denotes the $S$ matrix, $A_{-}$ and $A_{+}$ are normalization coefficients, and $k(x')$ is the local wave vector. Using the exact solution we calculate the $S$ matrix, and the reaction probability $P = 1-|S|^2$. As in this section we are interested only in the behavior of thermally-averaged reaction rates at high temperatures, where the phase shifts vary rapidly with collision energy, we may perform an averaging over the short-range phase. In this way we incorporate the effect of the shape resonances on the reaction rates in an average sense. Calculating reaction probability through the parabolic barrier and performing an average over uniformly distributed values of $\varphi$, we obtain a result valid for arbitrary short-range reaction probability $$\label{Ppar}
P(\ell,y) = 1-|S_\ell|^2=\frac{P^{\mathrm}{re}}{P^{\mathrm}{re} e^{-2\pi \varepsilon}+1}.$$ Here, $\varepsilon=E/\hbar\sqrt{\mu/k}$ denotes dimensionless energy measured with respect to the peak of the parabola. In the universal regime $y=1$ one can recover the WKB solution derived in [@Landau]. We can fit analytically the parabolic potential to the centrifugal barrier for arbitrary power-law potential, by equating the first and the second derivative in the maximum of the barrier. Then we change the zero of energy to the asymptotic zero of the physical potential. In the following we will work in 