Norm approximation for many-body quantum dynamics: focusing case in low dimensions

TL;DR

This paper proves that for low-dimensional focusing many-body quantum systems, the dynamics of the condensate and fluctuations can be effectively approximated by nonlinear Schrödinger and quadratic Hamiltonians, respectively, in the large particle limit.

Contribution

It establishes norm approximation results for focusing bosonic systems in low dimensions, extending the validity of Bogoliubov approximation for all interaction strengths in 1D and for subcritical interactions in 2D.

Findings

Effective description of condensate by nonlinear Schrödinger equation

Fluctuations governed by quadratic Hamiltonian from Bogoliubov approximation

Results valid for all interaction strengths in 1D and for 0<β<1 in 2D

Abstract

We study the norm approximation to the Schr\"odinger dynamics of $N$ bosons in $\mathbb{R}^d$ ($d=1,2$) with an interaction potential of the form $N^{d\beta-1}w(N^{\beta}(x-y))$. Here we are interested in the focusing case $w\le 0$. Assuming that there is complete Bose-Einstein condensation in the initial state, we show that in the large $N$ limit, the evolution of the condensate is effectively described by a nonlinear Schr\"odinger equation and the evolution of the fluctuations around the condensate is governed by a quadratic Hamiltonian, resulting from Bogoliubov approximation. Our result holds true for all $\beta>0$ when $d=1$ and for all $0<\beta<1$ when $d=2$.