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

Abstract

We consider a system of N bosons interacting through a singular twobody 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.

Central limit theorem for Bose gases interacting through singular potentials

“…As proven in [10,Proposition 8], any moments of the number of particles operator are approximately preserved with respect to conjugation with the Bogoliubov transformation T N,t . To be more precise for every fixed k ∈ N and δ > 0, there exists C > 0 such that…”

“…The proof of Theorem 1.2 is based on the norm approximation (1.29) from [10]. In the following we collect useful properties of the unitaries used therein.…”

“…The norm approximation (1.29) from [10] implies, that for every α < min{β/2, (1 − β)/2} there exists C > 0 such that…”

“…Besides the convergence of the one-particle reduced density γ N,t associated with ψ N,t , the norm approximation of ψ N,t has been studied for different settings of β ∈ (0, 1) in [10,22,26,27]. Our result is based on the norm approximation obtained in [10] covering β < 1 whose ideas we explain in the following.…”

“…Note that G N,t consists of terms which quadratic in creation and annihilation operators and of terms of higher order. Nevertheless, in [10,Lemma 5] it is shown that G N,t can be approximated through the generator G 2,N,t containing quadratic terms only.…”