error terms.

In the case $I$ is not the whole space but has boundary points $\{a,b\}=\p I$ (or just a single boundary point), we further introduce the auxiliary homogenous solutions $h_{\a}, h_{\b}, \tilde{h}_{\a}, \tilde{h}_{\b}$ of the problems $$\begin{aligned}
&\left(-k^{2}+\de_{yy} -2\de_{y} \frac{\dc^{(1)}}{\dc} -\frac{\dc^{(2)}}{\dc} \right)\tilde{h}_\bullet=0,\label{eq:homo1} \\
&\left(-k^{2}+\de_{yy} \right)h_\bullet=0,\label{eq:homo2}\end{aligned}$$ where $\bullet=\a,\b$, with boundary conditions $$\begin{aligned}
h_{\a}(\a)=\tilde{h}_{\a}(\a)=h_{\b}(\b)=\tilde{h}_{\b}(\b)&=1, \label{eq:boundh1}\\
h_{\a}(\b)=\tilde{h}_{\a}(\b)=h_{b}(\a)=\tilde{h}_{\b}(\a)&=0.\label{eq:boundh2}\end{aligned}$$ In particular, $$\begin{aligned}
&\left(-k^{2}+(\de_{y}-iktU')^{2} -2(\de_{y}-iktU') \frac{\dc^{(1)}}{\dc} -\frac{\dc^{(2)}}{\dc} \right)\e^{iktU}\tilde{h}_\bullet=0,\label{eq:homo3} \\
&\left(-k^{2}+(\de_{y}-iktU')^{2} \right)\e^{iktU}h_\bullet=0.\label{eq:homo4}
 \end{aligned}$$ We then have the following result.

\[lem:H1equation\] Let $F, \Psi$ be the solution to . Then $\dg F$ satisfies the following equations, where the terms past the first line only appear in the setting with boundary: $$\begin{aligned}
\label{eq:DyF}
\de_{t} \dg F &=  ik B \tPhi + (\dg B)ik \Psi\notag \\
&\quad+ B \dc(\a)\l  \dg F,\frac{ \e^{ikt(U(y)-U(\a))}}{tU'd} h_\a  \r  \e^{{ikt(U(y)-U(\a))}}\tilde{h}_{\a}\notag \\
&\quad-  B\dc(\b)\l  \dg F,\frac{ \e^{ikt(U(y)-U(\b))}}{tU'd} h_\b  \r    \e^{{ikt(U(y)-U(\b))}}\tilde{h}_{\b}\notag\\
&\quad+  B \dc(\a)\l  \de_y\left(\frac{h_\a}{tU'}\right)F,\e^{ikt(U(y)-U(\a))}  \r  \e^{{ikt(U(y)-U(\a))}}\tilde{h}_{\a}\notag \\
&\quad-  B\dc(\b)\l  \de_y\left(\frac{h_\b}{tU'}\right)F,\e^{ikt(U(y)-U(\b))}  \r      \e^{{ikt(U(y)-U(\b))}}\tilde{h}_{\b}\notag\\
&\quad+   \frac{\dc\omega^{in}}{tU'}\bigg|_{y=\a} B\e^{{ikt(U(y)-U(\a))}}\tilde{h}_{\a}+  \frac{\dc\omega^{in}}{tU'}\bigg|_{y=\b}B   \e^{{ikt(U(y)-U(\b))}}\tilde{h}_{\b}\end{aligned}$$ where $$\begin{aligned}
&\left(-k^{2}+(\de_{y}-iktU')^{2}-2 (\de_{y}-iktU') \frac{\dc^{(1)}}{\dc}-\frac{\dc^{(2)}}{d}\right) \tPhi \notag\\
&\qquad\qquad= \dg F +\left[ \left(\frac{U''}{U'}\dc\right)'- \frac{U''}{U'}\dc^{(1)}- 2 (\de_{y}-iktU') \frac{U''}{U'}\dc\right]\left(\de_y-iktU'\right)\Psi,\end{aligned}$$ with boundary conditions $\tPhi |_{y=a,b} =0$.

As a first step, we apply $\dg$ to and obtain $$\begin{aligned}
&\de_t \dg F = ikB \dg \Psi + ik (\dg B)\Psi,\label{eq:eqforDF} \\
&\left(-k^{2}+(\de_{y}-iktU')^{2}\right)\dg \Psi = 