Derivation of the two dimensional nonlinear Schrodinger equation from many body quantum dynamics

Kirkpatrick, Kay; Schlein, Benjamin; Staffilani, Gigliola

Derivation of the two dimensional nonlinear Schrodinger equation from many body quantum dynamics

Authors

Kirkpatrick, Kay

Schlein, Benjamin

Staffilani, Gigliola

Description

We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger 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.
29 pages, 3 figures; reference added, typos fixed, section 7.2 simplified. To appear, American Journal of Mathematics