Vol. 3, No. 4, 2021

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Beyond Bogoliubov dynamics

Lea Boßmann, Sören Petrat, Peter Pickl and Avy Soffer

Vol. 3 (2021), No. 4, 677–726

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.

Bose–Einstein condensate, perturbation theory, asymptotic expansion, quantum many-body dynamics, effective equations, Bogoliubov equation
Mathematical Subject Classification
Primary: 35Q40, 35Q55, 81Q05, 82C10
Received: 24 January 2021
Revised: 3 June 2021
Accepted: 21 August 2021
Published: 12 February 2022
Lea Boßmann
Fachbereich Mathematik
Eberhard Karls Universität Tübingen
Institute of Science and Technology Austria
Sören Petrat
Department of Mathematics and Logistics
Jacobs University Bremen
Faculty 3 - Mathematics and Computer Science
University of Bremen
Peter Pickl
Fachbereich Mathematik
Eberhard Karls Universität Tübingen
Mathematisches Institut
Avy Soffer
Department of Mathematics
Rutgers University
Piscataway, NJ
United States