  We consider a gas of N bosons with interactions in the mean-field scaling
regime. We review the proof of an asymptotic expansion of its low-energy
spectrum, eigenstates and dynamics, which provides corrections to Bogoliubov
theory to all orders in 1/N. This is based on joint works with S. Petrat, P.
Pickl, R. Seiringer and A. Soffer. In addition, we derive a full asymptotic
expansion of the ground state one-body reduced density matrix.
