For bosons in a harmonic oscillator-like trap we will use is a spherical (S) or an elliptical (E) harmonic trap in one, two and finally three dimensions, with the latter given by
$$ \begin{equation} V_{ext}(\mathbf{r}) = \Bigg\{ \begin{array}{ll} \frac{1}{2}m\omega_{ho}^2r^2 & (S)\\ \strut \frac{1}{2}m[\omega_{ho}^2(x^2+y^2) + \omega_z^2z^2] & (E) \tag{4} \end{array} \end{equation} $$where (S) stands for symmetric and
$$ \begin{equation} \hat{H} = \sum_i^N \left( \frac{-\hbar^2}{2m} { \bigtriangledown }_{i}^2 + V_{ext}({\bf{r}}_i)\right) + \sum_{i < j}^{N} V_{int}({\bf{r}}_i,{\bf{r}}_j), \tag{5} \end{equation} $$as the two-body Hamiltonian of the system.