for the full line Schrödinger operator, one typically introduces the right and left reflection coefficients $R(\lambda ),L(\lambda )$ and the transmission coefficient $T(\lambda )$ as functions of the momentum $\lambda $ (see e.g. [@Deift79]). These quantities (also called transition coefficients) can also be properly defined in much larger spectral situations through Wronskians and/or Titchmarsh-Weyl $m$-functions (see e.g. [@GNP97; @GS97]). Such extensions need not be unique. However, in our setting of step-like potentials decaying at $+\infty $, there is a natural candidate for the right reflection coefficient $R(\lambda )$.

\[def3.1\] Let $q(x)$ be real, locally integrable such that $q\in
L^{1}\left( \mathbb{R}_{+}\right) $ and $-\partial _{x}^{2}+q(x)$ is in the limit point case at $-\infty $. Denoting by $R_{n}(\lambda )$ the right reflection coefficient (which is necessarily well defined) from the potential $q_{n}=q|_{(-n,\infty )}$, we call the weak limit (if it exists) $$R(\lambda )\equiv \text{w-}\lim R_{n}(\lambda ),\;n\rightarrow \infty ,
\label{eq3.1}$$the right reflection coefficient from the potential $q$.

Note that one should not expect in  pointwise convergence as an explicit counterexample $q=$ $\chi _{-}$ readily shows. Uniform convergence in is not available in general even in the short-range setting [@Deift79].

As shown in [@Ry11], Lemma 5.4, the reflection coefficient introduced this way is well 