## On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with A Switchable Quadratic Trap    [PDF]

Xuwen Chen
We consider the dynamics of the 3D N-body Schr\"odinger equation in the presence of a switchable quadratic trap. We assume the pair interaction potential is N^{3{\beta}-1}V(N^{{\beta}}x). We justify the mean-field approximation and offer a rigorous derivation of the 3D cubic NLS with a switchable quadratic trap. In particular, this paper fills a gap in the literature between the work [29] by Lieb, Seiringer, Solovej, and Yngvason, which corresponds to the ground state when the trap is fully on, and the work of Erd\"os, Schlein and Yau [16] and Chen and Pavlovi\'c [5], which addresses the evolution after the trap has been turned off. Our proof is an adaptation and simplification of the argument in Chen and Pavlovi\'c [5] which solves the same problem in the absence of a trap. We also extend the range of {\beta} from (0,1/4) in Chen and Pavlovi\'c [5] to (0,2/7].
View original: http://arxiv.org/abs/1204.0125