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

We study the norm approximation to the Schrodinger dynamics of N bosons in 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 <= 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 Schrodinger 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. (C) 2019 Elsevier Inc. All rights reserved.