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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
R. Adami, C. Bardos, F. Golse and A. Teta, Towards a rigorous derivation of the cubic NLSE in dimension one. Asymptot. Anal. 40(2):93–108 (2004).
C. Bardos, L. Erdös, F. Golse, N. Mauser and H.-T. Yan, Derivation of the Schrödinger-Poisson equation from the quantum N-body problem. C. R. Math. Acad. Sci. Paris 334(6):515–520 (2002).
C. Bardos, F. Golse and N. Mauser, Weak coupling limit of the N particles Schrödinger equation. Methods Appl. Anal. 7(2):275–293 (2000).
A. Elgart, L. Erdös, B. Schlein and H.-T. Yau, Gross-pitaevskii equation as the mean field limit of weakly coupled Bosons. Math. Phys. preprint archive, mp-arc 04-333 (2004).
L. Erdös, Private communication.
L. Erdös, B. Schlein and H.-T. Yau, Derivation of the gross-pitaevskii equation for the dynamics of bose-einstein condensate. Math. Phys. preprint archive, mp-arc 04-319 (2004).
L. Erdös, B. Schlein and H.-T. Yau, Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems, mp-arc 05-280.
L. Erdös, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate, mp-arc 06-176.
L. Erdös and H.-T. Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6):1169–1205 (2001).
J. Ginibre and G. Velo, The classical field limit of scattering theory for non relativistic many-boson systems. I and II. Comm. Math. Phys. 66: 37–76 (1979), and 68:45–68 (1979).
K. Hepp, The classical limit for quantum mechanical correlation functions. Comm. Math. Phys. 35:265–277 (1974).
E.H. Lieb, R. Seiringer and J. Yngvason, The Quantum-mechanical Many Body Problem: The Bose Gas, http://arxiv.org/abs/math-ph/?math-ph%2F0405004.
E. H. Lieb, R. Seiringer and J. Yngvason, One-dimensional behavior of dilute, trapped Bose gases. Comm. Math. Phys. 244(2):347–393 (2004).
H. Spohn, Kinetic equations from hamiltonian dynamics. Rev. Mod. Phys. 52(3):600–640 (1980).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Adami, R., Golse, F. & Teta, A. Rigorous Derivation of the Cubic NLS in Dimension One. J Stat Phys 127, 1193–1220 (2007). https://doi.org/10.1007/s10955-006-9271-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9271-z