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

• Published in: Archive for rational mechanics and analysis Vol. 221; no. 2; pp. 631 - 676
• Main Authors: Chen, Xuwen, Holmer, Justin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2016
• Subjects: Classical Mechanics; Complex Systems; Derivation; Derivatives; Dynamical systems; Dynamics; Estimates; Fluid- and Aerodynamics; Mathematical and Computational Physics; Nonlinear dynamics; Physics; Physics and Astronomy; Schroedinger equation; Texts; Theoretical; Rigorous Derivation; Pitaevskii Equation; Marginal Density; Trace Class Operator; Wigner Measure
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-016-0970-6

We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by N β - 1 V ( N β ) . where ∫ 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 → ∞ 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.