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Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation

Nikolai Leopold, Chiara Saffirio

6/3/22 Published in : arXiv:2203.03031

We investigate the semiclassical limit from the semi-relativistic Hartree-Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form K(x)=\gamma\frac{1}{|x|^a}, a \in \left( \max \left\{ \frac{d}{2} -2 , - 1 \right\}, d-2 \right], d\in\{2,3\} and \gamma\in\mathbb{R}, with the convention K(x)=\gamma\log(|x|) if a=0. For mixed states, we show convergence in Schatten norms with explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [J. Stat. Phys. 172 (2), 398--433 (2018)] where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions.

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  • Quantum Systems

Global-in-time semiclassical regularity for the Hartree-Fock equation

Ground state energy of a Bose gas in the Gross-Pitaevskii regime

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