# Fluctuations of $N$-particle quantum dynamics around the nonlinear Schrödinger equation

### Christian Brennecke

Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland### Phan Thành Nam

Department of Mathematics, LMU Munich, Theresienstrasse 39, 80333 Munich, Germany### Marcin Napiórkowski

Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland### Benjamin Schlein

Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

## Abstract

We consider a system of $N$ bosons interacting through a singular two-body potential scaling with $N$ and having the form $N_{3β−1}V(N_{β}x)$, for an arbitrary parameter $β∈(0,1)$. We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose–Einstein condensation in terms of a cubic nonlinear Schrödinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.

## Cite this article

Christian Brennecke, Phan Thành Nam, Marcin Napiórkowski, Benjamin Schlein, Fluctuations of $N$-particle quantum dynamics around the nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1201–1235

DOI 10.1016/J.ANIHPC.2018.10.007