  We study the statistics of quasiparticle and quasihole levels in small
interacting disordered systems within the Hartree-Fock approximation. The
distribution of the inverse compressibility, given according to Koopmans'
theorem by the distance between the two levels across the Fermi energy, evolves
from a Wigner distribution in the non-interacting limit to a shifted Gaussian
for strong interactions. On the other hand the nature of the distribution of
spacings between neighboring levels on the same side of the Fermi energy
(corresponding to energy differences between excited states of the system with
one missing or one extra electron) is not affected by the interaction and
follows Wigner-Dyson statistics. These results are derived analytically by
isolating and solving the appropriate Hartree-Fock equations for the two
levels. They are substantiated by numerical simulations of the full set of
Hartree-Fock equations for a disordered quantum dot with Coulomb interactions.
We find enhanced fluctuations of the inverse compressibility compared to the
prediction of Random Matrix Theory, possibly due to localization of the
wavefunctions around the edge of the dot. The distribution of the inverse
compressibility calculated from the discrete second derivative with respect to
the number of particles of the Hartree-Fock ground state energy deviates from
the distribution of the level spacing across the Fermi energy. The two
distributions have similar shapes but are shifted with respect to each other.
The deviation increases with the strength of the interaction thus indicating
the breakdown of Koopmans' theorem in the strongly interacting limit.
