\hat{U}_0(+\infty, -\infty){\left| \Psi(-\infty) \right\rangle} + {\left| \Psi_1 \right\rangle} + {\left| \Psi_2 \right\rangle}, \label{Psi_at_plus_infty}\\
{\left| \Psi_1 \right\rangle} &=& -2\pi i{ \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_{n'} \int dk dk' \delta\left( E_{n'} + \frac{k'^2}{2M} - E_n - \frac{k^2}{2M} \right) \nonumber\\
&& \times {\left| n' k' \right\rangle} W_{n n'}(k-k') \langle k{\left| \psi_{in} \right\rangle}, \\
{\left| \Psi_2 \right\rangle} &=& -2\pi i{ \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_{n'', n'} \int dkdk'dk'' {\left| n'' k'' \right\rangle}\langle k {\left| \psi_{in} \right\rangle} \nonumber\\
&& \times \delta\left( E_{n''} + k''^2 /[2M] - E_n - k^2/[2M]\right) \nonumber\\
&&\times \frac{ W_{n' n''}(k'-k'') W_{n n'}(k-k')}{E_n + k^2/(2M) - E_{n'} - k'^2/(2M) + i0},\end{aligned}$$ where $W_{n n'}(k - k') = {\left\langle{k' n'}\right|} V_1 + V_2 {\left| n k \right\rangle}$ and ${\left| \Psi_{1,2} \right\rangle}$ are the first- and second-order corrections, respectively. Higher-order corrections can be derived in a similar manner, but what is important for our further analysis is that they are functions of $W$.

We simplify the matrix element $W$ by representing it as follows $$\begin{aligned}
\label{W_initial_expression}
W_{n n'}(k - k') &=& \int \frac{dR d\rho dq}{2\pi} e^{i(k-k')R}\phi_{n'}^*(\rho)\phi_n(\rho) \nonumber\\
&& \times [ V_1(q)\delta(R - \mu\rho/m_1 - q) \nonumber\\
&& + V_2(q)\delta(R + \mu\rho/m_2 - q)],\end{aligned}$$ where $\phi_n(\rho) = \langle \rho {\left| n \right\rangle}$. After trivial integration 