Reformulation of the Nagaoka Equations

The Green's-function equations describing an electron gas in the presence of a magnetic impurity have been approximated by Nagaoka by means of a truncation procedure that keeps correlations between the spins of the impurity and a conduction electron, but neglects higher correlations. Nagaoka's approximate equations are examined here and reformulated as a one-dimensional nonlinear equation. This equation is solved for temperatures not too close to $T_K$, a characteristic temperature of the system, and shown to have two solutions: a normal solution which exists at all temperatures and a condensed solution valid only below $T_K$. Perturbative corrections to the normal solution are presented.