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03/06/03 Colloquium
DR. MOURAD ISMAIL
UNIVERSITY OF SOUTH FLORIDA
UNIVERSITY OF SOUTH FLORIDA
A New Twist on an Old Problem of Heine and Stieltjes
Abstract: We survey the Heine-Stieljes problem of finding polynomial solutions to differential equations and its connection with electrostatic equilibrium problems. We then discuss the quantized version of this problem which is based on joint work, in progress, with Shao-shiung Lin and Shi-shyr Roan. Our quantized version turned out to be closely related to the algebraic Bethe Ansatz of the spin XXZ model of a finite size. We have demonstrated that solutions of the algebraic Bethe Ansatz give rise to the polynomial solutions of a second order -difference equation in terms of Askey-Wilson operator. The more general form of algebraic Bethe Ansatz and the mathematical problems relevant to the physics of XXZ model will be discussed. A similar correspondence between Bethe Ansatz of XXX model and the Sturm-Liouville type difference equation in terms of the Wilson operator has also been found.