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We present an exact expression for the evolution of the wavefunction of $N$interacting atoms in an arbitrarily time-dependent, $d$-dimensional parabolictrap potential $\omega(t)$. The interaction potential between atoms is taken tobe 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 modeexcitations. These excitations are relevant to the linear response of thesystem; they are universal in that their frequencies are independent of theinitial 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 numberof atoms $N$. The time evolution of the system for general $\omega(t)$ derivesentirely from the solution to the corresponding classical 1D single-particleproblem. An analytic expression for the frequency response of the $N$-atomcluster is given in terms of $\omega(t)$. We consider the important example ofa sinusoidally-varying trap perturbation. Our treatment, being exact, spans the`linear' and `non-linear' regimes. Certain features of the response spectrumare found to be insensitive to interaction strength and atom number.Comment: to appear in Phys. Rev. A. 16 pages, RevTex, 2 Postscript figure

Analytic results for the linear and nonlinear response of atoms in a trap with a model interaction