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

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 $d=2$.