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

By Kay Kirkpatrick, B. Schlein, Gigliola Staffilani. 2008

155 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5a7f5a607afd6bde8f0a762a3a4d9751a86098f6

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In Atomic and Molecular Physics, and Optics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Effective Evolution Equations from Quantum Dynamics (2015, background): “Results in this direction have been obtained in [1, 2, 20] in the one-dimensional setting, in [42] in the two-dimensional case and in [27] in three dimensions.”
Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, background): “This general scheme, first introduced in [36] for bounded interactions, was later extended to more singular potentials; see, for example, [1,6, 9, 10, 12, 19, 27].”
Mean–Field Evolution of Fermionic Systems (2013, background): “As a consequence, the limiting Hartree equation has a local nonlinearity (nonlinear Schrödinger equations with local nonlinearity in one and two dimensions have been derived from many body quantum mechanics in appropriate mean field limits in [1, 31]).”