Exponential decay of the number of excitations in the weakly interacting Bose gas

We consider $N$ trapped bosons in the mean-field limit with coupling constant ${\lambda }_N=1\mathrm{/}(N-1)$. The ground state of such systems exhibits Bose–Einstein condensation. We prove that the probability of finding $\mathrm{\ell }$ particles outside the condensate wave function decays exponentially in $\mathrm{\ell }$.