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We consider the focusing 3D quantum many-body dynamic which models a dilutebose gas strongly confined in two spatial directions. We assume that themicroscopic pair interaction is attractive and given by$a^{3\beta-1}V(a^{\beta}\cdot)$ where $\int V\leqslant 0$ and $a$ matches theGross-Pitaevskii scaling condition. We carefully examine the effects of thefine interplay between the strength of the confining potential and the numberof particles on the 3D $N$-body dynamic. We overcome the difficulties generatedby the attractive interaction in 3D and establish new focusing energyestimates. We study the corresponding BBGKY hierarchy which contains adiverging coefficient as the strength of the confining potential tends to$\infty $. We prove that the limiting structure of the density matricescounterbalances this diverging coefficient. We establish the convergence of theBBGKY sequence and hence the propagation of chaos for the focusing quantummany-body system. We derive rigorously the 1D focusing cubic NLS as themean-field limit of this 3D focusing quantum many-body dynamic and obtain theexact 3D to 1D coupling constant.

Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schr\"{o}dinger Equation from 3D