Bogoliubov theory for trapped Bose-Einstein condensates

We consider systems of NN particles interacting through a repulsive potential in the Gross-Pitaevskii regime. We prove complete Bose-Einstein condensation and we determine the form of the low-energy spectrum, in the limit of large NN. Our results confirm the predictions of Bogoliubov theory. This talk is based on joint works with C. Boccato, C. Brennecke and S. Cenatiempo.