On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup></mml:math> rate of convergence in the limit from the Hartree to the Vlasov–Poisson equation

Using a new stability estimate for the difference of square roots two solutions Vlasov–Poisson equation, we obtain convergence in L 2 norm Wigner transform solution Hartree equation with Coulomb potential to rate proportional ℏ. This improves ℏ 3/4-ε obtained [L. Lafleche, C. Saffirio: Analysis & PDE, appear]. Another reason interest this paper is method, reminiscent ones used prove mean-field limit from many-body Schrödinger towards Hartree–Fock mixed states.