\frac{1}{2}|g(y)|^{2} \sign(U'(y))U'''(y)\right]\dd y\notag\\
&= \int_J\left[ \left(k^2|U'(y)|- \frac{1}{2}\sign(U'(y))U'''(y) \right) |g(y)|^{2}+|U'(y)| |\de_y g(y)|^{2}\right]\dd y\notag\\
&\geq \gamma_{0}\int_J |U'(y)||\nabla_k g(y)|^2\dd y,
\label{eq:elli1}\end{aligned}$$ which is what we need to conclude the proof.

Localized potentials {#sub:localizedpote}
--------------------

In order to properly localize the streamfunction in , we introduce the following auxiliary problems. For any $j\in \J$, let $$\begin{aligned}
\label{eq:Ajs}
A_j(t): L^2(I_j) \rightarrow L^2(I_j)\end{aligned}$$ be a symmetric operator such that $t\mapsto A_j(t)$ is continuously differentiable. Define the localized potentials $\psi_j, \psi_{j,A}$ by $$\label{eq:local1}
\begin{cases}
-\Delta_k \psi_j = \chi_j \omega, \quad &\mbox{in } I_j , \\
\psi_{j}= 0, &\mbox{on } \de I_j, 
\end{cases}$$ and $$\label{eq:local2}
\begin{cases}
-\Delta_k \psi_{j,A} = A_j(t)\chi_j \omega, \quad &\mbox{in }  I_j , \\
\psi_{j,A}= 0, &\mbox{on }  \de I_j,
\end{cases}$$ respectively. When referring to the scattering formulation , we will make use of the scattered localized potentials $\Psi_j, \Psi_{j,A}$ defined by $$\label{eq:local1scat}
\begin{cases}
-E_{t} \Psi_j = \chi_j F, \quad &\mbox{in } I_j , \\
\Psi_{j}= 0, &\mbox{on } \de I_j, 
\end{cases}$$ and $$\label{eq:local2scat}
\begin{cases}
-E_{t} \Psi_{j,A} = A_j(t)\chi_j F, \quad &\mbox{in }  I_j , \\
\Psi_{j,A}= 0, &\mbox{on }  \de I_j.
\end{cases}$$ The following lemma relates a term that commonly arises in our computations with its localized version.

\[lem:local\_elliptic\] For every $k\geq 1$ we 