matter). Splitting $$q=q_{-}+q_{+},\ \ q_{\pm }=q|_{\mathbb{R}_{\pm }}  \label{split_q}$$induces the representation $$R=\frac{T_{+}^{2}R_{-}}{1-R_{-}L_{+}}+R_{+}$$where $\pm $ label scattering quantities associated with $q_{\pm }$. The functions $T_{+},L_{+},R_{-}$ can be analytically continued into $\mathbb{C}^{+}$ and $$A\equiv \frac{T_{+}^{2}R_{-}}{1-R_{-}L_{+}}$$has properties , . For $R_{+}$, which is independent of $n$, we use the representation [@Deift79], Theorem 2, $$R_{+}(\lambda )=\frac{T_{+}(\lambda )}{\lambda }G(\lambda )$$where $$G(\lambda )=\frac{1}{2i}\int_{-\infty }^{\infty }e^{-2i\lambda x}g(x)dx
\label{eq6'.1}$$with some $g$ obeying $$\left\vert g(x)\right\vert \leq \left\vert q(x)\right\vert +\limfunc{const}\int_{x}^{\infty }\left\vert q\right\vert .  \label{Est_on_g}$$Since $R_{+}(\lambda )$ is a reflection coefficient we have . Since $T_{+}$ is a transmission coefficient, $$T_{+}(\lambda )=\prod_{k=1}^{N}\frac{\lambda +i\varkappa _{k}}{\lambda
-i\varkappa _{k}}\cdot S(\lambda )=B\left( \lambda \right) ^{-1}S(\lambda )$$where $S$ is an outer function of $\mathbb{C}^{+}$: $\left\vert S(\lambda
)\right\vert \leq 1$, $\lambda \in \mathbb{C}^{+}$. This proves and .

The proposition is proven if we show . Due to we should demonstrate that $G$ admits a pseudo analytic extension the whole complex plane such that $$\left\vert \partial _{\overline{\lambda }}G\right\vert \lesssim \exp \left\{
-Q\left\vert {\operatorname{Im}}\lambda \right\vert ^{-\frac{\varepsilon }{1-\varepsilon }}\right\}  \label{eq6''.1}$$with some $Q>0$. There are a few explicit ways to construct pseudo analytic continuations (see e.g. [@Dyn76], [@BorDyn93], [@Tarama04]) producing different extensions. We modify the one used in [@Tarama04] to obtain a better $Q$ in \[eq6”.1\]. Note that $$G\left( \lambda \right) \eqsim 