Mean Field Evolution of Fermions with Coulomb Interaction

Abstract

We study the many body Schrödinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree–Fock dynamics. Our result holds under a condition on the solution of the Hartree–Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree–Fock evolution is trivial), but that we expect to hold more generally.

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Acknowledgements

We acknowledge the support of the NCCR SwissMAP. Furthermore, M.P. has been supported by the Swiss National Science Foundation through the grant “Mathematical aspects of many-body quantum systems”. C.S. was supported by the Forschungskredit UZHFK-15-108. B.S. is happy to acknowledge support from the Swiss National Science Foundation through the grant “Effective equations from quantum dynamics”.

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Correspondence to Marcello Porta.

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Porta, M., Rademacher, S., Saffirio, C. et al. Mean Field Evolution of Fermions with Coulomb Interaction. J Stat Phys 166, 1345–1364 (2017). https://doi.org/10.1007/s10955-017-1725-y

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Keywords

  • Many-body quantum dynamics
  • Mean field scaling
  • Semiclassical scaling
  • Hartree–Fock dynamics
  • Coulomb interaction