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
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.

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