coupling of the partial waves to the closed channels, given by the ${\mathfrak{U}_{\ell\ell}}({\epsilon})$’s in Eq. exhibits roots for all considered partial waves $\ell$. These roots ${{\epsilon}^{*}_{\ell n}}=\lfloor {{\epsilon}^{*}_{\ell n}}\rfloor+\Delta{{\epsilon}^{*}_{\ell n}}$, in the following called *decoupling energies*, depend on the partial wave $\ell$, and, in general, also on the number of open channels $n$. The decoupling energies determine the particular energy where the bound state of the closed channels decouples from *all* the open channels. Therefore, in this case, the physical [$K$ matrix ]{}can be written in the following form $$\begin{aligned}
  {\mathbf{K}_{oo}^{1D,phys}}|_{\lfloor{\epsilon}\rfloor+{\epsilon}^*}&=({\mathbf{K}^{3D}})_{\ell\ell}\;{\mathbf{F}_{\ell\ell}^{\mathbf{o}}}(\lfloor{\epsilon}\rfloor+{\epsilon}^{*})\nonumber\\
&=\mathbf{U}^T{\mathbf{K}^{3D}}\mathbf{U}|_{\lfloor{\epsilon}\rfloor +{\epsilon}^*}
\label{eq:kmatrix_decoupled}\end{aligned}$$ where $\lfloor{\epsilon}\rfloor$ is the threshold energy for the largest open channel. Equation shows the expected result, that the 3D scattering information which emerges close to the origin is transfered to the asymptotic regime without being affected by the closed channels of the trapping potential. In other words the colliding pair experiences effectively a free space collision in the presence of the waveguide geometry.

In particular by using the expression for ${\mathfrak{U}_{\ell\ell}}({\epsilon})$ in Eq. one can show, that for the cases $\ell=0$ and $1$, the decoupling energies do not depend on the number of open channels, i.e. $\Delta{\epsilon}^{*}_{\ell n}=\Delta{\epsilon}^*_\ell$, which means that the closed channels decouple from 