We derive rigorously the one-dimensional cubic nonlinear Schrödinger equation from a many-body quantum dynamics. The interaction potential is rescaled through a weak-coupling limit together with a short-range one. We start from a factorized initial state, and prove propagation of chaos with the usual two-step procedure: in the former step, convergence of the solution of the BBGKY hierarchy associated to the many-body quantum system to a solution of the BBGKY hierarchy obtained from the cubic NLS by factorization is proven; in the latter, we show the uniqueness for the solution of the infinite BBGKY hierarchy. © Springer Science+Business Media, LLC 2007.
Adami, R., Golse, F., Teta, A. (2007). Rigorous Derivation of the cubic NLS in dimension one. JOURNAL OF STATISTICAL PHYSICS, 127(6), 1193-1220 [10.1007/s10955-006-9271-z].
Rigorous Derivation of the cubic NLS in dimension one
ADAMI, RICCARDO;
2007
Abstract
We derive rigorously the one-dimensional cubic nonlinear Schrödinger equation from a many-body quantum dynamics. The interaction potential is rescaled through a weak-coupling limit together with a short-range one. We start from a factorized initial state, and prove propagation of chaos with the usual two-step procedure: in the former step, convergence of the solution of the BBGKY hierarchy associated to the many-body quantum system to a solution of the BBGKY hierarchy obtained from the cubic NLS by factorization is proven; in the latter, we show the uniqueness for the solution of the infinite BBGKY hierarchy. © Springer Science+Business Media, LLC 2007.
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