Analytical Results for Trapped Weakly Interacting Bosons in Two Dimensions

We consider a model of N two-dimensional bosons in a harmonic trap with translational and rotational invariant, weak two-particle interaction. We present in configuration space a systematical recursive method for constructing all wave functions with angular momentum L and corresponding energies and apply it to L\leq 6 for all N. The lower and the upper bounds for interaction energy are estimated. We analitically confirm the conjecture of Smith et al. that elementary symmetric polynomial is the ground state for repulsive delta interaction, for all N\geq L up to L\leq 6. Additionally, we find that there exist vanishing-energy solutions for L\geq N(N-1), signalizing the exclusive statistics. Finally, we consider briefly the case of attractive power-like potential r^k, k>-2, and prove that the lowest-energy state is still the one in which all angular momentum is absorbed by the center-of-mass motion.