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Bogoliubov Spectrum of Interacting Bose Gases
Author(s) -
Lewin Mathieu,
Nam Phan Thành,
Serfaty Sylvia,
Solovej Jan Philip
Publication year - 2015
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21519
Subject(s) - boson , eigenfunction , ground state , eigenvalues and eigenvectors , hamiltonian (control theory) , bose–einstein condensate , coulomb , hartree , quantum mechanics , physics , convergence (economics) , unitary state , mathematics , mathematical physics , electron , mathematical optimization , political science , law , economics , economic growth
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. © 2015 Wiley Periodicals, Inc.