Fluctuations around Hartree states in the mean-field regime

@article{Lewin2013FluctuationsAH,
  title={Fluctuations around Hartree states in the mean-field regime},
  author={Mathieu Lewin and Phan Th{\`a}nh Nam and Benjamin Schlein},
  journal={American Journal of Mathematics},
  year={2013},
  volume={137},
  pages={1613 - 1650}
}
We consider the dynamics of a large system of $N$ interacting bosons in the mean-field regime where the interaction is of order $1/N$. We prove that the fluctuations around the nonlinear Hartree state are generated by an effective quadratic Hamiltonian in Fock space, which is derived from Bogoliubov's approximation. We use a direct method in the $N$-particle space, which is different from the one based on coherent states in Fock space. 
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