, i.e., the point where we have the crossing of the functions $y=\omega -\Omega 
_{{\bf k}}$ and ${\rm Re}\Sigma ( {\bf k},\omega )$ , see Figs.\[f3\] (a) and \[f3\] (b). In Fig. \[f3\] (d) we show $A_p({\bf k}_1,\omega )$ calculated for $\eta =0.002$ (solid line) and $\eta =0.0005$ (dashed line) in order to study the scaling behavior of the peaks and their widths with respect to changes in $\eta $. Both peaks fit quite closely with a Lorentzian $(1/\pi )\left\{ Z({\bf k} _1)\eta /\left[ {\left( 
\omega -\epsilon ({\bf k}_1)\right) }^2+\eta ^2\right] \right\} $ , $\epsilon ({\bf k}_1)$ is the location of the peak, which in the limit $\eta \rightarrow 0$ becomes $Z({\bf k}_1)\delta 
(\omega -\epsilon ({\bf k}_1))$. It means that ${\rm Im}\Sigma 
({\bf k},\omega _p)\rightarrow 0$ in the same limit. Here and below we shall speak about the position $\epsilon ({\bf k})$ of such peaks (with imaginary part of pole close to zero) as about the energy of the quasiparticle.

Figs. \[f3\] (a) and \[f3\] (b) also demonstrate that the incoherent part of $A_p({\bf k},\omega )$ increases and the pole strength decreases with the increase of $\epsilon ({\bf k})$, $Z({\bf k}_1)=0.82,Z({\bf k} _2)=0.347$. Let us recall that $\Omega _{{\bf k}}$ 