VIOLATIONS OF THE STRING HYPOTHESIS IN THE SOLUTIONS OF THE BETHE-ANSATZ EQUATIONS IN THE XXX-HEISENBERG MODEL

We study the equations for the quasi-momenta in the Bethe ansatz for t he XXX-Heisenberg model. We show in a simple analytical fashion, that the usual ''string hypothesis'' incorrectly predicts the number of rea l solutions and the number of complex solutions for N > 21 in the sect or with two spins flipped, confirming the work of Essler et al. We als o introduce a new set of variables which allows the equations to be wr itten as a single polynomial equation in one variable. We consider in some detail the case of three spins flipped.