Anderson Model of Localized Magnetic Moments. II. "Derivation" of the Integral Equation

In this, the second in a series of papers on the Anderson model, we "derive" the integral equation satisfied by $G_d^{}{}_{}{}^{\sigma }\left(\omega \right)$, the Green's function of the $d$ electron of spin $\sigma$ (allowing the possibility of a weak external magnetic field). The "derivation" of the equation rests on two observations: first, the recognition that the theory is unrenormalizable, so that even the "leading" logarithmic terms are not treated correctly by any finite truncation of the equations of motion; second, the observation that the first three terms in the expansion of the "mass operator" in powers of [$\frac{\Delta }{U\xi (1-\xi )}$] $\ln \left(\frac{W}{T}\right)$ (i.e., all terms up to and including the ${\left\{\right[\frac{\Delta }{U\xi }(1-\xi )\left] \ln \right(\frac{W}{T}\left)\right\}}^3$ terms) form a geometric series, with the subsequent assumption that this behavior persists to all orders. One consequence of this assumption is that $T_C$ [the temperature at which the real part of the denominator of $G_d^{}{}_{}{}^{\sigma }\left(\omega \right)$ vanishes] is proportional to $e^{\frac{1}{J\rho }}$, in agreement with the value obtained in the $s-d$ exchange model.