Beyond Bogoliubov dynamics

Abstract

We consider a system of $N$ interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true $N$-body dynamics in norm to arbitrary precision. The $N$-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick’s theorem. We determine the $n$-point correlation functions of the excitations around the condensate, as well as the reduced densities of the $N$-body system, to arbitrary accuracy, given only the knowledge of the two-point correlation functions of a quasifree state and the solution of the Hartree equation. In this way, the complex problem of computing all $n$-point correlation functions for an interacting $N$-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point correlation functions.