C\int_{I_j}|U'(y)||\nabla_k \psi_j(y)||\nabla_k \psi(y)|\dd y.\end{aligned}$$ Thus, , and Lemma \[lem:weighted\_elliptic\] yield $$\begin{aligned}
 &\gamma_{0}\int_I |U'(y)||\nabla_k \psi(y)|^2\dd y\\
 &\qquad\qquad\leq \sum_{j\in \J}\l  \Delta_k\psi_j,|U'|\psi_j\r+ C\sum_{j\in \J}\int_{I_j}|U'(y)||\nabla_k \psi_j(y)||\nabla_k \psi(y)|\dd y\notag\\
 &\qquad\qquad\leq -\gamma_0\sum_{j\in \J}\int_{I_j} |U'(y)||\nabla_k \psi_j(y)|^2\dd y+C\sum_{j\in \J}\int_{I_j}|U'(y)||\nabla_k \psi_j(y)||\nabla_k \psi(y)|\dd y\notag\\
 &\qquad\qquad\leq C\sum_{j\in \J}\int_{I_j} |U'(y)||\nabla_k \psi_j(y)|^2\dd y +\frac{\gamma_{0}}2 \int_{I} |U'(y)| |\nabla_k \psi(y)|^{2}\dd y.\end{aligned}$$ Hence, collecting all of the above and applying Lemma \[lem:weighted\_elliptic\] once more we find $$\begin{aligned}
\frac{\gamma_{0}}2\int_I |U'(y)||\nabla_k \psi(y)|^2\dd y 
\leq C\sum_{j\in \J}\int_{I_j} |U'(y)||\nabla_k \psi_j(y)|^2\dd y,\end{aligned}$$ which is what we wanted. The proof is over.

With these results at hand, we can now prove the following localized energy inequality, holding for a fairly general collection of symmetric of operators.

\[prop:reduction\] Let $A(t):L^2\to L^2$ be the symmetric operator defined by $$\begin{aligned}
A(t)= \sum_{j\in\J} \chi_j A_j(t)\chi_j, \end{aligned}$$ with $A_j$ as in and localized potential as in -. Then it holds that $$\begin{aligned}
   \ddt \l \omega, A(t)\omega  \r \leq \Re\l \omega,  \dot A(t)\omega\r +C\eps_0\sum_{j\in \J} \int_{I_j} |U'(y)| \left[|\nabla_k  \psi_j(y)|^2 +k^2|\nabla_k \psi_{j,A}(y) |^2\right]\dd y.   
  \end{aligned}$$

By direct computations and using , we find that $$\begin{aligned}
\ddt \l \omega, A(t)\omega  \r
&=\Re\l \omega, \dot A(t)\omega  \r+2\Re\l ik B  \psi , A(t)\omega  \r-2\Re\l ik U \omega, A(t)\omega  \r\notag\\
&=\Re\l \omega, \dot A(t)\omega  \r+2\Re\l ik B  \psi , A(t)\omega  \r,\end{aligned}$$ thanks to 