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A rigorous derivation of Bogoliubov's spectrum of interacting Bose gases

Phan Thanh Nam
(Universite de Cergy-Pontoise)
Torsdag, 11 oktober, 2012, at 16:15-17:00, in Aud. D3 (1531-215)
In 1947 Bogoliubov predicted that the excitation spectrum of some certain N-body Bose gas can be approximated by that of the so-called Bogoliubov Hamiltonian in Fock space. We gives abstract conditions on which Bogoliubov's theory is valid. More precisely, assuming that the ground state energy is on the first order given by Hartree's theory, that the Hartree ground state is unique and non-degenerate, and that there is Bose-Einstein condensation on this state, we prove the convergence of the lower eigenvalues and eigenfunctions of the N-body Hamiltonian towards those of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. We then treat two applications: atoms with ``bosonic'' electrons, and trapped Coulomb gases. This is joint work with M. Lewin, S. Serfaty, and J. P. Solovej.
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