Bogoliubov spectrum of interacting bose gases

Mathieu Lewin, Phan Thành Nam, Sylvia Serfaty, Jan Philip Solovej

Research output: Contribution to journalArticle

Abstract

We study the large-N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that 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. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and nondegenerate, and that there is complete Bose-Einstein condensation on this state. Using our method we then treat two applications: atoms with "bosonic" electrons on one hand, and trapped two-dimensional and three-dimensional Coulomb gases on the other hand.

Original languageEnglish (US)
Pages (from-to)413-471
Number of pages59
JournalCommunications on Pure and Applied Mathematics
Volume68
Issue number3
DOIs
StatePublished - Mar 1 2015

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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