  A system of N two-dimensional weakly interacting bosons in a harmonic trap is
considered. When the two-particle potential is a delta function Smith and
Wilkin have analytically proved that the elementary symmetric polynomials of
particle coordinates measured from the center of mass are exact eigenstates. In
this study, we point out that their proof works equally well for an arbitrary
two-particle potential which possesses the translational and rotational
symmetries. We find that the interaction energy associated with the eigenstate
with angular momentum L is equal to aN(N-1)/2+(b-a)NL/2, where a and b are the
interaction energies of two bosons in the lowest-energy one-particle state with
zero and one unit of angular momentum, respectively. Additionally, we study
briefly the case of attractive quartic interactions. We prove rigorously that
the lowest-energy state is the one in which all angular momentum is carried by
the center of mass motion.
