Mean Field Derivation of DNLS from the Bose–Hubbard Model

We prove that the flow of the discrete nonlinear Schrödinger equation (DNLS) is the mean field limit of the quantum dynamics of the Bose–Hubbard model for N interacting particles. In particular, we show that the Wick symbol of the annihilation operators evolved in the Heisenberg picture converges, as N becomes large, to the solution of the DNLS. A quantitative L p -estimate, for any p ≥ 1 , is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states.