SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET(5)

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