Communications in Mathematical Physics

, 294:273

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


  • Manoussos G. Grillakis
    • Department of MathematicsUniversity of Maryland
    • Department of MathematicsUniversity of Maryland
  • Dionisios Margetis
    • Department of MathematicsUniversity of Maryland
    • Institute for Physical Science and TechnologyUniversity of Maryland
    • Center for Scientific Computation and Mathematical ModelingUniversity of Maryland

DOI: 10.1007/s00220-009-0933-y

Cite this article as:
Grillakis, M.G., Machedon, M. & Margetis, D. Commun. Math. Phys. (2010) 294: 273. doi:10.1007/s00220-009-0933-y


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.

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© Springer-Verlag 2009