Collot, C.; de Suzzoni, Anne-Sophie Stability of equilibria for a Hartree equation for random fields. (English. French summary) Zbl 1439.35438 J. Math. Pures Appl. (9) 137, 70-100 (2020). Summary: We consider a Hartree equation for a random field, which describes the temporal evolution of infinitely many fermions. On the Euclidean space, this equation possesses equilibria which are not localized. We show their stability through a scattering result, with respect to localized perturbations in the not too focusing case in high dimensions \(d \geq 4\). This provides an analogue of the results of M. Lewin and J. Sabin [Anal. PDE 7, No. 6, 1339–1363 (2014; Zbl 1301.35122)], and of T. Chen et al. [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35, No. 2, 393–416 (2018; Zbl 1383.81366)] for the Hartree equation on operators. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger and Gross-Pitaevskii equations. Cited in 5 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B25 Singular perturbations in context of PDEs 35P25 Scattering theory for PDEs 35R60 PDEs with randomness, stochastic partial differential equations 35Q40 PDEs in connection with quantum mechanics Keywords:Hartree equation; random fields; stability; scattering; nonlinear Schrödinger equation; Gross-Pitaevskii equation Citations:Zbl 1301.35122; Zbl 1383.81366 PDFBibTeX XMLCite \textit{C. Collot} and \textit{A.-S. de Suzzoni}, J. Math. Pures Appl. (9) 137, 70--100 (2020; Zbl 1439.35438) Full Text: DOI arXiv References: [1] Adler, R. J.; Taylor, J. E., Random Fields and Geometry (2009), Springer Science [2] Bardos, C.; Erdös Golse, L.; Mauser, N. F.; Yau, H. T., Derivation of the Schrödinger Poisson equation from the quantum n-body problem, C. R. 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