We consider the dynamics of a 2D Bose gas with an interaction potential of the form N 2-1w (N center dot ) for E (0,. 3 ). The interaction may be chosen to be negative and large, leading to the instability regime 2 where the corresponding focusing cubic nonlinear Schr & ouml,.dinger equation (NLS) may blow up in finite time. We show that to leading order, the N-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics.

We consider a gas of N weakly interacting bosons in the ground state. Such gases exhibit Bose-Einstein condensation. The binding energy is defined as the energy it takes to remove one particle from the gas. In this article, we prove an asymptotic expansion for the binding energy, and compute the first orders explicitly for the homogeneous gas. Our result addresses in particular a conjecture by Nam (Lett Math Phys 108(1):141-159, 2018), and provides an asymptotic expansion of the ionization energy of bosonic atoms.