Norm approximation for manybody quantum dynamics: focusing case in low dimensions
Abstract
We study the norm approximation to the Schrödinger dynamics of $N$ bosons in $\mathbb{R}^d$ ($d=1,2$) with an interaction potential of the form $N^{d\beta1}w(N^{\beta}(xy))$. Here we are interested in the focusing case $w\le 0$. Assuming that there is complete BoseEinstein condensation in the initial state, we show that in the large $N$ limit, the evolution of the condensate is effectively described by a nonlinear Schrödinger 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 $d=2$.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.09684
 arXiv:
 arXiv:1710.09684
 Bibcode:
 2017arXiv171009684T
 Keywords:

 Mathematical Physics;
 Condensed Matter  Quantum Gases;
 Mathematics  Analysis of PDEs
 EPrint:
 Advances in Mathematics 350, 547587 (2019)