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Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 1, February 2011
- pp. 91-130
- 10.1353/ajm.2011.0004
- Article
- Additional Information
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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.