Some exact analytic results for the linear and non-linear response of atoms in a trap with a model interaction

We present an exact expression for the evolution of the wavefunction of $N$ interacting atoms in an arbitrarily time-dependent, $d$-dimensional parabolic trap potential $\omega(t)$. The interaction potential between atoms is taken to be of the form $\xi/r^2$ with $\xi>0$. For a constant trap potential $\omega(t)=\omega_0$, we find an exact, infinite set of relative mode excitations. These excitations are relevant to the linear response of the system; they are universal in that their frequencies are independent of the initial state of the system (e.g. Bose-Einstein condensate), the strength $\xi$ of the atom-atom interaction, the dimensionality $d$ of the trap and the number of atoms $N$. The time evolution of the system for general $\omega(t)$ derives entirely from the solution to the corresponding classical 1D single-particle problem. An analytic expression for the frequency response of the $N$-atom cluster is given in terms of $\omega(t)$. We consider the important example of a sinusoidally-varying trap perturbation. Our treatment, being exact, spans the `linear' and `non-linear' regimes. Certain features of the response spectrum are found to be insensitive to interaction strength and atom number.