Derivation of the two-dimensional nonlinear Schrodinger equation from many body quantum dynamics
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Author(s)
Kirkpatrick, Kay
Schlein, Benjamin
Staffilani, Gigliola
Date Issued
February 2011
Journal
American Journal of Mathematics
Publisher
Johns Hopkins University Press
Citation
Kay Kirkpatrick, Benjamin Schlein, and Gigliola Staffilani. “Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics.” American Journal of Mathematics 133.1 (2011): 91-130.
Version
Author's final manuscript
Abstract
We derive rigorously, for both ${\Bbb R}^2$ and $[{-}L,L]^{\times 2}$, the cubic nonlinear Schr\"odinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1353/ajm.2011.0004
