The rescaled Schrödinger dynamics of N dense fermions with C^2 pair interactions converges to the time-dependent Hartree dynamics with explicit rate N^{-1/24}.
Commutator Estimates and Quantitative Local Weyl's Law for Schr\"odinger Operators with Non-Smooth Potentials
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We analyze semi-classical Schr\"odinger operators with potentials of class $C^{1,1/2}$ and establish commutator estimates for the associated projection operators in Schatten norms. These are then applied to prove quantitative versions of the local and phase space Weyl laws in $L^p$ spaces. We study both non-interacting, and interacting particle systems. In particular, we are able to treat the case of the minimizers of the Hartree energy in the case of repulsive singular pair interactions such as the Coulomb potential.
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Derivation of the time-dependent Hartree equations for strongly interacting dense fermionic systems
The rescaled Schrödinger dynamics of N dense fermions with C^2 pair interactions converges to the time-dependent Hartree dynamics with explicit rate N^{-1/24}.