1$ and $\tan\lambda\to 0$.

The solution of any scattering problem is given by the scattering matrix $S$, from which one can calculate all the relevant quantities [@Taylor1972]. In MQDT framework the $S$ matrix is given in terms of the $R$ matrix [@Mies1984a] $$\label{Smatr}
\mathbf{S}(E)=e^{i\boldsymbol{\xi}}\left[1+i\mathbf{R}(E)\right]\left[1-i\mathbf{R}(E)\right]^{-1}e^{i\boldsymbol{\xi}},$$ where $$\label{Rmatr}
\mathbf{R}(E)=\mathbf{C}^{-1}(E)\left(\mathbf{Y}^{-1}(E)-\tan\boldsymbol{\lambda}\right)^{-1}\mathbf{C}^{-1}(E)$$ Here $\boldsymbol{\xi}_{ij}=\xi_i\delta_{ij}$, $\mathbf{C}_{ij}=C_i\delta_{ij}$ and $\tan\boldsymbol{\lambda}_{ij}=\tan\lambda_i\delta_{ij}$. The pleasing aspect of this theory is that the $\mathbf{Y}(E)$ matrix remains analytic in energy across the thresholds and can usually be regarded as energy- and angular momentum insensitive [@Gao2001], so that $\mathbf{Y}(E)\approx \mathbf{Y}$ and the matrix elements do not depend on the partial wave. This results from the separation of length and energy scales. As a result, all energy and angular momentum dependence is encoded in $C$ and $\tan\lambda$ functions.

Collision rates in two channel model
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The formalism introduced in the previous section was general and suitable for any multichannel scattering problem. Let us now restrict our attention to a two-channel problem, where $p=1$ is the entrance channel and $p=2$ is the loss channel, both channels are assumed to be open. It is possible to include more channels in the analysis, but this simple case already exhibits interesting features. By choosing the reference potentials to accurately 