the separation of length scales is assumed, i.e. ${\beta_{6}}\ll{a_{\perp}}$. In short, this implies that the Hamiltonian has three distinct regions where (a) different potential contributions dominate and (b) different symmetries are obeyed by the corresponding Hamiltonian.

\(i) Starting in the inner region, where $r\sim {\beta_{6}}$ holds, the interatomic potential dominates. The two particles thus experience a free-space collision with total energy $E=\frac{\hbar^2 k^2}{2\mu}$.

\(ii) The effect of this collision on the wave function is best observed from the intermediate region $({\beta_{6}}\ll r\ll
{a_{\perp}})$, where both potential contributions are negligible. Hence, we can monitor the outcome of the collisional event in region (i) by a well defined phase shift ${\delta_{\ell}}$ for each partial wave. Due to the invariance under rotations $SO(3)$ we can arrange the full scattering information in a diagonal, energy dependent, K matrix ${\mathbf{K}^{3D}}$.

\(iii) As the asymptotic region $r\gg{a_{\perp}}$ is concerned, only the transverse confining potential contributes. The wave function is thus a direct product of a sine or cosine function in the $z$-direction and a 2D harmonic oscillator (HO) mode for the $\rho$-direction, i.e. $\ket{\psi}=\sum_{n}c_n\ket{q_{n};n;m}$, where $q_{n}$ denotes the channel momentum in $z$-direction, $n$ denotes the transversal oscillator mode and $m$ the magnetic quantum number. The total 