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The semi-classical limit of large fermionic systems

  • Søren Fournais
  • Mathieu Lewin
  • Jan Philip Solovej
Article

Abstract

We study a system of N fermions in the regime where the intensity of the interaction scales as 1 / N and with an effective semi-classical parameter \(\hbar =N^{-1/d}\) where d is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas–Fermi minimizers in the limit \(N\rightarrow \infty \). The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti–Hewitt–Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.

Mathematics Subject Classification

81V70 

Notes

Acknowledgements

M.L. and J.P.S acknowledge financial support from the European Research Council (Grant Agreements MNIQS 258023 and MASTRUMAT 321029). S.F. acknowledges support from a Danish research council Sapere Aude grant. This work was started when the authors were at the Centre Émile Borel of the Institut Henri Poincaré in Paris in 2013. Part of this work was done when S.F. was a visiting professor at the University Paris-Dauphine.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Søren Fournais
    • 1
  • Mathieu Lewin
    • 2
  • Jan Philip Solovej
    • 3
  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark
  2. 2.CEREMADE, CNRS University Paris-DauphinePSL Research University Place de Lattre de TassignyParisFrance
  3. 3.Department of MathematicsUniversity of CopenhagenCopenhagen ØDenmark

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