^2+L_s^{-2}\right]
^{1/2}$, $L_s$ being the spin-spin correlation length, $L_s\rightarrow
\infty $ in our case of a long range order state of the spin subsystem. Below this renormalization is taken into account empirically by the substitution $$\label{renorm}\Gamma ({\bf k},{\bf q})\rightarrow \tilde \Gamma ({\bf k},
{\bf q})=\Gamma ({\bf k},{\bf q})\sqrt{\left( 1+\gamma _{{\bf q}}\right) .}$$ The introduced vertex correction is proportional to $\left| {\bf q}-{\bf q}
_0\right| $ for ${\bf q}$ close to ${\bf q}_0$ . We have used also the following two functions for the vertex correction $\sqrt{1+\gamma _{{\bf q}
}^3}$ and $\sqrt{1+\gamma _{{\bf q}}^5}$ and have obtained the results similar to those presented below. Let us mention that the bare vertex leads the dramatic decrease of the QP bandwidth.

Solution of the integral equation
=================================

The equation (\[gineq\]) is usually solved by an iteration procedure. We propose here an alternative way, based on the continued fraction expansion of $G({\bf k},z)$

$$\label{cf1}G({\bf k},z)=\frac{b_0^2}{z-a_0-}\frac{b_1^2}{z-a_1-}\cdots \frac{
b_n^2}{z-a_n-}\cdots , \quad a_n=a_n({\bf k}),\quad b_n=b_n({\bf k}),$$

where $$b_0^2=\int_{-\infty }^{+\infty }A_p({\bf k},\omega )d\omega =K_{{\bf k}
},\quad a_0=\frac 1{b_0^2}\int_{-\infty }^{+\infty }\omega A_p({\bf k}
,\omega )d\omega =\Omega _{{\bf k}}.$$

The coefficients $b_n,a_n,n>0$ are related with the spectral density $ A_p({\bf k},\omega )$ via the set of orthogonal polynomials $P_n(\omega )$, 