Communications in Mathematical Physics

, Volume 225, Issue 2, pp 223-274

On the Point-Particle (Newtonian) Limit¶of the Non-Linear Hartree Equation

  • Jürg FröhlichAffiliated withTheoretical Physics, ETH-Hönggerberg, 8093 Zürich, Switzerland. E-mail: juerg@itp.phys.ethz.ch
  • , Tai-Peng TsaiAffiliated withCourant Institute, New York University, New York, NY 10012, USA.¶E-mail: ttsai@cims.nyu.edu; yau@cims.nyu.edu
  • , Horng-Tzer YauAffiliated withCourant Institute, New York University, New York, NY 10012, USA.¶E-mail: ttsai@cims.nyu.edu; yau@cims.nyu.edu

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Abstract:

We consider the nonlinear Hartree equation describing the dynamics of weakly interacting non-relativistic Bosons. We show that a nonlinear Møller wave operator describing the scattering of a soliton and a wave can be defined. We also consider the dynamics of a soliton in a slowly varying background potential Wx). We prove that the soliton decomposes into a soliton plus a scattering wave (radiation) up to times of order ɛ−1. To leading order, the center of the soliton follows the trajectory of a classical particle in the potential Wx).