this work, thus allowing inelastic collisions, where particles can be scattered into different channels. The latter are asymptotically defined by the transversal trap modes. Our approach is based on the fully analytical and non-perturbative description in terms of $K$ matrices, whose usefulness has already been demonstrated in a series of previous works [@granger2004; @giannakeas2012; @giannakeas2013; @hess2014]. Due to the seminal results of Bo Gao [@gao1998solvdw; @gao1998qdtvdw; @gao2001angular] who investigated the free-space scattering properties of neutral alkaline atoms possessing a van der Waals tail, analytical formulas for the (generalized) scattering length were derived, which are used in the present setup to adequately describe the scattering event on the interatomic scale.

The $K$ matrix approach provides a generalization of the works of Granger [*et al*]{} [@granger2004] and Kim [*et al*]{} [@kim2005] incorporating however [*all*]{} the higher partial waves and contributions from [*all*]{} the closed channels. Furthermore, going beyond the previous studies we derive the connection of the [*physical*]{} $K$-matrix with all the relevant scattering observables obtaining thus the [*full* ]{} multi-component scattering wave function. Using this formalism we study the universal properties of ultracold collisions in the $\ell$-wave single partial wave approximation ([$\ell$-SPWA]{}). Here, we find the existence of energies above the 