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\begin{lemma} \label{9}
\only<1>{</div><div>For notational simplicity we set </div><div>$ \hat{\phi}_j=\hat{\phi}(k_j)$ and $\omega_j=\omega(k_j)$ for $k_j \in \mathbb{R}^3, j=1, 2$.
Let
\begin{align*}
E_j=\frac{|k_j|^2}{2m}+\omega_j, \quad j=1, 2, \quad
E_{12}=\frac{|k_1+k_2|^2}{2m}+\omega_1+\omega_2.
\end{align*}It is proven that $a_2(\Lambda)$ can be expanded as
$$ a_2(\Lambda)= \frac{2}{3m}\sum_{j=1}^8{\rm I}_j(\Lambda)+\frac{E_2(\Lambda)}{m}{\rm I}_9(\Lambda)-a_1(\Lambda) {\rm I}_{10}(\Lambda)+a_1(\Lambda)^2, $$
}
\only<2>{</div><div>where ${\rm I}_j$ are given by~
\begin{align*}
%\allowdisplaybreaks
&{\rm I}_1(\Lambda)=\frac{1}{4}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2}(\frac{|k_1|^2}{E_1^3}+\frac{|k_2|^2}{E_2^3})(\frac{1}{E_1}+\frac{1}{E_2})\frac{1}{E_{12}},
\end{align*}
\begin{align*}
&
{\rm I}_2(\Lambda)=\frac{1}{8}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2}(\frac{|k_1|^2}{E_1^4}+\frac{|k_2|^2}{E_2^4})\frac{1}{E_{12}},
\end{align*}
\begin{align*}
&
{\rm I}_3(\Lambda)=\frac{1}{8}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2}(\frac{1}{E_1^2}+\frac{1}{E_2^2})(\frac{1}{E_1}+\frac{1}{E_2})\frac{(k_1, k_2)}{E_{12}^2},
\end{align*}
\begin{align*}
&
{\rm I}_4(\Lambda)=\frac{1}{4}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2}(\frac{|k_1|^2}{E_1^2}+\frac{|k_2|^2}{E_2^2})(\frac{1}{E_1}+\frac{1}{E_2})\frac{1}{E_{12}^2},
\end{align*}
}
\only<3>{</div><div>\begin{align*}
&
{\rm I}_5(\Lambda)=\frac{1}{4}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2E_1^2E_2^2}\frac{(k_1, k_2)}{E_{12}},
\end{align*}
\begin{align*}
&
{\rm I}_6(\Lambda)=\frac{1}{8}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2}(\frac{1}{E_1}+\frac{1}{E_2})^2\frac{|k_1|^2+|k_2|^2}{E_{12}^3},
\end{align*}
\begin{align*}
&
{\rm I}_7(\Lambda)=\frac{1}{4}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2}(\frac{1}{E_1}+\frac{1}{E_2})^2\frac{(k_1, k_2)}{E_{12}^3},
\end{align*}
\begin{align*}
&
{\rm I}_8(\Lambda)=\frac{1}{4}\int\!\!\!\!\int \!\! dk_1dk_2\frac{|\hat{\phi}_1|^2|\hat{\phi}_2|^2}{\omega_1\omega_2}(\frac{1}{E_1}+\frac{1}{E_2})\frac{(k_1, k_2)}{E_{12}^4},
\end{align*}
}
\only<4>{</div><div>\begin{align*}
&
{\rm I}_9(\Lambda)=\frac{1}{2}\int \frac{|\hat{\phi}(k)|^2|k|^2}{\omega(k)E(k)^4}dk,
\end{align*}
\begin{align*}
&
{\rm I}_{10}(\Lambda)=\frac{1}{2}\int \frac{|\hat{\phi}(k)|^2}{\omega(k)E(k)^2}dk.
\end{align*}
}
\end{lemma}
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\end{document}