2 n?k+ (2k? 2)! P2(2k?2) (0); n?4n X k=1

n?1= E? H0

A2: k

From what has been developed so far we see that separation of variables goes through for XXZ r-matrix algebras constructed in this way. In the previous article 8] for the case of spaces of constant curvature we essentially have the rational rmatrix algebra and it is possible to formulate using well de ned limiting procedures the cases of integrable systems for which some of the ei parameters are equal. What was also established previously was the construction of integrable systems given on the algebra with commutation relations (2.9)J f(ZjJ )`; (Zk 0 )mg=? JJ 0 (ZjJ+k?NJ )s`ms;

where`; m; s= 1; 2; 3; 0< j< NJ; 0< k< NJ 0; 0< J p and`ms is the usual totally antisymmetric tensor, and the vector ZjJ has the form (2.10)

0 P (pJ pJ i i j J=@ i P (pJ pJ Zj j i iP1 4 4

1 J J?i+ xi xj+1?i ) J J A+1?i? xi xj+1?i ) i J xJ i pi j+1?i 2+1

in the coordinate representation. Indeed, if we adopt the limiting procedure

AJ ! AJ+ J 1?1; j= 1;:::; NJ; J= 1;:::; p; j 1 j !(2.11) whereJ i?1= i (J 1? J 1 );`=2 j?1`?2 j+1?i NJ J+ XJ p1 ! i=2 NJ q X xJ ! aJ xJ+ J j j 1 i=2

pJ j

q

aJ j

i?1 pJ j+1?i i

i?1 xJ j+1?i i

!

;

;1

aJ= j

; k6=j (J 1?1? J 1?1 ) j k

and N1+:::+ Np= n+ 1, then the Hamiltonian H has the form (2.12)

n p 2 2 N X? 2+1 2 k? ) k pUi+ H0 Uk=1 (A(U 2 Ui U 2 ) H= ( n=1 Aj )?1 j 2 i j 6=i i? j i=1

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

6

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

with obvious separation equations. (We require J i= 0 and take the limit as the 0 J 1 ! 0 for h= 1;:::; NJ? 1, se

e 8].) The generating function for the constants h can be derived by applying these procedures to det L(u). We will, however, adopt a di erent and more general strategy. If we leave the matrix elements of L(u) in the form (1.5) and subject the resulting expression for det L(U ) 1? 2 det L(U )= 2 (2.13) where U= eu, and S+n 1 X=1

S3

!

2

n X S

3 U 2? A2?=1

2 n X X A Si !+ 2U 4 U?A i n n X X S !#2 2 2=1=1 2 2

S3+ U 2

=1

U 2? A2=1

3

;

= S1

iS2, to the transformations j= 1;:::; NJ; J= 1;:::; p; k= 0;:::; NJ;

AJ ! AJ+ J 1?1; j 1 j

NJ J S1 k0+ X(J 1 )k (J Sj )= Z J j?1 NJ?k; j=2

then we arrive at a general expression for the generating function det L(U ). The constants of the motion are obtained by the usual means of expanding the expression following from (2.13) in partial fractions and reading o the independant components. In the case of degenerate roots the expression can be readily modi ed. Accordingly we havep X J 1? 2 det L(U )= 1 2 J=1(ZNJ )3 22 2=1 1

!

2

(2.14)

+

0 p J? X@X NX 1+ 2U 6 4 i J j j! p J? X NX 1@ j1=1=0

2 !@ j Aj J (ZNJ?j )i A 2? A2@Aj U j

1

@Aj U? Aj J=1 j=0 j ! 0 p NJ?1 1 X@ J X?(ZNJ )3+ U 2 j=0 j ! J=12

1

!

2

J (ZNJ?j )3 j2

@@Aj

U? A2 j

1

!

13 J (ZNJ?j ) A5:3

From this expression constants of the motion can be deduced just as before. The separation of variables proceeds as usual in the case of the choice of coordinates as given in 8]. The expressions for the coordinates corresponding to multiple roots with signature N1; N2;:::; Np can be obtained from the generic case by the limiting procedures already outlined. In rational form the generic coordinates are (2.15)6=? x2= n=1(A2? Ui2 )] n=11Us] (kA2i AkA2 ): i j j s`6=i`? i

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

7

For the case of signature N1; N2;:::; Np the coordinates are given by the relationsj X i=1

xJ xJ+1?i= i j

@ r n=1 ((AJ )2? Ui2): 1 i@AJ AJ L6=J (2AL)NL 1 1 1 This gives a complete description of the separation of variables procedure for the signature N1; N2;:::; Np case. We illustrate these ideas with two examples. A. The case of signature 2,1 and dimension 3. In this case the generating function assumes the form?(Z 1) (Z 2) 1 1 (Z 1:Z 1+ Z 1:Z 1)? 1 det L(u)= 1 1 2 2 2 sinh(e1? e3 ) 1 2 1 2 sinh (u? e1 ) 2 1 2 1 2+(Z1 )1 (Z1 )1+ cosh(e1? e3 )(Z1 )3(Z1 )3g 1 1? 1 2 1 2+ Z1:Z1 coth(u? e1 ) sinh(e1? e ) (Z2 )1(Z1 )1+ (Z2 )2(Z1 )2 4 2 sinh (u? e1 ) 1 3?(Z 1) (Z 2) 1 1 2+ cosh(e1? e3 )(Z2 )3 (Z1 )3? 2 sinh (e1? e3 ) 1 3 1 3 1 2 1 2+ cosh(e1? e3 )((Z1 )2(Z1 )2+ (Z1 )1(Z1 )1?? 2 1 2 1? coth(u? e3 ) sinh(e1? e )? (Z1 )2(Z2 )2+ (Z1 )1(Z2 )1 1 3?(Z 2) (Z 1) 1 2 1+ cosh(e1? e3 )(Z1 )3 (Z2 )3+ sinh2(e1? e3 ) 1 3 1 3? 2 1 2 1+ cosh(e1? e3 ) (Z1 )1 (Z2 )1+ (Z1 )2 (Z2 )2: The constants of the motion can be deduced from the coe cents of independent functions of u. In the coordinate representation these

constants have the form? H1= x2p2+ p2 x2? 2x1 x2p1 p2+ sinh(e1? e ) x2p2+ x2 p2? 2x1 x3p1 p3; 1 2 1 2 1 3 3 1 1 3? H2= sinh(e2? e ) x1 x2p2+ p1 p2x2? (x1x3 p2 p3+ p1 p3 x2x3 ) cosh(e1? e3 ) 3 3 1 3? 21 (x1p3+ p1 x3)2; sinh (e1? e3 ) where we have used the notation x1= x1; x2= x1 and x3= x2, with similar 1 2 1 relations for the pi 's. The coordinates are given by the formulas? e1 ) x2=? sinh(u1sinh(e sinh(u2? e1 ); 1 1? e3 ) 2x1 x2=? sinh(u1? e1 ) sinh(u2? e1 ) cosh(e1? e3 )? sinh(e1? e ) sinh2(e1? e3 ) 1 3 (sinh(u1? e1 ) cosh(u2? e1 )+ sinh(u1? e3 ) cosh(u2? e3 )); x2= sinh(u1? e3 ) sinh(u2? e3 ): 3 sinh2 (e1? e3 )

r=1 L )NL] L6=J (A1 (2NJ )AJ r! 1

n Uk] k=1

(X j?2

(NJ? r? 3) r?2(NJ+ q? 2)] q=0

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

8

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

has the form 1 L(U )= (U 2? A2)6 32A8Z1:Z2 1 1 1+ (U 2? A2)5 16A2 5Z1:Z1+ (Z1)2 (Z2)2+ 2A1 ((Z1)3 (Z2)3+ (Z1 )1(Z2 )1)] 1 1 1+ (U 2? A2)4 2A4 33Z1:Z1+ 36Z1:Z2+ 8A2Z1:Z3+ 4A2Z2:Z3 1 1 1 1 1+ (U 2? A2)3 2A2 26A1Z1:Z2+ 8A2Z2:Z2 1 1 1+14A2Z1:Z3+ Z1:Z1+ 4A3Z3 Z2+ (Z1 )2 1 1 3 1+ (U 2? A2)2 2 A4Z3:Z3+ 6A3Z2:Z3 1 1?5?(Z1 )2+ (Z )2+ 6(Z ) (Z )+ 4(Z )2+ A2 1 1?6(Z )2 (2Z )+2 (1 ) (Z )3 2+ 4(Z1 ) (Z )2 3+ (Z )2+A1 Z2 1 1 1 2 2 1 2 2 3 1 3 1 1+ U 2? A2 2 A2Z3:Z3+ 2A1Z2:Z3+ (Z2)2+ (Z2 )2+ (Z1 )3(Z3)3: 1 2 1 1 The coordinates are given by 2 1 A4 U x2= 4 U1 U2? A1 U1+ U2+ 4U 1; 1 4 U1 2 1 U2 3 5 3 U U 1 2x1x2=? 8A U1 U2? A1 U1+ U2+ 8UAU; 8 1 2 1 1 2 3 2 U U 1 1 2x1 x3+ x2= 3UAU2+ 1 U1+ U2+ 8UAU: 2 8 2 8 2 1 1 2 1 3. The XYZ Magnet. These methods can be extended to the case of elliptic or XYZ r-matrix algebras. The only di erence is that in this case a solution of the problem via separation of variables is not yet known1 but the coalescing of indices goes through just as before. Indeed, the operator L(u) can be taken just as in (1.1). The non zero elements of the r-matrix in this case are? dn( u) r(u)11= r(u)44= cn(u); r(u)14= r(u)41= 1 2sn(u)u); sn( (3.1) We now make the ansatz 1 u) A(u)= cn(u) S3; B (u)= 2sn(u) (1+ dn(u))S?+ (1? dn(u))S+]; sn( 1 C (u)= 2sn(u) (1+ dn(u))S++ (1? dn(u))S?]:+ dn( r(u)23= r(u)32= 1 2sn(u)u):

B. The Case of Signature 3 and Dimension 3. The generating function

(3.2)1

See 12] where the variable separation has been done for the periodic classical XYZ-chain from which the system in question can be obtained through the limit.

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

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

10

E.G. KALNINS, V.B. KUZNETSOV AND WILLARD MILLER, JR.2 1 2 1 H1= k2 sn(e2? e ) (Z1 )1 (Z2 )1? k2cn(e1? e3 )(Z1 )3(Z2 )3 1 3 k2 cn(e1? e3 )dn(e1? e3 ) (Z 2 ) (Z 1 )+ dn(e1? e3 ) (Z 2) (Z 1);? 1 1 1 1 sn(e1? e3 ) sn(e1? e3 ) 1 3 1 3 H2=?H1;? 1 E (e e 1 2 2 2 H3= k2 sn(1?? 3e) ) H1+ k2 (Z1 )2? (Z2 )2? (Z1 )2? (Z1 )2? (Z1 )2 1 2 2 2 3 e1 3 2 1 2 1? 2k2 sn(e1? e3 )(Z1 )1 (Z1 )1+ 2dn(e1? e3 )(Z1 )3(Z2 )3; 1 1 1 1 H4= Z1:Z1; H5= 2Z1:Z2; 1 1 1 1 1 2 1 H6= Z2:Z2? (Z1 )2? (Z1 )2? k2(Z1 )2+ sn(e 2? e ) (Z1 )1(Z1 )1 1 2 3 1 3 cn(e1? e3 ) (Z 2) (Z 1);? 2 sn(e? e ) 1 3 2 3 1 3 2 2 H7= Z1:Z1:

where

We note that the ideas developed here also work in the case of separation of variables for spaces of constant Riemannian curvature, as developed in previous articles 6{8]. Indeed, in that case the rational r-matrix algebra is as before and the non zero elements of the r-matrix are (3.7)

r(u)11= r(u)44= r(u)23= r(u)32= 1:

The generating function of the constants of the motion for signature N1;::::; Np is then (3.8)

0 p NJ? X XX det L(u)=@3

1

k=1 J=1 j=0

(ZjJ+1 )k A (u? J e1 )NJ?j+ k

1

2

:

This is the generalisation of the generating function for separable coordinates on P spaces of constant curvature of dimension n= p=1 NJ+ 1. Indeed, if we use the j form (3.8) and if k= 0 for k= 1; 2; 3 then we have the generating function on the sphere for generic ellipsoidal coordinates, and if 1=?1=4, 2= 1=4, 3= 0 then we have the generating function of ellipsoidal coordinates in n-dimensional Euclidean space. As an example consider the system with signature 2,1. The generating function is thenZ:Z det L(u)= e1?e3:: 1) (? ((Z1?:eZ)22) (Z11 )2(Z21 )? (Ze121)?(eZ311 ) e 3+ u? e1 (u? e1 )2 1 (Z1 ):(Z 1 ) (Z 2 ):(Z 1 ) 1 1 2 2 1 2 (Z1 ):(Z1 )+ (Z1 ):(Z1 )? e11?e32? (e1?e3 )22+ (Z1 ):(Z1 ):+ (u? e )3 (u? e )4 u? e3 (u? e3 )2 1 12 1 ( 1) ( 2) 1 1

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

11

1. E.G

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