2015/09/30 by M. Grillakis, M. Machedon · 2 citations
Mathematics · Physics and Astronomy · #Boson #Cold Atom Physics and Bose-Einstein Condensates #Excitation #Field (mathematics) #Fock space #Function (biology) #Harmonic #Mean field theory #Random Matrices and Applications #Space (punctuation) #Spectral Theory in Mathematical Physics #math.AP
paper · pdf · doi:10.1080/03605302.2016.1255228
published as Communications in PDE, Vol 42, No 1, 24--67 (2017)
openalex created_date 2016/06/24 · arxiv created 2016/07/31 · openalex publication_date 2016/11/28 · arxiv updated 2017/06/06 · openalex updated_date 2026/08/05
We consider a large number of Bosons with interaction potential . In our earlier papers (Grillakis et al. in Comm. Math. Phys. (2010) and in Adv. Math. (2011), as well as Grillakis and Machedon in Comm. Math. Phys., (2013)) we considered a set of equations for the condensate ϕ and pair excitation function k and proved that they provide a Fock space approximation to the exact evolution of a coherent state for . In Grillakis and Machedon, J. Fixed Point Theory Appl., (2013), in the hope of treating higher values of β<1, we introduced a coupled refinement of our original equations. In that paper, we showed the coupled equations conserve the number of particles and energy. In the current paper, we prove that the coupled equations do indeed provide a Fock space approximation for , at least locally in time. In order to do that, we reformulate the coupled equations in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of those used by Chen and Holmer in J. Euro. Math. Soc. (2016) to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function k. While our earlier papers provide background material, the methods of this paper paper are mostly new, and the presentation is self-contained.