Bogoliubov Spectrum of Interacting Bose Gases

By Mathieu Lewin, Phan Thanh Nam, Sylvia Serfaty, J. P. Solovej. 2012

184 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a3dff1247f0eb03a37993968c81c55fab8406a9c

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Derivation of Hartree's theory for generic mean-field Bose systems (2013, background): “23), and if the unique Hartree ground state is non-degenerate, then it is shown by Lewin, Nam, Serfaty and Solovej in [37] that the energy can be expanded as E (N) = NeH(1) + e V B + o(1) where eB is the ground state energy of an effective operator in Fock space called the Bogoliubov Hamiltonian.”
Mean–Field Evolution of Fermionic Systems (2013, methodology): “A new and more general approach to the analysis of the excitations spectra of bosonic mean field systems was then obtained in [33].”
Effective Evolution Equations from Quantum Dynamics (2015, unknown): “tuations for data with ﬁxed number of particles. For N-particle initial data, a better approach to study ﬂuctuations around the Hartree dynamics has been proposed in [46], based on ideas developed in [47] to analyze the excitation spectrum of mean-ﬁeld Hamiltonians. Fix ϕ∈ L2(R3). Then we can write ”
Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, background): “Bogoliubov transformations play an important role to get a norm approximation for the dynamics of N -particle systems (adapting techniques developed in the timeindependent setting in [30]).”