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arXiv:math-ph/0111042 · Advances in Theoretical and Mathematical Physics

Derivation of the nonlinear Schrödinger equation from a many-body Coulomb system

László Erdős, Horng‐Tzer Yau

Published Jan 1, 2001• 254 citations
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Abstract

We consider the time evolution of iV 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 -> oo.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 Schrodinger equations.

DOI
10.4310/atmp.2001.v5.n6.a6

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We consider the time evolution of iV 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 -> oo.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 Schrodinger equations.
We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.
We provide an error bound for approximating the time evolution of N bosons by a generalized nonlinear Hartree equation. The bosons are assumed to interact via permutation symmetric bounded many-particle potentials and the initial wave-function is a product state. We show that the error between the actual evolution of a single particle derived from tracing out the full N-particle Schrodinger equation and the solution to the mean field approximate generalized nonlinear Hartree equation scales as 1/N for all times. Our result is a generalization of rigorous error bounds previously given for the case of bounded 2-particle potentials
We consider the dynamics of N interacting Bosons in three dimensions which are strongly confined in one or two directions. We analyze the mean field scaling and the nonlinear Schrödinger scaling of the interaction potential w and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit the dynamics of the N-particle system is described by a nonlinear equation in two or one dimensions. In the case of the mean field scaling this equation is the Hartree equation and for the second scaling the nonlinear Schrödinger equation. In both cases we obtain explicit bounds for the rate of convergence of the N-particle dynamics to the one-particle dynamics.
We study the evolution of a many-particle system whose wave function obeys the N-body Schroedinger equation under Bose symmetry. The system Hamiltonian describes pairwise particle interactions in the absence of an external potential. We derive apriori dispersive estimates that express the overall repulsive nature of the particle interactions. These estimates hold for a wide class of two-body interaction potentials which are independent of the particle number, N. We discuss applications of these estimates to the BBGKY hierarchy for reduced density matrices analyzed by Elgart, Erdos, Schlein and Yau.