non-trivial part is the relative motion Hamiltonian, which reads $$H=\frac{-\hbar^2}{2\mu}\Delta+\frac{1}{2}\mu\omega_{\perp}^2\rho^2+V_{LJ}(r),
\label{eq:Hamiltonian}$$ where $r=\sqrt{z^2+\rho^2}$ is the interparticle distance, with $z$ and $\rho$ describing the longitudinal and transversal degrees of freedom, respectively. $\mu$ denotes the reduced mass of the colliding pair and $\omega_\perp$ is the confinement frequency. Accordingly, the harmonic oscillator length scale is given by $a_{\perp}=\sqrt{\hbar /
 \mu \omega_{\perp}}$. The term $V_{LJ}(r)=\frac{C_{10}}{r^{10}}-\frac{C_{6}}{r^{6}}$ is the Lennard-Jones 6-10 potential indicating the short-range, two-body interatomic interactions. $C_{6}$ is the dispersion coefficient and it defines the van der Waals length scale via the relation $\beta_{6}=(2\mu C_{6}/\hbar^2)^{1/4}$. We regard $C_{10}$ as a parameter in order to tune the corresponding scattering lengths induced by the short-range potential. Among others, the particular choice $V_{LJ}(r)$ is motivated by the fact that we attribute to the two-body physics a realistic character avoiding the use of zero-range approximations. In addition, this particular type of interactions are analytically solvable by means of the generalized effective range theory [@gao2009single]. However, any other interatomic potential is also suited as long as the length scale $\beta_n$ associated with this potential is small compared to the oscillator length, i.e. $\beta_n\ll a_{\perp}$.

As in previous works [@granger2004; @giannakeas2012; @giannakeas2013; @hess2014] on the Hamiltonian given in Eq. , 