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

Abstract

We consider a system of $N$ bosons interacting through a singular two-body potential scaling with N and having the form $N^{3β-1}V(N^βx)$ , for an arbitrary parameter$β \in(0 ;1)$. 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.