Derivation of the nonlinear Schr\"odinger equation from a many body Coulomb system

By L. Erdős, Horng-Tzer Yau. 2001

270 citations recorded by the source corpus.

We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the correlation functions factorize in the limit $N \to \infty$. Furthermore, the limiting one particle density matrix satisfies the nonlinear Hartree equation. The key ingredients are the uniqueness of the BBGKY hierarchy for the correlation functions and a new apriori estimate for the many-body Schr\"odinger equations.

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

View Paper: https://www.semanticscholar.org/paper/b004f1c41f10653acdccebbcf70c9a5f07b5950d

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate (2004, background): “See the work of Hepp [10] and Spohn [26] for bounded potentials and [2, 5] for potentials with Coulomb type singularity.”
Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems (2005, unknown): “Keywords: Feynman diagrams, BBGKY hierarchy, dispersive estimates, propagation of chaos.”
Mean field dynamics of boson stars (2005, methodology): “Hence, although our general strategy is the same as in [ 4 ], we need here more refined estimates.”
Bogoliubov Spectrum of Interacting Bose Gases (2012, methodology): “19) is well understood in the time-dependent setting [58, 20, 1, 2, 50, 14], and in this case there has been recent interest in corrections to the Hartree equation [25, 26, 6], where our method might also apply.”
The quintic NLS as the mean field limit of a boson gas with three-body interactions (2008, background): “A related fundamental line of research addresses the derivation of the mean field dynamics for a dilute Bose gas, in a scaling regime where the interparticle interactions and the kinetic energy are comparable in magnitude (the Gross-Pitaevskii scaling).”
Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. (2009, methodology): “This approach has been highly successful, even for very singular potentials, in the work of Elgart, Erdős, Schlein and Yau [6, 7, 8, 9, 10, 11].”
Mean–Field Evolution of Fermionic Systems (2013, methodology): “In [9], the method of [40] was extended to prove the convergence (1.”
Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, background): “This general scheme, first introduced in [23] for bounded interactions, was later extended to potentials with Coulomb singularities in [11] (and in [4], for bosons with relativistic dispersions).”