Estudos sobre as equações de Bethe

Abstract

In this dissertation we made an analytic study of the Bethe Ansatz equations for the XXZ six vertex model with periodic boundary conditions. We had show that the Bethe Ansatz equations deduced from the algebraic and coordinate Bethe Ansatze are related by a conformal map. This allowed us to reduce the Bethe Ansatz equations to a system of polynomial equations. For the one, two and three magnon sectors, we succeeded in decouple these equations, so that the solutions could be expressed in terms of the roots of some self-inversive polynomials, Pa (z). Through new theorems deduced here about the distribution of the roots of self-inversive polynomials in the complex plane, we did a thorough analysis of the distribution of the Bethe roots for the two-magnon sector. This analysis allowed us to show that the Bethe Ansatz is indeed complete for this sector, except at some critical values of the anisotropy parameter A, in which the polynomials Pa (z) may have multiple roots. Finally, an unexpected connection between the Bethe Ansatz equations and the Salem polynomials was found and a new algorithm for search small Salem numbers was elaborated.