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.

arXiv comment75 pages; introduction improved and some errors in the propagation of regularity part corrected

References in corpus (11)

• Second-order corrections to mean field evolution for weakly interacting Bosons. I
• On the Mean Field and Classical Limits of Quantum Mechanics
• Rate of Convergence Towards Hartree Dynamics
• Mean field evolution of fermions with Coulomb interaction
• From the Hartree dynamics to the Vlasov equation
• Bogoliubov corrections and trace norm convergence for the Hartree dynamics
• On Mean Field Limits for Dynamical Systems
• The Vlasov-Poisson dynamics as the mean field limit of extended charges
• Empirical Measures and Quantum Mechanics: Application to the Mean-Field Limit
• Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
• Combined mean-field and semiclassical limits of large fermionic systems

Cited by in corpus (10)

• On Quantum Sobolev Inequalities
• Global-in-time Semiclassical Regularity for the Hartree-Fock Equation
• On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation
• 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