  We consider identical quantum bosons with weak contact interactions in a
two-dimensional isotropic harmonic trap, and focus on states at the Lowest
Landau Level (LLL). At linear order in the coupling parameter $g$, we exploit
the rich algebraic structure of the problem to give an explicit construction of
a large family of quantum states with energies of the form $E_0+gE_1/4+O(g^2)$,
where $E_0$ and $E_1$ are integers. As a result, any superposition of these
states evolves periodically with a period of at most $8\pi/g$ until, at much
longer time scales of order $1/g^2$, corrections to the energies of order $g^2$
become important and may upset this perfectly periodic behavior. We further
construct coherent-like combinations of these states that naturally connect to
classical dynamics in an appropriate regime, and explain how our findings
relate to the known time-periodic features of the corresponding weakly
nonlinear classical theory. We briefly comment on possible generalizations of
our analysis to other numbers of spatial dimensions and other analogous
physical systems.
