Large quantum systems: a mathematical and numerical perspective

Abstract : This thesis is devoted to the mathematical study of variational models for large quantum systems. The mathematical methods are that of nonlinear analysis, calculus of variations, partial differential equations, spectral theory, and numerical analysis.

The first part contains some results on finite systems. We study several approximations of the N-body Schrödinger for electrons in an atom or a molecule, and then the so-called Hartree-Fock-Bogoliubov model for a system of fermions interacting via the gravitational force.

In a second part, we propose a new method allowing to prove the existence of the thermodynamic limit of Coulomb quantum systems.

Then, we construct two Hartree-Fock-type models for infinite systems. The first is a relativistic theory deduced from Quantum Electrodynamics, allowing to describe the behavior of electrons, coupled to that of Dirac's vacuum which can become polarized. The second model describes a nonrelativistic quantum crystal in the presence of a charged defect. A new numerical method is also proposed.

The last part of the thesis is devoted to spectral pollution, a phenomenon which is observed when trying to approximate eigenvalues in a gap of the essential spectrum of a self-adjoint operator, for instance for periodic Schrödinger or Dirac operators.
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Habilitation à diriger des recherches
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Submitted on : Wednesday, June 10, 2009 - 7:46:15 PM
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Mathieu Lewin. Large quantum systems: a mathematical and numerical perspective. Mathematics [math]. Université de Cergy Pontoise, 2009. ⟨tel-00394205⟩

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