The Green's function in the problem of charge dynamics on a one-dimensional lattice with an impurity center

In the “tight-binding” approximation (the Hückel model), we consider the evolution of the charge wave function on a semi-infinite one-dimensional lattice with an additional energy $U$ at a single impurity site. In the case of the continuous spectrum (for $|U|<1)$ where there is no localized state, we construct the Green's function using the expansion in terms of eigenfunctions of the continuous spectrum and obtain an expression for the time Green's function in the form of a power series in $U$. It unexpectedly turns out that this series converges absolutely even in the case where the localized state is added to the continuous spectrum. We can therefore say that the Green's function constructed using the states of the continuous spectrum also contains an implicit contribution from the localized state.