The mean-field limit of the Lieb-Liniger model

We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $\end{document} bosons interacting pairwise on the line via the \begin{document}$ \delta $\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [ 3 ] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [ 65 , 66 , 67 ] and Knowles and Pickl [ 44 ]. To overcome difficulties stemming from the singularity of the \begin{document}$ \delta $\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$ N $\end{document} -body wave function in a single particle variable. By further exploiting the \begin{document}$ L^2 $\end{document} -subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of \begin{document}$ N $\end{document} -body initial states.