Since 1947 Bogoliubov treatment of Bose-Einstein condensation for
interacting bosons has represented the guide model to thinking about
Bose gases. Remarkably, such a theory predicts a linear excitation
spectrum (in sharp contrast with the quadratic dispersion of free
particles) and provides expressions for the thermodynamic functions
which are believed to be correct in the dilute limit. However, there
are only a few cases where the predictions of Bogoliubov theory can be
actually proved. In particular, one of the main mathematical issues is
to recover the physical intuition that the parameter that should
appear in the expressions of the physical quantities is the scattering
length of the interaction.
In this talk I will discuss how the predictions of Bogolibov theory
can be rigoursly obtained in the case of systems of N interacting
bosons trapped in a box with volume one and interacting through a weak,
repulsive potential with scattering length 1/N (Gross-Pitaevskii regime).
Joint work with C. Boccato, C. Brennecke and B. Schlein.