Mean Field Limits in Strongly Confined Systems

الأصل · EN

We consider the dynamics of N interacting bosons in three dimensions which are strongly confined in one or two directions. We analyze the two cases where the interaction potential w is rescaled by either N⁻¹w(·) or a³θ⁻¹w(aθ·) and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit N→ ∞ the dynamics of the N-particle system are described by a nonlinear equation in one or two dimensions. In the case of the scaling N⁻¹w(·) this equation is the Hartree equation and for the scaling a³θ⁻¹w(aθ·) 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.