Journal of Statistical Physics

, Volume 127, Issue 6, pp 1193–1220

Rigorous Derivation of the Cubic NLS in Dimension One

Article

DOI: 10.1007/s10955-006-9271-z

Cite this article as:
Adami, R., Golse, F. & Teta, A. J Stat Phys (2007) 127: 1193. doi:10.1007/s10955-006-9271-z

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.

Keywords

quantum mechanicsnonlinear schrödingergross-pitaevskiipropagation of chaosBBGKY hierarchy

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Riccardo Adami
    • 1
  • François Golse
    • 2
  • Alessandro Teta
    • 3
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità Milano BicoccaMilanoItaly
  2. 2.Département de Mathématiques et ApplicationsÉcole Normale Supérieure, Paris 45, rue d’UlmParis cedex 05France
  3. 3.Dipartimento di Matematica Pura ed ApplicataUniversità di L’AquilaCoppito di L’Aquila (AQ)Italy