\|\omega^{in}\|^2_{H^{1}}, \\
    \int |U'(y)| |\psi (t)|^{2} \dd y \leq \frac{C}{\l kt\r^4}\|\omega^{in}\|^2_{H^{2}}.
  \end{aligned}$$

Higher order stability can be stated in terms of modified differential operators. Let $$\begin{aligned}
\label{eq:newdiff}
\dc(y):=\sum_{j\in\J}\frac{\|U'\|_{L^\infty(I_j)}}{U'(y)}\chi_j(y), \qquad \dg:=\dc(y)\de_y. \end{aligned}$$ Our choice of $\dg$ here is determined by satisfying two opposing objectives. On the one hand, we want an operator that is very similar to the usual derivative $\de_y$. On the other hand, $\dg$ should have good commutation properties with $\de_{y}-iktU'$, which leads to considering $(U')^{-1}\de_y$.

Notation and conventions
------------------------

-   $C\geq 1$ is a generic positive constant independent of $k$.

-   $\l\cdot,\cdot\r$ is the $L^2$ scalar product in $y\in I$.

-   $\nabla_k=(ik,\de_y)$

-   $[A,B]=AB-BA$

-   $\l x\r=\sqrt{1+x^2}$

Localization and L2 stability {#sec:simple_case}
=============================

Let $\omega$ be a solution to . Then we claim that $\omega$ satisfies $$\begin{aligned}
\ddt \|\omega\|^2_{L^2} \leq 2 \int_I |B'(y)||\nabla_k\psi(y)|\dd y\leq C \int_I |U'(y)||\nabla_k\psi(y)|\dd y.\end{aligned}$$ Indeed, using the anti-symmetry of $ikU$ and $ikb$ and integrating by parts, we have $$\begin{aligned}
\ddt \|\omega\|^2_{L^2} &=2\Re\l ik B\psi, \omega\r=-2\Re\l ik B\psi, \Delta_k \psi\r
=-2\Re\l ik B\psi, \de_{yy} \psi\r=2\Re\l ik B'\psi, \de_y \psi\r,\end{aligned}$$ so the claim follows by the Cauchy–Schwarz inequality and . Notice that here we used that $\psi=0$ on $\de I$ in the integration by parts.

In 