Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I.

By M. Grillakis, M. Machedon, Dionisios Margetis. 2009

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Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential $${v(x)= \epsilon \chi(x) |x|^{-1}}$$, where $${\epsilon}$$ is sufficiently small and $${\chi \in C_0^{\infty}}$$ even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/460b63b6dc00b691383f1b96934c3cd52642929b

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Mean–Field Evolution of Fermionic Systems (2013, background): “Among many other important contributions to the mathematical understanding of the many body dynamics for bosonic systems in the mean field regime, let us also recall [17], [2], [35, 32], and the series of papers [21, 22, 23].”