problem, where the internal states are labeled by the index $p$ and the channels by the index $i=\{p\ell m\}$, where $\ell,m$ are the angular momentum quantum numbers, to shorten the notation. We assume that the long-range interaction between the particles is described by a power-law potential $-C_n/r^n$ ($n>3$), with which the characteristic length $R_n = \left(2 \mu C_n/\hbar^2 \right)^{1/(n-2)}$ and energy $E_n = \hbar^2/(2 \mu R_n^2)$ can be associated ($\mu$ is the reduced mass). The short range forces, including the interchannel couplings which are responsible for inelastic processes, are assumed to be limited to the short range, acting at distances $R_0\ll R_n$. The interaction matrix is then asymptotically diagonal $$W_{pp'}(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}\left(E^\infty _p+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}-\frac{C_n}{r^n}\right)\delta_{pp'}$$ where $E^\infty _p$ are the threshold energies for each channel and $\ell$ is the angular momentum quantum number.

We analyze this problem using MQDT, following its formulation by Mies [@Mies1984a; @Mies1984b; @Mies2000]. In this treatment one first chooses a reference potential $V_p$ in each channel. With each potential one can associate a pair of linearly independent solutions $\hat{f}(r,E)$ and $\hat{g}(r,E)$ that have local WKB-like normalization at short distances $$\begin{aligned}
\label{fghat}
\left.
\begin{array}{lll}
\hat{f}_i(r,E) & \cong & k_i(r)^{-1/2} \sin \beta_i (r),\phantom{\Big(}\\
\hat{g}_i(r,E) & \cong & k_i(r)^{-1/2} \cos \beta_i 