From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

Abstract

We obtain the combined mean-field and semiclassical limit from the $N$-body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree$\unicode{x2013}$Fock equation with singular pair interaction potentials of the form $\pm |x-y{|}^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the Hartree$\unicode{x2013}$Fock and Vlasov equations in Schatten norms. For $a\in (0,1/2)$, we obtain local-in-time results when $N^{-1/2}\ll h\le N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in [1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff.

75 pages; introduction improved and some errors in the propagation of regularity part corrected

References in corpus (1)

• Mean field evolution of fermions with Coulomb interaction

Cited by in corpus (10)

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• Bosonization of Fermionic Many-Body Dynamics
• Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law
• Dynamics of mean-field Fermi systems with nonzero pairing
• Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation
• Derivation of Vlasov equations for multi-species fermions
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate
• From Quantum Many-Body Systems to Ideal Fluids