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