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Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. (English) Zbl 1014.81063

Summary: 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 Schrödinger equations.

MSC:

81V70 Many-body theory; quantum Hall effect
35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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