Exact eigenstates for repulsive bosons in two dimensions

Abstract

We consider a model of N two-dimensional bosons in a harmonic potential with weak repulsive \ensuremath{\delta}-function interactions. We show analytically that, for angular momentum $L<~N,$ the elementary symmetric polynomials of particle coordinates measured from the center of mass are exact eigenstates with energy $N(N\ensuremath{-}L/2\ensuremath{-}1).$ Extensive numerical analysis confirms that these are actually the ground states, but we are currently unable to prove this analytically. The special case $L=N$ can be thought of as the generalization of the usual superfluid one-vortex state to Bose-Einstein condensates in a trap.