  When a Bose-Einstein condensed cloud of atoms is given some angular momentum,
it forms vortices arranged in structures with a discrete rotational symmetry.
For these vortex states, the Hilbert space of the exact solution separates into
a "primary" space related to the mean-field Gross-Pitaevskii solution and a
"complementary" space including the corrections beyond mean-field. Considering
a weakly-interacting Bose-Einstein condensate of harmonically-trapped atoms, we
demonstrate how this separation can be used to close the conceptual gap between
exact solutions for systems with only a few atoms and the thermodynamic limit
for which the mean-field is the correct leading-order approximation. Although
we illustrate this approach for the case of weak interactions, it is expected
to be more generally valid.
