Propagation of chaos for many-boson systems in one dimension with a point pair-interaction

We study the mean field limit of nonrelativistic quantum many-boson systems with delta potential in one-dimensional space. Such problem is related to the semiclassical limit of a second quantized Hamiltonian with an interaction given by a quartic Wick product. In this framework, we show that the evolution of coherent states is semiclassically given by squeezed coherent states under the action of a time-dependent affine Bogoliubov transformation. Results similar to those stated by Hepp [Comm. Math. Phys. 35 (1974), 265-277] and Ginibre-Velo [Comm. Math. Phys. 66 (1979), 37-76 and 68 (1979), 45-68] are proved. Furthermore, we show propagation of chaos for Schrodinger dynamics in the mean field limit using the argument of Rodnianski-Schlein [Comm. Math. Phys. 291 (2009), 31-61]. Thus, we provide a derivation of the cubic NLS equation in one dimension.