  We consider identical quantum bosons with weak contact interactions in a
two-dimensional isotropic harmonic trap. When the interactions are turned off,
the energy levels are equidistant and highly degenerate. At linear order in the
coupling parameter, these degenerate levels split, and we study the patterns of
this splitting. It turns out that the problem is mathematically identical to
diagonalizing the quantum resonant system of the two-dimensional
Gross-Pitaevskii equation, whose classical counterpart has been previously
studied in the mathematical literature on turbulence. Our purpose is to explore
the implications of the symmetries and energy bounds of this resonant system,
previously studied for the classical case, for the quantum level splitting.
Simplifications in computing the splitting spectrum numerically result from
exploiting the symmetries. The highest energy state emanating from each
unperturbed level is explicitly described by our analytics. We furthermore
discuss the energy level spacing distributions in the spirit of quantum chaos
theory. After separating the eigenvalues into blocks with respect to the known
conservation laws, we observe the Wigner-Dyson statistics within specific large
blocks, which leaves little room for further integrable structures in the
problem beyond the symmetries that are already explicitly known.
