some $Q_{f}>0.$

By [@BorDyn93], Theorem 3, the statement $f\in G^{\alpha }$ is equivalent to the statement that $f$ admits a pseudo analytic extension to the whole complex plane such that $$\left\vert \partial _{\overline{z}}f\right\vert \lesssim _{f}\exp \left\{
-Q\left\vert {\operatorname{Im}}z\right\vert ^{-\frac{1}{\alpha }}\right\}
\label{lambda_bar}$$with some $Q>0$.

In a similar manner one introduces local Gevrey classes $G_{\limfunc{loc}}^{\alpha }$.

Next, $\mathfrak{S}_{2}$ denotes the Hilbert-Schmidt class$$\mathfrak{S}_{2}=\left\{ A:\left\Vert A\right\Vert _{\mathfrak{S}_{2}}^{2}\equiv\func{tr}\left( A^{\ast }A\right) <\infty \right\}$$and $\mathfrak{S}_{1}$ is the trace class: $$\mathfrak{S}_{1}=\left\{ A:\left\Vert A\right\Vert _{\mathfrak{S}_{1}}\equiv\func{tr}\left( A^{\ast }A\right) ^{1/2}<\infty \right\} .$$ Note that $A\in\mathfrak{S}_1$ if and only if $A=A_1A_2$ with some $A_1,A_2\in\mathfrak{S}_2$.

Some other miscellaneous notation: $\chi _{S}\left( x\right) $ is the characteristic function of a set $S$, i.e. $$\chi _{S}\left( x\right) \equiv\left\{ 
\begin{array}{c}
1,x\in S \\ 
0,x\notin S\end{array}\right. .$$In particular $\chi _{\pm }\equiv\chi _{_{\mathbb{R}_{\pm }}}$is the Heaviside function of $\mathbb{R}_{\pm }$. We also write $$\left. f\right\vert _{S}=\chi _{S}f.$$

The notation $H_{q}\equiv -\partial _{x}^{2}+q(x)$ for the Schrödinger operator on $L^{2}\left( \mathbb{R}\right) $ will be frequently used.

The reflection coefficient and its analytic structure
=====================================================

In this section we define a suitable reflection coefficient and investigate its properties which will play a central role in our consideration. The results of this section may have some independent interest.

In the short-range scattering 