We consider the ground state and the low-energy excited states of a system of
$N$ identical bosons with interactions in the mean-field scaling regime. For
the ground state, we derive an Edgeworth expansion for the fluctuations of
bounded one-body operators, which yields corrections to a central limit theorem
to any order in $1/\sqrt{N}$ . For suitable excited states, we show that the
limiting distribution is a polynomial times a normal distribution, and that
higher order corrections are given by an Edgeworth-type expansion.
