$\Psi|_{\de I}=0$ and thus $$\begin{aligned}
  \de_{t} F|_{\de I}=0\end{aligned}$$ for all times and boundary values are preserved. That is, for all $t>0$ $$\begin{aligned}
  F(t)|_{\p I}= \omega^{in}|_{\p I}.\end{aligned}$$ We further remark that, since the spectral gap of $E_{t}$ involves $k^{2}$, $ik \Psi$ asymptotically scales as $|k|^{-1}$. Thus, and in view of condition \[H1\], estimates for larger $k$ are simpler to establish than for smaller $k$.

Our main result has a natural formulation frequency by frequency, which we state here below.

\[thm:L2\] There exists a symmetric positive operator $A(t):L^2 \rightarrow L^2$, with $C^1$ dependence on $t$, such that any solution $\omega$ of satisfies $$\begin{aligned}
    \frac1C\|\omega(t)\|_{L^2}^2 \leq\l \omega(t), A(t) \omega(t) \r_{L^2} \leq C \|\omega(t)\|_{L^2}^2, \qquad \forall t\geq0,
  \end{aligned}$$ for some constant $C\geq 1$, and $$\begin{aligned}
    \ddt \l \omega(t), A(t) \omega(t) \r_{L^2} +\frac12 \int_{I} |U'(y)| |\nabla_k \psi (t,y)|^{2}  \dd y \leq 0, \qquad \forall t\geq0.
  \end{aligned}$$ In particular, this implies $L^2$ stability and $$\begin{aligned}
    \int_{I} |U'(y)| |\nabla_k \psi (y)|^{2} \dd y  \in L^1_t(0,\infty).
  \end{aligned}$$ If further, the initial data is in $H^{1}$ or $H^{2}$, then also $$\begin{aligned}
    \|\e^{ikUt}\omega(t, y)\|_{H^{1}} \leq C \|\omega^{in}\|_{H^{1}}, \\
    \|\min(1, \dist(y, \p I)) \de_{yy}\e^{ikUt}\omega(t, y)\|_{L^{2}} \leq C \|\omega^{in}\|_{H^{2}},
  \end{aligned}$$ and $$\begin{aligned}
    \int |U'(y)| |\nabla_k \psi (t)|^{2}  \dd y \leq \frac{C}{\l kt\r^{2}} 