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\end{equation}  and similar equations for $G_{31}$, etc. This sequence of Green functions for $G_{n1}$ generates an infinite number of equations. Now, it comes the trick. Consider that site $2$ is in reality another semi-infinite chain coupled to site $1$. Then, the equation for $G_{21}$ becomes simply  \begin{equation}  G_{12}=g_{ch} t_{21} G_{11} G_{11}.  \end{equation}  By replacing this expression for $G_{12}$ in $G_{11}$ we can close the Dyson equation and to obtain explicitly $G_{11}$. Therefore  \begin{equation}  G_{11}=g_{11}+ g_{11}|t|^2 g_{ch} G_{11} G_{11}.  \end{equation}  Besides, we notice that $G_{11}$ corresponds to a semi-infinite chain itself, thus  \begin{equation}  g_{ch}=g_{11}+g_{11}|t|^2 g_{ch}^2  \end{equation}  With the notorious result  \begin{equation}  g_{ch}=\frac{\omega-\epsilon_0}{2t^2}\pm\frac{1}{2t^2} \sqrt{(\omega-\epsilon_0)^2-4t^2},  \end{equation}