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Annales Henri Poincaré

, Volume 19, Issue 4, pp 1167–1214 | Cite as

The Dirac–Frenkel Principle for Reduced Density Matrices, and the Bogoliubov–de Gennes Equations

  • Niels Benedikter
  • Jérémy Sok
  • Jan Philip Solovej
Open Access
Article

Abstract

The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose–Einstein condensation and quantum chemistry. We reformulate the Dirac–Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov–de Gennes and Hartree–Fock–Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov–de Gennes equations in energy space and discuss conserved quantities.

Notes

Acknowledgements

Open access funding provided by Institute of Science and Technology (IST Austria). The authors acknowledge support by ERC Advanced Grant 321029 and by VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). The authors would like to thank Sébastien Breteaux, Enno Lenzmann, Mathieu Lewin and Jochen Schmid for comments and discussions about well-posedness of the Bogoliubov–de Gennes equations.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.IST AustriaKlosterneuburgAustria
  2. 2.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland
  3. 3.QMath, Department of Mathematical SciencesUniversity of CopenhagenKøbenhavn ØDenmark

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