energy dependent $\ell$-th phase shift and the matrices ${\mathbf{F}_{\ell{\ell^{\prime}}}^{\mathbf{o}}}$ are given by $({\mathbf{F}_{\ell{\ell^{\prime}}}^{\mathbf{o}}})_{{n{n^{\prime}}}}=U_{\ell n}U_{{\ell^{\prime}}{n^{\prime}}}$, with $0\le n,{n^{\prime}}\le n_o-1$ and $n_o$ denoting the number of open channels. At this point we also introduce the generalized, energy dependent, scaled scattering length defined for all partial waves $\ell$, by ${\bar{a}_{\ell}}(E)^{2\ell+1}=(a_{\ell}(E)/{a_{\perp}})^{2\ell+1}=-\frac{\Delta_{\ell}(E)}{({a_{\perp}}k)^{2\ell+1}}$. As the local frame transformation $U_{\ell n}$ also depends on the energy, this also holds for the matrices ${\mathbf{F}_{\ell{\ell^{\prime}}}^{\mathbf{o}}}={\mathbf{F}_{\ell{\ell^{\prime}}}^{\mathbf{o}}}(E)$. Here, we also note that one should carefully distinguish the number of open channels $n_o$ from the actual quantum numbers of the corresponding transverse modes $n$, e.g. consider the single mode regime where only the population of the lowest transversal mode $\ket{q_0;0}$ is allowed, we have $n_o=1$.

The energy dependent $K$ matrix ${\mathbf{K}_{oo}^{1D,phys}}$, given in Eq. appropriately describes the scattering in a tight harmonic waveguide with several open transverse modes. The coupling of two arbitrary partial waves $\ell$ and ${\ell^{\prime}}$ belonging to the same symmetry class, i.e. $\ell-{\ell^{\prime}}=0\operatorname{mod}2$, is properly taken into account. Before proceeding let us recall that the effect of the closed channels on the scattering phase shift can conveniently be expressed as $$\begin{aligned}
  {\mathfrak{U}_{\ell{\ell^{\prime}}}}({\epsilon})&=\sum_{p=0}^{\ell+{\ell^{\prime}}}\frac{{c_{p}^{(\ell,{\ell^{\prime}})}}}{2({\epsilon}+\frac{1}{2})^\frac{p+1}{2}}{\zeta_{H}\big(-\frac{p-1}{2},n_{o}-{\epsilon}\bigr)},
\label{eq:closed_channel_umat}
\\
{c_{p}^{(\ell,{\ell^{\prime}})}} &=
(-1)^{\frac{\ell+{\ell^{\prime}}}{2}}\sqrt{(2\ell+1)(2{\ell^{\prime}}+1)}\nonumber\\
&\times \sum_{\nu=\max\{p,|\ell-{\ell^{\prime}}|\}}^{\ell+{\ell^{\prime}}}\Gamma(\ell,{\ell^{\prime}},\nu,p)\quad,
\label{eq:clossel_channel_expansion_coefficients_cll}
\\
\Gamma(\ell,{\ell^{\prime}},\nu,p)&=i^{p-1}2^{\nu-1}(2\nu+1)\nonumber\\
&\times\begin{pmatrix}v\\p\end{pmatrix}\begin{pmatrix}\frac{\nu+p-1}{2}\\\nu\end{pmatrix}\begin{pmatrix}\ell&{\ell^{\prime}}&\nu\\0&0&0\end{pmatrix}
\label{eq:clossel_channel_gamma}\end{aligned}$$ where $\zeta_{H}(s,a)$ denotes the Hurwitz 