SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET(8)

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.