additional contribution. In this section, we derive a simple model which incorporates this effect. To this end we approximate the centrifugal barrier by an inverted parabolic potential $V(x)=-\frac12 k x^2$, with $k>0$. For such a potential one can find an analytic solution, which is given in terms of parabolic cylinder functions. Considering asymptotic expansions of these solutions at large distances one can show that they have a WKB-like form. We perform the expansion of fully analytical solution far away from the barrier and identify the parts propagating to the left and to the right. After that we calculate the $S$ matrix with the following boundary conditions $$\begin{split}
\Psi(x) \stackrel{x \to -\infty}{\longrightarrow} &A_{-} \left(\exp\left[ - i \int_{-\infty}^x |k(x')| dx' - i \varphi \right]\right.\\
 &- \left.\frac{1-y}{1+y} \exp \left[ i \int_{-\infty}^x |k(x')| dx' + i \varphi \right] \right)
\end{split}$$ at large distances to the left of the barrier, and $$\begin{split}
\Psi(x) \stackrel{x \to \infty}{\longrightarrow} A_{+} \left( \exp\left[- i \int_x^{\infty} k(x') dx' \right]\right.\\ - \left. S \exp \left[ i \int_x^{\infty} k(x') dx' \right] \right)
\end{split}$$ at large distances to the right of the barrier. Here $y$ is as before the parameter describing the reactivity of the system [@Idziaszek2010], $\varphi$ is some arbitrary phase, $S$ 