Archive for Rational Mechanics and Analysis

, Volume 198, Issue 1, pp 273–330 | Cite as

Setting and Analysis of the Multi-configuration Time-dependent Hartree–Fock Equations

  • Claude Bardos
  • Isabelle Catto
  • Norbert Mauser
  • Saber Trabelsi


In this paper, we formulate and analyze the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations for molecular systems with pairwise interaction. This set of coupled nonlinear PDEs and ODEs is an approximation of the N-particle time-dependent Schrödinger equation based on (time-dependent) linear combinations of (time-dependent) Slater determinants. The “one-electron” wave-functions satisfy nonlinear Schrödinger-type equations coupled to a linear system of ordinary differential equations for the expansion coefficients. The invertibility of the one-body density matrix (full-rank hypothesis) plays a crucial rôle in the analysis. Under the full-rank assumption a fiber bundle structure emerges and produces unitary equivalence between different useful representations of the MCTDHF approximation. For a large class of interactions (including Coulomb potential), we establish existence and uniqueness of maximal solutions to the Cauchy problem in the energy space as long as the density matrix is not singular. A sufficient condition in terms of the energy of the initial data ensuring the global-in-time invertibility is provided (first result in this direction). Regularizing the density matrix violates energy conservation. However, global well-posedness for this system in L 2 is obtained with Strichartz estimates. Eventually, solutions to this regularized system are shown to converge to the original one on the time interval when the density matrix is invertible.


Cauchy Problem Density Matrix Strong Solution Mild Solution Hermitian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Claude Bardos
    • 1
    • 2
  • Isabelle Catto
    • 3
    • 4
  • Norbert Mauser
    • 5
  • Saber Trabelsi
    • 1
    • 2
  1. 1.Laboratoire J.-L. LionsUniversité Paris 7ParisFrance
  2. 2.Wolfgang Pauli InstituteViennaAustria
  3. 3.CNRS, UMR 7534ParisFrance
  4. 4.Université Paris-Dauphine, CEREMADEParisFrance
  5. 5.Wolfgang Pauli Institute, c/o Fak. f.MathematikUniv. WienViennaAustria

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