The excitation spectrum of two dimensional Bose gases in the Gross-Pitaevskii regime

We consider $N$ bosons in two dimensions, interacting through a potential whose scattegin length is exponentially small in the number of particles of the systems (Gross-Pitaevskii regime). For this system we provide an expression for the ground state energy and the low energy excitation spectrum of $N$, up to errors vanishing with $N$. Our result represents the first case where the validity of Bogoliubov predictions for the low energy spectrum of a dilute two dimensional Bose gas has been established in a scaling where the dimensionality of the system plays a role.