Affiliations: IRMAR, Université de Rennes I, UMR-CNRS 6625, Campus de Beaulieu, 35042 Rennes Cedex, France. E-mails: {zied.ammari, sebastien.breteaux}@univ-rennes1.fr
Abstract: 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 Schrödinger 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.
Keywords: classical limit, coherent state, Fock space, non-linear Schrödinger equation, mean field limit, non-autonomous Schrödinger equation
