Fluctuations around Hartree states in the mean-field regime

TL;DR

This paper analyzes the fluctuations of large bosonic systems around Hartree states in the mean-field regime, deriving an effective quadratic Hamiltonian via a direct N-particle space method, based on Bogoliubov's approximation.

Contribution

It introduces a new direct N-particle space method to derive the fluctuation dynamics, differing from traditional coherent state approaches.

Findings

Fluctuations are governed by an effective quadratic Hamiltonian.

The method provides a rigorous derivation of fluctuation dynamics.

Results are applicable to large bosonic systems in the mean-field limit.

Abstract

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.