Rigorous Derivation of the Cubic NLS in Dimension One |
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| Authors: | Riccardo Adami François Golse Alessandro Teta |
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| Affiliation: | (1) Dipartimento di Matematica e Applicazioni, Università Milano Bicocca, v. R. Cozzi 53, 20125 Milano, Italy;(2) Département de Mathématiques et Applications, école Normale Supérieure, Paris 45, rue d’Ulm, F75230 Paris cedex 05, France;(3) Dipartimento di Matematica Pura ed Applicata, Università di L’Aquila, Via Vetoio (Coppito 1), 67010 Coppito di L’Aquila (AQ), Italy |
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| 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. |
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| Keywords: | quantum mechanics nonlinear schr?dinger gross-pitaevskii propagation of chaos BBGKY hierarchy |
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