Model Hamiltonian

In order to describe a cold bosonic gas in a trap we use the model Hamiltonian of type (2.122)

$$\hat H=- \frac{\hbar^2}{2m}\sum_{i=1}^N\Delta_i+\sum_{i<j}V_... ...vec r}_i-{\vec r}_j\vert)+\sum_{i=1}^N V_{ext}({\vec r}_i) \;,$$ (3.1)

Our system consists of $N$ spinless bosons of mass $m$ interacting through the two-body interatomic potential $V_{int}(r)$ and is subject to the external field which is taken to be harmonic and axially symmetric:

$$V_{ext}(\r)=m(\omega_\perp^2 r_\perp^2 + \omega_z^2z^2)/2,$$ (3.2)

where $z$ is the axial coordinate, $r_\perp$ is the radial transverse coordinate and $\omega_z$, $\omega _\perp$ are the corresponding oscillator frequencies.

For the interatomic potential we use two different repulsive model potentials: the hard-sphere (HS) potential (1.48) and the soft-sphere (SS) potential (1.52). In the case of the HS potential the $s$-wave scattering length coincides with the radius of the sphere and in the case of the SS potential is given by (1.59). For finite $V_0$ one always has $R>a_{3D}$, while for $V_0\to+\infty$ the SS potential coincides with the HS one with $R=a_{3D}$. The height $V_0$ of the potential is fixed by the value of the range $R$ in units of the scattering length, for which we choose $R=5a_{3D}$. It is worth noticing that the HS and the SS model with $R=5a_{3D}$ represent two extreme cases for a repulsive interatomic potential. In the HS case, the energy is entirely kinetic, while for the SS potential $a\simeq(m/\hbar^2)\int_0^{\infty}V(r)r^2 dr$, according to Born approximation, and the energy is almost all potential. By comparing the results of the two model potentials we can investigate to what extent the ground-state properties of the system depend only on the $s$-wave scattering length and not on the details of the potential.