boundary, we find that for any homogeneous solution $h_\bullet$ we have $$\begin{aligned}
\l F, \e^{iktU}h_\bullet  \r &=   \l (-k^{2}+(\de_{y}-iktU')^{2}) \Psi, \e^{iktU}h_\bullet  \r\notag\\
    &=  \e^{iktU}h_\bullet  (\de_{y}-iktU')\Psi \big|_{y=\a}^{\b} -\e^{iktU} \Psi  (\de_{y}-iktU')  h_\bullet\big|_{y=\a}^{\b}\notag\\
    &=  \e^{iktU}h_\bullet \de_{y}\Psi \big|_{y=\a}^{\b}=\e^{iktU}h_\bullet \frac{1}{d}\dg\Psi \big|_{y=\a}^{\b}.
  \end{aligned}$$ Therefore, from - we conclude that $$\begin{aligned}
\label{eq:boundaryterms}
\dg\Psi|_{y=\a} =-\dc(\a)\l F, \e^{ikt(U(y)-U(\a))}h_\a  \r   , \qquad \dg\Psi|_{y=\b} =\dc(\b)\l F, \e^{ikt(U(y)-U(\b))}h_\b  \r,\end{aligned}$$ Thus, integrating by parts we obtain $$\begin{aligned}
\dg\Psi|_{y=\a} 
&=-\dc(\a)\l  \frac{h_\a}{iktU'}F,\de_y\e^{ikt(U(y)-U(\a))}  \r \notag \\
&=\dc(\a)\l  \de_y\left(\frac{h_\a}{iktU'}\right)F,\e^{ikt(U(y)-U(\a))}  \r  +\dc(\a)\l  \frac{h_\a}{iktU'}\de_yF,\e^{ikt(U(y)-U(\a))}  \r +\frac{\dc(\a)F(\a)}{iktU'(\a)} \notag\\
&=\dc(\a)\l  \de_y\left(\frac{h_\a}{iktU'}\right)F,\e^{ikt(U(y)-U(\a))}  \r  +\dc(\a)\l  \dg F,\frac{ \e^{ikt(U(y)-U(\a))}}{iktU'd} h_\a  \r +\frac{\dc(\a)F(\a)}{iktU'(\a)}.\end{aligned}$$ In a similar fashion, $$\begin{aligned}
\dg\Psi|_{y=\b} 
=-\dc(\b)\l  \de_y\left(\frac{h_\b}{iktU'}\right)F,\e^{ikt(U(y)-U(\b))}  \r  -\dc(\b)\l  \dg F,\frac{ \e^{ikt(U(y)-U(\b))}}{iktU'd} h_\b  \r +\frac{\dc(\b)F(\b)}{iktU'(\b)}.\end{aligned}$$ Going back to , we use and the above computations to obtain precisely , upon using that the boundary of $F(t)$ is preserved (c.f. Remark \[remark:boundary\_data\]), and thus finishing the proof.

Having established the modified equation, we show that we can localize estimates as in Section \[sec:simple\_case\]. Here, as in [@Zill6] we further split $\dg F$ into contributions $\beta_{\a}, \beta_{\b}$ with zero initial data and right-hand-side involving $\omega^{in}(\a),\omega^{in}(\b)$ and another contribution $F^{(1)}$ with initial data $\dg F|_{t=0}$ but simpler right-hand-side. This splitting allows us to separately treat the different time behavior and growth of boundary terms 