The mean-field limit of the Lieb-Liniger model

<p style='text-indent:20px;'>We consider the well-known Lieb-Liniger (LL) model for <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> bosons interacting pairwise on line via id="M2">\begin{document}$ \delta potential in mean-field scaling regime. Assuming suitable asymptotic factorization of initial wave functions and convergence microscopic energy per particle, we show that time-dependent reduced density matrices system converge trace norm to pure states given by solution one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate convergence. In contrast previous work [<xref ref-type="bibr" rid="b3">3</xref>] relying formalism second quantization coherent without explicit rate, our proof is based counting method Pickl rid="b65">65</xref>,<xref rid="b66">66</xref>,<xref rid="b67">67</xref>] Knowles rid="b44">44</xref>]. To overcome difficulties stemming from singularity id="M3">\begin{document}$ potential, introduce a new short-range approximation argument exploits Hölder continuity id="M4">\begin{document}$ $\end{document}</tex-math></inline-formula>-body function single particle variable. By further exploiting id="M5">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-subcritical well-posedness theory 1D NLS, can prove when limiting NLS has finite mass, but only very special class id="M6">\begin{document}$ states.</p>