energy $E$ distributes over these two degrees of freedom according to the relation $E=\hbar\omega_{\perp}(2n+|m|+1)+\frac{\hbar^2
  q_{n}^2}{2\mu}$.

We note however, that the azimuthal quantum number $m$ asymptotically associated to the $SO(2)$ rotations, is a good quantum number in both regimes and therefore is a fixed quantity throughout our analysis which is set to $m=0$ for what follows and is therefore also omitted in the labeling of the states. Hereafter, we drop the typical assumption in most of the existing literature that the total collision energy has to be sufficiently small such that only the energetically lowest transversal state can be populated, i.e. $\ket{\psi}\sim\ket{q_0;0}$ and thus we are going beyond previous studies by allowing inelastic collisions involving several transverse modes. Already from this expansion which is only invariant under $\mathcal{T}_z\otimes SO(2)$, where $\mathcal{T}_z$ denotes translation along the $z$-direction and $SO(2)$ rotations around that axis, we conclude that the mapping from region (ii) to (iii) cannot be accomplished by a unitary transformation. Thus, in order to transfer the scattering information between these two regions of different symmetry, an appropriate way is given by the local frame transformation $U_{ln}$ [@harmin1982nonhydrostark; @*harmin1982stark; @*harmin1985; @fano1981; @greene1987]. However, as already discussed previously [@granger2004; @giannakeas2013] the application of this 