operators $A_{j}(t)$ such that $\l \omega, A(t)\omega \r$ is a Lyapunov functional. Since on each $I_{j}$ the function $U(y)$ is bilipschitz, we can construct adapted Fourier multipliers (strictly speaking just multipliers in a convenient $L^{2}$ basis, c.f. [@Zill5] for other bases). We rely on the following result from [@Zill5].

\[lem:Bilipschitz\] Suppose that $U$ is bilipschitz on the interval $I_{j}$. There exists $A_{j}(t)$ such that $$\begin{aligned}
\int_{I_j} |U'(y)| \left[|\nabla_k  \psi_j(y)|^2 +k^2|\nabla_k \psi_{j,A}(y) |^2\right]\dd y &\leq -C \Re\l \chi_{j} \omega, \dot A_{j}(t)\chi_{j}\omega \r, 
  \end{aligned}$$ and $$\begin{aligned}
     \frac1C\|\chi_{j}\omega\|_{L^{2}}^{2}\leq \l \chi_{j} \omega, A_{j}(t)\chi_{j}\omega \r \leq C \|\chi_{j}\omega\|_{L^{2}}^{2}.
  \end{aligned}$$ Here, $C$ depends on $U$ only via $\frac{\max(|U'|)}{\min(|U'|)}$, and it is therefore independent of $j$. In particular, $$\begin{aligned}
\label{eq:derbound}
\sum_{j\in\J}\int_{I_j} |U'(y)| \left[|\nabla_k  \psi_j(y)|^2 +k^2|\nabla_k \psi_{j,A}(y) |^2\right]\dd y &\leq -C \sum_{j} \Re\l  \chi_{j}\omega, \dot A_{j}(t) \chi_{j}\omega \r.
  \end{aligned}$$

We introduce the new time variable $\tau=t \min(|U'|)$ and observe that is then given by $$\begin{aligned}
\begin{cases}
\displaystyle\de_{\tau}F  =ik\frac{B(y)}{\min (|U'|)}\Psi,   \quad &\mbox{in } I, \\
\displaystyle \left(-k^2+\left(\de_y-ikt\frac{U'(y)}{\min (|U'|)}\right)^2\right)\Psi  = F, &\mbox{in } I,\\
\Psi= 0, &\mbox{on } \de I.
\end{cases}\end{aligned}$$ We may thus interpret this as a new equation of type with $U,B$ replaced by $$\begin{aligned}
 U_{\star}:=\frac{U}{\min(|U'|)},\qquad  B_{\star}:=\frac{B}{\min(|U'|)},
  \end{aligned}$$ respectively. The flow $U_{\star}$ is then bilipschitz with constants $1$ 