Weak Edgeworth expansion for the mean-field interacting Bose gas

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 $1\mathrm{/}\sqrt{N}$ . 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.