×

Stability of equilibria for a Hartree equation for random fields. (English. French summary) Zbl 1439.35438

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.

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
PDFBibTeX XMLCite
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. Math., 334, 6, 515-520 (2002) · Zbl 1018.81009
[3] Bardos, C.; Golse, F.; Gottlieb, A. D.; Mauser, N. J., Mean field dynamics of fermions and the time-dependent Hartree-Fock equation (2002), arXiv preprint
[4] Benedikter, N.; Jakšić, V.; Porta, M.; Saffirio, C.; Schlein, B., Mean-field evolution of fermionic mixed states, Commun. Pure Appl. Math., 369, 12, 2250-2303 (2016) · Zbl 1352.81061
[5] Benedikter, N.; Porta, M.; Schlein, B., Mean-field evolution of fermionic mixed states, Commun. Math. Phys., 331, 3, 1087-1131 (2014) · Zbl 1304.82061
[6] Bez, N.; Hong, Y.; Lee, S.; Nakamura, S.; Sawano, Y., On the Strichartz estimates for orthonormal systems of initial data with regularity, Adv. Math., 354, Article 106736 pp. (2019) · Zbl 1423.35049
[7] Bove, A.; Da Prato, G.; Fano, G., An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction, Commun. Math. Phys., 37, 3, 183-191 (1974) · Zbl 0303.34046
[8] Bove, A.; Da Prato, G.; Fano, G., On the Hartree-Fock time-dependent problem, Commun. Math. Phys., 49, 1, 25-33 (1976)
[9] Chadam, J. M., The time-dependent Hartree-Fock equations with Coulomb two-body interaction, Commun. Math. Phys., 46, 2, 99-104 (1976) · Zbl 0322.35043
[10] Chen, T.; Hong, Y.; Pavlović, N., Global well-posedness of the nls system for infinitely many fermions, Arch. Ration. Mech. Anal., 224, 1, 91-123 (2017) · Zbl 1369.35075
[11] Chen, T.; Hong, Y.; Pavlović, N., On the scattering problem for infinitely many fermions in dimensions \(d = 3\) at positive temperature, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 35, 2, 393-416 (2018) · Zbl 1383.81366
[12] de Suzzoni, A. S., An equation on random variables and systems of fermions (2015), arXiv preprint
[13] Elgart, A.; Erdős, L.; Schlein, B.; Yau, H.-T., Nonlinear Hartree equation as the mean field limit of weakly coupled fermions, J. Math. Pures Appl., 83, 10, 1241-1273 (2004) · Zbl 1059.81190
[14] Frank, R. L.; Lewin, M.; Lieb, E. H.; Seiringer, R., Strichartz inequality for orthonormal functions, J. Eur. Math. Soc., 16, 7, 1507-1526 (2014) · Zbl 1307.35245
[15] Frank, R. L.; Sabin, J., Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates, Am. J. Math., 139, 6, 1649-1691 (2017) · Zbl 1388.42018
[16] Fröhlich, J.; Knowles, A., A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction, J. Stat. Phys., 145, 1, 23 (2011) · Zbl 1269.82042
[17] Giuliani, G.; Vignale, G., Quantum Theory of the Electron Liquid (2005), Cambridge University Press
[18] Guo, Z.; Hani, Z.; Nakanishi, K., Scattering for the 3d Gross-Pitaevskii equations, Commun. Math. Phys., 359, 1, 265-295 (2018) · Zbl 1393.35221
[19] Gustafson, S.; Nakanishi, K.; Tsai, T. P., Scattering theory for the Gross-Pitaevskii equation, Math. Res. Lett., 13, 2, 273-285 (2006) · Zbl 1119.35084
[20] Gustafson, S.; Nakanishi, K.; Tsai, T. P., Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré, 8, 7, 1303-1331 (2007) · Zbl 1375.35485
[21] Gustafson, S.; Nakanishi, K.; Tsai, T. P., Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math., 11, 04, 657-707 (2009) · Zbl 1180.35481
[22] Lewin, M.; Sabin, J., The Hartree equation for infinitely many particles, ii: dispersion and scattering in 2d, Anal. PDE, 7, 6, 1339-1363 (2014) · Zbl 1301.35122
[23] Lewin, M.; Sabin, J., The Hartree equation for infinitely many particles I. well-posedness theory, Commun. Math. Phys., 334, 1, 117-170 (2015) · Zbl 1312.35146
[24] Lindhard, J., On the properties of a gas of charged particles, Mat.-Fys. Medd. Danske Vid. Selsk., 28, 8 (1954) · Zbl 0059.22309
[25] Simon, B., \(P ( \phi )_2\) Euclidean (Quantum) Field Theory (2015), Princeton University Press
[26] Zagatti, S., The Cauchy problem for Hartree-Fock time-dependent equations, Ann. Inst. Henri Poincaré, 56, 357-374 (1992) · Zbl 0763.35089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.