Rate of Convergence towards Hartree Dynamics with Singular Interaction Potential

Abstract

We consider a system of $N$-Bosons with a two-body interaction potential $V\in L^2({\mathbb{R}}^3)+L^{\infty }({\mathbb{R}}^3)$, possibly singular than the Coulomb interaction. We show that, with $H^1({\mathbb{R}}^3)$ initial data, the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order $1/N$, for any fixed time. The $N$-dependence of the bound is optimal.

arXiv commentLemma 2.2 is modified. Due to the change, Theorm 1.1, other lemmas and propositions have been changed. Minor errors were corrected

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• On the time dependence of the rate of convergence towards Hartree dynamics for interacting Bosons
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
• Convergence towards the Vlasov-Poisson Equation from the $N$-Fermionic Schrödinger Equation
• Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential