\right\rangle}$, and eigenvalues, $E_n$, of the internal motion are the solutions of the problem $$\begin{aligned}
\label{InternalMotionEigenstates}
\left[ \frac{-1}{2\mu}\frac{d^2}{d\rho^2} + U(\rho) \right]{\left| n \right\rangle} = E_n{\left| n \right\rangle},\end{aligned}$$ where the index $n$ denotes bound and continuous states. Introducing ${\left| n k \right\rangle} \equiv {\left| n \right\rangle} \otimes {\left| k \right\rangle}$, where ${\left| k \right\rangle}$ is an eigenfunction of the free motion of the center of mass $\langle R {\left| k \right\rangle} \equiv \exp(ikR)/\sqrt{2\pi}$ – a plane wave and ${\left| n \right\rangle}$ is an eigenstate of the internal motion, the unperturbed propagator reads $$\begin{aligned}
\hat{U}_0(t_f, t_i) = { \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_n \int dk e^{-i\left(E_n + \frac{k^2}{2M}\right)(t_f - t_i)}{\left| n k \right\rangle}{\left\langle{k n}\right|}\end{aligned}$$ The total propagator is a solution of the Lippmann-Schwinger equation written in the “post” form $$\begin{aligned}
\hat{U}(t_f, t_i) &=& \hat{U}_0(t_f, t_i) \nonumber\\
&-& i \int_{t_i}^{t_f} dt \hat{U}_0(t_f, t) [V_1 + V_2] e^{\epsilon t} \hat{U}(t, t_i),\end{aligned}$$ where we set $\epsilon \to 0$.

Assuming that the initial condition ${\left| \Psi(t_i) \right\rangle} \equiv {\left| n \right\rangle}\otimes{\left| \psi_{in} \right\rangle}$, where ${\left| n \right\rangle}$ is one of the eigenstates (\[InternalMotionEigenstates\]) and ${\left| \psi_{in} \right\rangle}$ is a wave packet localized before the barriers \[see, e.g., Eq. (\[InitialCondition\])\], we obtain $$\begin{aligned}
{\left| \Psi(+\infty) \right\rangle} &\approx& 