Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrodinger Equation

We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by N beta-1V(N-beta.) where integral V <= 0. We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the 1D focusing cubic NLS with a quadratic trap as the N -> infinity limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.