over $R$, we obtain $$\begin{aligned}
\label{Simplified_W_general}
&& W_{n n'}(k - k') = \mathrsfs{F}_{n n'}\left( \frac{\mu}{m_1}[k-k']\right)\int \frac{dq}{2\pi} e^{i(k-k')q} V_1(q) \nonumber\\
&& \quad + \mathrsfs{F}_{n n'}\left( \frac{\mu}{m_2}[k'-k]\right)\int \frac{dq}{2\pi} e^{i(k-k')q} V_2(q),\end{aligned}$$ where the quantity, $$\begin{aligned}
\label{Formfactor_Def}
 \mathrsfs{F}_{n n'}(p) = \int d\rho e^{ip\rho} \phi_{n'}^*(\rho)\phi_n(\rho),\end{aligned}$$ is called the form factor, and it is well known in the scattering theory. Its physical interpretation is the probability amplitude of transferring a momentum $p$ from the center of mass to the interparticle degree of freedom by making the transition $n\to n'$.

Now we consider the case of identical particles: $m_1 = m_2 = m$ and $U(-\rho) = U(\rho)$. Then, there are two types of the eigenstates of the internal motion: even ($+$), $\phi_n(-\rho) = \phi_n(\rho)$, and odd ($-$), $\phi_n(-\rho) = -\phi_n(\rho)$. Since $ \mathrsfs{F}_{n n'}(-p) =  \mathrsfs{F}_{n n'}(p)$ \[$\mathrsfs{F}_{n n'}(-p) =  -\mathrsfs{F}_{n n'}(p)$\] in the case of $\phi_n$ and $\phi_{n'}$ being of the same (different) parity, Eq. (\[Simplified\_W\_general\]) takes the form

$$\begin{aligned}
\label{W_equivalent_particles}
W_{n n'}(k - k')  &=&  \left\{ 
\begin{array}{ccl}
\mathrsfs{F}_{n n'}\left( [k - k']/2\right) \int \frac{dq}{2\pi}e^{i(k-k')q} [V_1(q) + V_2(q)], &\mbox{if}& \mbox{$\phi_n$ and $\phi_{n'}$ have the same parity} \\
\mathrsfs{F}_{n n'}\left( [k - k']/2\right) \int \frac{dq}{2\pi}e^{i(k-k')q} [V_1(q) - V_2(q)], &\mbox{if}& \mbox{$\phi_n$ and $\phi_{n'}$ have different parities}, 
\end{array}
\right.\end{aligned}$$

