Seminarium KMMF "Teoria Dwoistości"

We consider the dynamics of N bosons in the mean-field limit. It is well known that to leading order the N-body Schroedinger dynamics can be approximated by the Hartree equation. This talk is about the Bogoliubov approximation, which is the next-to-leading order correction that also describes the fluctuations around the mean-field Hartree state. While the Hartree approximation gives convergence in the sense of reduced densities, the Bogoliubov approximation gives convergence in the sense of the L^2 norm of the N-body Hilbert space. Here, I will present a rigorous derivation of this approximation in the most simple mean-field scaling limit. We also consider the setting where both the volume and the density of the gas tend to infinity and the interaction is scaled with the inverse density. These results can be applied to the setting of a Bose gas with slight perturbations. Then the coupling constant is such that the self-interaction of the fluctuations is of leading order, which leads to a finite (non-zero) speed of sound in the gas.